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:1401.3703 · Physical Review B

Electromagnetic and gravitational responses of two-dimensional noninteracting electrons in a background magnetic field

Alexander G. Abanov, Andrey Gromov

Published Jul 29, 2014• 164 citations
arXivHTMLPDF

Abstract

We compute electromagnetic, gravitational, and mixed linear response functions of two-dimensional free fermions in an external quantizing magnetic field at an integer filling factor. The results are presented in the form of the effective action and as an expansion of currents and stresses in wave vectors and frequencies of the probing electromagnetic and metric fields. In addition to the well-studied $U(1)$ Chern-Simons and Wen-Zee terms we find a gravitational Chern-Simons term that controls the correction to the Hall viscosity due to the background curvature. We relate the coefficient in front of the term with the chiral central charge.

DOI
10.1103/physrevb.90.014435

Related papers

Topological electromagnetic responses of bosonic quantum Hall, topological insulator, and chiral semimetal phases in all dimensions

Matthew F. Lapa, Chao‐Ming Jian, Peng Ye +1 more

We calculate the topological part of the electromagnetic response of bosonic integer quantum Hall (BIQH) phases in odd (space-time) dimensions, and bosonic topological insulator (BTI) and bosonic chiral semimetal (BCSM) phases in even dimensions. To do this, we use the nonlinear sigma model (NLSM) description of bosonic symmetry-protected topological (SPT) phases, and the method of gauged Wess-Zumino (WZ) actions. We find the surprising result that for BIQH states in dimension $2m\ensuremath{-}1$ $(m=1,2,\ensuremath{\cdots})$, the bulk response to an electromagnetic field ${A}_{\ensuremath{\mu}}$ is characterized by a Chern-Simons term for ${A}_{\ensuremath{\mu}}$ with a level quantized in integer multiples of $m!$ (factorial). We also show that BTI states (which have an extra ${\mathbb{Z}}_{2}$ symmetry) can exhibit a ${\mathbb{Z}}_{2}$-breaking quantum Hall effect on their boundaries, with this boundary quantum Hall effect described by a Chern-Simons term at level $\frac{m!}{2}$. We show that the factor of $m!$ can be understood by requiring gauge invariance of the exponential of the Chern-Simons term on a general Euclidean manifold, and we also use this argument to characterize the electromagnetic and gravitational responses of fermionic SPT phases with $\text{U}(1)$ symmetry in all odd dimensions. We then use our gauged boundary actions for the BIQH and BTI states to (i) construct a bosonic analog of a chiral semimetal (BCSM) in even dimensions, (ii) show that the boundary of the BTI state exhibits a bosonic analog of the parity anomaly of Dirac fermions in odd dimensions, and (iii) study anomaly inflow at domain walls on the boundary of BTI states. In a series of Appendixes we derive important formulas and additional results. In particular, in Appendix A we use the connection between equivariant cohomology and gauged WZ actions to give a mathematical interpretation of the actions for the BIQH and BTI boundaries constructed in this paper.

arXiv:1611.03504•Physical review. B./Physical review. B•Jan 27, 2017•29 citations

Gauge invariance and current algebra in nonrelativistic many-body theory

Jürg Martin Fröhlich, Urban M. Studer

The main purpose of this paper is to further our theoretical understanding of the fractional quantum Hall effect, in particular of spin effects, in two-dimensional incompressible electron fluids subject to a strong, transverse magnetic field. As a prerequisite for an analysis of the quantum Hall effect, the authors develop a general formulation of the many-body theory of spinning particles coupled to external electromagnetic fields and moving through a general, geometrically nontrivial background. Their formulation is based on a Lagrangian path-integral quantization and is valid in arbitrary coordinates, including coordinates moving according to a volume-preserving flow. It is found that nonrelativistic quantum theory exhibits a fundamental, local U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, and the corresponding gauge fields are identified with physical, external fields. To illustrate the usefulness of their formalism, the authors prove a general form of the quantum-mechanical Larmor theorem and discuss some well-known effects, including the Barnett-Einstein-de Haas effect and superconductivity, emphasizing the implications of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance. They then consider two-dimensional, incompressible quantum fluids in more detail. Exploiting U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, they calculate the leading terms in the effective actions of such systems as functionals of the U(1) and SU(2) gauge fields, on large-distance and low-frequency scales. Among the applications of these results are a simple proof of the Goldstone theorem for spin waves and the linearresponse theory of two-dimensional, incompressible Hall fluids, including a Hall effect for spin currents and sum rules for the response coefficients. For two-dimensional, incompressible systems with broken parity and time-reversal symmetry, a particularly significant implication of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance is a duality between the physics inside the bulk of the system and the physics of gapless, chiral modes propagating along the boundary of the system. These modes form chiral $\stackrel{^}{\mathrm{u}}(1)$ and $\mathrm{s}\stackrel{^}{\mathrm{u}}(2)$ current algebras. The representation theory of these current algebras, combined with natural physical constraints, permits one to derive the quantization of the response coefficients, such as the Hall conductivity. A classification of incompressible Hall fluids is outlined, and many examples, including one concerning a superfluid $^{3}\mathrm{He}$-$\frac{A}{B}$ interface, are discussed.

Reviews of Modern Physics•Jul 1, 1993•298 citations

Field-theory approach to the quantum Hall effect

A. Cabo, Масуд Чайчиан

Fradkin's formulation of statistical field theory is applied to the Coulomb interacting electron gas in a magnetic field. The electrons are confined to a plane in normal three-dimensional space and also interact with the physical three-dimensional electromagnetic field. The magnetic-translation-group Ward identities are derived. By using them, it is shown that the exact electron propagator is diagonalized in the basis of the wave functions of the free electron in a magnetic field whenever the magnetic-translation-group symmetry is unbroken. The general tensor structure of the polarization operator is obtained and used to show that the Chern-Simons action always describes the Hall-effect properties of the system. A general proof of the Str\^eda formula for the Hall conductivity is presented. It follows that the coefficient of the Chern-Simons terms in the long-wavelength approximation is exactly given by this relation. Such a formula, expressing the Hall conductivity as a simple derivative, in combination with a diagonal form of the full propagator, allows us to obtain a simple expression for the filling factor and the Hall conductivity. Indeed, these results, after assuming that the chemical potential lies in a gap of the density of states, lead to the conclusion that the Hall conductivity is given without corrections by ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$=\ensuremath{\nu}${\mathit{e}}^{2}$/h, where \ensuremath{\nu} is the filling factor. In addition, it follows that the filling factor is independent of the magnetic field if the value of the chemical potential remains in the gap.

Physical review. B, Condensed matter•Nov 15, 1991•6 citations

Topological response of gapped fermions to a $\text{U}(1)$ Gauge field

Bruno Mera

We present a detailed path integral derivation of the topological response of gapped free fermions, in $2+1$ dimensions, to an external $\text{U}(1)$ Gauge field. The well-known Hall response is obtained by identifying the Chern-Simons term in the effective action with the correct coefficient. We extend the result to $2d+1$ dimensions in which the response is associated to a Chern-Simons term with a coefficient related to a characteristic class coming from topological band theory. We comment on the bulk-to-boundary principle which arises naturally when one considers the theory on a manifold with boundary.

arXiv:1705.04394•May 11, 2017•4 citations

Effective action of a (2+1)-dimensional system of nonrelativistic fermions in the presence of a uniform magnetic field: Dissipation effects

S. Sakhi

The effective action of nonrelativistic fermions in 2+1 dimensions is analyzed at finite temperature and chemical potential in the presence of a uniform magnetic field perpendicular to the plane. The method used is a generalization of the derivative expansion technique. The induced Chern-Simons term is computed and shown to exhibit Hall quantization. Effects of dissipation due to collisions are also analyzed.

arXiv:cond-mat/9401033•Physical review. B, Condensed matter•May 15, 1994•7 citations

Field Theory for Fractional Quantum Hall States

Tankut Can, Michael Laskin, Paul B. Wiegmann

We develop a field theory description of fractional quantum Hall (FQH) states. We show that in the leading approximation in a gradient expansion, Laughlin states are described by a Gaussian free field theory with a background charge which is identified with the anomalous viscosity of the states. The background charge increases the central charge of the corresponding conformal field theory above 1, similar to that of the theory of 2D quantum gravity. Gradient corrections to the Gaussian field theory arising from ultraviolet regularization reflect the gravitational anomaly. They are also related to the Liouville theory of quantum gravity. We show how the gradient expansion of the field action encodes the universal features of the FQH effect beyond that of Hall conductance. This method provides a more transparent and useful alternative for computing the gravitational anomaly and correlation functions of the FQHE in a curved space than the method of iterating the Ward identity employed by the authors in previous papers.

•Dec 30, 2014•2 citations
We compute electromagnetic, gravitational, and mixed linear response functions of two-dimensional free fermions in an external quantizing magnetic field at an integer filling factor. The results are presented in the form of the effective action and as an expansion of currents and stresses in wave vectors and frequencies of the probing electromagnetic and metric fields. In addition to the well-studied U(1)U(1)U(1) Chern-Simons and Wen-Zee terms we find a gravitational Chern-Simons term that controls the correction to the Hall viscosity due to the background curvature. We relate the coefficient in front of the term with the chiral central charge.
We calculate the topological part of the electromagnetic response of bosonic integer quantum Hall (BIQH) phases in odd (space-time) dimensions, and bosonic topological insulator (BTI) and bosonic chiral semimetal (BCSM) phases in even dimensions. To do this, we use the nonlinear sigma model (NLSM) description of bosonic symmetry-protected topological (SPT) phases, and the method of gauged Wess-Zumino (WZ) actions. We find the surprising result that for BIQH states in dimension $2m\ensuremath{-}1$ $(m=1,2,\ensuremath{\cdots})$, the bulk response to an electromagnetic field ${A}_{\ensuremath{\mu}}$ is characterized by a Chern-Simons term for ${A}_{\ensuremath{\mu}}$ with a level quantized in integer multiples of m!m!m! (factorial). We also show that BTI states (which have an extra Z2{\mathbb{Z}}_{2}Z2​ symmetry) can exhibit a Z2{\mathbb{Z}}_{2}Z2​-breaking quantum Hall effect on their boundaries, with this boundary quantum Hall effect described by a Chern-Simons term at level m!2\frac{m!}{2}2m!​. We show that the factor of m!m!m! can be understood by requiring gauge invariance of the exponential of the Chern-Simons term on a general Euclidean manifold, and we also use this argument to characterize the electromagnetic and gravitational responses of fermionic SPT phases with U(1)\text{U}(1)U(1) symmetry in all odd dimensions. We then use our gauged boundary actions for the BIQH and BTI states to (i) construct a bosonic analog of a chiral semimetal (BCSM) in even dimensions, (ii) show that the boundary of the BTI state exhibits a bosonic analog of the parity anomaly of Dirac fermions in odd dimensions, and (iii) study anomaly inflow at domain walls on the boundary of BTI states. In a series of Appendixes we derive important formulas and additional results. In particular, in Appendix A we use the connection between equivariant cohomology and gauged WZ actions to give a mathematical interpretation of the actions for the BIQH and BTI boundaries constructed in this paper.
The main purpose of this paper is to further our theoretical understanding of the fractional quantum Hall effect, in particular of spin effects, in two-dimensional incompressible electron fluids subject to a strong, transverse magnetic field. As a prerequisite for an analysis of the quantum Hall effect, the authors develop a general formulation of the many-body theory of spinning particles coupled to external electromagnetic fields and moving through a general, geometrically nontrivial background. Their formulation is based on a Lagrangian path-integral quantization and is valid in arbitrary coordinates, including coordinates moving according to a volume-preserving flow. It is found that nonrelativistic quantum theory exhibits a fundamental, local U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, and the corresponding gauge fields are identified with physical, external fields. To illustrate the usefulness of their formalism, the authors prove a general form of the quantum-mechanical Larmor theorem and discuss some well-known effects, including the Barnett-Einstein-de Haas effect and superconductivity, emphasizing the implications of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance. They then consider two-dimensional, incompressible quantum fluids in more detail. Exploiting U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, they calculate the leading terms in the effective actions of such systems as functionals of the U(1) and SU(2) gauge fields, on large-distance and low-frequency scales. Among the applications of these results are a simple proof of the Goldstone theorem for spin waves and the linearresponse theory of two-dimensional, incompressible Hall fluids, including a Hall effect for spin currents and sum rules for the response coefficients. For two-dimensional, incompressible systems with broken parity and time-reversal symmetry, a particularly significant implication of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance is a duality between the physics inside the bulk of the system and the physics of gapless, chiral modes propagating along the boundary of the system. These modes form chiral $\stackrel{^}{\mathrm{u}}(1)$ and $\mathrm{s}\stackrel{^}{\mathrm{u}}(2)$ current algebras. The representation theory of these current algebras, combined with natural physical constraints, permits one to derive the quantization of the response coefficients, such as the Hall conductivity. A classification of incompressible Hall fluids is outlined, and many examples, including one concerning a superfluid 3He^{3}\mathrm{He}3He-AB\frac{A}{B}BA​ interface, are discussed.
Fradkin's formulation of statistical field theory is applied to the Coulomb interacting electron gas in a magnetic field. The electrons are confined to a plane in normal three-dimensional space and also interact with the physical three-dimensional electromagnetic field. The magnetic-translation-group Ward identities are derived. By using them, it is shown that the exact electron propagator is diagonalized in the basis of the wave functions of the free electron in a magnetic field whenever the magnetic-translation-group symmetry is unbroken. The general tensor structure of the polarization operator is obtained and used to show that the Chern-Simons action always describes the Hall-effect properties of the system. A general proof of the Str\^eda formula for the Hall conductivity is presented. It follows that the coefficient of the Chern-Simons terms in the long-wavelength approximation is exactly given by this relation. Such a formula, expressing the Hall conductivity as a simple derivative, in combination with a diagonal form of the full propagator, allows us to obtain a simple expression for the filling factor and the Hall conductivity. Indeed, these results, after assuming that the chemical potential lies in a gap of the density of states, lead to the conclusion that the Hall conductivity is given without corrections by ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$=\ensuremath{\nu}e2{\mathit{e}}^{2}e2/h, where \ensuremath{\nu} is the filling factor. In addition, it follows that the filling factor is independent of the magnetic field if the value of the chemical potential remains in the gap.
Topological response of gapped fermions to a U(1)\text{U}(1)U(1) Gauge field
We present a detailed path integral derivation of the topological response of gapped free fermions, in 2+12+12+1 dimensions, to an external U(1)\text{U}(1)U(1) Gauge field. The well-known Hall response is obtained by identifying the Chern-Simons term in the effective action with the correct coefficient. We extend the result to 2d+12d+12d+1 dimensions in which the response is associated to a Chern-Simons term with a coefficient related to a characteristic class coming from topological band theory. We comment on the bulk-to-boundary principle which arises naturally when one considers the theory on a manifold with boundary.
We develop a field theory description of fractional quantum Hall (FQH) states. We show that in the leading approximation in a gradient expansion, Laughlin states are described by a Gaussian free field theory with a background charge which is identified with the anomalous viscosity of the states. The background charge increases the central charge of the corresponding conformal field theory above 1, similar to that of the theory of 2D quantum gravity. Gradient corrections to the Gaussian field theory arising from ultraviolet regularization reflect the gravitational anomaly. They are also related to the Liouville theory of quantum gravity. We show how the gradient expansion of the field action encodes the universal features of the FQH effect beyond that of Hall conductance. This method provides a more transparent and useful alternative for computing the gravitational anomaly and correlation functions of the FQHE in a curved space than the method of iterating the Ward identity employed by the authors in previous papers.