# Dual Symmetry in Bent-Core Liquid Crystals and Unconventional Superconductors

^{1}

^{2}

^{*}

## Abstract

**:**

_{c}superconductors by considering additional secondary pseudo-proper order parameters. These systems exhibit a remarkable analogy relating their symmetry groups, lists of phases, and an infinite set of physical tensors. This analogy lies upon an internal dual structure shared by the two theories. We study the dual operator transforming rotations into translations in liquid crystals, and gauge symmetries into rotations in superconductors. It is used to classify the bent-core line defects, and to analyze the electronic gap structure of lamellar d-wave superfluids.

## 1. Introduction

_{P}phase predicted in 1992 [10] and then observed in bent-core systems [10,11], EL describes the commonly observed B2 phase [12,13], and we have proposed that C is an unmodulated approximation of the intriguing B7 phase found in 1999 [2]. In these phases the polarization and tilt waves are "parallel", in a sense explained below. However, we shall show that when one permits the two waves to be "non parallel", i.e., relatively shifted along the smectic normal or relatively rotated, four additional phases can be stabilized. The reason of this unusual extension of the polymorphism, which is characteristic of transitions with continuous-symmetry breakdowns, has been described within the context of superconductivity in Ref. [14].

_{c}lamellar superconductors or unconventional superfluid films. This analogy relies upon the fact that the two theories have the same "image group" [16], that is, the same set of matrices associated with the symmetries of the parent, isotropic or normal, phases. The main consequence of this peculiarity is that both systems have the same list of phases with analog symmetry groups, the same theoretical phase diagrams, and the same thermodynamic variations of the primary and secondary order parameters. At first sight the analogy is rather formal since a common matrix can be associated with distinct actual symmetry elements in each system. For instance, a gauge symmetry in superconductors yields the same matrix as a translation in liquid crystals. Thus, we do not expect a common behavior of the same physical quantities. However, the analogy is more subtle for it relates in fact the behaviors of distinct, but analog, tensors. For instance, the macroscopic polarization normal to the smectic planes exhibits exactly the same thermodynamic behavior as the linear magneto-electric suceptibility in superconductors. More precisely, both cancel in analog sets of ordered phases, with the same critical exponents.

## 2. The Vector-Wave Model

_{x}, p

_{y}, ${\phi}_{x}$ and ${\phi}_{y}$ are the real amplitudes and phases of the wave. The characteristic features of $\overrightarrow{\mathrm{P}}$(z) may be more conveniently expressed in terms of the following complex amplitudes:

_{3}, where O(3) is the orthogonal group generated by rotations and inversion, and T

_{3}is the 3D continuous translation group. It is spanned by an infinite set of waves propagating along all the directions of space. However, since for stabilizing ordered smectic-type phase one needs to consider only the two parallel wave vectors $\overrightarrow{k}$=k${\overrightarrow{e}}_{z}$ and -$\overrightarrow{k}$ appearing in Eq. (1), one can restrict the symmetry analysis to the subgroup ${D}_{\infty h}\times {T}_{3}$ of O(3)$\times $T

_{3}leaving the set {$\overrightarrow{k}$ and -$\overrightarrow{k}$} invariant. ${D}_{\infty h}\times {T}_{3}$ is generated by the rotations ${C}_{\phi}$ around z, the space inversion I, one mirror plane ${\sigma}_{x}$ parallel to $\overrightarrow{k}$ and the translations T

_{t}parallel to Oz. The action of these symmetries on the complex amplitudes (${\eta}_{1},{\eta}_{2},{\eta}_{1}{}^{*},{\eta}_{2}{}^{*}$) is given by the matrices:

_{1}, a

_{2}and ${\rho}_{1},{\rho}_{2}$, which describe the shape of $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{P}}$ separately, and, on the other hand, ${\phi}_{R}^{P}-{\phi}_{R}^{A},{\phi}_{T}^{P}-{\phi}_{T}^{A}$ which describe the relative orientation and z-position of the two waves. ${\phi}_{R}$ is naturally associated with the subgroup generated by rotations and the space inversion I since it transforms as:

_{2x}).

_{1}22.

_{1}/m.

_{1}ma.

_{1}.

_{P}=R, B2=EL, C’ or EL’, B6=R’ or R", B7= C, C’ or EL’, Sm0=EL’, and B8 is a subphase of EL’ obtained with an additional homogeneous order parameter.

_{3.}For non-zero wave vectors, the tensor waves are classified according to the 2D-rotation little group SO(2) of $\overrightarrow{k}$.

_{x},V

_{y},V

_{z}), splits into one 2D-vector (${\Gamma}_{1}^{}$:V

_{x},V

_{y}) plus one even 2D-scalar (${\Gamma}_{0}^{+}$:V

_{z}).

_{L}=${D}_{\infty h}\times {T}_{1}$, which is isomorphic to the abstract group O(2)$\otimes $O(2). The first copy of the 2D orthogonal group O(2) is ${C}_{\infty \mathrm{v}}$, generated by the rotations about Oz and the mirror plane ${\sigma}_{x}$. The second copy contains the group T

_{1}of translations along Oz (mod. 2$\pi $/k since we consider only waves with wave vectors p$\overrightarrow{k}$) and the space inversion I. Thus the image group can be rewritten as:

_{L}. C

_{S}={e,${\sigma}_{x}$} and C

_{i}={e,I} (rigorously speaking the actual image group is rather G

_{L}/{e,${C}_{\pi}{T}_{\pi /k}$} since ${C}_{\pi}{T}_{\pi /k}$ is represented by the identity matrix in Eq. (3)).

_{L}. The irreducible representations of O

_{T}(2) and O

_{R}(2) are both labeled ${\Gamma}_{0}^{+}$, ${\Gamma}_{0}^{-}$ and $\Gamma $

_{n}(n positive integer). For O

_{R}(2), $\Gamma $

_{0}

^{+}and $\Gamma $

_{0}

^{–}represent 2D-scalars, respectively even and odd with respect to ${\sigma}_{x}$, and $\Gamma $

_{n}represents a 2D-tensor of rank n. In O

_{T}(2), $\Gamma $

_{0}

^{+}and $\Gamma $

_{0}

^{–}represent homogeneous quantities respectively even and odd with respect to I, and $\Gamma $

_{p}represents a wave with wave vector p$\overrightarrow{k}$. Then the full set of tensor waves can be classified according to the following irreducible representations of O

_{R}(2) $\otimes $ O

_{T}(2):

_{n}$\otimes $${\Gamma}_{0}^{+}$: homogeneous tensor of rank n, symmetric under I (A${}_{\mathrm{x}}^{\left(0\right)}$, A${}_{\mathrm{y}}^{\left(0\right)}$ for n=1).

_{n}$\otimes $${\Gamma}_{0}^{-}$: homogeneous tensor of rank n, odd under I (P${}_{\mathrm{x}}^{\left(0\right)}$, P${}_{\mathrm{y}}^{\left(0\right)}$ for n=1).

_{p}: scalar wave with wave vector p$\overrightarrow{k}$ (density wave or P

_{z}(z) for p=1).

_{p}: pseudo-scalar wave with wave vector p$\overrightarrow{k}$ (e.g., A

_{z}(z) for p=1).

_{n}$\otimes $$\Gamma $

_{p}: tensor wave with rank n and wave vector p$\overrightarrow{k}$ (e.g., order parameters $\overrightarrow{\mathrm{A}}$(z) and $\overrightarrow{\mathrm{P}}$(z) for n=p=1).

^{(0)}and $\overrightarrow{\mathrm{P}}$

^{(0)}are homogeneous axial and polar vectors, whereas $\overrightarrow{\mathrm{A}}$(z) and $\overrightarrow{\mathrm{P}}$(z) are axial and polar transverse vector waves with wave vector $\overrightarrow{k}$. The transverse components of the axial vector $\overrightarrow{\mathrm{A}}$

^{(0)}represent the tilt vector. Its longitudinal component A${}_{\mathrm{z}}^{\left(0\right)}$ can be interpreted as the component t

_{xxz}+t

_{yyz}of the second-order electroclinic tensor t

_{ijk}defined by $\delta $A${}_{\mathrm{i}}^{\left(0\right)}$=t

_{ijk}E

_{j}E

_{k}, which describes the action of high electric fields on the homogeneous tilt vector. The dielectric and optic tensors {${\epsilon}_{xx}-{\epsilon}_{yy},{\epsilon}_{xy}$} are examples of $\Gamma $

_{2}$\otimes $${\Gamma}_{0}^{+}$, whereas the first-order electroclinic tensor {${\epsilon}_{\mathrm{xx}}^{\mathrm{c}}-{\epsilon}_{\mathrm{yy}}^{\mathrm{c}},{\epsilon}_{\mathrm{xy}}^{\mathrm{c}}$}, where $\delta $${\mathrm{A}}_{\mathrm{i}}^{\left(0\right)}={\epsilon}_{\mathrm{ij}\text{}}^{\mathrm{c}}{\mathrm{E}}_{\mathrm{j}}$, provides an example of $\Gamma $

_{2}$\otimes $${\Gamma}_{0}^{-}$.

**P**(z).

**P**(z) = n

_{L}+A

_{T}Cos(2kz) + B

_{T}Sin(2kz)

_{L}the scalar ${\rho}_{1}^{2}+{\rho}_{2}^{2}$, and $[{A}_{T},{B}_{T}]=2{\rho}_{1}{\rho}_{2}[\mathrm{cos}({\varphi}_{1}-{\varphi}_{2}),-\mathrm{sin}({\varphi}_{1}-{\varphi}_{2})]$ transforms as ${\Gamma}_{2}^{}\otimes {\Gamma}_{0}^{+}$. To visualize M(z) we consider M and kz as the radial and angular polar coordinates, respectively, in an abstract plane where the function M(z) defines then a closed orthorhombic curve (Figure 2a). It is invariant under a rotation ${\mathrm{C}}_{\phi}$, and it rotates under a translation T

_{t}. In the EL-phase the curve has the shape of a double balloon which becomes circular in the C-phase, and which splits into two parts in the R-phase. The splitting results from the cancellation of

**P**(z) at periodic positions, denoted by “nodes” of the wave, along z in R. The presence of a node is associated with specific symmetry elements that we shall discuss below. Let us notice that a "dual" description of the angular properties of the wave is provided by the ${\Gamma}_{2}^{}\otimes {\Gamma}_{0}^{+}$–type tensor: $[{A}_{R},{B}_{R}]=2{\rho}_{1}{\rho}_{2}[-\mathrm{cos}({\varphi}_{1}+{\varphi}_{2}),\mathrm{sin}({\varphi}_{1}+{\varphi}_{2})]$. Finally, these two tensors and corresponding figures can be similarly defined with the secondary wave $\overrightarrow{\mathrm{A}}$(z) in order to distinguish graphically primed and unprimed phases.

_{z}(z) with period $\lambda $), and anticlinicity (wave A

_{z}(z) with period $\lambda $), on the other hand. Then, the ordered phases can only be transversely antiferro- and longitudinally ferro- electric or -clinic. Likewise, only even harmonics of the density wave can condense, so that the smectic character of the phases corresponds always to a bilayer ordering (smectic period = $\lambda $/2).

_{xxz}+t

_{yyz}). But, in contrast to C’, no homogeneous polarization arises along the helix axis. Conversely, at the R$\to $R” transition the system becomes longitudinally ferroelectric, but without any second-order electroclinic effect.

_{xxz}+t

_{yyz}). Finally, at the R’$\to $EL’ transition the system becomes chiral, polar along Oz with first-order electroclinicity, whereas at the R”$\to $EL’ transition the chirality is accompanied by a longitudinal pseudo-scalar wave with period $\lambda $/2 and a second-order electroclinic tensor.

_{(R)}$\otimes $$\Gamma $

_{(T)}(R, T=n, ${}_{0}^{+}$ or ${}_{0}^{-}$) is permitted, then the “dual” tensor $\Gamma $

_{(T)}$\otimes $$\Gamma $

_{(R)}is also permitted. Furthermore, if $\Gamma $

_{(R)}$\otimes $$\Gamma $

_{(T)}is permitted in R’ then $\Gamma $

_{(T)}$\otimes $$\Gamma $

_{(R)}is permitted in R”. Finally, all tensors of the symmetric type ($\Gamma $

_{(T)}$\otimes $$\Gamma $

_{(T)}) and only them, are permitted in C. The following section is devoted to explain and formalize these regularities that we shall denote by the rotation/translation dual character of the vector-wave model.

## 3. The Dual Symmetry

_{L}(Eq. 7), in which the translation subgroup T

_{1}$\times $C

_{i}is isomorphic to the rotation subgroup ${C}_{\infty \mathrm{v}}$, (ii) in the fact that the tensor-wave representation associated with the order parameter $\overrightarrow{\mathrm{P}}$(z) is ${\Gamma}_{1}\otimes {\Gamma}_{1}$, in which the indices have the same unit value (n=p=1) for its tensor (rotations) and wave (translations) aspects. This duality relates different symmetry operations (e.g., rotations ${C}_{\varphi}$ to translations ${T}_{\varphi /k}$), different tensor fields (e.g., ${\Gamma}_{1}\otimes {\Gamma}_{2}$ to ${\Gamma}_{2}\otimes {\Gamma}_{1}$) different components of the order parameter (${\phi}_{T}$ to ${\phi}_{R}$) and different phases (e.g. R’ to R”). It permits one to explain also various specific properties of “self-dual” objects such as ${C}_{\varphi}{T}_{\varphi /k}$, ${\Gamma}_{1}\otimes {\Gamma}_{1}$or the C-phase.

_{L}might be used as well as the definition of a possible duality operator). Furthermore, D permutes also the elements of O

_{R}(2) with those of O

_{T}(2):

_{L}, then one finds an “extended” group ${\tilde{G}}_{L}$ which contains “dual-free” elements g (belonging to G

_{L}) together with dual combinations gD of D with dual-free operations. ${\tilde{G}}_{L}$ is not a symmetry group in the usual sense because D is a classificatory operation, which has a descriptive role, and not a dynamical symmetry constraining the physical properties of the system. However, the free energy of the vector-wave model is invariant under D, so that the list of low-symmetry phases and their thermodynamic properties have well-defined dual features (e.g., dual pairs of phases appear in symmetric parts of the theoretical phase diagram). This makes a classification of the phases based on the duality operator consistent. We denote by ${\tilde{G}}_{ph}$ the extended symmetry group of the phase "ph" (=C, R, EL, R’…), defined as the subgroup of ${\tilde{G}}_{L}$ leaving $\overrightarrow{\mathrm{P}}$(z) and $\overrightarrow{\mathrm{A}}$(z) invariant.

^{−1}). Thus, it turns out to be more convenient to distinguish between “dual-free groups” (containing only dual-free operations) and “dual groups” (containing dual operations). The unprimed phases (R, C, EL) have dual groups while the primed phases (C’, R’, R”, EL’) have dual-free groups. In a phase with a dual group it is always possible to find one domain (by setting the two Goldstone phases to zero) in which D is present. Accordingly, we use the Shubnikov’s notation applied to this special domain for denoting the extended symmetry groups of the unprimed phase as ${\tilde{G}}_{ph}={G}_{ph}{1}^{\prime}$whereas in the primed phase ${\tilde{G}}_{ph}={G}_{ph}$.

_{L}. It commutes only with the operations ${\mathrm{C}}_{\varphi}{\mathrm{T}}_{\mathrm{t}}$ provided that $\varphi -\mathrm{kt}=\mathrm{n}\pi $ (n integer) and with the twofold axes U

_{2x}and U

_{2y}(or with a different pair of perpendicular twofold axes if one chooses another definition for D). We denote such operations as “self-dual” (they form a group coinciding with that of a cholesteric). For the non self-dual operations g one can thus define their dual conjugate DgD (DgD=g when g is self-dual, because D

^{2}is the identity operation). This permits one to make a more subtle classification of the low-symmetry groups that splits them into “self-dual groups” (which contain only self-dual operations), “globally self-dual groups” (such that D${\tilde{G}}_{ph}$D=${\tilde{G}}_{ph}$) and non-self-dual groups. In the latter case ${\tilde{G}}_{ph}$ is associated with its “dual-conjugate” D${\tilde{G}}_{ph}$D. Any dual group is automatically globally self-dual. C, C’, EL and EL’ are self-dual, R is globally self-dual, and R’ and R” are non-self-dual phases which are mutually dual conjugated. The analogy between duality and time reversal is correct only for the self-dual groups, since in this case D is present in all the domains of the dual phases. C and EL are then analogous to paramagnetic structures whereas C’ and EL’ are analogous to ferromagnetics.

_{xx}–$\epsilon $

_{yy},$\epsilon $

_{xy}) (second-rank homogeneous tensor) is dual-conjugated with the second harmonic of the density wave.

_{C}forbids any homogeneous tensor (except scalars) and, according to the previous duality rule, any scalar wave. Thus, C is macroscopically 2D-isotropic and not smectic: It is optically uniaxial, on the one hand, and it does not give rise to Bragg peaks in normal (non-resonant) x-ray diffraction, on the other hand. Moreover, the order-parameter modulus n

_{L}(${\Gamma}_{0}^{+}\otimes {\Gamma}_{0}^{+}$) and the chirality index $\chi $ (${\Gamma}_{0}^{-}\otimes {\Gamma}_{0}^{-}$) are permitted while the macroscopic longitudinal polarization P${}_{\mathrm{z}}^{\left(0\right)}$ (${\Gamma}_{0}^{+}\otimes {\Gamma}_{0}^{-}$) and tilt vector A${}_{\mathrm{z}}^{\left(0\right)}$ (${\Gamma}_{0}^{-}\otimes {\Gamma}_{0}^{+}$) vanish. Indeed, the helical symmetry of C makes it “maximally self-dual” because it allows only self-dual tensor waves (${\Gamma}_{a}^{}\otimes {\Gamma}_{a}^{}$) and, furthermore, all the self-dual tensor waves. For instance, in the x–y plane a single harmonic (with wave vector $\overrightarrow{k}$) of the polarization $\overrightarrow{\mathrm{P}}$(z) and tilt $\overrightarrow{\mathrm{A}}$(z) waves are allowed, hence making the C-phase perfectly helielectric. Similarly, a single harmonic (with wave vector 2$\overrightarrow{k}$) of the 2D-optic tensor ($\epsilon $

_{xx}(z)–$\epsilon $

_{yy}(z),$\epsilon $

_{xy}(z)) is permitted, yielding the same rotatory-power effect than in a cholesteric phase, and the same optic gap features (within a much shorter wavelength range).

_{xx}–$\epsilon $

_{yy},$\epsilon $

_{xy}), which makes the phase biaxial. R is also characterized by the onset of all the odd harmonics of $\overrightarrow{\mathrm{P}}$ (${\Gamma}_{1}^{}\otimes {\Gamma}_{2p+1}^{}$), each one giving one resonant Bragg peak at (2p+1)$\overrightarrow{k}$. Simultaneously, the first harmonic of all the odd-rank tensor waves (${\Gamma}_{2n+1}^{}\otimes {\Gamma}_{1}^{}$) are present in R.

^{n}, whereas the modulus of the tensor with rank 2n vanishes as (T–T”)

^{n}at the R”$\to $R transition temperature T”.

## 4. Dual classification of Line Defects

#### 4.1. Circular phases

_{−},$\lambda $

_{+})-lines involve spatial variations of the direction of $\overrightarrow{k}$. They cannot be induced by transitions from defect-free nematics or in $\overrightarrow{k}$-locked configurations, but only in the isotropic liquid or in nematics with preexisting disclinations.

_{2}) while it becomes polar and monoclinic (C

_{2}) in C, and triclinic (C

_{1}) in C’. Thus, although the disclinations are similar in these three phases, four types can be found in cholesterics, while only three types exist in C, and one in C’:

_{+}and $\lambda $

_{−}. Like $\chi $ they can be observed in circular phases appearing in nematics provided that a ±$\pi $-disclination line preexists. A nematic wedge disclination (angle 2$\pi $m) transforms in $\lambda $

_{+}(for m=1/2) or $\lambda $

_{-}(for m=-1/2) at the Nematic$\to $C transition. $\lambda $

_{−}and $\lambda $

_{+}become topologically unstable in C’.

_{−}and $\lambda $

_{+}yields the following merging effects: Two lines of the same type annihilate each other. Two lines of different types combine in such a way as to form a single defect of the third type.

#### 4.2 Linear Phases

(iii/+)(iii/-)=(iv/+)(iv/-)={n,0} (or {-n,1})

(iii/+)(iv/+)=(iii/-)(iv/-)={n+1/2,1/2} (or {-n-1/2,3/2})

(iii/+)(iv/-)=(iii/-)(iv/+)={n+1/2,3/2} (or {-n-1/2,1/2})

{n,0}{iii/+}={n,1}{iii/-}={n+1/2,1/2}{iv/+}={n+1/2,3/2}{iv/-}={iii/+}

{n,0}{iv/+}={n,1}{iv/-}={n+1/2,3/2}{iii/+}={n+1/2,1/2}{iii/-}={iv/+}

{n,0}{iii/-}={n,1}{iii/+}={n+1/2,3/2}{iv/+}={n+1/2,1/2}{iv/-}={iii/-}

{n,0}{iv/-}={n,1}{iv/+}={n+1/2,1/2}{iii/+}={n+1/2,3/2}{iii/-}={iv/-}

#### 4.3 Elliptic Phases

_{mod}=1/Q and $\lambda $=1/K<<$\lambda $

_{mod}are incommensurate. The plane containing the polarization of two successive molecules precesses slowly around Oz. This helical structure differs from that in C and C’ in two respects: (i) The length scale of the elliptic helical modulation is much larger than the molecular size (typically within the optic range) while the scale of the pitch $\lambda $ in the circular structures ranges within molecular length. (ii) The helical symmetry in C and C’ is continuous (perfect helix) while it is discrete in EL and EL’. However, the symmetry groups of EL and EL’ are not broken by the incommensurate modulations. Indeed, Eq. (10) shows that the modulated space groups are generated by the screw axis ${T}_{\lambda /2}{C}_{\pi -Q\lambda /2}$ (which becomes 2

_{1}when Q=0) and U

_{2x}. The groups with Q=0 and Q≠0 are isomorphic. Accordingly, the classification of line defects is not modified by the modulation though their spatial structures are changed.

## 5. Analogy With d-Wave Superconductivity

_{3}(p-wave pairing) [26,27,28], the heavy fermions (p or d-wave pairing) [30] and the high-T

_{c}oxide superconductors (d-wave pairing) [31,32]. We shall show now that the theory of d-wave 2D superconductors is strongly analogous to the theory of bent-core liquid crystals.

_{1z}and S

_{2z}be the spins of two fermions at positions $\overrightarrow{\mathrm{r}}$

_{1}and $\overrightarrow{\mathrm{r}}$

_{2}, and $\psi $(r, $\theta $,S

_{1z},S

_{2z}) their wave function. r and $\theta $ are the polar coordinates of the relative position $\overrightarrow{\mathrm{r}}$

_{1}-$\overrightarrow{\mathrm{r}}$

_{2}. In a d-wave the orbital momentum is L=2, and the spin is in the singlet state S(S

_{1z},S

_{2z}) [15]:

_{1z}, S

_{2z}) = {D+ exp(2i θ) + D

_{−}exp(-2i θ)} S(S

_{1z}, S

_{2z}).

_{+}=|D

_{+}|exp(i $\varphi $

_{+}), D

_{−}=|D

_{−}|exp(i $\varphi $

_{−}), D

_{+}* and D

_{−}* are the four complex components of the order parameter. They transform according to an irreducible representation of the normal phase symmetry group, which contains the gauge transformations g${}_{\alpha}$, the continuous 2D rotations C${}_{\phi}$, as well as the discrete time reversal T and mirror plane $\sigma $

_{x}operations. The corresponding matrices are given by:

_{+},D

_{+}*,D

_{–},D

_{–}*}. These matrices generate the “image group”:

_{G}(2) ⊗ O

_{R}(2)

_{G}(2) contains gauge and time reversal transformations whereas O

_{R}(2) is the 2D rotation group C${}_{\infty}$

_{v}. As in the vector wave model O

_{G}(2) and O

_{R}(2) are isomorphic copies of the 2D orthogonal group O(2) (see Eq. (7)). One sees immediately that the order parameter (12) transforms as the irreducible representation $\Gamma $

_{1}

^{(G)}⊗ $\Gamma $

_{1}

^{(R)}of O

_{G}(2)⊗O

_{R}(2). The analogy between the bent-core and d-wave models results from, (i) the group isomorphism of their high-symmetry phases (see Eqs. 7 and 13), (ii) the fact that in both cases the order parameter spans the irreducible representation $\Gamma $

_{1}⊗$\Gamma $

_{1}of these groups. Consequently, the d-wave model [15] exhibits a rotation/gauge duality completely analog to the translation/rotation duality of bent-core mesophases.

_{x}is equivalent to the superfluid mirror $\sigma $

_{x}. This yields the following identification between the order-parameters:

_{1z},S

_{2z}). This can be achieved in many different ways, one of them being described in Ref. [14] within the context of s-wave superconductivity. Thus, the four unprimed phases, which have already been calculated with a single wave function in Ref. [15], can then be complemented by four additional primed phases. Their properties are summarized in Table 2a and Table 2b, and compared with their bent-core analogs. The gauge part of the symmetry groups are not indicated in these tables. Moreover, in systems containing a mirror plane parallel to x,y (for instance in high-T

_{c}superconductors), this plane is never broken by the order parameter, and it must be included in the ordered symmetry groups. For instance, the magnetic symmetry of phase II becomes 4mmm1’.

_{n}$\otimes $$\Gamma $

_{p}can be associated with an observable only when n=0 since only gauge-invariant quantities are measurable. Thus, we have only to consider the quantities $\Gamma $${}_{0}^{+}$$\otimes $$\Gamma $${}_{0}^{+}$, $\Gamma $${}_{0}^{+}$$\otimes $$\Gamma $${}_{0}^{-}$, $\Gamma $${}_{0}^{+}$$\otimes $$\Gamma $

_{p}, which represent scalars, pseudo scalars and p-rank tensors invariant under time reversal, on the one hand, and their antisymmetric time-reversal counterparts $\Gamma $${}_{0}^{-}$$\otimes $$\Gamma $${}_{0}^{+}$, $\Gamma $${}_{0}^{-}$$\otimes $$\Gamma $${}_{0}^{-}$, $\Gamma $${}_{0}^{-}$$\otimes $$\Gamma $

_{p}, on the other hand. Their values in the ordered phases can be deduced from Table 1 on using any one of the previous analogies (being careful that a tensor of rank p in liquid crystal is associated with a tensor of rank 2p in superconductors!).

_{S}, z projection of the orbital momentum L

_{z}and z projection of an axial vector A

_{z}. The scalar $\sigma $ antisymmetric under time-reversal is the response coefficient associated with a second-order magneto-electric effect: $\overrightarrow{\mathrm{P}}$ = $\sigma $ $\overrightarrow{\mathrm{E}}\times \overrightarrow{\mathrm{B}}$. In structures where $\sigma $ is finite a polarization $\overrightarrow{\mathrm{P}}$ perpendicular to the electric field $\overrightarrow{\mathrm{E}}$ appears when a magnetic field $\overrightarrow{\mathrm{B}}$ is applied. This effect happens for instance in unconventional s+s’ superconductors [21].

_{xy}, $\epsilon $

_{xx}-$\epsilon $

_{yy}], and a second-rank tensor antisymmetric under time-reversal [2$\tau $

_{xy}, $\tau $

_{xx}–$\tau $

_{yy}]. The 3D tensor $\tau $ is the response function of the linear magneto-electric effect $\overrightarrow{\mathrm{P}}$=$\tau $$\overrightarrow{\mathrm{B}}$ where $\overrightarrow{\mathrm{P}}$ is the polarization and $\overrightarrow{\mathrm{B}}$ the magnetic field.

_{ij}) response (M

_{ijkl}) to a bilinear magneto-electric excitation: u

_{ij}=M

_{ijkl}B

_{k}E

_{l}. Where A’=M

_{1111}–M

_{2211}–M

_{1122}+M

_{2222}, and B’=M

_{1112}–M

_{2212}–M

_{1121}+M

_{2221}.

_{z}and a homogeneous longitudinal axial vector A

_{z}. Note again that according to the first analogy liquid-crystal tensors of rank r are associated with superconducting tensors of rank 2r. For instance, the fourth-rank elastic coefficients in the superconducting system are associated with the second-rank optical tensor in the liquid crystal.

_{S}=|D

_{+}|

^{2}+|D

_{−}|

^{2}represents the Cooper pair density ($\Gamma $${}_{0}^{+}$$\otimes $$\Gamma $${}_{0}^{+}$).

_{z}=|D

_{+}|

^{2}–|D

_{−}|

^{2}represents the 2D orbital momentum ($\Gamma $${}_{0}^{-}$$\otimes $$\Gamma $${}_{0}^{+}$).

_{+}*D

_{−}+ D

_{−}*D

_{+}, B = i (D

_{+}*D

_{−}+ D

_{−}*D

_{+}) form a quadrivalent tensor ($\Gamma $${}_{0}^{+}$$\otimes $$\Gamma $

_{2}), which can be used to characterize the angular variation of the pair wave function. Indeed, let us write the gauge-invariant norm of $\psi $ as:

_{x}) is related to the paramagnetic feature of phase II (due to the time reversal T). Let us consider one domain of phase II defined by $\phi $

_{+}=$\phi $

_{−}=0. Its symmetry group is generated by g${}_{\pi}$C${}_{\pi /2}$, T and $\sigma $

_{x}. g${}_{\pi}$C${}_{\pi /2}$ changes the sign of the wave function after a 90° rotation, therefore $\psi $($\theta $) must vanish along four directions. This is the symmetry origin of the four nodes occurring in the gap. In order to determine the vanishing directions let us consider the symmetry g${}_{\pi}$C${}_{\pi /2}$$\sigma $

_{x}=g${}_{\pi}$$\sigma $

_{xy}, which changes the sign of the wave function together with a mirror plane directed along x+y, which transforms $\theta $ into $\pi $/2-$\theta $: g${}_{\pi}$$\sigma $

_{xy}$\psi $($\theta $)=−$\psi $($\pi $/2−$\theta $). The invariance of $\psi $ under g${}_{\pi}$$\sigma $

_{xy}provides immediately the four nodes directions: $\psi $($\pi $/4)=$\psi $(3$\pi $/4)=$\psi $(5$\pi $/4)=$\psi $(7$\pi $/4)=0. The zeroes of the vector-waves result from the same arguments. According to the second analogy the generators of the R-phase are C${}_{\pi}$T${}_{\lambda /2}$, $\sigma $

_{x}and I. C${}_{\pi}$T${}_{\lambda /2}$ means that the transverse vector-waves reverse their directions after a translation of $\lambda $/2. $\sigma $

_{x}indicates that $\overrightarrow{\mathrm{P}}$(z) is polarized in the (y-z)-plane and $\overrightarrow{\mathrm{A}}$(z) in the (x,z)-plane. Since the transverse axial and polar vector-waves are linearly polarized and reverse their directions after $\lambda $/2, they must vanish on a lattice of points separated by $\lambda $/2. To determine the positions of these zeroes, let us consider the mirror plane C${}_{\pi}$T${}_{\lambda /2}$I normal to Oz and located at z=$\lambda $/4. This plane forces $\overrightarrow{\mathrm{A}}$ to vanish at z=$\lambda $/4 and 3$\lambda $/4. On the contrary, the zeroes of $\overrightarrow{\mathrm{P}}$ are determined by the location of the inversion centers I located at z=0 and $\lambda $ (analogy 2 makes apparently $\psi $ and $\overrightarrow{\mathrm{A}}$ closer than $\psi $ and $\overrightarrow{\mathrm{P}}$, because of the sign change D

_{−}$\leftrightarrow $−$\eta $

_{2}* in the correspondences (13), a $\pi $/2 phase shift appears between $\overrightarrow{\mathrm{P}}$(z) and $\psi $($\theta $) as we introduce the additional correspondence kz$\to $$\theta $).

## 6. Conclusion

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**Figure 1.**

**(a)**Group-subgroup relationships between the ordered phases.

**(b)**Scheme of the molecular organization in the various stable phases of the vector-wave model. The difference between R and R’ is rather subtle: (i) The direction of the polarization in R’ is not fixed by symmetry, so that the molecular plane can rotate when the temperature is changed. (ii) The molecular fluctuations are less symmetric in R’, which cannot appear in the figure since we represent only the mean orientation at the positions of maximum density.

**Figure 2.**

**(a)**Graphic representation of the unprimed bent-core phases. The angular coordinate represents the position z in the unit cell (a complete rotation in the graphic plane is associated with a translation of one unit cell $\lambda $), whereas the radial coordinate represents the modulus of the polarization at position z. In the circular C phase the polarization modulus does not vary with z. In EL the modulus varies periodically yielding a bean-shape curve, but it vanishes nowhere. In R the small axis of the bean vanishes along directions corresponding to the nodes of the polarization wave.

**(b)**Angular variations of the wave function modulus in the superconducting states. It is isotropic in phase I, orthorhombic without nodes in phase III, and exhibits four nodes in phase II.

**Figure 3.**Line defects in C. The “equal phase” surfaces (analogous to the smectic layers in R and EL) are represented by continuous lines, and the polarization by arrows when it is parallel to the figure and circles when it is normal to it.

**(a)**$\chi $-line (2$\pi $-disclination) with a core parallel to $\overrightarrow{\mathrm{k}}$.

**(b)**$\chi $-line with a core normal to $\overrightarrow{\mathrm{k}}$ (jelly-roll configuration). A domain wall between two oppositely wound domains is represented by a thick line.

**(c)**(λ−) line (–π-disclination with core normal to $\overrightarrow{\mathrm{k}}$).

**(c)**(λ+) line (+π-disclination with core normal to $\overrightarrow{\mathrm{k}}$).

**Figure 4.**Line defects in R. The straight lines indicate the edge of the smectic layers.

**(a)**Dispiration {1/2,1/2} with core parallel to $\overrightarrow{\mathrm{k}}$.

**(b)**An equivalent defect with core normal to $\overrightarrow{\mathrm{k}}$.

**(c)**Pure {0,1} 2$\pi $ -disclination. The arrows and circles indicate the components of the polarization $\overrightarrow{\mathrm{P}}$ at the center of the layers respectively parallel and normal to the figure.

**(d)**Pure {1,0} dislocation.

**(e)**–$\pi $ -disclination of type (iv,-).

**(f)**–$\pi $ -disclination of type (iii,-).

**(g)**+$\pi $-disclination of type (iv,+).

**(h)**+$\pi $-disclination of type (iv/-).

**Table 1.**First line: Tensor-wave types. Second line: Physical meaning of the waves. The superscript (0) indicates that the tensor is homogeneous. For each tensor wave, the list of phases in which they take non-zero values is indicated in the third line.

$\Gamma $_{0}^{+}$\otimes $ $\Gamma $_{0}^{–} | $\Gamma $_{0}^{–}$\otimes $ $\Gamma $_{0}^{+} | $\Gamma $_{0}^{–} $\otimes $ $\Gamma $_{0}^{–} | $\Gamma $_{0}^{+} $\otimes $ $\Gamma $_{p}$\Gamma $ _{n}$\otimes $ $\Gamma $_{0}^{+} | $\Gamma $_{0}^{–} $\otimes $ $\Gamma $_{p} | $\Gamma $_{n} $\otimes $ $\Gamma $_{0}^{–} | $\Gamma $_{n} $\otimes $ $\Gamma $_{p} |

polarization P${}_{\mathrm{z}}^{\left(0\right)}$ | tilt A${}_{\mathrm{z}}^{\left(0\right)}$ | chiral index $\chi $ | P_{z}(z) n=1A${}_{\mathrm{x}}^{\left(0\right)}$,A${}_{\mathrm{y}}^{\left(0\right)}$p=1 | A_{z}(z) p=1 | P${}_{\mathrm{x}}^{\left(0\right)}$,P${}_{\mathrm{y}}^{\left(0\right)}$n=1 | $\overrightarrow{\mathrm{A}}$(z) or $\overrightarrow{\mathrm{P}}$(z) n=p=1 |

C’ R” EL’ | C’ R’ EL’ | C C’ EL EL’ | n even: R,R’,R” EL,EL’ | n even: R’,EL,EL’ | n even: R”,EL,EL’ | n+p even: R,EL,EL’,R’,R” n=p: C,C’ |

**Table 2a.**Symmetry and properties of the superconducting unprimed phases. Their analog mesophases are indicated in rows 4 (first analogy) and 5 (second analogy).

Unprimed phases | Normal | I | II | III |

Magnetic groups | ∞m1’ | ∞m’. | 4mm1’ | 4m’m’ |

Properties | non magnetic non chiral | magnetic non chiral | non magnetic non chiral | magnetic non chiral |

First analogy | Isotropic | C | R | EL |

Second analogy | Isotropic | C | R | EL |

**Table 2b.**Symmetry and properties of the superconducting primed phases. Their analog mesophases are indicated in rows 4 (first analogy) and 5 (second analogy).

Primed phases | I’ | II’ | II" | III’ |

Magnetic groups | ∞ | 4mm | 41’ | 4 |

Properties | magnetic chiral | magnetic non chiral | non magnetic chiral | magnetic chiral |

First analogy | C′ | R′ | R″ | EL′ |

Second analogy | C’ | R" | R’ | EL’ |

**Table 3.**Superconducting tensors and their liquid crystal analogs. The type of tensor wave is indicated in the first column. The first number indicates the wave vector and the second number indicates the tensor rank of the waves.

Tensors | Superconductor | First analogy | Second analogy |

[0^{+},0^{+}] | Pair density, n_{S} | n_{L} | n_{L} |

[0^{+},0^{−}] | Bilinear Mag-elec. $\epsilon $_{1} | Polarization P_{z} | Axial-vector, A_{z} |

[0^{−},0^{+}] | Linear Mag-elec. $\tau $_{1} | Tilt-vector A_{z} | Polarization P_{z} |

[0^{−},0^{−}] | Magnetization L_{z} | Chirality, $\chi $ | Chirality $\chi $ |

[0^{+},1] | Biaxiality [$\epsilon $ _{xy}, $\epsilon $_{xx}-$\epsilon $_{yy}] | Tilt (T _{x}, T_{y}) | Smecticity Density wave ( k) |

[0^{−},1] | Linear Mag.electric [$\tau $ _{xy}, $\tau $_{xx}-$\tau $_{yy}] | Polarization (P _{x},P_{y}) | A_{z} e^{±ikz} |

[0^{+},2] | Tetragonality Elastic [A,B] | Biaxiality Optic $\epsilon $ _{xy}, $\epsilon $_{xx}-$\epsilon $_{yy} | Smecticity Density wave (2 k) |

[0^{−},2] | 3^{d} order Mag.elect.[A’B’] | Electro-clinic. [${\epsilon}_{xx}^{C}-{\epsilon}_{yy}^{C},{\epsilon}_{xy}^{C}$] | A_{z} e^{±2ikz} |

© 2010 by the author; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Mettout, B.; Lorman, V.
Dual Symmetry in Bent-Core Liquid Crystals and Unconventional Superconductors. *Symmetry* **2010**, *2*, 15-39.
https://doi.org/10.3390/sym2010015

**AMA Style**

Mettout B, Lorman V.
Dual Symmetry in Bent-Core Liquid Crystals and Unconventional Superconductors. *Symmetry*. 2010; 2(1):15-39.
https://doi.org/10.3390/sym2010015

**Chicago/Turabian Style**

Mettout, Bruno, and Vladimir Lorman.
2010. "Dual Symmetry in Bent-Core Liquid Crystals and Unconventional Superconductors" *Symmetry* 2, no. 1: 15-39.
https://doi.org/10.3390/sym2010015