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Torsion, Parity-odd Response and Anomalies in Topological States
Mesoscale and Nanoscale Physics · Submitted 25 Jul 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Finally, using dimensional reduction from $d=4+1$, we derive the effective action for a $d=3+1$ time-reversal invariant topological insulator in the presence of torsion and curvature, and discuss its various physical consequences.
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The abstract, sentence by sentence
We study the response of a class of topological systems to electromagnetic and gravitational sources, including torsion and curvature. By using the technology of anomaly polynomials, we derive the parity-odd response of a massive Dirac fermion in $d=2+1$ and $d=4+1$, which provides a simple model for a topological insulator. We discuss the covariant anomalies of the corresponding edge states, from a Callan-Harvey anomaly-inflow, as well as a Hamiltonian spectral flow point of view. We also discuss the applicability of our results to other systems such as Weyl semi-metals. Finally, using dimensional reduction from $d=4+1$, we derive the effective action for a $d=3+1$ time-reversal invariant topological insulator in the presence of torsion and curvature, and discuss its various physical consequences.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the response of a class of topological systems to electromagnetic and gravitational sources, including torsion and curvature. | contribution 0.84 | 0.87 | 0.06 | asserted 0.88 |
| s1 | By using the technology of anomaly polynomials, we derive the parity-odd response of a massive Dirac fermion in $d=2+1$ and $d=4+1$, which provides a simple model for a topological insulator. | contribution 0.48 | 0.96 | 0.23 | asserted 1.00 |
| s2 | We discuss the covariant anomalies of the corresponding edge states, from a Callan-Harvey anomaly-inflow, as well as a Hamiltonian spectral flow point of view. | contribution 0.76 | 0.83 | 0.08 | asserted 0.91 |
| s3 | We also discuss the applicability of our results to other systems such as Weyl semi-metals. | contribution 0.44 | 0.69 | 0.05 | asserted 0.54 |
| s4 | Finally, using dimensional reduction from $d=4+1$, we derive the effective action for a $d=3+1$ time-reversal invariant topological insulator in the presence of torsion and curvature, and discuss its various physical consequences. | result 0.59 | 0.96 | 0.38 | asserted 1.00 |
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The paper
Original abstract
We study the response of a class of topological systems to electromagnetic and gravitational sources, including torsion and curvature. By using the technology of anomaly polynomials, we derive the parity-odd response of a massive Dirac fermion in $d=2+1$ and $d=4+1$, which provides a simple model for a topological insulator. We discuss the covariant anomalies of the corresponding edge states, from a Callan-Harvey anomaly-inflow, as well as a Hamiltonian spectral flow point of view. We also discuss the applicability of our results to other systems such as Weyl semi-metals. Finally, using dimensional reduction from $d=4+1$, we derive the effective action for a $d=3+1$ time-reversal invariant topological insulator in the presence of torsion and curvature, and discuss its various physical consequences.
Human review
How the literature uses it
19 arXiv papers cite this in 45 sentences
Papers citing it · 29 sentences without a confident label
Builds on it 1 paper
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We extend the consideration of [ref] to the case, when both emergent vielbein and emergent gauge field are present, and restrict ourselves to the case, when the emergent vielbein does not depend on time while $e_0^i = 0$ for $i = 1,2,3$.
Emergent gravity and chiral anomaly in Dirac semimetals in the presence of dislocations · General expression for the anomaly · 1.00
Uses its method or tool 1 paper
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Following [ref] we represent anomaly as a result of the spectral flow that follows from the explicit solution of Pauli equation.
Emergent gravity and chiral anomaly in Dirac semimetals in the presence of dislocations · General expression for the anomaly · 0.91
Compares with it 2 papers
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As a result the pattern of the spectral flow and the pumping of quasiparticles from vacuum that we discuss in the present paper differs somehow from that of [ref] .
Emergent gravity and chiral anomaly in Dirac semimetals in the presence of dislocations · Introduction · 0.99
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[ref] coupled to the vector field strength.
On Nieh-Yan Transport · Descent formalism for the Nieh-Yan anomaly · 0.80
Cites it as background 8 papers
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The chiral anomaly in Weyl and Dirac semimetals was discussed in a number of papers (see [ref] and references therein).
Emergent gravity and chiral anomaly in Dirac semimetals in the presence of dislocations · Introduction · 0.99
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Recently, there has been an increasing interest in condensed matter in what has been called (perhaps misleadingly) torsional (or Nieh-Yan) anomaly [ref] .
On chiral responses to geometric torsion · Introduction · 0.96
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In Section [ref] we show how the residual gauge symmetries of the model are easily handled via the DFM framework, and we also clarify the status of the so-called Nieh-Yan-Weyl (NYW) symmetry [ref] .
Unconventional Supersymmetry via the Dressing Field Method · Introduction · 0.81
What it cites · 31 sentences without a confident label
Builds on 6 papers
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In recent work [ref] we were able to resolve a similar problem in 2+1-d fermionic insulators whereby torsional terms in the effective action of time-reversal breaking topological insulators were shown to correspond to Hall viscosity transport (see [ref] and the review [ref] for a detailed exposition to Hall viscosity in various systems) along with concomitant anomalies on the interface between topological phases, and we will now apply our techniques to the 3+1-d case.
Hall viscosity from effective field theory · Introduction · 0.89
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In recent work [ref] we were able to resolve a similar problem in 2+1-d fermionic insulators whereby torsional terms in the effective action of time-reversal breaking topological insulators were shown to correspond to Hall viscosity transport (see [ref] and the review [ref] for a detailed exposition to Hall viscosity in various systems) along with concomitant anomalies on the interface between topological phases, and we will now apply our techniques to the 3+1-d case.
Kubo formulas for viscosity: Hall viscosity, Ward identities, and the relation with conductivity · Introduction · 0.89
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In recent work [ref] we were able to resolve a similar problem in 2+1-d fermionic insulators whereby torsional terms in the effective action of time-reversal breaking topological insulators were shown to correspond to Hall viscosity transport (see [ref] and the review [ref] for a detailed exposition to Hall viscosity in various systems) along with concomitant anomalies on the interface between topological phases, and we will now apply our techniques to the 3+1-d case.
"Hall viscosity" and intrinsic metric of incompressible fractional Hall fluids · Introduction · 0.89
Uses the method or tool of 1 paper
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We can first consider the gapless boundary modes of the 4+1-d topological insulator which will be standard 3+1-d chiral fermions as would be found in the bulk of a Weyl semi-metal, and second, we can dimensionally reduce the 4+1-d insulator to obtain a time-reversal invariant strong topological insulator in 3+1-d using the framework set forth by Ref. [ref] .
Topological Field Theory of Time-Reversal Invariant Insulators · Informal preliminaries · 0.80
Compares with 2 papers
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Since the anomaly polynomial is a total derivative, we may read off the parity odd effective action from the above as the corresponding Chern-Simons form S_odd[e,ω,A] = i/2∫_M_3(ζ_H e^A∧T_A+σ_H A∧dA. + .κ_H (ω^(-)∧dω^(-)+2/3ω^(-)∧ω^(-)∧ω^(-))) Expanding \(S_odd\) to linear order in torsion, we find S_odd[e,ω,A] = i/2∫_M_3(σ_H A∧dA+κ_H(∧d+2/3∧∧). + .ζ_H e^A∧T_A-κ_H e^A∧T_A+⋯). which is the same action that was derived in [ref] by a more direct computation.
Torsional Anomalies, Hall Viscosity, and Bulk-boundary Correspondence in Topological States · $d=2+1$ · 0.98
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Equation [ref] is similar to the relation between Hall viscosity and spin presented in [ref] .
Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and p_x+ip_y paired superfluids · $d=2+1$ · 1.00
Cites as background 19 papers
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We will be a little imprecise here, but the overall picture is correct (for more detail see Ref. [ref] ).
Topological Field Theory of Time-Reversal Invariant Insulators · 3+1-d Topological Insulator via Dimensional reduction · 0.85
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(see [ref] for more details).
Torsional Anomalies, Hall Viscosity, and Bulk-boundary Correspondence in Topological States · Formal preliminaries · 1.00
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Recently, in condensed matter, effects that are essentially connected to torsion have been brought to the forefront and include things like the Hall viscosity in Chern insulators [ref] and the properties of dislocations and disclinations in topological phases [ref] .
Classification of Two Dimensional Topological Crystalline Superconductors and Majorana Bound States at Disclinations · Informal preliminaries · 0.89
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