Equipped with the recently recognized symmetry structure of mixed states of matter and "strong-weak spontaneous symmetry breaking" (SW-SSB), we develop a quantum particle-vortex duality of emergent model-F and model-A hydrodynamics of 2d boson/rotor systems. This duality relation demonstrates that classical hydrodynamics of 2d bosons can be described in terms of the charge symmetry $U(1)_c$, but also equivalently in terms of the dual (emergent) 1-form symmetry $U(1)^{(1)}_e$, as well as $U(1)_v$ associated with the conservation of vortices. This duality provides a bridge between the hydrodynamics of matter and magnetohydrodynamics.
Equipped with the recently recognized symmetry structure of mixed states of matter and "strong-weak spontaneous symmetry breaking" (SW-SSB), we develop a quantum particle-vortex duality of emergent model-F and model-A hydrodynamics of 2d boson/rotor systems. This duality relation demonstrates that classical hydrodynamics of 2d bosons can be described in terms of the charge symmetry U(1)c, but also equivalently in terms of the dual (emergent) 1-form symmetry U(1)e(1), as well as U(1)v associated with the conservation of vortices. This duality provides a bridge between the hydrodynamics of matter and magnetohydrodynamics.
We explore the onset of spontaneous strong-to-weak symmetry breaking (SW-SSB) under U(1)-symmetric (i.e., charge-conserving) open-system dynamics. We define this phenomenon for quantum states and classical probability distributions, and explore it in three complementary models, one of which exhibits nontrivial quantum coherence at short times. Our main conclusions are as follows. In one dimension, the strong symmetry is not spontaneously broken at any finite time; however, correlators probing strong-to-weak symmetry breaking develop order on length scales that grow linearly in time, parametrically faster than charge diffusion. We provide numerical evidence for this scaling in multiple distinct probes of SW-SSB, and derive it from a field-theory analysis. Moreover, we relate this scaling to the problem of inferring the charge inside a subregion by measuring its surroundings, and construct explicit decoding protocols that illustrate its origin. In two dimensions, field theory and numerical simulations support a finite-time Berezinskii-Kosterlitz-Thouless-like SW-SSB transition. Within continuum hydrodynamics, by contrast, SW-SSB happens at infinitesimal time in two or more dimensions. The SW-SSB transition time can thus be interpreted as marking the emergence of a continuum hydrodynamic description, or (more precisely) the timescale beyond which non-hydrodynamic information such as discrete particle worldlines can no longer be inferred. We support this picture by analyzing a model in which we exploit SW-SSB to derive a classical stochastic hydrodynamic description from the underlying quantum dynamics.
Inspired by the hunt for new phases of matter in quantum mixed states, it has recently been proposed that the equivalence of microcanonical and canonical ensembles in statistical mechanics is a manifestation of strong-to-weak spontaneous symmetry breaking (SWSSB) in an underlying many-body quantum description. Here, we build an effective field theory for SWSSB of a global U(1) symmetry; the answer exactly reproduces the Schwinger-Keldysh effective field theory of diffusion for the conserved charge. We conclude that hydrodynamics can be understood as a theory of superfluidity for the broken strong symmetry: a nonvanishing susceptibility is a measurable order parameter for SWSSB, the diffusion mode is the Goldstone boson of the spontaneously broken continuous symmetry, and a generalization of Goldstone's theorem implies that the diffusion mode is always long lived. This perspective provides a transparent physical explanation for the unusual reparameterization symmetries, which are a necessary ingredient of Schwinger-Keldysh effective field theories for normal fluids.
We study an effective field theory of a vortex lattice in a two-dimensional neutral rotating superfluid. Utilizing particle-vortex dualities, we explore its formulation in terms of a U(1) gauge theory coupled to elasticity that at low energies reduces to a compact Lifshitz theory augmented with a Berry phase term encoding the vortex dynamics in the presence of a superflow. Utilizing elasticity- and Lifshitz-gauge-theory dualities, we derive dual formulations of the vortex lattice in terms of a traceless symmetric scalar-charge theory and demonstrate low-energy equivalence of our dual gauge theory to its elasticity-gauge theory dual. We further discuss a multipole symmetry of the vortex lattice and the multipole one-form symmetries of its dual gauge theory. We also study its topological crystalline defects, where the multipole one-form symmetry plays a prominent role. It classifies the defects, explains their restricted mobility, and characterizes descendant vortex phases, which includes a vortex-supersolid phase. Using the dual gauge theory, we also develop a mean-field theory for the quantum melting transition from a vortex crystal to a vortex supersolid.
Quantum liquids in two dimensions represent interesting dynamical quantum systems for several reasons, among them the possibility of the existence of infinite hidden symmetries, such as conformal symmetry or the symmetry associated with area preserving diffeomorphisms. It is known that when the symmetry algebra is large enough, symmetry may fully prescribe the dynamics. However, the way this is borne out in two-dimensional hydrodynamics, both classical and quantum, is not fully understood. Here we take a step towards clarifying this issue, by focusing on a particular example, namely that of a two-dimensional phase space liquid which emerges when one considers the Calogero model, a many-body one-dimensional system interacting through an inverse square law potential. We demonstrate how the symmetry algebra of conserved quantities of the one-dimensional system is expressed in terms of the incompressible Euler hydrodynamics of point vortices in phase space. Due to a formal relation between quantum hydrodynamics and classical stochastic hydrodynamics, which is inherent in the method of stochastic quantization, the ideas and methods developed here may also have application in the study of stochastic classical hydrodynamics, after suitable modifications.
Strong-to-weak spontaneous symmetry breaking (SW-SSB) has recently emerged as a useful framework for studying phases of matter in open systems, quantum or classical. Beginning with the simple idea of extending symmetry breaking to general mixed states, and the familiar equivalence between canonical and grand-canonical ensembles in statistical mechanics, the concept has grown into a unifying perspective connecting many different ideas in physics, including topological orders, emergent hydrodynamics, and information-theoretic characterization of phases of matter. This review provides a bird's-eye view of some of these recent developments.
Many of the most vexing phenomena observed in condensed matter physics involve strongly correlated systems near quantum critical points in two spatial dimensions. Examples range from the sharing of critical exponents (superuniversality) among quantum Hall plateau transitions to the emergence of a charge-vortex symmetry near the field-tuned superconductor-insulator transition in thin films to the appearance of “anomalous” metallic states which evade localization. These problems elude theoretical understanding because they lack small parameters; standard perturbative approaches constitute poor, often uncontrolled approximations. This thesis concerns itself with the development of techniques reaching beyond the perturbative paradigm, focusing especially on duality, the idea that two seemingly different theories are actually one and the same. While a duality is most useful when it relates a strongly interacting theory to a weakly interacting one, even dualities between strongly coupled theories can reveal emergent symmetries and exotic gapped phases that may be obscure in one theory but not its dual. Recently, starting from a relativistic analogue of flux attachment, an entire web of quantum field theory dualities was proposed connecting a wide variety of quantum critical states in two spatial dimensions. This web of dualities has led to a surge of progress on a wide variety of condensed matter problems, several examples of which are presented in this thesis. One of the major challenges that comes with the proposal of new dualities is to “derive” them. In Chapter 2, we construct simple, explicit derivations of many members of the web of dualities in (particle-hole symmetric) models of relativistic current loops. In relativistic theories, flux attachment necessarily involves transmutation of both statistics and spin, an operation which can be made precise in the context of loop models. We further show that while non-relativistic theories are invariant under attachment of even numbers of flux quanta due to the periodicity of statistics, this symmetry is completely lost in relativistic theories due to the presence of spin, clarifying the interpretation of earlier loop models in which this symmetry appeared to be present. Motivated in part by this lack of statistical periodicity in relativistic flux attachment, in Chapter 3 we turn our attention to the metallic, “composite Fermi liquid” states occuring in quantum Hall systems at filling fraction ν = 1/2n. Famously, the state at ν = 1/2 is known to display particle-hole symmetry, which has recently led to the proposal of a manifestly particle-hole symmetric theory of relativistic, or Dirac, composite fermions that features prominently in the web of dualities. Surprisingly, however, an analogous “reflection symmetry” has also been observed in transport experiments about ν = 1/4. To explain this symmetry, we propose a series of relativistic composite fermion theories for the compressible states at ν = 1/2n, in which the reflection symmetry is incorporated as a mean field time-reversal symmetry of the composite fermions. These theories consist of electrically neutral Dirac fermions attached to 2n flux quanta via an emergent Chern-Simons gauge field. While not possessing an explicit particle-hole symmetry, these theories reproduce the known Jain sequence states proximate to ν = 1/2n, and we show that such states can be related by the observed reflection symmetry, at least at mean field level. In Chapter 4, we describe how duality can be used to access exotic gapped phases, in particular non- Abelian quantum Hall states. Using proposed non-Abelian bosonization dualities in two spatial dimensions, which morally relate U(N)k and SU(k)−N Chern-Simons-matter theories, we present pairing scenarios for which non-Abelian quantum Hall states can be obtained starting from theories of Abelian composite particles. The advantage of these dualities is that regions of the phase diagram which may be obscure on one side of the duality can be accessed by condensing local operators on the other side. Starting from parent Abelian states, we use this approach to construct Landau-Ginzburg theories of non-Abelian states through a pairing mechanism. In particular, we obtain the bosonic Read-Rezayi sequence at fillings ν = k/(kM +2) by starting from k layers of bosons at ν = 1/2 with M Abelian fluxes attached and then condensing k-clusters of the dual non-Abelian bosons. We further extend this constructions to obtain generalizations of the Read-Rezayi states with emergent global symmetries. In Chapter 5, we describe a context in which non-perturbative ideas from duality and perturbative results in the language of the renormalization group (RG) can inform one another: the interplay of quenched disorder and strong interaction effects near quantum critical points. In particular, we focus on the problem of quenched disorder at the superfluid-insulator transition of the O(N) model in the large-N limit. While a random mass is strongly relevant at the free fixed point, its effect is screened by the strong interactions of the Wilson-Fisher fixed point in this model, enabling a perturbative RG study of the interplay of disorder and interactions about this fixed point. In contrast to the spiralling flows obtained in earlier double-ε expansions, we show that the theory flows directly to a quantum critical point characterized by finite disorder and interactions, with critical exponents in remarkable agreement with numerical studies of the superfluid-Mott glass transition. With these results in hand, we apply duality to discuss the possible implications of this result for the dual Abelian Higgs and Chern-Simons-Dirac fermion theories when N = 1.