Topology and nematic ordering. I. A gauge theory

Hydrodynamic theory of p-atic liquid crystals

We formulate a comprehensive hydrodynamic theory of two-dimensional liquid crystals with generic p-fold rotational symmetry, also known as p-atics, of which nematics (p=2) and hexatics (p=6) are the two best known examples. Previous hydrodynamic theories of p-atics are characterized by continuous O(2) rotational symmetry, which is higher than the discrete rotational symmetry of p-atic phases. By contrast, here we demonstrate that the discrete rotational symmetry allows the inclusion of additional terms in the hydrodynamic equations, which, in turn, lead to novel phenomena, such as the possibility of flow alignment at high shear rates, even for p>2. Furthermore, we show that any finite imposed shear will induce long-ranged orientational order in any p-atic liquid crystal, in contrast to the quasi-long-ranged order that occurs in the absence of shear. The induced order parameter scales like a nonuniversal power of the applied shear rate at small shear rates.

Scalar gauge-Higgs models with discrete Abelian symmetry groups

We investigate the phase diagram and the nature of the phase transitions of three-dimensional lattice gauge-Higgs models obtained by gauging the Z_{N} subgroup of the global Z_{q} invariance group of the Z_{q} clock model (N is a submultiple of q). The phase diagram is generally characterized by the presence of three different phases, separated by three distinct transition lines. We investigate the critical behavior along the two transition lines characterized by the ordering of the scalar field. Along the transition line separating the disordered-confined phase from the ordered-deconfined phase, standard arguments within the Landau-Ginzburg-Wilson framework predict that the behavior is the same as in a generic ferromagnetic model with Z_{p} global symmetry, p being the ratio q/N. Thus, continuous transitions belong to the Ising and to the O(2) universality class for p=2 and p≥4, respectively, while for p=3 only first-order transitions are possible. The results of Monte Carlo simulations confirm these predictions. There is also a second transition line, which separates two phases in which gauge fields are essentially ordered. Along this line we observe the same critical behavior as in the Z_{q} clock model, as it occurs in the absence of gauge fields.

Coupling ultracold matter to dynamical gauge fields in optical lattices: From flux attachment to ℤ2 lattice gauge theories

From the standard model of particle physics to strongly correlated electrons, various physical settings are formulated in terms of matter coupled to gauge fields. Quantum simulations based on ultracold atoms in optical lattices provide a promising avenue to study these complex systems and unravel the underlying many-body physics. Here, we demonstrate how quantized dynamical gauge fields can be created in mixtures of ultracold atoms in optical lattices, using a combination of coherent lattice modulation with strong interactions. Specifically, we propose implementation of ℤ2 lattice gauge theories coupled to matter, reminiscent of theories previously introduced in high-temperature superconductivity. We discuss a range of settings from zero-dimensional toy models to ladders featuring transitions in the gauge sector to extended two-dimensional systems. Mastering lattice gauge theories in optical lattices constitutes a new route toward the realization of strongly correlated systems, with properties dictated by an interplay of dynamical matter and gauge fields.

Topological order, emergent gauge fields, and Fermi surface reconstruction

This review describes how topological order associated with the presence of emergent gauge fields can reconstruct Fermi surfaces of metals, even in the absence of translational symmetry breaking. We begin with an introduction to topological order using Wegner's quantum [Formula: see text] gauge theory on the square lattice: the topological state is characterized by the expulsion of defects, carrying [Formula: see text] magnetic flux. The interplay between topological order and the breaking of global symmetry is described by the non-zero temperature statistical mechanics of classical XY models in dimension D  =  3; such models also describe the zero temperature quantum phases of bosons with short-range interactions on the square lattice at integer filling. The topological state is again characterized by the expulsion of certain defects, in a state with fluctuating symmetry-breaking order, along with the presence of emergent gauge fields. The phase diagrams of the [Formula: see text] gauge theory and the XY models are obtained by embedding them in U(1) gauge theories, and by studying their Higgs and confining phases. These ideas are then applied to the single-band Hubbard model on the square lattice. A SU(2) gauge theory describes the fluctuations of spin-density-wave order, and its phase diagram is presented by analogy to the XY models. We obtain a class of zero temperature metallic states with fluctuating spin-density wave order, topological order associated with defect expulsion, deconfined emergent gauge fields, reconstructed Fermi surfaces (with 'chargon' or electron-like quasiparticles), but no broken symmetry. We conclude with the application of such metallic states to the pseudogap phase of the cuprates, and note the recent comparison with numerical studies of the Hubbard model and photoemission observations of the electron-doped cuprates. In a detour, we also discuss the influence of Berry phases, and how they can lead to deconfined quantum critical points: this applies to bosons on the square lattice at half-integer filling, and to quantum dimer models.

Generic first-order phase transitions between isotropic and orientational phases with polyhedral symmetries

Polyhedral nematics are examples of exotic orientational phases that possess a complex internal symmetry, representing highly nontrivial ways of rotational symmetry breaking, and are subject to current experimental pursuits in colloidal and molecular systems. The classification of these phases has been known for a long time; however, their transitions to the disordered isotropic liquid phase remain largely unexplored, except for a few symmetries. In this work, we utilize a recently introduced non-Abelian gauge theory to explore the nature of the underlying nematic-isotropic transition for all three-dimensional polyhedral nematics. The gauge theory can readily be applied to nematic phases with an arbitrary point-group symmetry, including those where traditional Landau methods and the associated lattice models may become too involved to implement owing to a prohibitive order-parameter tensor of high rank or (the absence of) mirror symmetries. By means of exhaustive Monte Carlo simulations, we find that the nematic-isotropic transition is generically first-order for all polyhedral symmetries. Moreover, we show that this universal result is fully consistent with our expectation from a renormalization group approach, as well as with other lattice models for symmetries already studied in the literature. We argue that extreme fine tuning is required to promote those transitions to second-order ones. We also comment on the nature of phase transitions breaking the O(3) symmetry in general cases.

Hierarchy of orientational phases and axial anisotropies in the gauge theoretical description of generalized nematic liquid crystals

The paradigm of spontaneous symmetry breaking encompasses the breaking of the rotational symmetries O(3) of isotropic space to a discrete subgroup, i.e., a three-dimensional point group. The subgroups form a rich hierarchy and allow for many different phases of matter with orientational order. Such spontaneous symmetry breaking occurs in nematic liquid crystals, and a highlight of such anisotropic liquids is the uniaxial and biaxial nematics. Generalizing the familiar uniaxial and biaxial nematics to phases characterized by an arbitrary point-group symmetry, referred to as generalized nematics, leads to a large hierarchy of phases and possible orientational phase transitions. We discuss how a particular class of nematic phase transitions related to axial point groups can be efficiently captured within a recently proposed gauge theoretical formulation of generalized nematics [K. Liu, J. Nissinen, R.-J. Slager, K. Wu, and J. Zaanen, Phys. Rev. X 6, 041025 (2016)2160-330810.1103/PhysRevX.6.041025]. These transitions can be introduced in the model by considering anisotropic couplings that do not break any additional symmetries. By and large this generalizes the well-known uniaxial-biaxial nematic phase transition to any arbitrary axial point group in three dimensions. We find in particular that the generalized axial transitions are distinguished by two types of phase diagrams with intermediate vestigial orientational phases and that the window of the vestigial phase is intimately related to the amount of symmetry of the defining point group due to inherently growing fluctuations of the order parameter. This might explain the stability of the observed uniaxial-biaxial phases as compared to the yet to be observed other possible forms of generalized nematic order with higher point-group symmetries.

Classification of point-group-symmetric orientational ordering tensors

The concept of symmetry breaking has been a propelling force in understanding phases of matter. While rotational-symmetry breaking is one of the most prevalent examples, the rich landscape of orientational orders breaking the rotational symmetries of isotropic space, i.e., O(3), to a three-dimensional point group remain largely unexplored, apart from simple examples such as ferromagnetic or uniaxial nematic ordering. Here we provide an explicit construction, utilizing a recently introduced gauge-theoretical framework, to address the three-dimensional point-group-symmetric orientational orders on a general footing. This unified approach allows us to enlist order parameter tensors for all three-dimensional point groups. By construction, these tensor order parameters are the minimal set of simplest tensors allowed by the symmetries that uniquely characterize the orientational order. We explicitly give these for the point groups {C_{n},D_{n},T,O,I}⊂SO(3) and {C_{nv},S_{2n},C_{nh},D_{nh},D_{nd},T_{h},T_{d},O_{h},I_{h}}⊂O(3) for n,2n∈{1,2,3,4,6,∞}. This central result may be perceived as a road map for identifying exotic orientational orders that may become more and more in reach in view of rapid experimental progress in, e.g., nanocolloidal systems and novel magnets.

Global defect topology in nematic liquid crystals

We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in R3, showing that the distinct homotopy classes have a 1–1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group has representatives that are planar and characterize the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterization. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of τ lines in cholesterics.

Topology, delocalization via average symmetry and the symplectic Anderson transition

A field theory of the Anderson transition in two-dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortexlike topological defects is essential for localization. The sign of vortex fugacity determines the Z(2) topological class of the localized phase. There are two distinct fixed points with the same critical exponents, corresponding to transitions from a metal to an insulator and a topological insulator, respectively. The critical conductivity and correlation length exponent of these transitions are computed in an N=1-[symbol: see text] expansion in the number of replicas, where for small [symbol: see text] the critical points are perturbatively connected to the Kosterlitz-Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.

Memory and topological frustration in nematic liquid crystals confined in porous materials

Orientational ordering is key to functional materials with switching capability, such as nematic liquid crystals and ferromagnetic and ferroelectric materials. We explored the confinement of nematic liquid crystals in bicontinuous porous structures with smooth surfaces that locally impose normal orientational order on the liquid crystal. We find that frustration leads to a high density of topological defect lines permeating the porous structures, and that most defect lines are made stable by looping around solid portions of the confining material. Because many defect trajectories are possible, these systems are highly metastable and efficient in memorizing the alignment forced by external fields. Such memory effects have their origin in the topology of the confining surface and are maximized in a simple periodic bicontinuous cubic structure. We also show that nematic liquid crystals in random porous networks exhibit a disorder-induced slowing-down typical of glasses that originates from activated collisions and rearrangements of defect lines. Our findings offer the possibility to functionalize orientationally ordered materials through topological confinement.

Role of topological defects in the phase transition of a modified XY model: a Monte Carlo study

Monte Carlo simulation has been performed on a classical two-dimensional XY model with a modified form of interaction potential to investigate the role of topological defects on the phase transition exhibited by the model. In simulations in a restricted ensemble without defects, the system appears to remain ordered at all temperatures. Suppression of topological defects on the square plaquettes in the modified XY model leads to complete elimination of the phase transition observed in this model.

Anisotropic behavior of water in ferroelectric liquid crystals

The outcome of water addition in ferroelectric liquid crystal (FLC) has been investigated in uniform and defect-free homogeneous and homeotropically aligned monodomain sample cells from electro-optical and dielectric spectroscopic measurements. The lagging in optical response between nonconducting (spatially variable switching) and conducting (conventional switching) portions of water added FLC sample cell has been observed by frequency-dependent electro-optical studies. The bias-dependent water related new relaxation peak near the conventional Goldstone mode relaxation process has been observed only in the homogeneous alignment and not in the homeotropic one. Further, the significant increment in dielectric anisotropy as well as faster diffusion of water along long molecular axis than short molecular axis has also been monitored. These studies strongly suggest that the distribution of water is anisotropic in FLC medium and could be the reason for new relaxation peak in the water added FLC sample.

Isotropic-to-nematic transition in confined liquid crystals: an essentially nonuniversal phenomenon

Computer simulations are presented of the isotropic-to-nematic transition in a liquid crystal confined between two parallel plates a distance H apart. The plates are neutral and do not impose any anchoring on the particles. Depending on the shape of the pair potential acting between the particles, we find that the transition either changes from first order to continuous at a critical film thickness H=H(x) , or that the transition remains first order irrespective of H . This demonstrates that the isotropic-to-nematic transition in confined geometry is not characterized by any universality class, but rather that its fate is determined by microscopic details. The resulting capillary phase diagrams can thus assume two topologies: one where the isotropic and nematic branches of the binodal meet at H=H(x), and one where they remain separated. For values of H where the transition is strongly first order the shift Deltaepsilon of the transition temperature is in excellent agreement with the Kelvin equation. Not only is the relation Deltaepsilon proportional, variant 1/H recovered but also the prefactor of the shift is in quantitative agreement with the independently measured bulk latent heat and interfacial tension.

Growth of tactoidal droplets during the first-order isotropic to nematic phase transition of F-actin

We report on the observed characteristics of the first-order phase transition of F-actin from the isotropic state to the nematic liquid-crystalline state. Solutions of short average filament length F-actin at appropriate concentrations phase separate to form tactoidal droplets. These tactoids are the result of the minimization of their free energy and show a bipolar director field connecting two opposite poles. The tactoids are shown to form through two distinct mechanisms: nucleation and growth and spinodal decomposition. Both mechanisms produce tactoids with final domain sizes that are of the same order of magnitude. Additionally, analysis of the system shows several features of metastability. The solution can exist in a variety of steady states near equilibrium and can be easily perturbed, settling in one prescribed by the path followed in phase space.

Isotropic to nematic liquid crystalline phase transition of F-actin varies from continuous to first order

We report that the properties of the isotropic to nematic liquid crystalline phase transition of F-actin depend critically on the average filament length. For average filament lengths longer than 2 microm, we confirm previous findings that the phase transition is continuous in both alignment and concentration. For average filament lengths shorter than 2 microm, we show for the first time a first order transition with a clear discontinuity in both alignment and concentration. Tactoidal droplets of coexisting isotropic and nematic phases, differing in concentration by approximately 30%, form over the course of hours and appear to settle into near equilibrium metastable states.

Orientational order parameter of the nematic liquid crystalline phase of F-actin

We measured the orientational order parameter of F-actin traversing the isotropic-nematic phase transition using a combination of techniques, including fluorescence imaging, local birefringence measurements, and small-angle x-ray scattering. The order parameter approaches a saturated value of 0.75 for actin concentrations above the region of the isotropic-nematic phase transition. This result implies a significant extent of misalignment and consequently entanglement among long actin filaments, even in the nematic phase. We determine the specific birefringence of completely aligned F-actin to be Deltan(0)=2.3 x 10(-5) ml/mg. At concentrations slightly below the isotropic-nematic transition, nonzero values of the order parameter are detected for hours following an initial alignment, indicating extremely slow rotational kinetics of F-actin in the entangled networks.

The interplay between viscoelastic and thermodynamic properties determines the birefringence of F-actin gels

F-actin gels of increasing concentrations (25-300 microM) display in vitro a progressive onset of birefringence due to orientational ordering of actin filaments. At F-actin concentrations <100 microM, this birefringence can be erased and restored at will by sonication and gentle flow, respectively. Hence, the orientational ordering does not result from a thermodynamic transition to a nematic phase but instead is due to mechanical stresses stored in the gels. In contrast, at F-actin concentrations > or =100 microM, gels display spontaneous birefringence recovery, at rest, which is the sign of true nematic ordering, in good agreement with statistical physics models of the isotropic/nematic transition. Well-aligned samples of F-actin gels could be produced and their small-angle x-ray scattering patterns are quite anisotropic. These patterns show no sign of filament positional short-range order and could be modeled by averaging the form factor with the Maier-Saupe nematic distribution function. The derived nematic order parameter S of the gels ranged from S = 0.7 at 300 microM to S = 0.4 at 25 microM. Both birefringence and small-angle x-ray scattering data indicate that, even in absence of cross-linking proteins, spontaneous cooperative alignment of actin filaments may arise in motile regions of living cells where F-actin concentrations can reach values of a few 100 microM.

Nematics with quenched disorder: pinning out the origin of memory

Memory effects and glassy behavior have been repeatedly observed in disordered nematic liquid crystals but the connection between these effects and the system topology remained unrevealed. We present an analysis of the local and global topology of the nematic ordering in the presence of quenched disorder and we show that nematics with quenched disorder can be mapped into a system of pinned defect lines and that the memory of the system stems from the pinning of these strings.

Phase transitions in two planar lattice models and topological defects: a Monte Carlo study

Monte Carlo simulation has been performed in the planar P2 and P4 models to investigate the effects of the suppression of topological defects on the phase transition exhibited by these models. Suppression of the 1/2 defects on the square plaquettes in the P2 model leads to complete elimination of the phase transition observed in this model. However, in the P4 model, on suppressing the single 1/2 defects on square plaquettes, the otherwise first order phase transition changes to a second order one which occurs at a higher temperature, and this is due to the presence of a large number of 1/2 pair defects which are left within the square plaquettes. When we suppressed these charges too, complete elimination of the phase transition was observed.

Continuous isotropic-nematic liquid crystalline transition of F-actin solutions

The phase transition from the isotropic (I) to nematic (N) liquid crystalline suspension of F-actin of average length ł> or =3 microm was studied by local measurements of optical birefringence and protein concentration. Both parameters were detected to be continuous in the transition region, suggesting that the I-N transition is higher than first order. Thus we report experimental evidence for a continuous I-N transition for a suspension of rodlike macromolecules. Our findings are consistent with a recent theory by Lammert, Rokhsar, and Toner [Phys. Rev. Lett. 70, 1650 (1993)], predicting that the I-N transition may become continuous due to suppression of disclinations.

Coarsening dynamics of biaxial nematic liquid crystals

We study the coarsening dynamics of two- and three-dimensional biaxial nematic liquid crystals, using Langevin dynamics. Unlike previous work, we use a model with no a priori relationship among the three elastic constants associated with director deformations. Biaxial nematics possess four topologically distinct classes of defects, three of which have half-integer charge, while the fourth, which plays a minor role in coarsening, is of integer charge. We find a rich variety of coarsening behavior, including the presence of one, two, or three of the half-integer classes at late times, depending on the relative values of the elastic constants and the resulting energetics of the decay channels of the defects. The morphology of the defect tangle in three dimensions when all three classes are present is particularly interesting. Rather than forming independent defect loops (as occurs when only one or two of the classes are present), the defect lines meet at junction points which are distributed uniformly throughout the system. As the system coarsens some pairs of neighboring junction points approach each other and annihilate, allowing the formation of nonintersecting loops each formed from a single defect class. These loops then shrink independently during the very final stages of the coarsening sequence.

Disclination loop behavior near the nematic-isotropic transition

We investigate the behavior of disclination loops in the vicinity of the first-order nematic-isotropic transition in the Lebwohl-Lasher and related models. We find that two independent measures of the transition temperature, the free energy, and the distribution of disclination line segments, give essentially identical values. We also calculate the distribution function D(p) of disclination loops of perimeter p and fit it to a quasiexponential form. Below the transition, D(p) falls off exponentially, while in the neighborhood of the transition, it decays with a power-law exponent approximately equal to 2.5, consistent with a "blowout" of loops at the transition. In a modified Lebwohl-Lasher model with a strongly first-order transition we are able to measure a jump in the disclination line tension at the transition, which is too small to be measured in the Lebwohl-Lasher model. We also measure the monopole charge of the disclination loops and find that in both the original and modified Lebwohl-Lasher models, there are large loops that carry monopole charge, while smaller isolated loops do not. Overall, the nature of the topological defects in both models is very similar.

Transparent nematic phase in a liquid-crystal-based microemulsion

Complex fluids are usually produced by mixing together several distinct components, the interactions between which can give rise to unusual optical and rheological properties of the system as a whole. For example, the properties of microemulsions (composed of water, oil and surfactants) are determined by the microscopic structural organization of the fluid that occurs owing to phase separation of the component elements. Here we investigate the effect of introducing an additional organizing factor into such a fluid system, by replacing the oil component of a conventional water-in-oil microemulsion with an intrinsically anisotropic fluid--a nematic liquid crystal. As with the conventional case, the fluid phase-separates into an emulsion of water microdroplets (stabilized by the surfactant as inverse micelles) dispersed in the 'oil' phase. But the properties are further influenced by a significant directional coupling between the liquid-crystal molecules and the surfactant tails that emerge (essentially radially) from the micelles. The result is a modified bulk-liquid crystal that is an ordered nematic at the mesoscopic level, but which does not exhibit the strong light scattering generally associated with bulk nematic order: the bulk material here is essentially isotropic and thus transparent.