Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2005.02882

Nonlinear parameter-gauge coupling approach to generalization of generalized Thouless pumps and $-1$-form anomaly

Yuan Yao

Published May 6, 2020• 0 citations
arXivHTMLPDF

Abstract

We study the nontrivial topology of the parameter space of general $U(1)$-symmetric fermionic non-degenerately gapped system and its consequences on the transport properties in arbitrary dimensions. By a nonlinear parameter-gauge topological response theory, we find that such nontrivial topology can impose quantization constraints on the charge transport in the presence of background fluxes or, more generally, instantons in general dimensions and our result generalizes the Thouless pump and its higher dimensional generalizations. We also show that these nontrivial transport properties are related to an unconventional quantum anomaly, which generalizes $-1$-form anomalies. This anomaly imposes non-perturbative ingappabilities of various types of spatial interfaces or time-dependent system evolution.

DOI
10.48550/arxiv.2005.02882

Related papers

Generalized Thouless pumps in (1+1) -dimensional interacting fermionic systems

Shuhei Ohyama, Ken Shiozaki, Masatoshi Sato

The Thouless pump is a phenomenon in which $U(1)$ charges are pumped from an edge of a fermionic system to another edge. The Thouless pump has been generalized in various dimensions and for various charges. In this paper we investigate the generalized Thouless pumps of fermion parity in both trivial and nontrivial phases of $(1+1)$-dimensional interacting fermionic short-range entangled (SRE) states. For this purpose, we use matrix product states (MPSs). MPSs describe many-body systems in $(1+1)$ dimensions and can characterize SRE states algebraically. We prove fundamental theorems for fermionic MPSs (fMPSs) and use them to investigate the generalized Thouless pumps. We construct nontrivial pumps in both the trivial and nontrivial phases, and we show the stability of the pumps against interactions. Furthermore, we define topological invariants for the generalized Thouless pumps in terms of fMPSs and establish consistency with existing results. These are invariants of the family of SRE states that are not captured by the higher dimensional Berry curvature. We also argue a relation between the topological invariants of the generalized Thouless pump and the twist of the $K$ theory in the Donovan-Karoubi formulation.

arXiv:2206.01110•Physical review. B./Physical review. B•Oct 17, 2022•10 citations

Symmetry-protected topological phases and quantum anomalies

Chang-Tse Hsieh

We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.

•Jul 11, 2017•0 citations

Higher-dimensional generalizations of the Thouless charge pump

Anton Kapustin, Lev Spodyneiko

We define and study analogs of the Thouless charge pump for many-body gapped systems in dimension $D$. We show how to attach a topological invariant to a $D$-dimensional family of such systems, provided all of them have an on-site $U(1)$ symmetry. For a large class of families we argue that this topological invariant is an integer. In the case of gapped systems of free fermions in two dimensions, the invariant can be expressed in terms of the curvature of the Bloch-Berry connection. We also obtain a new formula for the Thouless charge pump in 1d which involves only static linear response and is analogous to the Streda formula for Hall conductivity.

arXiv:2003.09519•Mar 20, 2020•14 citations

Anomalous continuous symmetries and quantum topology of Goldstone modes

Naren Manjunath, Dominic V. Else

We consider systems in which a continuous symmetry $G$, which may be anomalous, is spontaneously broken to an anomaly-free subgroup $H$ such that the effective action for the Goldstone modes contains topologically nontrivial terms. If the original system has trivial $G$ anomaly, then it is known that the possible topological terms are fully determined by SPT or SET invariants of the residual $H$ symmetry. Here we address the more general setting in which the $G$ symmetry has an anomaly. We argue that in general, the appropriate concept to consider is the ``compatibility relation'' between the Goldstone invariants and the $G$ anomaly. In the case where the Goldstone modes can be gapped out to obtain invertible families (i.e., without any topological order), we give an explicit mathematical scheme to construct the desired compatibility relation. We also address the case where gapping out the Goldstone modes leads to a family of topologically ordered states. We discuss several examples including the canonical Thouless pump, the quantum Hall ferromagnet, pumps arising from breaking $\mathrm{U}(1)$ symmetry at the boundary of topological insulators in two and three dimensions, and pumps classified by the higher Chern number.

arXiv:2410.05268•Physical review. B./Physical review. B•Mar 21, 2025•5 citations

Thouless Pumps and Bulk-Boundary Correspondence in Higher-Order Symmetry-Protected Topological Phases

Julian F. Wienand, Friederike Horn, Monika Aidelsburger +2 more

The bulk-boundary correspondence relates quantized edge states to bulk topological invariants in topological phases of matter. In one-dimensional symmetry-protected topological systems, quantized topological Thouless pumps directly reveal this principle and provide a sound mathematical foundation. Symmetry-protected higher-order topological phases of matter (HOSPTs) also feature a bulk-boundary correspondence, but its connection to quantized charge transport remains elusive. Here, we show that quantized Thouless pumps connecting C_{4}-symmetric HOSPTs can be described by a tuple of four Chern numbers that measure quantized bulk charge transport in a direction-dependent fashion. Moreover, this tuple of Chern numbers allows to predict the sign and value of fractional corner charges in the HOSPTs. We show that the topologically nontrivial phase can be characterized by both quadrupole and dipole configurations, shedding new light on current debates about the multipole nature of the HOSPT bulk. By employing corner-periodic boundary conditions, we generalize Restas's theory to HOSPTs. Our approach provides a simple framework for understanding topological invariants of general HOSPTs and paves the way for an in-depth description of future dynamical experiments.

arXiv:2111.02491•Physical Review Letters•Jun 17, 2022•32 citations

Topological gauge theory for mixed Dirac stationary states in all dimensions

Ze-Min Huang, Xiao-Qi Sun, Sebastian Diehl

We derive the universal real-time $\text{U}(1)$ topological gauge field action for mixed Dirac states of fermions, described by a density matrix weighted by a gapped Dirac operator, in all dimensions. The key prerequisites are charge quantization and charge conservation; in fact, charge quantization is the root of the action's topological properties. The gauge action encodes nonquantized linear responses as expected for mixed states, but also quantized nonlinear responses, which are accessible in experiment. Our construction furthermore demonstrates how the physical pictures of anomaly inflow and bulk-boundary correspondence extend to nonequilibrium systems. Finally, we demonstrate the independence of the gauge action of the underlying equilibrium or nonequilibrium nature of dynamics stabilizing the state.

arXiv:2109.06891•Physical review. B./Physical review. B•Dec 19, 2022•14 citations
Nonlinear parameter-gauge coupling approach to generalization of generalized Thouless pumps and −1-1−1-form anomaly
We study the nontrivial topology of the parameter space of general U(1)U(1)U(1)-symmetric fermionic non-degenerately gapped system and its consequences on the transport properties in arbitrary dimensions. By a nonlinear parameter-gauge topological response theory, we find that such nontrivial topology can impose quantization constraints on the charge transport in the presence of background fluxes or, more generally, instantons in general dimensions and our result generalizes the Thouless pump and its higher dimensional generalizations. We also show that these nontrivial transport properties are related to an unconventional quantum anomaly, which generalizes −1-1−1-form anomalies. This anomaly imposes non-perturbative ingappabilities of various types of spatial interfaces or time-dependent system evolution.
The Thouless pump is a phenomenon in which U(1)U(1)U(1) charges are pumped from an edge of a fermionic system to another edge. The Thouless pump has been generalized in various dimensions and for various charges. In this paper we investigate the generalized Thouless pumps of fermion parity in both trivial and nontrivial phases of (1+1)(1+1)(1+1)-dimensional interacting fermionic short-range entangled (SRE) states. For this purpose, we use matrix product states (MPSs). MPSs describe many-body systems in (1+1)(1+1)(1+1) dimensions and can characterize SRE states algebraically. We prove fundamental theorems for fermionic MPSs (fMPSs) and use them to investigate the generalized Thouless pumps. We construct nontrivial pumps in both the trivial and nontrivial phases, and we show the stability of the pumps against interactions. Furthermore, we define topological invariants for the generalized Thouless pumps in terms of fMPSs and establish consistency with existing results. These are invariants of the family of SRE states that are not captured by the higher dimensional Berry curvature. We also argue a relation between the topological invariants of the generalized Thouless pump and the twist of the KKK theory in the Donovan-Karoubi formulation.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
We define and study analogs of the Thouless charge pump for many-body gapped systems in dimension DDD. We show how to attach a topological invariant to a DDD-dimensional family of such systems, provided all of them have an on-site U(1)U(1)U(1) symmetry. For a large class of families we argue that this topological invariant is an integer. In the case of gapped systems of free fermions in two dimensions, the invariant can be expressed in terms of the curvature of the Bloch-Berry connection. We also obtain a new formula for the Thouless charge pump in 1d which involves only static linear response and is analogous to the Streda formula for Hall conductivity.
We consider systems in which a continuous symmetry GGG, which may be anomalous, is spontaneously broken to an anomaly-free subgroup HHH such that the effective action for the Goldstone modes contains topologically nontrivial terms. If the original system has trivial GGG anomaly, then it is known that the possible topological terms are fully determined by SPT or SET invariants of the residual HHH symmetry. Here we address the more general setting in which the GGG symmetry has an anomaly. We argue that in general, the appropriate concept to consider is the ``compatibility relation'' between the Goldstone invariants and the GGG anomaly. In the case where the Goldstone modes can be gapped out to obtain invertible families (i.e., without any topological order), we give an explicit mathematical scheme to construct the desired compatibility relation. We also address the case where gapping out the Goldstone modes leads to a family of topologically ordered states. We discuss several examples including the canonical Thouless pump, the quantum Hall ferromagnet, pumps arising from breaking U(1)\mathrm{U}(1)U(1) symmetry at the boundary of topological insulators in two and three dimensions, and pumps classified by the higher Chern number.
The bulk-boundary correspondence relates quantized edge states to bulk topological invariants in topological phases of matter. In one-dimensional symmetry-protected topological systems, quantized topological Thouless pumps directly reveal this principle and provide a sound mathematical foundation. Symmetry-protected higher-order topological phases of matter (HOSPTs) also feature a bulk-boundary correspondence, but its connection to quantized charge transport remains elusive. Here, we show that quantized Thouless pumps connecting C_{4}-symmetric HOSPTs can be described by a tuple of four Chern numbers that measure quantized bulk charge transport in a direction-dependent fashion. Moreover, this tuple of Chern numbers allows to predict the sign and value of fractional corner charges in the HOSPTs. We show that the topologically nontrivial phase can be characterized by both quadrupole and dipole configurations, shedding new light on current debates about the multipole nature of the HOSPT bulk. By employing corner-periodic boundary conditions, we generalize Restas's theory to HOSPTs. Our approach provides a simple framework for understanding topological invariants of general HOSPTs and paves the way for an in-depth description of future dynamical experiments.
We derive the universal real-time U(1)\text{U}(1)U(1) topological gauge field action for mixed Dirac states of fermions, described by a density matrix weighted by a gapped Dirac operator, in all dimensions. The key prerequisites are charge quantization and charge conservation; in fact, charge quantization is the root of the action's topological properties. The gauge action encodes nonquantized linear responses as expected for mixed states, but also quantized nonlinear responses, which are accessible in experiment. Our construction furthermore demonstrates how the physical pictures of anomaly inflow and bulk-boundary correspondence extend to nonequilibrium systems. Finally, we demonstrate the independence of the gauge action of the underlying equilibrium or nonequilibrium nature of dynamics stabilizing the state.