arXiv subjects

Anton Kapustin

Publications and source records attributed to Anton Kapustin.

At least 19 recordsLinked to original sources

arXiv2026

A mathematical theory of topological invariants of quantum lattice systems

We show that Hall conductance and its non-abelian and higher-dimensional analogs are obstructions to promoting a symmetry of a state to a gauge symmetry. To do this, we define a local Lie algebra over a Grothendieck site as a pre-cosheaf of Lie algebras with additional properties and propose that a gauge symmetry should be described by such an object. We show that infinitesimal symmetries of a gapped state of a quantum spin system form a local Lie algebra over a site of semilinear sets and use it to construct topological invariants of the state. Our construction applies to lattice systems on arbitrary asymptotically conical subsets of a Euclidean space including those which cannot be studied using field theory.

math-ph↗

arXiv2026

Chaos and thermalization in Clifford-Floquet dynamics

We study the ergodic properties of a unitary Floquet dynamics arising from the repeated application of a translationally-invariant Clifford Quantum Cellular Automata to an infinite system of qubits in d dimensions. One expects that if the QCA does not exhibit any periodicity, a generic initial state of qubits will thermalize, that is, approach the infinite-temperature state. We show that this is true for many classes of states, both pure and mixed. In particular, this is true for all initial states that are short-range entangled and close to the equilibrium state. We also point out a subtle distinction between weak and strong thermalization.

quant-ph↗

arXiv2026

On thermalization in many-body classical Floquet systems

It is expected that a generic closed many-body system prepared in a well-behaved initial state and subjected to a periodic drive will eventually thermalize, i.e. approach the state of maximal entropy. This property, while compatible with and even demanded by the physical intuition, is much stronger than ergodicity or mixing and is difficult to justify mathematically. We describe an infinite set of classical many-body Floquet systems of algebraic origin for which thermalization of very general initial states can be proved. For example, we show that a Gibbs state of any sufficiently uniform local differentiable Hamiltonian heats up to infinite temperature at long times. We show that in agreement with the physical intuition, the only obstruction to thermalization is the existence of local observables which are periodic in time.

cond-mat.stat-mech↗

arXiv2025

Higher symmetries and anomalies in quantum lattice systems

We define an 't Hooft anomaly index for a group acting on a 2d quantum lattice system by finite-depth circuits. It takes values in the degree-4 cohomology of the group and is an obstruction to on-siteability of the group action. We introduce a 3-group (modeled as a crossed square) describing higher symmetries of a 2d lattice system and show that the 2d anomaly index is an obstruction for promoting a symmetry action to a morphism of 3-groups. This demonstrates that 't Hooft anomalies are a consequence of a mixing between ordinary symmetries and higher symmetries. Similarly, to any 1d lattice system we attach a 2-group (modeled as a crossed module) and interpret the Nayak-Else anomaly index as an obstruction for promoting a group action to a morphism of 2-groups. The meaning of indices of Symmetry Protected Topological states is also illuminated by higher group symmetry.

math-ph↗

arXiv2025

Higher symmetries, anomalies, and crossed squares in lattice gauge theory

We examine higher-form symmetries of quantum lattice gauge theories through the lens of homotopy theory and operator algebras. We show that in the operator-algebraic approach both higher-form symmetries and 't Hooft anomalies arise from considering restrictions of symmetry transformations to spatial regions. The data of these restrictions are naturally packaged into a higher group. For example, for gauge theories in two spatial dimensions, this information is encoded in a crossed square of groups, which is an algebraic model of a 3-group. In general, we propose that higher groups appear in lattice models and QFT as crossed n-cubes of groups via a nonabelian version of the Cech construction.

hep-th↗

arXiv2024

Interplay between two mechanisms of resistivity

Mechanisms of resistivity can be divided into two basic classes: one is dissipative (like scattering on phonons) and another is quasi-elastic (like scattering on static impurities). They are often treated by the empirical Matthiessen rule, which says that total resistivity is just the sum of these two contributions, which are computed separately. This is quite misleading for two reasons. First, the two mechanisms are generally correlated. Second, computing the elastic resistivity alone masks the fundamental fact that the linear-response approximation has a vanishing validity interval at vanishing dissipation. Limits of zero electric field and zero dissipation do not commute for the simple reason that one needs to absorb the Joule heat quadratic in the applied field. Here, we present a simple model that illustrates these two points. The model also illuminates the role of variational principles for non-equilibrium steady states.

cond-mat.stat-mech↗

arXiv2024

Quantization of the higher Berry curvature and the higher Thouless pump

We show that for families of 1d lattice systems in an invertible phase, the cohomology class of the higher Berry curvature can be refined to an integral degree-3 class on the parameter space. Similarly, for families of U(1)-invariant 2d lattice systems in an invertible phase, the higher Thouless pump can be refined to an integral degree-2 class on the parameter space. We show that the 2d Thouless pump can be identified with an excess Berry curvature of a flux insertion.

math-ph↗

arXiv2024

A soluble model of a Non-Equilibrium Steady State: the van Kampen objection and other lessons

A simple model of charge transport is provided by a classical particle in a smooth random potential and a dissipative coupling to the environment in the form of Markovian noise and friction. The corresponding Non-Equilibrium Steady State (NESS) can be determined analytically when both the disorder and dissipation are weak. We use it to illuminate some foundational issues in non-equilibrium statistical mechanics. We show that Linear Response Theory has a nonempty regime of validity only in the presence of a dissipative coupling to the environment, thereby validating van Kampen's objection. We also show that the Principle of Minimum Entropy Production does not determine the NESS beyond linear order in the electric field, while entropy maximization fails to produce the correct NESS already at linear order.

cond-mat.stat-mech↗

arXiv2024

Anomalous symmetries of quantum spin chains and a generalization of the Lieb-Schultz-Mattis theorem

For any locality-preserving action of a group $G$ on a quantum spin chain one can define an anomaly index taking values in the group cohomology of $G$. The anomaly index is a kinematic quantity, it does not depend on the Hamiltonian. We prove that a nonzero anomaly index prohibits any $G$-invariant Hamiltonian from having $G$-invariant gapped ground states. Lieb-Schultz-Mattis-type theorems are a special case of this result when $G$ involves translations. In the case when the symmetry group $G$ is a Lie group, we define an anomaly index which takes values in the differentiable group cohomology as defined by J.-L. Brylinski and prove a similar result.

math-ph↗

arXiv2023

The Universal Time-Dependent Ginzburg-Landau theory

We study the hydrodynamics of superconductors within the framework of Schwinger-Keldysh Effective Field Theory. We show that in the vicinity of the superconducting phase transition the most general leading-order EFT satisfying the local Kubo-Martin-Schwinger condition is described by a version of the Time-Dependent Ginzburg-Landau (TDGL) equations augmented with stochastic terms. This version of TDGL is applicable in the gapless regime independently of any microscopic details. Within this approach, it is possible to include systematically the effects of non-uniform temperature and heat conductivity, as well as explicit or spontaneous breaking of time reversal. We also introduce a thermal version of the Josephson relation and use it to construct an exotic hydrodynamics describing a phase of matter where heat can flow without dissipation.

cond-mat.supr-con↗

arXiv2022

Hohenberg-Mermin-Wagner-type theorems and dipole symmetry

We study the possibility of spontaneous symmetry breaking in systems with both charge and dipole symmetries. For $d$-dimensional systems at a positive temperature, we show that charge symmetry cannot be spontaneously broken for $d\leq 4$, while dipole symmetry cannot be spontaneously broken for $d\leq 2$. For $T=0$, we show that charge symmetry cannot be spontaneously broken for $d\leq 2$ if the compressibility is finite. We also show that continuum systems with a dipole symmetry have infinite inertial mass density.

cond-mat.str-el↗

arXiv2022

Local Noether theorem for quantum lattice systems and topological invariants of gapped states

We study generalizations of the Berry phase for quantum lattice systems in arbitrary dimensions. For a smooth family of gapped ground states in d dimensions, we define a closed (d+2)-form on the parameter space which generalizes the curvature of the Berry connection. Its cohomology class is a topological invariant of the family. When the family is equivariant under the action of a compact Lie group G, topological invariants take values in the equivariant cohomology of the parameter space. These invariants unify and generalize the Hall conductance and the Thouless pump. A key role in these constructions is played by a certain differential graded Frechet-Lie algebra attached to any quantum lattice system. As a by-product, we describe ambiguities in charge densities and conserved currents for arbitrary lattice systems with rapidly decaying interactions.

math-ph↗

arXiv2022

Microscopic formulas for thermoelectric transport coefficients in lattice systems

A macroscopic description of thermoelectric phenomena involves several tensorial transport coefficients. Textbook microscopic Kubo formulas for them are plagued with ambiguities in the definitions of the current operators and the magnetization. We derive a version of these formulas for lattice systems which is free from ambiguities but contains additional terms compared to the textbook results. For symmetric components of thermoelectric tensors, we identify a large class of lattice systems for which the additional terms vanish with a natural choice of the energy current. To eliminate ambiguities in the skew-symmetric components, one needs to interpret them as relative quantities: only their differences for pairs of materials are well-defined.

cond-mat.str-el↗

arXiv2022

Piezo-superconductivity: new effects in non-centrosymmetric superconductors

We study novel effects in non-centrosymmetric superconductors arising from their unique coupling of Cooper-pair condensate and elasticity. We show that although the much discussed Lifshitz coupling is not observable in a uniform bulk state, it strikingly endows dislocations with a fractional magnetic flux. We also predict a generation of voltage-free strain by a DC current in a P and T-breaking Josephson junction. Viewing superconductors through the lens of higher-form symmetries we identify the Lifshitz coupling as a chemical potential for the approximately conserved winding number, drawing an analogy with pyroelectric insulators.

cond-mat.supr-con↗

arXiv2021

A classification of phases of bosonic quantum lattice systems in one dimension

We study invertible states of 1d bosonic quantum lattice systems. We show that every invertible 1d state is in a trivial phase: after tensoring with some unentangled ancillas it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. If an invertible state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries, even after adding unentagled ancillas. We show that in the case of a finite unitary symmetry G the only obstruction is an index valued in degree-2 cohomology of $G$. We show that two invertible $G$-invariant states are in the same phase if and only if their indices coincide.

quant-ph↗

arXiv2021

Non-relativistic Geometry and the Equivalence Principle

We describe a geometric and symmetry-based formulation of the equivalence principle in non-relativistic physics. It applies both on the classical and quantum levels and states that the Newtonian potential can be eliminated in favor of a curved and time-dependent spatial metric. It is this requirement that forces the gravitational mass to be equal to the inertial mass. We identify the symmetry responsible for the equivalence principle as the remnant of time-reparameterization symmetry of the relativistic theory. We also clarify the transformation properties of the Schroedinger wave-function under arbitrary changes of frame.

gr-qc↗

arXiv2021

Nernst and Ettingshausen effects in gapped quantum materials

We investigate whether there could exist topological invariants of gapped 2D materials related to dissipationless thermoelectric transport at low temperatures. We give both macroscopic and microscopic arguments showing that thermoelectric transport coefficients vanish in the limit of zero temperature and thus topological invariants arise only from the electric Hall conductance and the thermal Hall conductance. Our arguments apply to systems with arbitrarily strong interactions. We also show that there is no analog of the Thouless pump for entropy.

cond-mat.str-el↗

arXiv2020

Hall conductance and the statistics of flux insertions in gapped interacting lattice systems

We study charge transport for zero-temperature infinite-volume gapped lattice systems in two dimensions with short-range interactions. We show that the Hall conductance is locally computable and is the same for all systems which are in the same gapped phase. We provide a rigorous versions of Laughlin's flux-insertion argument which shows that for short-range entangled systems the Hall conductance is an integer multiple of e^2/h. We show that the Hall conductance determines the statistics of flux insertions. For bosonic short-range entangled systems, this implies that the Hall conductance is an even multiple of e^2/h. Finally, we adapt a proof of quantization of the Thouless charge pump to the case of infinite-volume gapped lattice systems in one dimension.

math-ph↗