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+Geometric theory of topological defects: methodological
+developments and new trends
+Sébastien Fumeron‡, Bertrand Berche‡, and Fernando Moraes†
+‡Laboratoire de Physique et Chimie Théoriques,
+UMR Université de Lorraine - CNRS 7019, 54000 Nancy, France and
+†Departamento de Física, Universidade Federal Rural
+de Pernambuco, 52171-900, Recife, PE, Brazil
+Abstract
+Liquid crystals generally support orientational singularities of the director field known as topo-
+logical defects. These latter modifiy transport properties in their vicinity as if the geometry was
+non-Euclidean. We present a state of the art of the differential geometry of nematic liquid crystals,
+with a special emphasis on linear defects. We then discuss unexpected but deep connections with
+cosmology and high-energy-physics, and conclude with a review on defect engineering for transport
+phenomena.
+1
+arXiv:2301.01050v1  [cond-mat.soft]  3 Jan 2023
+
+I.
+INTRODUCTION
+One of Pierre-Gilles de Gennes’s greatest breakthrough was to realize that methods and
+concepts borrowed from superconductivity also apply to describe smectic-A phases [1]. His
+work is a striking example of cross-fertilization between different areas of physics and it
+highlights how progress arises at the crossroads of various scientific fields. In an article
+that has not been translated in English [2], he took the example of line singularities as a
+common denominator between liquid crystals, quark physics and superconductors [3]. The
+same observation was also made by William Brinkman and Patricia Cladis [4], and most
+notably by the two-Nobel-Prize winner John Bardeen in a review written as a plea for
+interdisciplinarity: “Line defects in three-dimensional systems, quantized vortex lines or flux
+lines, and dislocations account for similarities of behavior in superconductors, liquid crystals,
+and, it is hoped, color confinement of quarks“[5].
+In this spirit, the objective of this paper is to perform an in-depth survey of geometrical
+methods useful for investigating topological defects and to describe some of its modern
+applications, either as a playground to test fundamental ideas in high-energy physics or
+gravitational physics, or as high-performance tools to taylor transport phenomena from soft
+matter devices. We will be mainly concerned with nematic liquid crystals and topological
+line defects.
+In section II, we provide a self-contained introduction to phase transitions, geometry,
+topology and optics in such systems. We show how a metric description of defect lines in
+terms of Riemann manifolds naturally arises in nematics, before addressing the question of
+analogue gravity which may be less familiar to the liquid crystals community.
+Section III is an introduction to some of the ideas borrowed to cosmology which can
+be dealt with liquid crystals, such as the Kibble mechanism, which rules the formation of
+defects in the early universe but also in nematics. We review several outstanding problems
+involving line singularities, such as cosmic strings, wormholes and bouncing cosmologies,
+and discuss their connections with disclinations in liquid crystals.
+Section IV eventually discusses applications of the geometric formalism previously in-
+troduced for the description of acoustics, optics and heat transfer in the presence of such
+defects. The main idea is to show how the curvature carried by the topological defects
+can be used to design specific propagation patterns, a possible step forward towards the
+2
+
+defect-engineering of transport phenomena.
+II.
+TOPOLOGICAL DEFECTS IN NEMATIC LIQUID CRYSTALS
+A.
+Basics of the isotropic-nematic phase transition
+a.
+Historical milestones
+The story of liquid crystal has met with a shaky start. The
+first observations reported of what is today understood as a liquid crystal belong to the
+realm of biology. Georges-Louis Buffon (1707-1788) and later Rudolf Virchow (1854) and
+Carl Mettenheimer (1855) reported about the strange behavior of lecithins, a family of
+phospholipid substances contained in plants (wheat, rye...) and in animals (yolks, myelin
+- the insulating coating of nerve fibres - ...) [6]. When suspended in water, lecithins form
+birefringent tubular structures like a Iceland spar but that writhed like eels. Julius Planer
+in 1861 [7] and most importantly Friedrich Reinitzer in 1888 discovered similar optical
+behaviors with cholesterol compounds. Reinitzer extracted cholesteryl esters from carrots
+and made an unanticipated observation: contrary to what was known in crystallography,
+cholesteryl benzoate displays two melting points [8]. The lower one occurs at about 145.5◦C
+and correspond to the melting of the solid phase into a turbid fluid. The higher melting
+point corresponds to the clarification of the milky liquid beyond 178.5◦C.
+Such behavior left Reinitzer skeptical: had he discovered a genuinely new behavior of
+matter or was this simply the result of impure and incompletely melted crystals? Reinitzer
+wrote to Otto Lehmann, a leading physicist known for designing the first “crystallization
+microscope”: this latter consists in a microscope equipped with crossed polarizers and a
+thermal deck (a small Bunsen burner and two cooling blasts) to observe how crystals behave
+when the temperature varies. Lehmann reproduced and improved Reinitzer’s observations,
+and promoted these substances as new forms of matter, half-between liquids and crystals. In
+1889, he coined the term “liquid crystals” to account for his discovery (somehow pulling the
+sheet back towards him as he even claimed priority over Reinitzer [9]). Soon afterwards, he
+kept changing its name (including “flowing crystals”, “crystalline liquids”...), which reveals
+his difficulties to grasp the real nature of what he found.
+The years that followed Lehmann’s breakthrough have been critical. On one hand, the
+subject became more and more discussed in the scientific community, even drawing the
+3
+
+attention of future Nobel Prize laureates such as Max Born (in 1916, he conceived the
+first molecular theory of liquid crystals but predicted a generic ferroelectric behavior that
+turned out to be incorrect [10]), Jacobus Henricus van’t Hoff and Walther Nernst (Lehmann
+himself was an unlucky nominee from 1913 to 1922). On the other hand, the subject became
+highly controversial: partly because Nernst and most especially Gustav Tammann led a
+vivid opposition against liquid crystals (suspecting them of being nothing more than poorly
+prepared colloidal mixtures), partly because of Lehmann’s personality (a sparkling mix of
+pretentiousness and mysticism [9]).
+Liquid crystals have stayed a controversial subject, since the decisive contributions of
+Rudolf Schenk (1905), Daniel Vorländer (1907) and George Friedel (1922) to get a clear view
+on this subject. Schenk led a thorough study of the clearing point and unambiguously showed
+the observed properties (density, viscosity) could not be related to mixtures.
+Vorländer
+elucidated the mysterious anisotropic behavior exibited by the fluid: from a microscopic
+standpoint, liquid crystals consist in self-organized assemblies of rod-like molecules.
+As
+such, they exibit both the birefringence property expected from anisotropic uniaxial media
+and the ability to flow. Finally, Friedel realized that such substances should properly be
+understood as new full-blown phases of matter, that he named mesophases. Initially divided
+into three broad families (nematic, cholesteric and smectic), liquid crystals have now been
+enriched by many new mesophases, including columnar phases, cubatic phases, blue phases
+I, II and III...
+b.
+Mesogenic behavior
+The recipe for a molecule to be nematogenic (i.e. to have the
+ability to organize into a nematic phase) is rather simple: take a rod-like molecule and deck
+it with 1) a flexible outer part (an aliphatic chain), 2) a rigid core (phenyl groups most
+generally) and 3) a chemical group bearing a permanent dipole (for instance, a carbonitrile
+group as in 5CB). The resulting substance is a thermotropic nematic, a sensitive compromise
+between the attractive Van der Waals interactions that align rigid cores on average along the
+same direction (anisotropy) and the thermal agitation of the aliphatic chains increasing the
+mean steric hindrance (fluidity). Nematics can also be lyotropic: the nematogens display
+an amphiphilic structure (they have both hydrophilic and hydrophobic parts), the control
+parameter being the concentration of molecules in a solvent such as water. The frontier
+between thermotropic and lyotropic liquid crystalline behavior being not strict, nematogens
+can also behave as amphotropic media.
+4
+
+In the perspective of section III, let us focus on thermotropic nematics. In that case, there
+are different models accounting adequately for the phase transition. The Lebwohl-Lasher
+model [11] is the paradigmatic model in this context, in a sense the liquid crystal analogue
+of the Ising model [12]: the nematogen molecules are represented only by their direction and
+they occupy fixed positions on the sites of a cubic lattice. The different sites of the lattice
+interact only between nearest neighbors through a potential that favors configurations where
+the neighboring molecules point in the same direction. On the contrary, the Maier-Saupe
+approach is a mean-field theory: interactions between particles are replaced by an effective
+field experienced by all particles at the same time. This model considers only London forces
+between instantaneous molecular dipoles and ignores repulsive interactions [13]. This also
+favors alignment of the molecules in the same direction. We will briefly mention that for
+lyotropic nematics, Lars Onsager described the phase transition of an assembly of rod-like
+sticks as an entropic process driven [14]: due to purely steric effects, orientational entropy
+loss is more than offset by positional entropy gain which triggers the transition.
+Depending on the temperature range, three (or more) phases can be observed. At low
+temperatures, the steric hindrance of aliphatic chains is minimal and the nematogens get
+close enough for attractive forces to drive the system. As the dipole-dipole interactions
+prevail over thermal agitation, the assembly of rod-like molecules organize into a molecular
+crystal. This latter displays both a positional and rotational order for each molecule. On
+the contrary, at high temperatures, Van der Waals interactions are dominated by thermal
+effects and the nematogens form an isotropic fluid phase: both positional and orientational
+order are lost. Within the intermediate range of temperature, the two effects are of the same
+order and different kinds of mesophases may appear. In the nematic mesophase, only the
+orientational order is preserved: locally, the nematogens tend to align on average along a
+common direction, which defines the director field n. In usual nematics, the orientational
+order is preserved at long distance, as the correlation length is typically about a few µm,
+compared to the nematogen length around a few nanometers.
+As the phase transition
+involves nucleation, these domains wherein nematogens share a common orientation form
+submicronic bubbles (or spherulites). Then they grow in size and eventually they meet and
+mingle, sometimes leaving relics in the form of long threads.
+c.
+Order parameter
+Within a nematic, a particular molecule does generally not point
+exactly in the direction n and the degree of orientational ordering of the mesophase can thus
+5
+
+be assessed by looking how well the nematogens are aligned along the director field. The
+quadrupolar scalar order parameter S defined by Tsvetkov (1942) provides a quantitative
+criterion to characterize the nematic order: it is normalized (S = 0 in the isotropic fluid phase
+and S = 1 for perfectly aligned rods), and ranges from 0.3 < S < 0.8 in usual nematics (in
+practice, 0.3 < S < 0.4 for thermotropic liquid crystals, whereas 0.6 < S < 0.8 for lyotropic
+ones). The order parameter can be refined to include informations on the local orientation
+of n (Landau – de Gennes tensorial order parameter) or to encompass phase involving more
+complex-shaped mesogens (higher-order mutipole-multipole correlation functions).
+For thermotropic nematics, S can be taken as a function depending only on the temper-
+ature. The behavior of S at the transition can essentially discriminate between two main
+families of phase transitions (in the sense defined by Lev Landau in 1937 [15]): first-order
+phase transitions, for which the order parameter displays a jump at the transition control
+parameter (this class also involves latent heats and nucleation processes), and second-order
+phase transitions, for which the order parameter varies continuously at the transition (this
+class involves pretransitional effects and scaling behaviors). Experimentally, for most com-
+pounds (5CB, MBBA, 5CN...), the isotropic-nematic phase transition is identified as weakly
+first-order phase transition in three dimensions. It combines small discontinuities of S [16],
+nucleation [17] and low latent heats [18], but pretransitional effects of the dielectric proper-
+ties [18].
+B.
+From symmetry to topology
+a.
+Homotopy theory
+The existence of orientational and/or positional orders impart
+each phase about the transition with a specific set of symmetries. As a rule, the higher-
+temperature phase is generally the less-ordered one and its symmetry group is larger. In
+the isotropic-nematic case, the isotropic fluid phase and the nematic liquid crystal are both
+statistically invariant under any translation in space. But for the orientational part, the
+two phases do not share the same symmetry group. Indeed, the isotropic fluid phase is
+statistically invariant under any rotation in three dimensions, that is, under the elements of
+the group SO(3), while in the mesophase the director field plays the role of a symmetry axis,
+restricting the symmetry to statistical invariance under the elements of the group SO(2).
+But for energetic reasons, the dipoles borne by the nematogens tend to align anticollinear,
+6
+
+such that the assembly of rods is unchanged when inverting heads and tails (this dimeric
+structure was confirmed early by X-ray diffraction experiments in 5CB and 7CB [19]). Hence,
+the full symmetry group of the nematic phase is O(2).
+Many important features of a phase transition with a spontaneous symmetry-breaking
+are encompassed within the topology of an abstract object, called the order parameter space
+M. For a phase transition with a symmetry-breaking pattern G → H, the order parameter
+space is a manifold defined from the coset M = G/H. The toolbox of algebraic topology
+(Poincaré’s former analysis situs) can then be used to seek the algebraic invariants (numbers,
+groups, rings. . . ) of M and to classify this space into equivalence classes.
+Among the many entry points, homotopy theory is of particular interest to determine the
+presence of singularities. Two topological spaces are homotopic if they can be mapped into
+each other by a continuous deformation where bijectivity is not necessarily preserved (i.e.
+gluing, shrinking or fattening the space is allowed). Homotopy groups, denoted generically
+as πk(M), have been extensively studied in condensed matter physics, mainly in the pio-
+neering works of Kleman, Lavrentovich, Michel, Toulouse and Volovik [20–26] and they are
+associated to different kinds of topological properties for M. For instance, π1(M) tests the
+simple-connectedness of the order parameter space. Indeed, consider first M = R × R. It
+is simply-connected as all closed loops (dimension 1) are homotopic to a point (dimension
+0): therefore π1(M) = I and the order parameter space is simply connected. Conversely,
+for M = R∗ × R∗, there are two equivalence classes of closed loops: those not encircling
+the origin, which are homotopic to a point, and those encircling the origin which cannot be
+shrunk into a point. Therefore π1(M) ̸= I and the order parameter space is not simply con-
+nected. The 0D-hole has thus changed the homotopy content of π1(M). Interestingly, the
+dimensionality of the manifold is crucial here: a loop trying to lasso a 0D-hole can succeed in
+2D but will always fail in 3D. In its most general form, the fundamental result of homotopy
+analysis states that in dimension n, if the k th homotopy group πk(M) ̸= I, then holes of
+dimension n − 1 − k appear: such singularities are called topological defects. Strictly speak-
+ing, a defect is topological when the singular configuration of the order parameter cannot
+be transformed continuously into a uniform configuration. This process depends not only
+on the order parameter configuration but also on the dimensionality of the order parameter
+space (due to the possibility to “escape in the third dimension”). Therefore, there is also a
+more flexible use for the terminology “topological defect”, referring to the singularity asso-
+7
+
+ciated to any non-trivial homotopy content of the order parameter manifold, whatever its
+topological stability is. In the remainder of this article, we will stick to that latter meaning.
+b.
+Zoology of topological defects in nematics
+For the isotropic-nematic phase, the order
+parameter space is given by M = SO(3)/O(2) ≡ S2/Z2: the resulting space, called the real
+projective plane RP 2, can be pictured as a 2-sphere having its antipodal points identified.
+The manifold corresponding to an immersion of the real projective plane in 3D space is
+called a Boy surface and its topology is encompassed into its first four homotopy groups: for
+uniaxial nematics in 3D, these are π0(RP 2) = I (no domain wall), π1(RP 2) = Z2 (existence
+of linear defects), π2(RP 2) = Z (existence of point defects) and π3(RP 2) = Z (existence of
+textures).
+FIG. 1. Left: Class N = 0 of closed loops homotopic to a point in the order parameter space.
+Right: Class N = 1 of closed loops consisting of paths connecting two antipodal points, which are
+not homotopic to a point.
+Linear defects (or “disclinations” in Frank’s terminology) come from a breaking of the ro-
+tational symmetry group and they are probably the most widespread singularities observed
+in nematics. In optical microscopy, they appear as thread-like structures used by Friedel to
+coin the term nematic (from the greek νηµα="thread"). In polarizing microscopy, disclina-
+tions give rise to the beautiful Schlieren patterns, where dark brushes connect at singular
+8
+
+A'=A
+O
+T,(RP2)=Z2=[0,1)
+unstable
+stablepoints corresponding to the line defects viewed end on. The content of the first homotopy
+group (or Poincaré group) is Z2 = 0, 1, which means that there are two equivalence classes
+for closed loops. The trivial class N = 0 corresponds to defects that are not topologically
+stable (they can relax into a uniform configuration), whereas the second non-trivial class
+N = 1 corresponds to defects that cannot be removed (see Fig. 1). For reasons we will
+clarify later, we retain the terminology of wedge disclinations to the trivial class and the
+terminology of Mœbius disclinations to the non-trivial class. A disclination can stay almost
+straight or form loops. It is generally associated to other disclinations within dipoles (edge
+dislocations), amorphous networks (blue phases), etc. In that case, they have the possibility
+to interconnect [27] and they combine according to the algebra of Z2, that is 0 + 0 = 0,
+0 + 1 = 1 and 1 + 1 = 0. An extensive review on linear defects in the general context of ill-
+ordered condensed matter can be found in [28]. Since our main concern here is disclinations
+we refer the reader interested in point defects and textures (including the exotic skyrmions
+and hopfions) to the the very complete reviews [17] and [29], respectively.
+C.
+Optics in the presence of linear defects
+a.
+Director field of axial disclinations
+A region characterized by a given director field
+can undergo orientational distorsions as a result of external constraints. As n is a unit
+(headless) vector, the distorsions always occur in a plane orthogonal to the director field,
+i.e. δn.n = 0. From a Taylor expansion, one can rewrite the deformed state as the sum of
+three main elastic modes: a splay term in ∇.n, a twist term in n.(∇ × n) and a bend term
+in |n × (∇ × n)|. The Frank-Oseen free energy density is the elastic cost of orientational
+oscillations around n:
+fV = 1
+2K1(∇.n)2 + 1
+2K2(n.(∇ × n))2 + 1
+2K3(n × (∇ × n))2.
+(1)
+Equilibrium state corresponds to configurations such that fV is extremal. Elastic constants
+are of order E0/L, with the interaction energy about E0 ≈ 0.1 eV and L ≈ 1 nm, it is
+customary to perform the one-constant approximation for which K1 ≈ K2 ≈ K3 = K =
+10−11 N. This assumption is fair for most ordinary nematics: for instance, in the case of
+5CB at 298 K, one measures K1 = 6.2 pN, K2 = 6 pN and K3 = 8.2 pN [30, 31].
+The simplest class of linear defects consists in axial disclinations and they were firsly
+9
+
+considered by Oseen [32] and Frank [33] (for the class of perpendicular disclinations, proposed
+by de Gennes, see for instance [34]). Orientation of the director field is ill-defined along a
+line (say the z−axis) and n lies in a plane orthogonal to the defect axis (in our example,
+the x − y plane). In cylindral coordinates, let ψ(r, θ) be the angle between the director field
+and the radial unit vector. Then the Euler-Lagrange equations corresponding to a minimum
+of fV simply writes as ∆ψ = 0. The solutions representing disclination lines are given by
+ψ(θ) = mθ + ψ0, where m is the defect strength or topological charge (a priori in R) and ψ0
+a constant phase term. Around a closed loop, the total change in ψ is thus 2mπ. For the
+director field to be well-valued, this variation is tied by the hodograph rule coming from the
+Z2 symmetry of the nematic phase:
+�
+θ=2π
+dψ = 2mπ = kπ
+(2)
+where k ∈ Z. Hence, the director field writes as
+n =
+�
+�
+�
+�
+�
+cos(mθ + ψ0)
+sin(mθ + ψ0)
+0
+�
+�
+�
+�
+� ,
+(3)
+with the topological charge constrained to be integer and half-integer, i.e. m = ±1/2, ±1,
+±3/2,. . .
+Disclinations with integer strengths are topologically removable and belong to the N = 0
+homotopy class: defects m = +1 and m = −1 are topologically equivalent and can be trans-
+formed into one another by continuous deformations. They appear in optical microscopy
+as thick lines and their core is not singular (possibility to escape into the third dimension).
+Disclinations with half-integer strengths are not topologically removable and belong to the
+N = 1 homotopy class. In this latter case, fibring over a circle about the defect line by
+a line segment containing the director which is met at that point, gives a Mœbius ribbon
+which twists along the loop an odd number of times [26] (on the contrary, for the N = 0
+disclination, one gets an ordinary ribbon with two sides). They appear in optical microscopy
+as thin lines and they display a singular core structure. As the free energy density varies in
+m2 and therefore, it is energetically more favorable for a wedge disclination to decay into two
+Mœbius disclinations, as prescribed by the combination 0 = 1+1. It must be remarked that
+besides |m|, other topological invariants (such as the self-linking number, Poincaré-Hopf’s
+10
+
+index...) are needed to characterize the topology of a linear defect, as a disclination can
+globally self-connect, entangle with itself...
+b.
+The secrets of Fermat-Grandjean principle
+In the geometrical optics limit, light
+propagates along paths that can be traveled within the least time. In the case of isotropic
+media, this variational formulation takes the form of the well-known Fermat’s principle,
+established by Pierre de Fermat in 1662. In anisotropic uniaxial media, the constitutive re-
+lations involve a dielectric tensor that displays two different principal permittivities, namely
+ε⊥ and ε∥ (in a nematic, ε∥ corresponds to the permittivity in the direction of the director
+field, whereas ε⊥ is the permittivity orthogonally to it). Fresnel’s equation then provides
+two modes inside such material: the ordinary mode, behaving similarly as in an isotropic
+medium with refractive index n2 = ε⊥, and the extraordinary mode which experiences a
+direction-dependent refractive ray index given by [35]:
+Ne(r) =
+�
+ε⊥ cos2 β(r) + ε∥ sin2 β(r)
+(4)
+where β is the angle between n and the local tangent vector T. In 1919, Grandjean extended
+Fermat’s principle to uniaxial media and he showed that the energy carried by extraordinary
+light rays propagates along paths obeying [8]
+δ
+��
+Ne(r)dℓ
+�
+= 0
+(5)
+where ℓ is the curvilinear abscissa that parameterizes a ray.
+Because the director field changes from point to point, a nematic generally displays a
+varying refractive index and hence, extraordinary light beams propagate into the medium
+along curves (see Fig. 2). In the case of planar axial disclinations, the direction of n and
+consequently β is known at each point. In that case, it can be shown that the integrand in
+Fermat-Grandjean’s principle can be generally rewritten as [36, 37]
+N 2
+e (r)dℓ2 =
+�
+ε⊥ cos2 [(m − 1)θ + ψ0] + ε∥ sin2 [(m − 1)θ + ψ0]
+�
+dr2
++
+�
+ε⊥ sin2 [(m − 1)θ + ψ0] + ε∥ cos2 [(m − 1)θ + ψ0]
+�
+r2dθ2
+−
+�
+ε∥ − ε⊥
+�
+sin2 [2(m − 1)θ + 2ψ0] rdrdθ + dz2
+(6)
+In a seminal work [38], Walter Gordon pointed out the formal analogy between light
+propagation inside a moving dielectric and light propagation inside a non-Euclidean geome-
+try. This idea was developed by many authors eversince [39–45], as it elegantly replaces the
+11
+
+FIG. 2. Light paths and director fields in the presence of a planar disclination (Up left: m =
+1, ψ0 = π/2.
+Down left: m = −1, ψ0 = π/2.
+Up right: m = 1/2, ψ0 = π/4.
+Down right:
+m = −1/2, ψ0 = 0). Taken from [36].
+resolution of Fermat’s principle in a material medium by the search for the minimum-length
+lines (or geodesics) of an empty curved space. The main asset of that point of view is that
+one can use the toolbox of differential geometry to understand how the defect modifies trans-
+port phenomena in its vicinity. To illustrate how this works, let us consider the example of
+a (m = 1, ψ0 = π/2)-disclination (see Fig. 2). For this defect, Eq. (6) leads to the following
+line element:
+ds2
+3d = N 2
+e (r)dℓ2 = dr2 + α2r2dθ2 + dz2,
+(7)
+where α2 = ε∥/ε⊥. The line element is a fundamental quantity in differential geometry
+and it simply consists in a generalization of Pythagoras’ theorem for computing distances in
+arbitrary geometries. Here, instead of the familiar Euclidean line element ds2
+3d = dr2+r2dθ2+
+dz2, the term in α2 means that the circumference of a closed unit circle about the defect is no
+longer 2π but 2πα instead: in other words, there is a mismatch angle (called Frank angle)
+of value 2π(1 − α) compared to flat geometries. It is customary in differential geometry
+12
+
+to rewrite the line element as ds2
+3d = gijdxidxj, where Einstein’s summation convention on
+repeated indices is used. g is called the metric tensor and it corresponds to a positive definite
+quadratic form. The curvature scalar [46] as computed from the metric is:
+R = 2π(1 − α)
+αr
+δ2 (r)
+(8)
+whereas the torsion tensor is identically zero.
+An alternate approach to describe the influence of defects in optics, also from differential
+geometry, consists in using the formalism introduced by Paul Finsler in his 1918 thesis,
+for which there is no quadratic constraint on the geometry as in the Riemannian case [47].
+As a matter of fact, the arc length is given by a Finsler function F such that ds3d =
+F (x, y, z, dx, dy, dz) instead of ds3d =
+�
+gijdxidxj. In the case of anisotropic media, F(r) =
+Ne(r)dℓ and the metric corresponds to the Hessian of the ray index [48]. Ought to the
+particularly simple dependency of the ray index with respect to coordinates, this formalism
+turns out to be fully equivalent to Riemann’s approach (see discussion at the end of [36]).
+A line element of exactly the same form as (7) appears in the geometric theory of defects
+in elastic media [49], related to the strain field associated to wedge disclinations as will be
+described in Section II D. Such line defects can be formed by either inserting or removing
+a wedge of material of angle 2π(1 − α) with subsequent identification of the edges. In the
+case of a removal wedge disclination of axis z (α < 1), the geometry surrounding the defect
+is conical and can be easily pictured from the Volterra cut-and-weld process of Fig. 3. In
+other words, a disclination can be pictured as a Riemann manifold, for which curvature is
+only located on the disclination axis and vanishes everywhere else.
+c.
+Discussion
+From the elasticity point of view (as opposed to optics) the description
+of an axial wedge disclination by the geometry (7), or more generally by (6), calls for
+several remarks (it is important to stress here that ε∥ and ε⊥ now are related to elastic, not
+optical, anisotropy). First, a liquid crystal consists in an assembly of rod-like molecules and
+modeling it as a continuous medium is not self-explanatory. Rigorously, the continuum limit
+for nematoelasticity should come as a coarse grained approximation of molecular dynamics
+and it should fail at the atomic scale. n(r) is defined statistically, as the average common
+direction of the nematogens at each “point” in space. The “point” actually refers to a small
+volume of space that includes enough molecules for the averaging process to be physically
+significant. Hence, in practice, it means that the “point-volume” has to be large enough
+13
+
+FIG. 3. Volterra cut-and-weld process for a (wedge) disclination along z.
+compared to the molecular scale a (typically a ≈ 20 Å) and that the variations of the
+director field must occur at much larger scales than a.
+Only then, the distorted liquid
+crystal can be described as a continuous medium, as discussed by Oseen [32], Zöcher [50]
+and Frank [33].
+A second caveat is related to the status of (7), which obviously possesses non-vanishing
+curvature as in three-dimensional gravity.
+Yet, the nematic actually lives in a three-
+dimensional Euclidean space, which means that the background geometry is flat. How to
+reconcile these two standpoints? Following the analysis from De Wit [51], the state described
+by (7) does exist in the flat space, but only in an imaginary space where the medium is
+relaxed: gij comes from the projection of this imaginary space onto the physical flat space,
+in a similar way as a stereographic map projection transfers the geometric properties on
+a 2-sphere (the Earth, with its meridians and parallels) onto a flat plane while deforming
+them (Wulff net). It turns out that the geometric description of defects thus requires two
+metrics: 1) The physical flat metric, δij, will be used to perform operations on tensors such
+as raising/lowering indices... 2) The effective metric gij, which contains the elastic informa-
+tion, will be used to determine the kinematics of low energy perturbations (geodesics, first
+14
+
+disclination axis
+Frank angle
+2=2元(1-α）
+(α<1: removal)integrals...).
+Third, one may naturally wonder what really happens on the defect axis and how to
+refine our zero-width model.
+In soft matter and more especially nematics, defect cores
+are very narrow as well but they still belong to the realm of continuum mechanics. As
+discussed in [8], a disclination line can accurately be described by a “two-phase model”: the
+core consists in a tubular region, filled with the nematogens in isotropic phase (vanishing
+order-parameter), and surrounded by the nematic phase (non-zero order parameter). This
+approach is consistent with exact solutions obtained from the minimization of the Landau
+– Ginzburg – de Gennes free energy. Yet, the last word has probably not been said about
+disclination cores: observations made on lyotropic chromonic liquid crystals revealed that
+the core region has several unexpected features (asymmetric non-circular interfaces between
+the nematic and the isotropic phases, azimuthal and radial dependencies for the phase and
+amplitude of the order parameter...) compared to classic two-phase models [52].
+D.
+Analogue gravity: lessons and pitfalls
+a.
+Physics as geometry
+The geometric description of transport near linear defects does
+not restrict to optics near axial disclinations. Since the pioneering works by Bilby [53] and
+Kröner [54] in the 1950s, this approach has been extended to elasticity theory as well [55, 56].
+In the noteworthy set of works [49, 57, 58], Katanaev proposed a general framework based
+on Riemann-Cartan manifolds for dislocations and disclinations in elastic media but only
+considered the strain tensor field as the relevant degree of freedom. An expression of the
+effective metric gij in the medium rest frame can be obtained in the case of linear elasticity
+as
+gij = δij + 2εij
+(9)
+where εij denotes the strain tensor. Compared to ordinary elasticity theory (OET), the
+geometric theory of defects is in principle more accurate (ordinary elasticity only reproduces
+the first-order approximation of the geometric theory of defects [49]) and it is more versatile
+(changing the kind of defect only requires changing the metric, instead of a complicated set
+of boundary conditions in ordinary elasticity theory). Moreover, the geometric approach
+is also likely to encompass many other kinds of linear defects of interest in liquid crystals
+physics, such as screw dislocations in smectic A and C [59, 60] (in that case, the defect
+15
+
+must be described in terms of a Riemann-Cartan manifold, for which torsion is only located
+on the dislocation axis and vanishes everywhere else [61]), dispirations in antiferroelectric
+SmCA and the dimeric SmC2 [61, 62], edge dislocations in smectics [63, 64] (which are merely
+disclination dipoles)...
+The preceding examples testify that in many condensed matter systems, the effective
+degrees of freedom are represented by specific field excitations that propagate over effective
+Riemann-Cartan manifolds. Geometrization of physics is not a new idea. In Plato’s Timaeus,
+an attempt was made to describe the world in terms of only five regular polyhedra and ever
+since, geometrization of physics has been a dream pursued by many figures in science,
+including René Descartes, Bernhard Riemann, William K. Clifford [65] (for an updated
+account, see [66])... The most successful step forward in merging geometry and physics was
+made in the twentieth century by Albert Einstein with the theory of general relativity: the
+gravitational interaction turns out to be nothing more than a manifestation of the spacetime
+curvature. The possible implications of that theory did not escape the attention of influencial
+physicists such as Hermann Weyl [67], Arthur Stanley Eddington [68] and more especially
+John A. Wheeler. In the seminal paper Classical physics as geometry [69], Wheeler and
+Charles W. Misner borrow tools from cohomology, differential geometry, exterior algebra and
+topology to fully merge gravitation, electrodynamics and geometry. Provided spacetime is
+multiply-connected, Misner and Wheeler showed that similarly to mass, classical charge can
+also be seen as a byproduct of the spacetime geometry. A particularly meaningful example
+is the low-dimensional gravitational model proposed by Gerard ’t Hooft in the context of
+quantum gravity [70] (see below III C): in 2+1 dimensions, ’t Hooft showed that gravitating
+point particles can elegantly be described as conical point-like singularities of space-time,
+each deficit angle being related to the particle’s total energy. Today, this kind of ideas has
+spread out to the point where it has become an area of research on its own: analogue gravity
+(for an extensive review, see [71] and more recently [72]).
+b.
+Pitfalls
+Despite appealing to classical fields, analogue gravity is tricky and must be
+handled with great care. Indeed, it relies simultaneously on two different manifolds: 1) the
+background gravitational metric – which is experienced by all fields (generally Minskowski’s)
+– is the outcome of Einstein-Cartan equations. It is a tensor, used for instance to raise and
+lower indices of tensors, and as such, it is a covariant quantity, and 2) the effective metric –
+which is experienced only by the fields coupled to matter – does not obey Einstein-Cartan
+16
+
+gravitational equations.
+Its purpose is limited (for instance, to determine the geodesics
+followed by the coupled-fields excitations, as it is not a covariant quantity.
+Indeed, the
+effective metric is derived from physical quantities which are defined in a privileged frame,
+the medium rest-frame, and that are not invariant under Lorentz transformations.
+As can be seen from (9), the effective metric superimposes the background Minkowski
+metric and a correction taking into account the couplings between field and matter. In the
+original experiment led by Hippolyte Fizeau, the changes in the velocity field of water (and
+hence of the Gordon metric itself) were obviously ruled by the Navier-Stokes equations (for
+the velocity field) instead of Einstein-Cartan equations. In other words, effective spacetimes
+are generally stationary. Ref [73] pointed out that textures in nematic liquid crystals can
+indeed be described by the space sector of an Einstein-like equation, with the elastic-stress
+tensor replacing the energy-momentum tensor. The relevance of the effective metric is there-
+fore restricted to calculations of properties related to the kinematic properties of the fields
+coupled to matter. This encompasses as we said the geodesics of low-energy excitations
+but also the less obvious cases of Unruh effect or Hawking radiation which are purely kine-
+matic phenomena [74]. Therefore, the analogy between gravitation and condensed matter
+is strictly kinematic but not dynamical. To rephrase Wheeler, analog spacetime tells matter
+how to move... but matter does not tell analog spacetime how to curve.
+What is the purpose of analogue gravity? In cosmology, putting a theory into test is
+always a thorny challenge. In 1992, the great epistemologist Karl Popper already pointed
+out that the “major theoretical problem in cosmology is how the theory of gravitation may be
+further tested and how unified field theories may be further investigated” [75]. If the plentiful
+harvest of low-energy observations (baryonic oscillation spectroscopy, gravitational wave in-
+terferometry, mapping of the cosmic background ...) answered many questions, theoretical
+models involving (trans)planckian scales bloomed even faster, for which experimental con-
+firmations seem almost impossible – even in principle – to reach. A possible way out of this
+conundrum is to take advantage of the richness and flexibility of condensed matter. Within
+certain limits, analogues of gravity can be used to simulate different types of cosmological
+objects (signature transitions events, cosmic strings...) and to investigate the transport of
+bosonic and fermionic quasiparticles in nontrivial spacetimes. The next section reviews a
+series of works dealing with non-standard cosmological models that can be investigated from
+their liquid crystal counterparts.
+17
+
+III.
+UNRAVELING THE UNIVERSE WITH LIQUID CRYSTALS: COSMOLOGY
+IN THE LABORATORY
+A.
+Phase transitions in cosmology
+a.
+Thermal history of the universe
+In many senses, cosmology consists in thermody-
+namics applied to the largest expanding closed system: our universe. Our current under-
+standing of cosmic history is indeed based on the Standard Hot Big Bang Model and it
+originates in the pioneering works of three founding fathers: Albert Einstein, Alexander
+Friedman and George Lemaître. In essence, this model states that about 13.8 billion-years
+ago, the Universe was in an extremely hot dense state, consisting in a quark-gluon plasma,
+and that it has expanded ever since. In the framework of grand unified theory (GUT), the
+four fundamental interactions (the gravitational interaction, the electromagnetic interaction
+and two lesser known forces, the weak nuclear interaction – responsible for radioactive β de-
+cays – and the strong nuclear interaction – which ensures the cohesion of the atomic nuclei)
+were then assumed to be unified at energy scales estimated at about 1016 GeV.
+Each interaction is associated with internal (or gauge) symmetries: for instance, at today’s
+energy scales, the electromagnetic force displays gauge invariance under the elements of U(1),
+the unitary group of dimension 1 (for an accessible review on gauge theories see for instance
+[76]). Above 1016 GeV, the group G containing the internal symmetries of grand unified
+superforce is not known for sure and many candidates with exotic names are considered,
+such SU(5), SU(6), SU(7), SU(8), SU(9), SO(10), SO(14), E6... [77] The universe expansion
+played the role of a gigantic Joule-Thomson expansion, which caused a large temperature
+drop driving cosmological phase transitions. For example, the last of these transitions is
+the electroweak phase transition, occurring at energy scales about 102 GeV. It marks the
+splitting of the electroweak force into an electromagnetic part, described by Maxwell’s theory
+(1865), and the weak nuclear part, the first theory of which being Fermi’s theory (1933).
+This transition involves a spontaneous gauge symmetry breaking: the high temperature
+gauge symmetry group SU(3)c × SU(2)L × U(1)Y broke into SU(3)c × U(1)em [77].
+Let us now examine the topology of vacuum manifold (that is the set of field configurations
+minimizing the free energy modulo gauge transformations), which is the equivalent of the
+order parameter space M in condensed matter physics. In [78], Jeannerot et al determined
+18
+
+the homotopy content corresponding to all eligible groups G likely to decay below 1016GeV
+into SU(3)c × SU(2)L × U(1)Y . Their conclusion leaves no doubt concerning the formation
+of cosmic strings:
+. ..among the SSB schemes which are compatible with high energy physics
+and cosmology, we did not find any without strings after inflation.
+(if one assumes that the universe is topologically multi-connected, cosmic strings and
+monopoles may appear – not single but pairwise –, whereas two-dimensional defects –
+domain walls – cannot form at all [79]).
+b.
+Kibble-Zurek mechanism
+Cosmic inflation is a period of extremely fast expansion
+of the Universe scale factor (typically a factor 1026 within 10−32 seconds) that presumably
+happened at the very beginning of the universe [80]. From the point of view of statistical
+physics, inflation is nothing more than a quench and as such, it is likely to favor the formation
+of topological defects. In 1976 [81], Tom Kibble introduced a three-step mechanism (later
+refined by Zurek [82] who included the sensitivity to the quench speed) to describe the details
+of this quench.
+Basically, the Kibble-Zurek mechanism (KZM) consists in a nucleation
+process very similar to what happens at the isotropic-nematic phase transition, but instead
+of having an order locally described by the director field n, it is here described by the phase
+of a complex scalar field generically called a Higgs field – or an inflaton, because it needs
+not be the Higgs field responsible for the later breaking of electroweak symmetry. First,
+ordered protodomains (analog to the nematic spherulites) with no correlation between each
+other are formed and at the scale of a whole protodomain, the fast temperature drop due
+to inflation causes the Higgs field to locally take a non-vanishing vacuum expectation value
+and hence to make a phase choice. Then the protodomains grow in size until they coalesce.
+But as they were not correlated, the choices for the Higgs phase (technically, its vacuum
+expectation value) do not match in general, and line singularities of the Higgs appear when
+the boundaries of protodomains finally meet. These linear singularities are called cosmic
+strings.
+Besides this qualitative predictions, the KZM also makes quantitative predictions such
+as the scaling dynamics of the cosmic string network, the average density of defects, cor-
+relations between defects and antidefects... In the 1990s, several works [27, 83–87] showed
+that the KZM, originally developped for cosmology, was also perfectly describing line defects
+19
+
+in nematics with the very same scaling coefficients. For instance, this model predicts that
+in 2D the density of strings scales as ρ ∼ (t/τq)α with a critical exponent αth = 0.5, and
+measurements done by [27] with 5CB indeed gave αth = 0.51 ± 0.04. To sum up: defects
+consist in regions that cannot relax into the new vacuum or equivalently that are unable to
+make the transition into the new ordered phase, and they occur during phase transitions
+in cosmology and in liquid crystal physics that seem to belong to the same universality
+class. But the family resemblance goes further. Networks of cosmic strings and networks of
+disclinations also share similar intersection processes: 1) when two line defects intertwine,
+they may reconnect the other way as they cross (intercommutation) [27, 88] and 2) when
+one line defect self-intersects, it creates a loop [89, 90].
+c.
+An almost perfect analogy?
+Last but not least of these common points: the geometry.
+Nambu-Goto strings, which are the simplest cosmic defects one may expect in cosmology,
+consist in linear concentrations of energy and as such, they are considered as infinitely
+thin objects (as the thickness of a cosmic string is estimated at 10−28 cm, this is a fair
+approximation[91]). As required by thermal field theory and general relativity, the geometry
+around a Nambu-Goto string is described by the Vilenkin’s line element [88]:
+ds2 = −dt2 + dr2 + (1 − 4Gµ)2 r2dθ2 + dz2
+(10)
+where µ is the string energy density estimated at about 10 million billion tons per meter
+(we adopt hereafter the customary unit system of cosmology where c = 1). The space part
+of this element is identical to (7): it is a conical geometry corresponding to a removed Frank
+angle [92] (typically, for a GUT scale string, this angle is a few seconds of arc). The reader
+interested in the classical gauge theory of string interactions in curved spacetimes can refer
+to Ref.[93]. From the standpoint of the soft-matter physicist, Nambu-Goto strings can be
+understood as the cosmic counterparts of wedge disclinations. How to make sense of such
+incredible similarity? For the most part, this question is still open, but a noticeable attempt
+to address it was done in [73]: in essence, the reason is that equations of nematoelasticity
+have the form as the spatial sector of Einstein’s field equations, with the elastic-stress tensor
+playing the role of the energy momentum tensor.
+As the analogy between gravity and nematoelasticity does not concern time components,
+one expects that the dynamics of a cosmic defect cannot be directly mapped with those
+of a disclination. There are other discrepancies between cosmic and elastic defects that
+20
+
+one must bear in mind to avoid fallacies. Obviously, the motion of disclinations is classical
+(typically a few µm per second) whereas cosmic strings are ultra-relativistic. Dissipation
+mechanisms for cosmic strings are due to radiation of gravitational waves, while those in
+liquid crystals are friction-dominated. What is the outcome on the dynamics of the defects?
+In cosmology, monopoles annihilate in pairs (Langacker-Pi mechanism), but they do not
+annihilate fast and early enough to avoid that the Universe becomes monopole dominated
+(which is why inflation is necessary, as it drives monopoles very far away from each other).
+On the contrary, elastic hedgehogs in a nematic annihilate rapidly according to a scaling
+law. At a more fundamental level, this is linked to the fact that in high energy physics,
+broken symmetries are gauged (or internal) whereas in liquid crystals, broken symmetries
+are geometrical: in the first case, one is dealing with “gauged defects” and in the second
+case, one is dealing with “global defects”.
+B.
+Beyond cosmic wedge disclinations
+a.
+The way out of an observational dead-end
+Cosmic wedge disclinations exist either as
+stable infinite straight lines (their equation of state simply equates the string energy density
+to its tension µ = T) or as closed loops that radiate away gravitational waves until they
+vanish. When moving, strings happen to distort spacetime such that at all scales, matter
+accretes along its wake into sheet-like structures.
+They may account for the formation
+of large-scale structures in our universe (including the Great Wall) and they have several
+expected observable signatures such as the Kaiser-Stebbins effect [94, 95] (an asymmetric
+Doppler shift giving rise to anisotropies of the cosmic microwave background), gravitational
+lensing [96] (not in the form of an Einstein ring, but as a double image instead), geometric
+phase (Aharonov-Bohm effect but with a cosmic string replacing the flux tube [97])... Up to
+now, data collected by the PLANCK mission (2014) only settle upper bounds on the string
+parameter µ [98] and in 2020, observations of the stochastic gravitational wave background
+(NANOGrav experiment) may have provided with first evidences for cosmic strings [99–101].
+The non-conclusive observations of Nambu-Goto strings call for the search of refined mod-
+els for linear defects. In fact, the zero-width approximation and the straightness of cosmic
+strings are probably too coarse to account for realistic defects. Hiscock [102] and indepen-
+dently Gott [103] suggest to smoothen this singularity by introducing two string models with
+21
+
+a core structure of constant curvature: the flower-pot model (with zero curvature) and the
+ballpoint-pen model (with non-vanishing curvature). In the Gott-Hiscock thick cosmic string
+spacetime, the metric tensor is piecewise-defined and it must obey matchings conditions at
+the core radius [104]: the extrinsic curvature of the boundary should be the same whether
+measured in the interior or exterior region (O’Brien-Synge-Lichnerowicz jump condition).
+In contrast, thanks to experiments [52, 105] and molecular simulations [106, 107], much is
+known about the NLC disclination core. In particular, there is strong evidence for biaxiality
+and that strength +1 disclinations are in fact bound pairs of strength +1/2 ones, which
+may be manipulated by electrical fields [108]. We note that this rich structure may serve
+as an inspiration for novel cosmic string core models. In the same line, instead of being
+perfectly straight, linear defects can present cusps, kinks and wiggles: the averaged effect
+of these perturbations increases the linear mass density µ and decreases the string tension
+T, as prescribed by the equation of state µ T = µ2
+0 [109–111]. Compared to straight string,
+the geometry remains conical but the deficit angle is larger than in the straight string case,
+which increases polarization anisotropies of the cosmic background radiation [112].
+There are many other ways to dress a Nambu-Goto string such that it may account
+for observational results [113]. From an extension of Volterra process to 3+1 dimensions,
+Puntingam and Soleng showed that there was only 10 ways to modify a Minkowski spacetime
+into different pseudo-Riemann–Cartan geometries with respect to the Poincaré group. For
+example, a cosmic linear defect can display chirality [114–119]: in that case, the defect
+carries torsion along its axis and one gets the cosmic counterpart of a screw dislocation in
+a smectic liquid crystal. Twisted Nambu-Goto strings (or cosmic dispirations), consisting
+in spacetimes with delta function-valued curvature and torsion distributions have also been
+considered, as they combine both rotational and translational anholonomy [120–123]: as
+mentioned earlier, their effects on light could be tested from experiments done with elastic
+dispirations SmCA and SmC2.
+b.
+Going further
+Rotating disclinations are not likely to be stable but it is worth
+mentioning here that their cosmic counterparts have been long predicted in the literature
+[124]. A metamaterial analogue of the rotating cosmic string spacetime has been proposed
+[125], as well as a superfluid vortex analogy [126]. One of the most interesting properties of
+the spinning string is its association to closed timelike curves which may find applications
+in time travel [127]. Incidentally, parallel cosmic strings moving in opposite directions have
+22
+
+been suggested [128] as a prototype time machine. Related to, but not really a model for
+spinning strings, is the case of a hyperbolic nematic-based metamaterial with a disclination
+which was addressed in [129]. Along the same line, in [130] it was proposed a disclination
+model for the compactified Milne model of a cyclic universe. More details on this model in
+Section III C.
+Among the gauge strings, a very interesting possibility is the semilocal string [131]. Like
+Dirac’s string of magnetic dipoles, semilocal strings end on gauge monopoles. They are
+analogous to real disclinations in liquid crystals which, due to the finite size of a liquid
+crystalline sample, must end somewhere (hedgehog, 2D disclination on the liquid surface or
+on receptacle wall) or else, form a loop. Apropos, disclination loops in active nematics have
+very complex dynamics (including chaos) and may present recombination episodes [90].
+Defects in liquid crystals have inspired many other proposals in cosmology. GUT allows
+for discrete gauge symmetry groups, the standard Z2 parity and one Z3 parity, which are the
+only anomaly free groups that remain unbroken at low energy [96, 132]. The corresponding
+cosmic strings are generically called Zn-cosmic strings.
+Based on the known physics of
+Moebius disclinations, which are commonly observed in nematics, Satiro and Moraes have
+investigated some cosmological outcomes of Z2 cosmic strings [133]: in particular, they
+showed that Z2-cosmic strings display both positive and negative mass density regions.
+Bearing in mind the back and forth interplay between cosmology and soft matter, one
+cannot avoid to mention the latest works of Maurice Kleman, who imported homotopy
+theory from condensed matter to astrophysics and cosmology [134–136]. In particular, he
+conjectured and classified new families of cosmic defects (such as r-cosmic forms) allowed in
+a four-dimensional maximally symmetric spacetime [136].
+C.
+Black holes and Early universe
+a.
+Black holes, white holes, wormholes
+This is sometimes referred to as the “cosmology
+in the laboratory” game plan and it covers topics such as classical black holes [137]-[138],
+Hawking radiation [139]-[140], wormholes [141]-[142]... For instance, in Haller’s approxima-
+tion, the hydrodynamics of a nematic liquid crystal radially flowing down a drainhole is
+23
+
+experienced by light beams as the equatorial section of the Schwarzschild’s metric
+ds2 = −
+�
+1 − 2M
+r
+�
+dt2 +
+dr2
+�
+1 − 2M
+r
+� + r2(dθ2 + sin2 θdφ2)
+(11)
+for a specific velocity profile. The ordinary and extraordinary indexes of the NLC depend on
+the scalar order parameter of the liquid crystal. So, it was possible to taylor those indexes
+to get the proper optical metric. In order to achieve this, the Beris-Edwards hydrodynamic
+theory wass used to connect the order parameter with the velocity of the liquid crystal flow
+at each point. This was done in Ref. [143].
+More recently, an optical analogue of a wormhole threaded by a cosmic string was de-
+scribed in [142].
+Wormholes are solutions of Einstein’s equations that connect different
+regions of the spacetime. For instance, a spherically symmetric wormhole can be obtained
+by joining two Schwartzschild black hole spacetimes by a spherical hole carved around each
+singularity.
+Wormholes are usually represented by “embedding diagrams”, which are 2D
+slices of the 4D structure immersed in Euclidean 3D space. The embedding diagram of the
+notorious Morris-Thorne [144] wormhole is obtained by taking a t = const., θ = π/2 section
+of the spherically symmetric spacetime described by the metric
+ds2 = −c2dt2 +
+dr2
+1 − b2
+0/r2 + r2(dθ2 + sin2 θdφ2).
+(12)
+The restricted metric, ds2 =
+dr2
+1−b2
+0/r2 +r2dφ2, can be embedded in a 3D Euclidean space with
+metric ds2 = dz2 + dr2 + r2dφ2 such that z = z(r) is the equation of the embedded surface
+of revolution. For metric (12) the result is the catenoid.
+A thin nematic film on a catenoid with director field aligned either circularly or radially
+(see Fig. 4) has an optical metric given by [142]
+ds2 = dτ 2 + α2(τ 2 + b2
+0)dφ2,
+(13)
+where α = no/ne for the circular case, and α = ne/no for the radial one. The coordinate τ is
+the arc length of the catenary that under rotation gives rise to the surface. The parameter
+b0 is the radius of the wormhole “throat”. For α = 1, Eq. (13) reduces to the catenoid
+metric. It is clear from Eq. (13) that, asymptotically (τ >> b0), the optical metric of the
+disclination is recovered. This is also evident from the top view of the catenoids of Fig. 4.
+This optical model simulates the conical spacetime of a Morris-Thorne wormhole threaded
+by a cosmic string. The geodesics as obtained in [142] are represented in Fig. 5.
+24
+
+FIG. 4. Director field for circular and radial +1 disclinations on the catenoid, respectively. Taken
+from [142].
+(a)
+(b)
+(c)
+FIG. 5.
+Assorted geodesics for the circularly decorated catenoid.
+The blue lines represent the
+isotropic case α = 1. The red and black lines represent, respectively, circular (deficit angle) and
+radial (surplus angle) disclinations with α = 0.85 for (a) and (b), and with α = 0.98 for (c). Taken
+from [142].
+From Fig. 5 it is clear that the two parts of the wormhole joined by its throat act as a
+black hole/white hole pair.
+b.
+Road to quantum gravity
+A major contemporary challenge in physics is to find an
+extension of General Relativity able to describe gravity at all energy scales, in particular at
+the very beginning of the universe. This is the mission devoted to quantum gravity theories,
+which have the daunting task of reconciling Einstein’s general relativity and quantum field
+theory. Despite promising attempts including superstring theories, M-theory or quantum
+loop gravity, no proposal is entirely satisfactory up to now, and even so, the energy scales
+required to test these theories are far beyond our current scientific capabilities. A way out
+of this gridlock is to rely on simpler models that capture the essential features of quantum
+gravity but remain connected to low-energy-physics systems, i.e. analogue gravity. The
+rare pearl was first introduced in a seminal paper by Deser, Jackiw and ’t Hooft [145]: 2+1
+25
+
+gravity with point-particle sources.
+The main point is that there is no gravitational degrees of freedom in three dimensions
+[146], which drastically simplifies general relativity (now an exactly solvable model [147]).
+Within this framework, the geometry surrounding a point-particle is a conical singularity,
+the mismatch angle being proportional to the particle’s mass. In other words, conical de-
+fects represent point particles coupled to gravity in 2+1 spacetimes. After Katanaev [57]
+first pointed out that the theory of linear disclinations was isomorphic to the 2+1-gravity,
+Kholodenko [148] used the apparatus of quadratic differentials to establish the connection
+between Deser, Jackiw and ’t Hooft model and defects in liquid crystals. In essence, the exis-
+tence of massive particles considered as field singularities is directly related to the topology of
+the underlying manifold (the Euler characteristic) and to the emergence of the induced sad-
+dles: this means that 2+1 Einstein’s equations are strictly equivalent to the Poincaré-Hopf
+theorem (see section 5.2 in [148]), the Hopf quantization rule making the direct connection
+between particles masses Mi and the defect topological charge m, 4GMi = m [149].
+The 2+1 gravity model can therefore be experimentally investigated from a network of
+parallel disclinations lines in a 3D nematic sample. Geometry of disclinations networks has
+been theoretically investigated in the literature, sometimes allowing for analytical expres-
+sions for the metric tensor [150–152]. Several authors have shown the possibility to design
+arrays of topological linear defects from photopatterning techniques [153–157] and even to
+manipulate them [158, 159]. If this last point opens the possibility to emulate collisions be-
+tween particles in the 2+1 model, it is even more interesting for the extension of the Deser,
+Jackiw and ’t Hooft model to 3+1 dimensions [70]: matter particles are represented by a
+gas of piecewise straight string segments that are likely to collide with a higher frequency.
+The strings display both positive and negative mass densities, i.e. they are associated to
+α < 1 and α > 1 Frank angles, which makes liquid-crystal-based experiments particularly
+promising to investigate such models. This model may also have deep connections with
+Regge calculus in quantum gravity, where the smooth curved spacetime is replaced by a
+piecewise-flat simplicial manifold. This is like the triangulation of a surface in 3D where the
+local curvature is described by the dihedral angle between adjacent triangles (the triangle is
+a 2D simplex). The effect of gluing the edges of the simplexes generate a network of cone-like
+singularities (Regge cones) which are analogs to wedge disclinations [160, 161] (see Fig. 6).
+26
+
+FIG. 6. Triangulation of a sphere at the Itapetinga radiotelescope (Brazil). Covering a curved
+surface by the triangles generates dihedral angles all around the sphere.
+c.
+Non-standard cosmology
+Cosmology at transplanckian scales is a thorny problem,
+both theoretically and of course observationally. For instance has the universe popped up
+from a unique singular event, the Big-Bang? And if so, how not to wonder what could have
+happened before it and how to design experiments to test these theories? Today, many
+high-energy physics theories such as quantum loop gravity and supertring theories entice
+the search for cyclic universe models, that is an endless repetition of big crunches followed
+by big bounces, along the same line of thought as the Stoics’ concept of palingenesia. A
+safe transition has been proposed [162, 163], where the singularity is nothing more than the
+temporary collapse of a fifth dimension, the three space dimensions remain large and time
+keeps flowing smoothly. A toy model for the geometry of this transition is the compactified
+2D Milne universe MC [164, 165]. The Milne universe metric is given by
+ds2 = −dt2 + t2dχ2 + t2 sinh2 χ
+�
+dθ2 + sin2 θdφ2�
+(14)
+and it was proposed by E.A. Milne in 1933. It represents a homogeneous, isotropic and
+expanding model for the universe with a negative curvature. In order to compactify the
+27
+
+Milne universe on hypersurfaces of fixed solid angle, let the variable χ acquire some period
+denoted as 2πκ: here, 0 < κ ∈ R1 is a constant parameter for compactifications. After
+reparametrization, the line element corresponding to the compactified Milne universe finally
+writes as
+ds2 = −dt2 + κ2t2dφ2
+(15)
+where t ∈ R1. As can be seen from the disclination line element (7), the presence of κ2 in
+the above metric indicates a conical singularity of the curvature at the origin (see Fig. 7).
+The passage through the initial singularity has several unusual features [130, 166]: the
+singularity acts as a filter for classical particles and a phase-eraser for quantum ones. The
+timelike geodesics of (15) reveal that depending on their angular momentum J, particles have
+two ways of crossing the singularity: 1) For non-vanishing J, a two-step dynamics consisting
+in an inward stable motion before the singularity, followed by outward stable motion on the
+other side, with a memory loss of the particle kinematical properties (quantum mechanically,
+this effect simply comes from strong oscillations of the phase of the wave function at the
+singularity. 2) For J = 0, a one-step dynamics consisting in a straight line through the
+collapse: yet such trajectories are very unstable since small perturbations in the value of J
+causes large deviations on the trajectories.
+Probing how particles behave in MC can be tested in the laboratory from hyperbolic
+liquid crystal metamaterials (HLCM): this means that the permittivity along the director
+axis ε∥ < 0 and the permittivity perpendicular to the director axis ε⊥ > 0 are of opposite
+sign [167]. Such media can be made from a host nematic liquid crystal that includes an
+admixture of metallic nanorods [168] or coated core-shell nanospheres [169]. To retrieve the
+Kleinian double-cone geometry, the HCLM must be endowed with an hyperbolic disclination:
+the line element writes as ds2 = ε⊥dρ2 − ε∥ρ2dφ2 + ε⊥dz2, which after a rescaling becomes
+ds2 = −γ2r2φ2 + dr2 + dz2. This line element is relevant only by the extraordinary modes
+and for radial injection conditions (planar trajectories z = Cst), the geometry experienced
+by extraordinary rays is perfectly identical to that of the compactified Milne universe.
+A stable configuration for the director field may be obtained from a cylindrical shell
+of HLCM with homeotropic anchoring at the boundaries. In the geometrical optics limit,
+extraordinary light paths turn out to be Poinsot’s spirals as for the compactified Milne
+universe. The practical realization sets limits to the efficiency of such analog device for
+the classical particles. First, the analysis holds only within a limited frequency bandwidth
+28
+
+FIG. 7. Timelike geodesics with the radial time t given in units of t0 and κ = 1/3. The blue and
+green (orange and red) lines are moving away (towards) from (to) the singularity. Furthermore,
+particles following the trajectories in the first (third) and second (fourth) quadrants are spinning
+clockwise (counterclockwise). The blank point at the origin is just to emphasize that the curves do
+not reach the singularity. Taken from [130].
+due to the resonant nature of the used core-shell spheres. Second, as previous phenomena
+concern the extraordinary mode, an efficient optical absorber should include a filter to shut
+off the ordinary wave. Finally, it should be noticed that the present model concerns optics
+inside a bulk hyperbolic material: to design a perfect optical analog, the hyperbolic medium
+must be impedance matched to avoid sizable reflections at the interfaces. The analogy was
+also extended to quantum particles by investigating light in the scalar wave approximation
+in the same device.
+29
+
+t
+3
+f+, J <0
+f+, J > 0.
+f-, J >0.
+J<O
+BigBounce
+2
+f, J <0.
+-1
+-2
+Big Crunch
+3
+X
+-2
+-1
+0
+1
+3
+J>0
+3
+J=0FIG. 8. Geodesics of the channel of defects in the potential landscape. Taken from [129].
+IV.
+TAYLORING TRANSPORT WITH LIQUID CRYSTALS: DEFECT-ENGINEERED
+MATERIALS
+A.
+Acoustics
+a.
+Ballistic guiding
+The geometric description of linear defects revealed wedge discli-
+nations carry curvature along their axis: a positive Frank angle corresponds to a conical
+geometry that focuses incoming light rays, in similar fashion to what happens with a con-
+verging lens. Conversely, a negative Frank angle corresponds to a saddle-like geometry that
+scatters geodesics, in the same fashion as a diverging lens. It is worth noticing that the effect
+of a disclination does not limit to the eikonal approximation, as it also diffracts incoming
+waves: from the present geometric approach, computing the differential scattering cross sec-
+tion of a wedge disclination showed good agreements with theoretical results obtained by
+Grandjean from standard acoustics [170, 171].
+The effect of a single wedge disclination on rays being clarified, one can legitimately won-
+der if a well-suited arrangement of such defects can be used to taylor the propagation of
+sound (we recall that photopatterning techniques now allow for the practical realization of
+almost any kind of arrays). In [151], a channel of disclination dipoles was considered, con-
+30
+
+V(x,y)
+10sisting in two infinite rows made of alternate disclinations separated by distance 2a (a kind
+of von Kármán alley), the distance between the rows being 2b. The positive disclinations
+are at points located at (na, (−1)nb), n ∈ Z, while negative disclinations have coordinates
+(na, (−1)(n+1)b), n ∈ Z. As always done in geometric models, the bulk medium is consid-
+ered in the continuum limit (i.e. limit of a vanishing lattice spacing). The corresponding
+background geometry writes as
+ds2 = −c2dt2 + e−4V (x,y) �
+dx2 + dy2�
++ dz2 = gµνdxµdxν
+(16)
+where c is the local speed of wave packet and V is the acoustic potential given by [151]:
+V (x, y) =
+|F|
+4π ln
+��
+cosh2 � π
+2a(y − b)
+�
+− cos2 � πx
+2a
+�
+cosh2 � π
+2a(y − b)
+�
+− sin2 � πx
+2a
+�
+� �
+cosh2 � π
+2a(y + b)
+�
+− sin2 � πx
+2a
+�
+cosh2 � π
+2a(y + b)
+�
+− cos2 � πx
+2a
+�
+��
+(17)
+where |F| is the absolute value of the Frank angle that characterizes the ±F defects. The
+sound paths are attracted by the positive defects while repelled by the negative ones (see
+Fig. 8 and 9). Adjusting the defect strengths along with parameters governing the geometry
+of a cell (namely a,b) opens the possibility to taylor material properties of the sheets, but
+this will only be achieved numerically considering the complexity of analytical expressions.
+Sound paths are yet very sensitive to the shooting angle. Hence, a thorough optimization of
+the distribution of defects deserves an additional treatment of chaos involving the statistical
+tools of dynamic hamiltonian systems.
+b.
+Acoustic rectification
+Differential geometry models for acoustics originates from
+[172] where vorticity effects in isotropic fluids were investigated. In nematics, for a pla-
+nar horizontal configuration, the acoustic metric experienced by the extraordinary mode is
+formally identical to (6) with the substitutions ε⊥ ↔ ρv2/C33 and ε∥ ↔ ρv2/C11, where ρ
+is the mass density, v is the velocity of sound in the isotropic phase, C33 and C11 are elas-
+tic constants respectively along the director’s direction and in directions orthogonal to it.
+When the nematic medium is confined inside a capillary tube (radius R) with homeotropic
+boundary conditions, the director field tends to be radially oriented everywhere but on the
+central axis where an orientational singularity lies, but to reduce the elastic energy of this
+configuration, the director escapes in the third dimension [63, 173] and the nematic relaxes
+31
+
+FIG. 9. Different geodesics, shot from the origin, in the channel of disclinations geometry. The
+positive disclinations correspond to red contours while negative disclinations correspond to blue
+contours. Depending on the shooting angle, the propagation of phonons may be guided by the
+street of topological defects. Taken from [129].
+into a funnel-shaped configuration, known as the escaped radial disclination (ERD) where
+the delta-distributed Ricci scalar becomes an extended smooth one [174].
+In general, rectification effects come from an asymmetry of the system in the direction
+along which the transport phenomenon occurs. Usually, this is generally achieved by relying
+on gradients of physical properties (e.g. pore density [175], distribution of compositional
+defects [176]...), asymmetric geometries [177, 178]... all examples of situations implying hard
+and non-flexible systems. Liquid crystals naturally provide a stable but flexible configuration
+corresponding to such asymmetry, the ERD. In the spirit of a low-cost soft-matter-based
+solution, satisfying levels of acoustic rectification have been obtained by combining the
+asymmetry of the ERD configuration to that of a container consisting in conical frustum
+of varying radius R(z) [179]. The inner surface prepared to produce the desired anchoring
+angle[180]. The anchoring angle adapted to maintain the ERD configuration. α depends on
+the surface geometry (radius), on the liquid crystal nature (elastic constant K, saddle-play
+32
+
+0.5
+0.5 constant K24) and on the surface treatment. Reference [179] considered acoustic waves in a
+frequency range between 20 Hz and 20 kHz (the average human audible range) propagating
+in 5CB for the ERD configuration. The rectification parameter used to estimate the acoustic
+device’s efficiency is the percent standard deviation of the lowest variation on the acoustic
+intensity
+Acoustic rectification(%) =
+����
+∆Ibt − ∆Itb
+min (∆Ibt, ∆Itb)
+���� × 100
+(18)
+where ∆Ibt = It − Ib is the acoustic intensity variation if the wave comes from the bot-
+tom to the top of the conical frustum and ∆Itb = Ib − It is the analogous variation for the
+counterpropagating case. Numerical simulations show a rectification effect for a longitudinal
+plane wave propagating along the conical frustum axis (see Figure 10). The optimization
+of the device parameters (geometry of the conical frustum, anchoring conditions the rectifi-
+cation effects...) allows to reach rectification levels up to 1300% for a continuous frequency
+bandwidth [181].
+FIG. 10. Left: Conical frustum with an ERD. Middle: Pressure field for the incoming waves from
+bottom to top. Right:Pressure field for the incoming waves from top to bottom. Taken from [181].
+33
+
+mPa
+mPa
+7.47237
+0.58721
+7.47236
+0.5872
+7.47235
+7.47234
+0.58719
+a)
+20
+Jm
+b)
+20
+Ars
+7.47233
+0.58718
+50
+50
+7.47232
+50
+50
+0.58717
+s) <0
+0
+μm
+-50-50
+m
+0
+-
+Aim
+7.47231
+-50-50
+μm
+0.58716
+7.4723
+0.58715B.
+Optics
+a.
+Waveguiding and light concentration
+Manipulation of light has become a major issue
+in a large number of applications ranging from solar energy harvesting to optical sensing.
+The beam steering technique relies on a modulation of refractive indices to guide light
+in a given direction ([182–185]). More recently, the possibility to continuously deflect in
+arbitrary directions and/or to simultaneously focus/defocus an incoming light beam has
+been demonstrated from multiple stacked nematic liquid crystal cells—building blocks [186].
+Another possibility to guide light beams is to use optical waveguides with nematic cores
+in radial escaped disclination configurations: the defocusing due to natural diffraction is
+compensated by the converging lens effect resulting form negative birefringence [187–191].
+Light focusing has long been suspected of suffering severe limitations, the Rayleigh crite-
+rion forbidding beam sizes below half of a wavelength. Recently, the advent of metamaterials
+provides new hopes for overcoming the diffraction limit using superlenses [192]. Another
+promising possibility is to use an hyperbolic liquid crystal metamaterial (HLCM), obtained
+from an admixture of metallic nanoobjects to nematics [193]. As seen in the previous sec-
+tion, the permittivity along the director axis ε∥ and the permittivity perpendicular to the
+director axis ε⊥ have opposite signs. For an orthoradial director field, the effective metric
+writes as gij = diag (1, −α2ρ2, 1) and in the eikonal approximation, the light paths are the
+aforementioned Poinsot spirals: the hyperbolic defect behaves as a sink for light paths [194].
+The smaller the value of α, the stronger is the spiraling behavior, 1/α corresponding to the
+defect vorticity. Concentration of light by an hyperbolic disclination extends beyond the
+geometrical optics limit. In the scalar wave approximation, the complex amplitude Φ of the
+wave is governed by the generalized form of the d’Alembert equation, involving the Laplace-
+Beltrami operator instead of the ordinary Laplace operator. The wave equation writes as
+a modified Bessel differential equation of imaginary order iℓ/α and the solutions are linear
+combinations of the modified Bessel functions of first and second kind, respectively. The
+intensity distribution for the propagating fields shows 1) that the electromagnetic field con-
+centrates along the axis of the device, and 2) that the bigger the value of the frequency, the
+smaller the light rings are (see Fig. 11).
+34
+
+FIG. 11. Examples of intensity profiles, representing the transverse field distributions concentrated
+in the vicinity of the hyperbolic defect. Taken from [194].
+b.
+Optical vorticity
+Active beam shaping has also emerged as a major trend in modern
+optics. The idea is to taylor the amplitude, the phase or the polarization of an optical
+wavefront from real-time driven systems that ideally must be compact enough, highly-flexible
+but yet have low manufacturing costs. Ought to their high response functions, liquid crystal-
+based devices naturally fulfill all these requirements and have emerged as promising low-
+cost and easy-to-manufacture alternatives to MEMS, moving opto-mechanical devices and
+photonic crystals [195, 196].
+In nematics, the orbital angular momentum carried by light beams can be tuned from
+q-plates. A q-plate consists in an inhomogeneous liquid crystal cell endowed with a discli-
+nation of topological charge q (for details regarding the dielectric tensor, see [197]). When
+an electromagnetic wave propagates inside the medium, its components acquire designed
+phase shifts (Pancharatnam-Berry phase) that trigger spin-to-angular momentum conver-
+sions. This results in light beams displaying optical vorticity, i.e. the wavefronts are helical
+and the intensities profiles distribute in doghnut-like shapes [198–203]. [204] demonstrated
+from FDTD simulations that disclination lines can transform the state of polarization of
+beams propagating along their axis. Umbilics ending disclination lines are also identified as
+35
+
+structures generating optical vortex arrays at predetermined wavelength [205].
+Nematics are not the only contenders in the contest for optical vorticity. In Ref. [206]
+the Raman-Nath diffraction was used to generate optical vortices from edge dislocations in
+the stripe pattern of a cholesteric liquid crystal (cholesterics are chiral nematics for which
+the director whirls around a well-defined direction). Cholesterics were also considered in
+[207], where it has been theoretically and experimentally demonstrated that optical vortices
+were generated from a Bragg-reflection-based device, therefore likely to operate at multiple
+wavelengths. Recently, charged particles crossing a cholesteric plate were reported to radiate
+purely twisted photons [208]. Chiral nematics are also likely to generate defect textures of
+a more complex kind than that optical dislocations [209]. Beside cholesterics, smectics also
+showed their potentialities for optical vorticity. [210] produced an optical vortex from focal
+conic domains, whereas in [166], screw dislocations in smectics were shown to imprint their
+torsion onto wavefronts (see also [211, 212] for a solid-state-oriented context).
+C.
+Heat transfer
+a.
+Principles of thermal design
+Manipulation of heat flux raises intensive research ef-
+forts because of the abundant wealth of potential applications including thermal shielding or
+stealth of objects, concentrated photovoltaics or thermal information processing (heat-flux
+modulators, thermal diodes, thermal transistors and thermal memories). These prospects
+come from the possibility of designing energy paths in a fashion similar to that of light in
+transformation optics. To do so, the first step is to understand the main peculiarities of
+heat transfer in the presence of a non-Euclidean geometry. Generally speaking, diffusion of
+a passive scalar (for instance the temperature field) can be seen as a collection of Markov
+processes obeying the stochastic Fokker-Planck equation. In the case of Brownian motion,
+the Fokker-Planck equation reduces to the well-known parabolic heat equation [213]. When
+considering diffusion processes in the presence of a non-Euclidean space, the problem is ad-
+dressed, as already discussed, by replacing the Laplace operator with the Laplace-Betrami
+operator ∆LB [214]:
+∂T
+∂t = D∆LBT.
+(19)
+Here, D is the diffusivity and its value depends on the material properties. Ought to the
+form of the metric of a wedge disclination, heat conduction locally occurs as in a monoclinic-
+36
+
+like crystal with no internal source [215]: the heat flux vectors are no longer perpendicular
+to the isothermal surfaces, which are bent depending on the value of the Frank angle. In
+other words, disclinations in nematics generate thermal lensing effects.
+FIG. 12. Top: Temperature field (a) and heat flux field (b) for a radial director field (homeotropic
+anchoring) and α =
+�
+C33/C11 = 0.5 Bottom: Temperature field (a) and heat flux field (b) for an
+orthoradial director field (parallel anchoring) such that α =
+�
+C11/C33 = 2. Taken from [216]
+Once the basic effects of single line defects are understood, the next step is to taylor them
+to guide heat. To do so, let us consider a hollow cylinder, inside which there is the core region
+where one aims at controlling the conductive heat flux. The cylinder is inserted inside a
+conducting solid sandwiched between two heated vertical plates. The host material consists
+37
+
+(a)
+(b)
+(a)
+(b)of a homogenous isotropic medium, whereas the intermediate thick cylinder consists of a
+nematic liquid crystal in a disclination-like configuration (no disclination core). For thermal
+management, mesophases with low melting and high clearing temperatures are required: a
+range of about 100 K can be reached by using eutectic liquid crystal mixtures (or “guest-
+host systems”). Numerical simulations [216] confirm the possibility of a strong heat guiding
+phenomenon: depending on the value of elastic constants C33 (along the director) and C11
+(along any direction perpendicular to the director), the device can either cloak the core region
+from the heat flux or concentrate heat there (see Fig. 12). Switching from the concentrator
+to the cloaking device is achieved by an electric-field-driven bistable anchoring with dye-
+doped mematics (sufficiently high values of the electrical potential difference between the
+two sides of the hollow cylinder were indeed shown to induce stable anchoring transitions
+between homeotropic and parallel states [217]). To avoid thermoconvective instabilities in
+the annulus domain, the device must be thin enough and the heat flux and temperature levels
+must be moderate. For instance, using 5CB and MBBA, if the temperature is about a few
+tens of degrees (thermoelectric applications) and the external radius of a few centimeters,
+the device handles heat flux that typically varies from 5 W/m2 (repeller) to 103 W/m2
+(concentrator).
+b.
+Thermal diodes
+The previous study is now refined in order to investigate thermal
+rectification. Thermal rectification is a very active subject in nanoscience and solid-state
+physics, as testified by the abundant litterature dealing with this subject (extensive reviews
+treating thermal rectification can be found in [218, 219]. More seldomly is heat conduction
+rectification considered from soft-matter-based devices. In liquid crystals, the macroscopic
+thermal properties of the nematic phase depend on temperature according to Haller’s ap-
+proximation [220]:
+λ∥(T) = λ0 + λ1 × (T − TNI) + λ1∥ × (T − TNI)α∥
+(20)
+λ⊥(T) = λ0 + λ1 × (T − TNI) + λ1⊥ × (T − TNI)α⊥
+(21)
+where λ0, λ1, λ1∥, λ1⊥, α∥ and α⊥ are material-depending constants, whereas TNI and TC
+are, respectively, the nematic-to-isotropic temperature and the clearing-point temperature
+of the liquid crystal. As discussed before, liquid crystals naturally provide a stable and
+flexible configuration corresponding to such asymmetry, the ERD, which already turns out
+to provide high levels of acoustic rectification. To achieve high rectification levels, the same
+38
+
+conical frustum of varying radius R(z) with anchoring conditions can be used. In analogy
+with the acoustic case discussed earlier, the rectification parameter used to estimate the
+thermal diode efficiency can be defined as
+Thermal rectification(%) =
+����
+∆Ti − ∆Td
+∆Td
+���� × 100
+(22)
+where ∆Td = Td,h − T0 is the difference between Td,h, the high temperature on one base
+produced by the heat pumped in the cylinder when working in the direct setup (i.e. when
+the heat is flowing from the narrow region to the wider one, i.e. the −z direction), and
+T0, the temperature at the other base, which is also the initial temperature.
+Similarly,
+∆Ti = Ti,h − T0 when working in the inverse setup.
+Numerical simulations show thermal rectification rates around 1266% [221]. On the shape
+parameters, alterations on the ratio Rr ∈ [0, 28; 0, 75] produced a percentage variation on
+the thermal rectification around 1273%, while modifications of the height h ∈ [50; 75] µm
+and on the larger radius Rl ∈ [50; 70] µm produced percentage changes lower than 5%.
+This indicates that the anisotropy of the conical frustum tube has a strong influence on the
+rectification. Other non-geometrical parameters such as the anchoring angle (in the range
+[0; 90◦]) and the inward pumped heat flux (in the range [5; 10] kW/m2) give percentage
+variations on the rectification around, respectively, 3, 8 and 1, 7%.
+Such characteristics
+enable this improved thermal diode to be miniaturized, applied on well-determined areas,
+while robust against variations of the inward pumped heat flux.
+The identical forms of the geometry experienced by light and by sound strongly suggests
+that devices using liquid crystals may be used to manipulate simultaneously optical and
+thermodynamical transport. Indeed, Ref. [222] reports the control of both electromagnetic
+propagation and heat flow by a liquid crystal device similar to the one depicted in Fig. 12,
+while Ref. [223] uses an escaped disclination configuration to rectify at the same time both
+heat and light, thus a thermo-optical diode.
+V.
+CONCLUSION AND PERSPECTIVES
+Since their early discovery in the XIXth century, liquid crystals have been the magic bullet
+in physics and engineering. Being in-between anisotropic solids and isotropic fluids, they
+extended our conception of condensed matter to an area where geometry and topology can
+39
+
+FIG. 13. Left: Isothermal surfaces of a liquid crystalline thermal diode in (up) inverse thermal
+setup and (bottom) direct thermal setup. The frustum diode has larger radius Rl = 70 µm, ratio
+between the radii is Rr = Rsm/Rl = 0, 28, the height is h = 50 µm, anchoring is 60◦ and the
+inward heat flux is Q = 5 kW/m2 on the base with the higher temperature and T0=296 K. Right:
+Rectification rate versus temperature T0 of the base for different larger radii Rl. Taken from [179].
+be almost as useful as in General Relativity. An orientationally ordered fluid, as the nematic
+liquid crystal, is a vivid representation of a Riemannian manifold where the director field
+can be associated to a local vector basis (triad or dreibein). The relative rotation of the
+director/triad associated to neighboring points indicates the presence of curvature. Bound-
+ary conditions like vessel shape, immersed objects, anchoring angle, etc., impose restrictions
+to the effective geometry whose eventual incompatibility with the nematic order (ground
+state or zero curvature everywhere) leads to the appearance of topological defects which
+accommodate the incompatibilities.
+This geometric view of the elastic distortions in the NLC is complemented by the optical
+effective geometry that appears naturally by comparing Fermat’s law of least time to the
+geodesic variational principle. Similar effective geometries can be obtained for acoustics
+and heat transport as well. In all these cases the defects, besides being the consequence of
+40
+
+μm
+ 309
+0
+308.58
+306.77
+307.67
+40
+305.87
+304.97
+304.06
+303.16
+20
+1.270 
+301.35
+0
+7 300
+1.260
+-50
+0
+50
+Rectification [%]
+1.250
+R, = 50 μm
+R; = 57 jm
+R; = 63 jm
+1.240 -
+R=70m
+μm
+ 301
+1.230
+300.64
+300.57
+1.220 
+40
+300.5
+300.44
+300.37
+1.210 
+300.3
+295
+296
+297
+298
+299
+300
+TOE
+20
+300.23
+300.17
+Te [K]
+300.1
+0
+300.03
+V 300
+-50
+0
+50the topology (boundary conditions), are the source of the geometry. One might then say
+that, as soon as there is real or effective curved geometry to describe a physical system with
+orientational order, one can expect defects. And, if there is curved geometry, one can relate
+NLC to gravitation and cosmology. In this article we reviewed, not only this relationship,
+but also the physical applications obtained with the help of the geometric tools.
+Many
+open problems both in gravitation and cosmology and in NLC certainly may benefit from
+the analogies derived by the (partially) common geometry. For instance, the experimental
+knowledge about the inner structure of disclinations may be an inspiration for cosmic string
+core models. Active matter, being a dynamic medium, may be described by time-depending
+metrics. Furthermore, being a dissipative medium, active matter (or its effective geometry)
+might obey a geometric flow like Ricci’s [224] in its way to the equilibrium.
+ACKNOWLEDGMENTS
+For the purpose of Open Access, a CC-BY public copyright licence,
+, has been
+applied by the authors to the present document and will be applied to all subsequent versions
+up to the Author Accepted Manuscript arising from this submission.
+[1] Pierre Gilles de Gennes. An analogy between superconductors and smectics a. Solid State
+Communications, 10(9):753–756, 1972.
+[2] Pierre Gilles de Gennes. Cinquante ans de recherches et de découvertes en sciences physiques.
+Revue du Palais de la découverte. Numéro spécial, (36):43–56, 1989.
+[3] The cases of disclination lines in nematics, gluon flux tubes and vortices in superconductors are
+discussed pp. 48-49 of [2], where he wrote: “I found quite extraordinary the deep relationship
+there is between molecular liquids such as nematics, at one end, and the building block of
+matter at the subnuclear scale.“.
+[4] PE Cladis and WF Brinkman. Defects in liquid crystals. Physics Today, 35(5):48–54, 1982.
+[5] John Bardeen. Unity of concepts in the structure of matter. Annual Review of Materials
+Science, 10(1):1–19, 1980.
+[6] Peter Palffy-Muhoray. The diverse world of liquid crystals. Physics today, 60:54, 2007.
+41
+
+BY[7] L Lisetski. What was observed by julius planer in 1861? Condensed Matter Physics, 2010.
+[8] Patrick Oswald and Pawel Pieranski. Nematic and cholesteric liquid crystals: concepts and
+physical properties illustrated by experiments. CRC press, 2005.
+[9] Michel Mitov.
+Liquid-crystal science from 1888 to 1922:
+Building a revolution.
+ChemPhysChem, 15(7):1245–1250, 2014.
+[10] M Born. On anisotropic fluids. the test of a theory of fluid crystals and of electrical kerr
+effects in fluids. Sitz. d. Phys. Math, 25:614, 1916.
+[11] Paul A Lebwohl and Gordon Lasher. Nematic-liquid-crystal order—a monte carlo calculation.
+Physical Review A, 6(1):426, 1972.
+[12] T Ising, R Folk, R Kenna, B Berche, and Yu Holovatch. The fate of ernst ising and the fate
+of his model. Journal of Physical Studies, 21(3), 2017.
+[13] Wilhelm Maier and Alfred Saupe.
+Eine einfache molekulare theorie des nematischen
+kristallinflüssigen zustandes. Zeitschrift für Naturforschung A, 13(7):564–566, 1958.
+[14] Lars Onsager. The effects of shape on the interaction of colloidal particles. Annals of the
+New York Academy of Sciences, 51(4):627–659, 1949.
+[15] Lev Davidovich Landau and EM Lifshitz. Statistical Physics, volume 5. Elsevier, 2013.
+[16] Subramanyan Chandrasekhar. Liquid crystals. Cambridge University Press, 1992.
+[17] Maurice Kleman and Oleg D Lavrentovich. Topological point defects in nematic liquid crys-
+tals. Philosophical Magazine, 86(25-26):4117–4137, 2006.
+[18] Bert Van Roie, Jan Leys, Katleen Denolf, Christ Glorieux, Guido Pitsi, and Jan Thoen.
+Weakly first-order character of the nematic-isotropic phase transition in liquid crystals. Phys-
+ical Review E, 72(4):041702, 2005.
+[19] Alan J Leadbetter, RM Richardson, and CN Colling. The structure of a number of nemato-
+gens. le Journal de Physique Colloques, 36(C1):C1–37, 1975.
+[20] GE Volovik and VP Mineev. Line and point singularities in superfluid 3he. JETP lett, 24(11),
+1976.
+[21] M Kléman, L Michel, and G Toulouse. Classification of topologically stable defects in ordered
+media. Journal de Physique Lettres, 38(10):195–197, 1977.
+[22] Louis Michel. Topological classification of symmetry defects in ordered media. In Group
+Theoretical Methods in Physics, pages 247–258. Springer, 1978.
+[23] Louis Michel.
+Symmetry defects and broken symmetry. configurations hidden symmetry.
+42
+
+Reviews of Modern Physics, 52(3):617, 1980.
+[24] GE Volovik and OD Lavrentovich. Topological dynamics of defects: boojums in nematic
+drops. Zh Eksp Teor Fiz, 85(6):1997–2010, 1983.
+[25] Mikhail V Kurik and OD Lavrentovich. Defects in liquid crystals: homotopy theory and
+experimental studies. Soviet Physics Uspekhi, 31(3):196, 1988.
+[26] Maurice Kléman. Defects in liquid crystals. Reports on Progress in Physics, 52(5):555, 1989.
+[27] Isaac Chuang, Ruth Durrer, Neil Turok, and Bernard Yurke. Cosmology in the laboratory:
+Defect dynamics in liquid crystals. Science, 251(4999):1336–1342, 1991.
+[28] Maurice Kleman and Jacques Friedel. Disclinations, dislocations, and continuous defects: A
+reappraisal. Reviews of Modern Physics, 80(1):61, 2008.
+[29] Ivan I Smalyukh. Knots and other new topological effects in liquid crystals and colloids.
+Reports on Progress in Physics, 83(10):106601, 2020.
+[30] PP Karat and NV Madhusudana. Elasticity and orientational order in some 4’-n-alkyl-4-
+cyanobiphenyls: Part ii. Molecular Crystals and Liquid Crystals, 40(1):239–245, 1977.
+[31] MJ Bradshaw, EP Raynes, JD Bunning, and TE Faber. The frank constants of some nematic
+liquid crystals. Journal de Physique, 46(9):1513–1520, 1985.
+[32] CW Oseen. The theory of liquid crystals. Transactions of the Faraday Society, 29(140):883–
+899, 1933.
+[33] Frederick C Frank. I. liquid crystals. on the theory of liquid crystals. Discussions of the
+Faraday Society, 25:19–28, 1958.
+[34] Michael J Stephen and Joseph P Straley.
+Physics of liquid crystals.
+Reviews of Modern
+Physics, 46(4):617, 1974.
+[35] Lev Davidovich Landau, JS Bell, MJ Kearsley, LP Pitaevskii, EM Lifshitz, and JB Sykes.
+Electrodynamics of continuous media, volume 8. Elsevier, 2013.
+[36] Caio Sátiro and Fernando Moraes. Lensing effects in a nematic liquid crystal with topological
+defects. The European Physical Journal E, 20(2):173–178, 2006.
+[37] Caio Sátiro and Fernando Moraes. On the deflection of light by topological defects in nematic
+liquid crystals. The European Physical Journal E, 25(4):425–429, 2008.
+[38] Walter Gordon.
+Zur lichtfortpflanzung nach der relativitätstheorie.
+Annalen der Physik,
+377(22):421–456, 1923.
+[39] Pham Mau Quan. Inductions électromagnétiques en relativité générale et principe de fermat.
+43
+
+Archive for Rational Mechanics and Analysis, 1(1):54–80, 1957.
+[40] Jerzy Plebanski. Electromagnetic waves in gravitational fields. Physical Review, 118(5):1396,
+1960.
+[41] James Evans and Mark Rosenquist. “f= ma”optics. American Journal of Physics, 54(10):876–
+883, 1986.
+[42] Kamal K Nandi and Anwarul Islam. On the optical–mechanical analogy in general relativity.
+American Journal of Physics, 63(3):251–256, 1995.
+[43] James Evans, Kamal K Nandi, and Anwarul Islam. The optical-mechanical analogy in general
+relativity: exact newtonian forms for the equations of motion of particles and photons. General
+Relativity and Gravitation, 28(4):413–439, 1996.
+[44] Paul M Alsing. The optical-mechanical analogy for stationary metrics in general relativity.
+American Journal of Physics, 66(9):779–790, 1998.
+[45] Ulf Leonhardt and Thomas G Philbin.
+General relativity in electrical engineering.
+New
+Journal of Physics, 8(10):247, 2006.
+[46] Mikio Nakahara. Geometry, topology and physics. IOP Publishing Bristol and Philadelphia,
+2003.
+[47] Shiing-Shen Chern.
+Finsler geometry is just riemannian geometry without the quadratic
+equation. Notices of the American Mathematical Society, 43(9):959–963, 1996.
+[48] A Joets and R Ribotta. A geometrical model for the propagation of rays in an anisotropic
+inhomogeneous medium. Optics communications, 107(3-4):200–204, 1994.
+[49] Mikhail O Katanaev. Geometric theory of defects. Physics-Uspekhi, 48(7):675, 2005.
+[50] H Zocher. The effect of a magnetic field on the nematic state. Transactions of the Faraday
+Society, 29(140):945–957, 1933.
+[51] Roland De Wit. A view of the relation between the continuum theory of lattice defects and
+non-euclidean geometry in the linear approximation. International Journal of Engineering
+Science, 19(12):1475–1506, 1981.
+[52] Shuang Zhou, Sergij V Shiyanovskii, Heung-Shik Park, and Oleg D Lavrentovich. Fine struc-
+ture of the topological defect cores studied for disclinations in lyotropic chromonic liquid
+crystals. Nature Communications, 8(1):1–7, 2017.
+[53] Bruce Alexander Bilby, R Bullough, and Edwin Smith. Continuous distributions of dislo-
+cations: a new application of the methods of non-riemannian geometry. Proceedings of the
+44
+
+Royal Society of London. Series A. Mathematical and Physical Sciences, 231(1185):263–273,
+1955.
+[54] Ekkehart Kröner.
+Kontinuumstheorie der versetzungen und eigenspannungen, volume 5.
+Springer, 1958.
+[55] MA Krivoglaz.
+Physics of defects edited by r. balian, m. kléman and j.-p. poirier.
+Acta
+Crystallographica Section A: Foundations of Crystallography, 39(5):821–823, 1983.
+[56] Aida Kadić and Dominic GB Edelen.
+A gauge theory of dislocations and disclinations.
+Springer, 1983.
+[57] MO Katanaev and IV Volovich. Theory of defects in solids and three-dimensional gravity.
+Annals of Physics, 216(1):1–28, 1992.
+[58] MO Katanaev and IV Volovich. Scattering on dislocations and cosmic strings in the geometric
+theory of defects. Annals of Physics, 271(2):203–232, 1999.
+[59] Peter S Pershan. Dislocation effects in smectic-a liquid crystals. Journal of Applied Physics,
+45(4):1590–1604, 1974.
+[60] M Kléman and L Lejček. Screw dislocations in the smectic c phase of liquid crystals. Philo-
+sophical Magazine A, 42(5):671–682, 1980.
+[61] A De Padua, Fernando Parisio-Filho, and Fernando Moraes. Geodesics around line defects
+in elastic solids. Physics Letters A, 238(2-3):153–158, 1998.
+[62] Yoichi Takanishi, Hideo Takezoe, Atsuo Fukuda, and Junji Watanabe. Visual observation of
+dispirations in liquid crystals. Physical Review B, 45(14):7684, 1992.
+[63] RB Meyer, B Stebler, and ST Lagerwall. Observation of edge dislocations in smectic liquid
+crystals. Physical Review Letters, 41(20):1393, 1978.
+[64] Fernando Moraes. Geodesics around a dislocation. Physics Letters A, 214(3-4):189–192, 1996.
+[65] William Kingdon Clifford. On the space-theory of matter. In The Concepts of Space and
+Time, pages 295–296. Springer, 1976.
+[66] Emilio Elizalde. Bernhard riemann, a (rche) typical mathematical-physicist?
+Frontiers in
+Physics, 1:11, 2013.
+[67] Hermann Weyl.
+Eine neue erweiterung der relativitätstheorie.
+Annalen der Physik,
+364(10):101–133, 1919.
+[68] Arthur Stanley Eddington. Space, time and gravitation: An outline of the general relativity
+theory. Cambridge University Press, 1920.
+45
+
+[69] Charles W Misner and John A Wheeler. Classical physics as geometry. Annals of physics,
+2(6):525–603, 1957.
+[70] Gerard t Hooft. A locally finite model for gravity. Foundations of Physics, 38(8):733–757,
+2008.
+[71] Carlos Barceló, Stefano Liberati, and Matt Visser.
+Analogue gravity.
+Living reviews in
+relativity, 14(1):1–159, 2011.
+[72] Maxime J Jacquet, Silke Weinfurtner, and Friedrich König. The next generation of analogue
+gravity experiments, 2020.
+[73] M Simões and M Pazetti. Liquid-crystals cosmology. EPL (Europhysics Letters), 92(1):14001,
+2010.
+[74] Matt Visser. Essential and inessential features of hawking radiation. International Journal
+of Modern Physics D, 12(04):649–661, 2003.
+[75] Karl Raimund Popper. In search of a better world: lectures and essays from thirty years.
+Taylor and Francis, 1994.
+[76] John David Jackson and Lev Borisovich Okun. Historical roots of gauge invariance. Reviews
+of Modern Physics, 73(3):663, 2001.
+[77] Patrick Peter and Jean-Philippe Uzan. Primordial cosmology. Oxford University Press, 2009.
+[78] Rachel Jeannerot, Jonathan Rocher, and Mairi Sakellariadou. How generic is cosmic string
+formation in supersymmetric grand unified theories. Physical Review D, 68(10):103514, 2003.
+[79] Jean-Philippe Uzan and Patrick Peter. The no-defect conjecture in cosmic crystallography.
+Physics Letters B, 406(1-2):20–25, 1997.
+[80] Alan H Guth. Inflationary universe: A possible solution to the horizon and flatness problems.
+Physical Review D, 23(2):347, 1981.
+[81] Thomas WB Kibble. Topology of cosmic domains and strings. Journal of Physics A: Math-
+ematical and General, 9(8):1387, 1976.
+[82] Wojciech H Zurek. Cosmological experiments in condensed matter systems. Physics Reports,
+276(4):177–221, 1996.
+[83] Mark J Bowick, L Chandar, Eric A Schiff, and Ajit M Srivastava. The cosmological kibble
+mechanism in the laboratory: string formation in liquid crystals. Science, 263(5149):943–945,
+1994.
+[84] Sanatan Digal, Rajarshi Ray, and Ajit M Srivastava. Observing correlated production of
+46
+
+defects and antidefects in liquid crystals. Physical Review Letters, 83(24):5030, 1999.
+[85] Tom Kibble. Phase-transition dynamics in the lab and the universe. Physics Today, 60(9):47,
+2007.
+[86] H Mukai, PRG Fernandes, BF De Oliveira, and GS Dias. Defect-antidefect correlations in a
+lyotropic liquid crystal from a cosmological point of view. Physical Review E, 75(6):061704,
+2007.
+[87] R Repnik, A Ranjkesh, V Simonka, M Ambrozic, Z Bradac, and S Kralj. Symmetry breaking
+in nematic liquid crystals: analogy with cosmology and magnetism.
+Journal of Physics:
+Condensed Matter, 25(40):404201, 2013.
+[88] Alexander Vilenkin. Cosmic strings and domain walls. Physics reports, 121(5):263–315, 1985.
+[89] Robert H Brandenberger. Topological defects and structure formation. International Journal
+of Modern Physics A, 9(13):2117–2189, 1994.
+[90] Guillaume Duclos, Raymond Adkins, Debarghya Banerjee, Matthew SE Peterson, Minu
+Varghese, Itamar Kolvin, Arvind Baskaran, Robert A Pelcovits, Thomas R Powers, Aparna
+Baskaran, et al. Topological structure and dynamics of three-dimensional active nematics.
+Science, 367(6482):1120–1124, 2020.
+[91] Compare, for instance, with the Bohr radius a0 = 5.29 × 10−9 cm.
+[92] However, no isolated cosmic string corresponds to the case of an added Frank angle (saddle-
+like geometry). To our knowledge, the existence of a negative-mass cosmic string has been
+considered only when the defect is associated to a wormhole [? ? ].
+[93] Bertrand Berche, Sébastien Fumeron, and Fernando Moraes.
+Classical kalb-ramond field
+theory in curved spacetimes. Phys. Rev. D, 105:105026, May 2022.
+[94] Nick Kaiser and Albert Stebbins.
+Microwave anisotropy due to cosmic strings.
+Nature,
+310(5976):391–393, 1984.
+[95] Alexander Vilenkin. Looking for cosmic strings. Nature, 322(6080):613–614, 1986.
+[96] Alexander Vilenkin and E Paul S Shellard.
+Cosmic strings and other topological defects.
+Cambridge University Press, 1994.
+[97] Valdir B Bezerra.
+Gravitational analogue of the aharonov-bohm effect in four and three
+dimensions. Physical Review D, 35(6):2031, 1987.
+[98] Peter AR Ade, N Aghanim, C Armitage-Caplan, M Arnaud, M Ashdown, F Atrio-Barandela,
+J Aumont, C Baccigalupi, AJ Banday, RB Barreiro, et al. Planck 2013 results. xxv. searches
+47
+
+for cosmic strings and other topological defects. Astronomy & Astrophysics, 571:A25, 2014.
+[99] Wilfried Buchmuller, Valerie Domcke, and Kai Schmitz.
+From nanograv to ligo with
+metastable cosmic strings. Physics Letters B, 811:135914, 2020.
+[100] Simone Blasi, Vedran Brdar, and Kai Schmitz. Has nanograv found first evidence for cosmic
+strings? Physical Review Letters, 126(4):041305, 2021.
+[101] John Ellis and Marek Lewicki. Cosmic string interpretation of nanograv pulsar timing data.
+Physical Review Letters, 126(4):041304, 2021.
+[102] William A Hiscock. Exact gravitational field of a string. Physical Review D, 31(12):3288,
+1985.
+[103] J Richard Gott III.
+Gravitational lensing effects of vacuum strings-exact solutions.
+The
+Astrophysical Journal, 288:422–427, 1985.
+[104] Bruce Allen and Adrian C Ottewill. Effects of curvature couplings for quantum fields on
+cosmic-string space-times. Physical Review D, 42(8):2669, 1990.
+[105] Shanju Zhang, Eugene M Terentjev, and Athene M Donald. Nature of disclination cores in
+liquid crystals. Liquid Crystals, 32(1):69–75, 2005.
+[106] Denis Andrienko and Michael P Allen. Molecular simulation and theory of a liquid crystalline
+disclination core. Physical Review E, 61(1):504, 2000.
+[107] Saša Harkai, Kaushik Pal, and Samo Kralj. Manipulation of m= 1 topological disclination
+line core structure. Journal of Molecular Structure, 1234:130162, 2021.
+[108] Adam L Susser, Saša Harkai, Samo Kralj, and Charles Rosenblatt. Transition from escaped
+to decomposed nematic defects, and vice versa. Soft matter, 16(20):4814–4822, 2020.
+[109] Patrick Peter. Comments on some metric properties of cosmic strings having a non-degenerate
+stress–energy tensor. Classical and Quantum Gravity, 11(1):131, 1994.
+[110] Brandon Carter.
+Transonic elastic model for wiggly goto-nambu string.
+Physical review
+letters, 74(16):3098, 1995.
+[111] Frankbelson dos S. Azevedo, Fernando Moraes, Francisco Mireles, Bertrand Berche, and
+Sébastien Fumeron. Wiggly cosmic string as a waveguide for massless and massive fields.
+Phys. Rev. D, 96:084047, Oct 2017.
+[112] Levon Pogosian and Tanmay Vachaspati. Cosmic microwave background anisotropy from
+wiggly strings. Physical Review D, 60(8):083504, 1999.
+[113] We will consider here only models likely to have a liquid crystal analog, which -to our
+48
+
+knowledge- should exclude superconducting cosmic strings.
+[114] DV Gal’Tsov and PS Letelier. Spinning strings and cosmic dislocations. Physical Review D,
+47(10):4273, 1993.
+[115] Laércio Dias and Fernando Moraes. Effects of torsion on electromagnetic fields. Brazilian
+Journal of Physics, 35:636–640, 2005.
+[116] Jianhua Wang, Kai Ma, Kang Li, and Huawei Fan. Deformations of the spin currents by
+topological screw dislocation and cosmic dispiration. Annals of Physics, 362:327–335, 2015.
+[117] RLL Vitória and K Bakke. Rotating effects on the scalar field in the cosmic string spacetime,
+in the spacetime with space-like dislocation and in the spacetime with a spiral dislocation.
+The European Physical Journal C, 78(3):1–6, 2018.
+[118] Soroush Zare, Hassan Hassanabadi, and Marc de Montigny. Duffin–kemmer–petiau oscillator
+in the presence of a cosmic screw dislocation. International Journal of Modern Physics A,
+35(30):2050195, 2020.
+[119] Faizuddin Ahmed. Quantum influence of topological defects on a relativistic scalar particle
+with cornell-type potential in cosmic string space-time with a spacelike dislocation. Advances
+in High Energy Physics, 2020, 2020.
+[120] VA De Lorenci and ES Moreira Jr. Renormalized scalar propagator around a dispiration.
+Physical Review D, 67(12):124002, 2003.
+[121] Huabing Cai, Zhi Wang, and Zhongzhou Ren. Radiative properties of a static two-level atom
+in a cosmic dispiration spacetime. Classical and Quantum Gravity, 35(15):155016, 2018.
+[122] HF Mota, ER Bezerra de Mello, and K Bakke. Scalar casimir effect in a high-dimensional
+cosmic dispiration spacetime. International Journal of Modern Physics D, 27(12):1850107,
+2018.
+[123] KEL de Farias, EAF Bragança, and HF Santana Mota. Scalar self-interaction in the spacetime
+of a cosmic dispiration. Physics Letters B, 822:136660, 2021.
+[124] Basilis C Xanthopoulos. A rotating cosmic string. Physics Letters B, 178(2-3):163–166, 1986.
+[125] Tom G Mackay and Akhlesh Lakhtakia. Towards a metamaterial simulation of a spinning
+cosmic string. Physics Letters A, 374(23):2305–2308, 2010.
+[126] Grigorij E Volovik.
+Superfluid analogies of cosmological phenomena.
+Physics Reports,
+351(4):195–348, 2001.
+[127] BjOm Jensen. Notes on spinning strings. Classical and Quantum Gravity, 9(1):L7, 1992.
+49
+
+[128] J Richard Gott III. Closed timelike curves produced by pairs of moving cosmic strings: Exact
+solutions. Physical Review Letters, 66(9):1126, 1991.
+[129] Sébastien Fumeron, Bertrand Berche, Fernando Santos, Erms Pereira, and Fernando Moraes.
+Optics near a hyperbolic defect. Physical Review A, 92(6):063806, 2015.
+[130] David Figueiredo, Fernando Moraes, Sébastien Fumeron, and Bertrand Berche. Cosmology
+in the laboratory: An analogy between hyperbolic metamaterials and the milne universe.
+Physical Review D, 96(10):105012, 2017.
+[131] Mark Hindmarsh. Semilocal topological defects. Nuclear Physics B, 392(2):461–489, 1993.
+[132] Luis E Ibanez and Graham G Ross. Discrete gauge symmetries and the origin of baryon
+and lepton number conservation in supersymmetric versions of the standard model. Nuclear
+Physics B, 368(1):3–37, 1992.
+[133] Caio Satiro, AM de M. Carvalho, and Fernando Moraes. An asymmetric family of cosmic
+strings. Modern Physics Letters A, 24(18):1437–1442, 2009.
+[134] Maurice Kleman. Forms of matter and forms of radiation. arXiv preprint arXiv:0905.4643,
+2009.
+[135] Maurice Kleman and Jonathan M Robbins. Tubes of magnetic flux and electric current in
+space physics. Solar physics, 289(4):1173–1192, 2014.
+[136] Maurice Kleman. Some aspects of defect theory in spacetime. In THE THIRTEENTH MAR-
+CEL GROSSMANN MEETING: On Recent Developments in Theoretical and Experimental
+General Relativity, Astrophysics and Relativistic Field Theories, pages 2531–2533. World Sci-
+entific, 2015.
+[137] Luis Javier Garay, JR Anglin, J Ignacio Cirac, and P Zoller. Sonic black holes in dilute
+bose-einstein condensates. Physical Review A, 63(2):023611, 2001.
+[138] Yaron Kedem, Emil J Bergholtz, and Frank Wilczek.
+Black and white holes at material
+junctions. Physical Review Research, 2(4):043285, 2020.
+[139] William George Unruh.
+Experimental black-hole evaporation?
+Physical Review Letters,
+46(21):1351, 1981.
+[140] Juan Ramón Muñoz de Nova, Katrine Golubkov, Victor I Kolobov, and Jeff Steinhauer.
+Observation of thermal hawking radiation and its temperature in an analogue black hole.
+Nature, 569(7758):688–691, 2019.
+[141] Jordi Prat-Camps, Carles Navau, and Alvaro Sanchez.
+A magnetic wormhole.
+Scientific
+50
+
+reports, 5(1):1–5, 2015.
+[142] Frankbelson dos S Azevedo, José Diêgo M de Lima, Antônio de Pádua Santos, and Fernando
+Moraes. Optical wormhole from hollow disclinations. Physical Review A, 103(2):023516, 2021.
+[143] Erms R Pereira and Fernando Moraes. Flowing liquid crystal simulating the schwarzschild
+metric. Central European Journal of Physics, 9(4):1100–1105, 2011.
+[144] Michael S Morris and Kip S Thorne. Wormholes in spacetime and their use for interstellar
+travel: A tool for teaching general relativity. American Journal of Physics, 56(5):395–412,
+1988.
+[145] Stanley Deser, Roman Jackiw, and G’t Hooft. Three-dimensional einstein gravity: dynamics
+of flat space. Annals of Physics, 152(1):220–235, 1984.
+[146] Indeed, Einstein tensor and the curvature tensor are equivalent in 2+1 dimensions. This
+means that in source-free regions, the curvature tensor simply identifies with the empty space
+solution of Einstein’s equations.
+[147] Edward Witten. 2+ 1 dimensional gravity as an exactly soluble system. Nuclear Physics B,
+311(1):46–78, 1988.
+[148] Arkady L Kholodenko. Use of quadratic differentials for description of defects and textures
+in liquid crystals and 2+ 1 gravity. Journal of Geometry and Physics, 33(1-2):59–102, 2000.
+[149] Arkady L Kholodenko. Use of meanders and train tracks for description of defects and textures
+in liquid crystals and 2+ 1 gravity. Journal of Geometry and Physics, 33(1-2):23–58, 2000.
+[150] Patricio S Letelier. Spacetime defects: von kármán vortex street like configurations. Classical
+and Quantum Gravity, 18(17):3639, 2001.
+[151] Sébastien Fumeron, Bertrand Berche, Fernando Moraes, Fernando AN Santos, and Erms
+Pereira.
+Geometrical optics limit of phonon transport in a channel of disclinations.
+The
+European Physical Journal B, 90(5):1–8, 2017.
+[152] Bertrand Berche, Sébastien Fumeron, and Fernando Moraes. On the energy of topological
+defect lattices. Condensed Matter Physics, 23(2):1–7, 2020.
+[153] Mengfei Wang, Yannian Li, and Hiroshi Yokoyama. Artificial web of disclination lines in
+nematic liquid crystals. Nature communications, 8(1):1–7, 2017.
+[154] Yubing Guo, Miao Jiang, Sajedeh Afghah, Chenhui Peng, Robin LB Selinger, Oleg D Lavren-
+tovich, and Qi-Huo Wei. Photopatterned designer disclination networks in nematic liquid
+crystals. Advanced Optical Materials, 9(16):2100181, 2021.
+51
+
+[155] Yuji Sasaki, Junnosuke Takahashi, Shunsuke Yokokawa, Takuho Kikkawa, Ryota Mikami,
+and Hiroshi Orihara. A general control strategy to micropattern topological defects in ne-
+matic liquid crystals using ionically charged dielectric surface. Advanced Materials Interfaces,
+8(11):2100379, 2021.
+[156] Inge Nys, Brecht Berteloot, Jeroen Beeckman, and Kristiaan Neyts. Nematic liquid crys-
+tal disclination lines driven by a photoaligned defect grid.
+Advanced Optical Materials,
+10(4):2101626, 2022.
+[157] Saša Harkai, George Cordoyiannis, Adam L Susser, Bryce S Murray, Andrew J Ferris, Brigita
+Rožič, Zdravko Kutnjak, Charles Rosenblatt, and Samo Kralj. Manipulation of mechanically
+nanopatterned line defect assemblies in plane-parallel nematic liquid crystals. Liquid Crystals
+Reviews, pages 1–25, 2022.
+[158] Zongdai Liu, Dan Luo, and Kun-Lin Yang. Flow-driven disclination lines of nematic liquid
+crystals inside a rectangular microchannel. Soft Matter, 15(28):5638–5643, 2019.
+[159] Jinghua Jiang, Kamal Ranabhat, Xinyu Wang, Hailey Rich, Rui Zhang, and Chenhui Peng.
+Active transformations of topological structures in light-driven nematic disclination networks.
+Proceedings of the National Academy of Sciences, 119(23):e2122226119, 2022.
+[160] Rafael Sorkin. Time-evolution problem in regge calculus. Physical Review D, 12(2):385, 1975.
+[161] Rafael Sorkin.
+Erratum: time-evolution problem in regge calculus.
+Physical Review D,
+23(2):565, 1981.
+[162] Justin Khoury, Burt A Ovrut, Nathan Seiberg, Paul J Steinhardt, and Neil Turok. From big
+crunch to big bang. Physical Review D, 65(8):086007, 2002.
+[163] Paul J Steinhardt and Neil Turok. Cosmic evolution in a cyclic universe. Physical Review D,
+65(12):126003, 2002.
+[164] Gary T Horowitz and Alan R Steif. Singular string solutions with nonsingular initial data.
+Physics Letters B, 258(1-2):91–96, 1991.
+[165] Przemysław Małkiewicz and Włodzimierz Piechocki. Probing the cosmological singularity
+with a particle. Classical and Quantum Gravity, 23(23):7045, 2006.
+[166] Sébastien Fumeron, Erms Pereira, and Fernando Moraes. Generation of optical vorticity from
+topological defects. Physica B: Condensed Matter, 476:19–23, 2015.
+[167] David Figueiredo, Felipe A Gomes, Sébastien Fumeron, Bertrand Berche, and Fernando
+Moraes.
+Modeling kleinian cosmology with electronic metamaterials.
+Physical Review D,
+52
+
+94(4):044039, 2016.
+[168] Jie Xiang and Oleg D Lavrentovich.
+Liquid crystal structures for transformation optics.
+Molecular Crystals and Liquid Crystals, 559(1):106–114, 2012.
+[169] Grzegorz Pawlik, Karol Tarnowski, Wiktor Walasik, Antoni C Mitus, and IC Khoo. Liquid
+crystal hyperbolic metamaterial for wide-angle negative–positive refraction and reflection.
+Optics letters, 39(7):1744–1747, 2014.
+[170] Erms Pereira and Fernando Moraes.
+Diffraction of light by topological defects in liquid
+crystals. Liquid Crystals, 38(3):295–302, 2011.
+[171] Erms Pereira, Sébastien Fumeron, and Fernando Moraes. Metric approach for sound propa-
+gation in nematic liquid crystals. Physical Review E, 87(2):022506, 2013.
+[172] Uwe R Fischer and Matt Visser.
+Riemannian geometry of irrotational vortex acoustics.
+Physical review letters, 88(11):110201, 2002.
+[173] PE Cladis and M Kleman. Non-singular disclinations of strength s=+ 1 in nematics. Journal
+de Physique, 33(5-6):591–598, 1972.
+[174] Sébastien Fumeron, Fernando Moraes, and Erms Pereira. Retrieving the saddle-splay elastic
+constant k24 of nematic liquid crystals from an algebraic approach. The European Physical
+Journal E, 39(9):1–11, 2016.
+[175] M Criado-Sancho, FX Alvarez, and D Jou.
+Thermal rectification in inhomogeneous
+nanoporous si devices. Journal of Applied Physics, 114(5):053512, 2013.
+[176] Riccardo Dettori, Claudio Melis, Riccardo Rurali, and Luciano Colombo. Thermal rectifica-
+tion in silicon by a graded distribution of defects. Journal of Applied Physics, 119(21):215102,
+2016.
+[177] D Sawaki, W Kobayashi, Y Moritomo, and I Terasaki. Thermal rectification in bulk materials
+with asymmetric shape. Applied Physics Letters, 98(8):081915, 2011.
+[178] Zhongwei Zhang, Yuanping Chen, Yuee Xie, and Shengbai Zhang. Transition of thermal
+rectification in silicon nanocones. Applied Thermal Engineering, 102:1075–1080, 2016.
+[179] José Guilherme Silva, Sébastien Fumeron, Fernando Moraes, and Erms Pereira. High ther-
+mal rectifications using liquid crystals confined into a conical frustum. Brazilian Journal of
+Physics, 48(4):315–321, 2018.
+[180] Whereas homeotropic anchoring of nematics on flat surfaces is well mastered [8], anchor-
+ing nematics with an arbitrary angle on a curved surface is generally not trivial and it is
+53
+
+particularly sensitive to the saddle-splay constant K24 [? ].
+[181] Eduardo Viana, Fernando Moraes, Sebastien Fumeron, and Erms Pereira. High rectification in
+a broadband subwavelength acoustic device using liquid crystals. Journal of Applied Physics,
+125(20):204503, 2019.
+[182] AF Fray and D Jones. Large-angle beam deflector using liquid crystals. Electronics Letters,
+11:358, 1975.
+[183] A Sasaki and T Ishibashi. Liquid-crystal light deflector. Electronics Letters, 10(15):293–294,
+1979.
+[184] Shin Masuda, Sounosuke Takahashi, Toshiaki Nose, Susumu Sato, and Hiromasa Ito. Liquid-
+crystal microlens with a beam-steering function. Applied optics, 36(20):4772–4778, 1997.
+[185] Boris Apter, Eldad Bahat-Treidel, and Uzi Efron. Continuously controllable, wide-angle liquid
+crystal beam deflector based on the transversal field effect in a three-electrode cell. Optical
+Engineering, 44:054001, 2005.
+[186] Urban Mur, Miha Ravnik, and David Seč. Controllable shifting, steering, and expanding of
+light beam based on multi-layer liquid-crystal cells. Scientific Reports, 12(1):1–13, 2022.
+[187] H Lin, P Palffy-Muhoray, and MA Lee. Liquid crystalline cores for optical fibers. Molecular
+Crystals and Liquid Crystals, 204(1):189–200, 1991.
+[188] H Lin and Palffy-Muhoray. Te and tm modes in a cylindrical liquid-crystal waveguide. Opt.
+Lett., 19:722–724, 1992.
+[189] Miha Čančula, Miha Ravnik, and Slobodan Žumer. Liquid microlenses and waveguides from
+bulk nematic birefringent profiles. Optics Express, 24(19):22177–22188, 2016.
+[190] Miha Čančula, Miha Ravnik, and Slobodan Žumer. Nematic topological line defects as optical
+waveguides. In Emerging Liquid Crystal Technologies X, volume 9384, page 938402. SPIE,
+2015.
+[191] Etienne Brasselet. Singular optics of liquid crystal defects. Liquid Crystals: New Perspectives,
+P Pieranki and MH Godinho, Wiley, pages 1–70, 2021.
+[192] Xiang Zhang and Zhaowei Liu. Superlenses to overcome the diffraction limit. Nature mate-
+rials, 7(6):435–441, 2008.
+[193] The losses due to metallic components do not jeopardize the performances of the device,
+as they might be offset by using gain media as pointed in [168] (highly doped oxides with
+lower dissipation levels have also been considered ). The low-loss limit for metamaterials can
+54
+
+reached by working within the terahertz waveband.
+[194] Frankbelson dos S Azevedo, David Figueiredo, Fernando Moraes, Bertrand Berche, and
+Sébastien Fumeron. Optical concentrator from a hyperbolic liquid-crystal metamaterial. EPL
+(Europhysics Letters), 124(3):34006, 2018.
+[195] Oleg D Lavrentovich. Liquid crystals, photonic crystals, metamaterials, and transformation
+optics. Proceedings of the National Academy of Sciences, 108(13):5143–5144, 2011.
+[196] Xiaobing Shang, Dieter Cuypers, Tigran Baghdasaryan, Michael Vervaeke, Hugo Thienpont,
+Jeroen Beeckman, Kristiaan Neyts, Quan Li, Chao Wu, Hongqiang Li, et al. Active optical
+beam shaping based on liquid crystals and polymer micro-structures. Crystals, 10(11):977,
+2020.
+[197] P Vaveliuk, F Moraes, S Fumeron, O Martinez-Matos, and ML Calvo.
+Structure of the
+dielectric tensor in nematic liquid crystals with topological charge.
+J. Opt. Soc. Am. A,
+27(6):1466, 2010.
+[198] Lorenzo Marrucci, C Manzo, and D Paparo. Pancharatnam-berry phase optical elements
+for wave front shaping in the visible domain: Switchable helical mode generation. Applied
+Physics Letters, 88(22):221102, 2006.
+[199] Lorenzo Marrucci. Rotating light with light: Generation of helical modes of light by spin-to-
+orbital angular momentum conversion in inhomogeneous liquid crystals. In Liquid Crystals
+and Applications in Optics, volume 6587, pages 56–66. SPIE, 2007.
+[200] Lorenzo Marrucci. Generation of helical modes of light by spin-to-orbital angular momen-
+tum conversion in inhomogeneous liquid crystals. Molecular Crystals and Liquid Crystals,
+488(1):148–162, 2008.
+[201] Sergei Slussarenko, Anatoli Murauski, Tao Du, Vladimir Chigrinov, Lorenzo Marrucci, and
+Enrico Santamato. Tunable liquid crystal q-plates with arbitrary topological charge. Optics
+express, 19(5):4085–4090, 2011.
+[202] Lorenzo Marrucci. The q-plate and its future. Journal of Nanophotonics, 7(1):078598, 2013.
+[203] Andrea Rubano, Filippo Cardano, Bruno Piccirillo, and Lorenzo Marrucci. Q-plate technol-
+ogy: a progress review. JOSA B, 36(5):D70–D87, 2019.
+[204] Miha Čančula, Miha Ravnik, and Slobodan Žumer. Generation of vector beams with liquid
+crystal disclination lines. Physical Review E, 90(2):022503, 2014.
+[205] Etienne Brasselet. Tunable optical vortex arrays from a single nematic topological defect.
+55
+
+Physical review letters, 108(8):087801, 2012.
+[206] D Voloschenko and OD Lavrentovich.
+Optical vortices generated by dislocations in a
+cholesteric liquid crystal. Optics Letters, 25(5):317–319, 2000.
+[207] Junji Kobashi, Hiroyuki Yoshida, and Masanori Ozaki. Polychromatic optical vortex gen-
+eration from patterned cholesteric liquid crystals. Physical review letters, 116(25):253903,
+2016.
+[208] OV Bogdanov, PO Kazinski, PS Korolev, and G Yu Lazarenko. Generation of hard twisted
+photons by charged particles in cholesteric liquid crystals. Physical Review E, 104(2):024701,
+2021.
+[209] Jin-Sheng Wu and Ivan I Smalyukh.
+Hopfions, heliknotons, skyrmions, torons and both
+abelian and nonabelian vortices in chiral liquid crystals. Liquid Crystals Reviews, pages 1–35,
+2022.
+[210] Baeksik Son, Sejeong Kim, Yun Ho Kim, K Käläntär, Hwi-Min Kim, Hyeon-Su Jeong, Siy-
+oung Q Choi, Jonghwa Shin, Hee-Tae Jung, and Yong-Hee Lee. Optical vortex arrays from
+smectic liquid crystals. Optics express, 22(4):4699–4704, 2014.
+[211] V Yu Bazhenov, MV Vasnetsov, and MS Soskin. Laser beams with screw dislocations in their
+wavefronts. Jetp Lett, 52(8):429–431, 1990.
+[212] Péter Salamon, Nándor Éber, Yuji Sasaki, Hiroshi Orihara, Ágnes Buka, and Fumito Araoka.
+Tunable optical vortices generated by self-assembled defect structures in nematics. Physical
+Review Applied, 10(4):044008, 2018.
+[213] Grigorios A Pavliotis. Stochastic processes and applications: diffusion processes, the Fokker-
+Planck and Langevin equations, volume 60. Springer, 2014.
+[214] Matteo Smerlak. Tailoring diffusion in analog spacetimes. Physical Review E, 85(4):041134,
+2012.
+[215] Sébastien Fumeron, Fernando Moraes, and Erms Pereira. Thermal and shape topological
+robustness of heat switchers using nematic liquid crystals. The European Physical Journal E,
+41(2):1–9, 2018.
+[216] Sébastien Fumeron, E Pereira, and F Moraes. Principles of thermal design with nematic
+liquid crystals. Physical Review E, 89(2):020501, 2014.
+[217] Jin Ki Kim, Khoa Van Le, Surajit Dhara, Fumito Araoka, Ken Ishikawa, and Hideo Takezoe.
+Heat-driven and electric-field-driven bistable devices using dye-doped nematic liquid crystals.
+56
+
+Journal of Applied Physics, 107(12):123108, 2010.
+[218] Geoff Wehmeyer, Tomohide Yabuki, Christian Monachon, Junqiao Wu, and Chris Dames.
+Thermal diodes, regulators, and switches: Physical mechanisms and potential applications.
+Applied Physics Reviews, 4(4):041304, 2017.
+[219] MY Wong, CY Tso, TC Ho, and HH Lee. A review of state of the art thermal diodes and
+their potential applications. International Journal of Heat and Mass Transfer, 164:120607,
+2021.
+[220] Ivan Haller. Thermodynamic and static properties of liquid crystals. Progress in solid state
+chemistry, 10:103–118, 1975.
+[221] Djair Melo, Ivna Fernandes, Fernando Moraes, Sébastien Fumeron, and Erms Pereira. Ther-
+mal diode made by nematic liquid crystal. Physics Letters A, 380(38):3121–3127, 2016.
+[222] Wallysson KP Barros and Erms Pereira. Concurrent guiding of light and heat by transforma-
+tion optics and transformation thermodynamics via soft matter. Scientific reports, 8(1):1–11,
+2018.
+[223] Silvio J Santos Jr, Jair Andrade, and Erms Pereira. Simultaneous rectification of heat and
+light using liquid crystal. Journal of Applied Physics, 124(9):094501, 2018.
+[224] Bennett Chow and Dan Knopf. The ricci flow: an introduction. Mathematical surveys and
+monographs, 110, 2008.
+57
+