Broken Lorentz symmetry and violation of the Wiedemann-Franz law in topological insulators

A. Daria Dumitriu-I.; Alessandro Principi; Feng Liu

Broken Lorentz symmetry and violation of the Wiedemann-Franz law in topological insulators

T0 review · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A two-dimensional topological insulator with a Newtonian mass term should show a thermal Hall conductivity $\kappa_{xy}=\pi T/6+7\pi^3 T^3/(15m_N^2)$, so the quantized thermal Hall effect is a low-temperature limit and the Wiedemann-Franz…

desk verdict A clean, internally consistent calculation of a T^3 correction to the thermal Hall conductivity, but the effect lives or dies on a covariantization choice that the paper never pins down. read the letter →

arxiv 2411.13649 v3 pith:7JOK4KIH submitted 2024-11-20 cond-mat.mes-hall

Feng Liu , A. Daria Dumitriu-I. , Alessandro Principi This is my paper · ORCID

classification cond-mat.mes-hall

keywords thermalHalleffecttopologicalinsulatorWiedemann-FranzlawNewtonianmassbrokenLorentzsymmetryLuttingergravityboundaryfreeenergyconductivity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies a two-dimensional topological insulator coupled to a static non-uniform gravitational field, which is Luttinger's way of representing a temperature gradient. It derives the effective free energy of the edge states directly from the bulk Hamiltonian and claims that a Newtonian mass term, which breaks Lorentz symmetry, produces a bulk thermal current that is absent for massive Dirac fermions. The resulting thermal Hall conductivity contains a temperature-cubed correction, meaning the Wiedemann-Franz law holds only in the low-temperature limit. A sympathetic reader would care because this correction is not obtainable from gravitational Chern-Simons theory and could make the Newtonian mass observable in thermal transport.

What carries the argument

The machinery is the extended Hamiltonian $H_{\mathrm{ex}}=H+H_{\mathrm{src}}$, where the source term is engineered so that the energy current can be expressed as a functional derivative of $H_{\mathrm{ex}}$ even though the physical Hamiltonian breaks Lorentz symmetry; the source term itself vanishes at the Luttinger metric. From this extended Hamiltonian the authors derive a one-dimensional boundary effective free energy $F[\zeta,\chi]=\frac{\pi T^2}{12}\int dx\,\frac{\zeta}{1+\zeta}-\frac{7\pi^3 T^4}{60}\int dx\,\frac{\chi}{(1+\zeta)^4}$, where $\zeta$ and $\chi$ are combinations of the metric and vielbein deviations. Varying this free energy with respect to the metric gives the boundary current, the continuity equation converts that boundary current into a bulk current, and the energy magnetization together with the generalized Streda formula produces $\kappa_{xy}$.

What would settle it

Compute directly, on a lattice version of the same continuum model, the bulk thermal current in linear response to a temperature gradient without using the edge continuity argument; if the bulk current does not equal $-7\pi^3 T^4/(30m_N^2)\,\partial_2\phi$ to leading order, the central claim fails. Experimentally, measure $\kappa_{xy}(T)$ in a two-dimensional topological insulator with known Newtonian mass: the prediction is $\kappa_{xy}/T=\pi/6+7\pi^3 T^2/(15m_N^2)$ at low temperatures.

Watch

Extended reading notes

Core claim

The central claim is that for a generic topological insulator with a Newtonian mass term $1/m_N$, the boundary energy current is $j_E^{\mathrm{bdry}}=-\pi T^2/12-7\pi^3 T^4/(60m_N^2)$ in the long-wavelength limit, up to gravitational-field factors. Conservation of energy then forces a bulk thermal current $j_E^1(0)=-7\pi^3 T^4/(30m_N^2)\,\partial_2\phi$, and the thermal Hall conductivity becomes $\kappa_{xy}=\pi T/6+7\pi^3 T^3/(15m_N^2)$. The first term is the universal quantized value associated with the edge central charge; the second term breaks the Wiedemann-Franz law and, the paper argues, cannot be obtained from the gravitational Chern-Simons action. In the limit $m_N\to\infty$, Lorentz symmetry is effectively restored and the correction vanishes.

Load-bearing premise

The argument stands on the choice of the energy (pseudo-)current defined through the extended Hamiltonian: if a different legitimate current were adopted for a system without Lorentz symmetry, the bulk current and the $T^3$ correction would change or vanish.

Editorial extensions

If this is right

Universal quantization of the thermal Hall conductivity is only a low-temperature limit; at finite temperature the ratio $\kappa_{xy}/T$ grows as $T^2$ with a coefficient fixed by the Newtonian mass.
The Wiedemann-Franz law is violated because a purely thermal bulk current appears without a companion charge current.
The temperature dependence $\kappa_{xy}-\pi T/6\propto T^3/m_N^2$ provides a transport route to extract the Newtonian mass of a topological insulator's band structure.
The gravitational Chern-Simons approach is incomplete for generic topological insulators; the correction comes from Lorentz-symmetry breaking that this effective field theory does not capture.
The extended-Hamiltonian boundary-theory method is claimed to generalize to other models beyond the specific continuum Hamiltonian treated here.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the paper's current definition is accepted, then any material with a parabolic correction to a Dirac band should show a detectable $T^3$ correction in thermal Hall measurements at temperatures where $T^2\sim m_N^2/7\pi^2$.
The result highlights an ambiguity inherent to non-Lorentz-invariant systems: because the physical energy current is not uniquely fixed by symmetry, the $T^3$ term is tied to the particular current defined through the extended Hamiltonian; a direct bulk lattice calculation of the linear-response heat current would be a decisive check.
The same boundary-free-energy technique could be adapted to interacting edge theories or to higher-dimensional topological phases, where Lorentz breaking is generically present and gravitational Chern-Simons descriptions are not available.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper's concrete result is κxy = πT/6 + 7π^3T^3/(15m_N^2), and that T^3 term is genuinely new: the massive-Dirac companion paper has no bulk current, and gravitational Chern-Simons effective actions only give the universal term. The derivation is also careful. The Matsubara sums produce the standard πT^2/12 and 7π^3T^4/60 coefficients, the zeta-resummation reproduces Eq. (9), and the two routes to κxy—bulk current plus magnetization, and the Streda formula—agree. No parameter is fitted to the headline result. I believe the algebra is right.

The soft spot is exactly what the reader flagged, and it is load-bearing rather than cosmetic. The energy current in a Lorentz-breaking system is not unique. The action in Eq. (1) couples the Newtonian mass term to gravity through the spatial-vielbein combination η^{ab}e^a_μe^b_ν. That is a choice. It is the choice that generates the χ(x2) term in the boundary Hamiltonian, and every non-universal piece of the final answer flows from that χ term. If instead you covariantize the Newtonian term with the spatial inverse metric g^{ij}∇_i∇_j, Luttinger's metric has g^{ij} flat, so no χ term appears and the T^3 correction vanishes. The stress-test note correctly identifies this. The paper does not justify why the vielbein coupling is the physical one for a topological insulator, nor does it derive the action from a lattice model. The bulk current is then inferred from the edge via continuity, and the boundary condition M^z_E(0)=0 is imported from Ref. [32]; those steps are standard but inherit the same current-definition ambiguity.

The Wiedemann-Franz claim is also asserted without an explicit Lorentz ratio, which is a minor gap, not a fatal one. The central argument holds up as a calculation conditional on the extended-Hamiltonian current definition; it does not hold up as an unconditional prediction.

Who should read this: people working on thermal Hall transport in topological systems and anyone who uses Luttinger's trick in Lorentz-breaking models. The method itself—the extended Hamiltonian construction—is worth taking seriously and could be applied elsewhere. But I would not treat the T^3 term as a firm result until the coupling ambiguity is resolved or derived from a microscopic lattice Hamiltonian.

Serious referee: yes. The paper is coherent, careful, and makes a checkable claim about a real ambiguity in how to define energy currents without Lorentz symmetry. A referee can do useful work here. I would set the bar at heavy revision, not desk rejection.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the T^3 correction is obtained from explicit Matsubara sums, and the only self-citation is a non-load-bearing comparison baseline.

full rationale

The central result, kappa_xy = pi T/6 + 7 pi^3 T^3 / (15 m_N^2) (Eq. 13), is obtained by a direct perturbative calculation: the boundary free energy F[zeta, chi] (Eq. 9 and SM Eq. S53) is computed by summing Matsubara frequencies, with the coefficient 7 pi^3 T^4 / 60 arising from the contour integrals in SM Eq. S52. No parameter is fitted to kappa_xy, and the leading pi T/6 term reproduces the known quantized thermal Hall conductance, providing an external consistency check. The extended Hamiltonian H_src (Eq. 3, SM Eq. S16) is introduced so that the functional derivative of the partition function reproduces the energy current already defined from the action (SM Eq. S4); this is a definitional consistency requirement of the linear-response construction, not a reduction of the predicted current to an assumed value. The self-citation [22] is used only for comparison ('in contrast to the case of massive Dirac fermions [22]', 'relative to the massive Dirac fermions model [22]') and supplies no premise for the new T^3 term. The boundary condition M^z_E(0)=0 and the Streda formula are taken from external references [32] and [18,33]. The sensitivity of the result to the choice of vielbein coupling for the Newtonian mass term, and the possibility that an alternative covariantization yields no correction, is a physical modelling assumption rather than a circular step; the paper computes the consequences of a stated action, and the Conclusion explicitly acknowledges the model dependence ('our derivation is based on the choice of a specific Hamiltonian'). No equation in the derivation chain is equal to the target result by construction.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central result is a pure parameter-free derivation given the model: no constants are fitted to data, and the cutoff-dependent pieces are dropped as temperature-independent. The only input controlling the new T^3 term is the Newtonian mass m_N, a coupling of the model Hamiltonian whose value the paper leaves unspecified. The derivation depends on several domain assumptions: Luttinger's equivalence of gravitational potential and temperature gradient, the action-based definition of the energy current and its reproduction by the extended Hamiltonian, the anomaly-inflow continuity relation, the vanishing of the bulk energy magnetization at the boundary, the k=0 edge wavefunction approximation, and the assumption that the metric varies only along the edge. All are standard tools or explicitly stated, except the energy-current definition, which is the paper's own choice and carries the physical weight of the result. No new physical entities are introduced.

free parameters (2)

m (relativistic Dirac mass) = m < 0 in the sample (model input)
Mass of the 2D topological insulator in Eq. (1); its negative value fixes the Chern number C = -1 and the edge decay lengths lambda1,2 = (1 +/- sqrt(1+4m))/2 (SM Eq. S28). Standard model parameter, not fitted.
m_N (Newtonian mass) = set to 1 in the derivation; restored in Eqs. (10)-(14) as an unspecified scale
Coupling of the Lorentz-breaking term (1/m_N)(grad psi-bar)(grad psi) in Eq. (1). The headline T^3 term in kappa_xy scales as 1/m_N^2 and vanishes as m_N tends to infinity. The paper leaves its physical value open, so the magnitude of the new effect is not predicted.

assumptions (7)

domain assumption Luttinger's equivalence between a static non-uniform gravitational potential and a temperature gradient in linear response
Invoked in the introduction and Fig. 1 caption; all temperature-gradient physics is computed as the response to the metric perturbation phi.
ad hoc to paper The energy current of the Lorentz-breaking model is defined by the vielbein functional derivative of the action (SM Eq. S4) and reproduced by the extended Hamiltonian H_src (Eq. 3)
SM S1 and Eqs. (S11)-(S19). This is the paper's own construction; the chi(x2) term in Eq. (7) and therefore the entire T^3 term originate from H_src.
domain assumption Anomaly-inflow continuity relation j^1_E(+0) = -d2 j^bdry_E (Eq. 11), taken from Ref. [21]
Converts the boundary energy current into the claimed bulk thermal current; without this relation the bulk effect does not exist in the calculation.
domain assumption Bulk energy magnetization vanishes at the boundary, M^z_E(0) = 0, from Ref. [32]
Used in SM Eq. (S56) to relate the boundary current to the bulk magnetization M^z_E(+infinity); imported without derivation.
domain assumption k=0 approximation for the edge wavefunction (SM S5): dispersion E(k) = -k + O(k^4) and spinor corrections suppressed by m
Required for the factorized boundary Hamiltonian in SM S2; if the wavefunction's k-dependence couples to chi, the T^3 coefficient changes.
domain assumption Near the boundary the metric depends only on x2, so x1 and x2 decouple in the boundary Hamiltonian
Used in SM S2 and the Fig. 1 setup to factorize psi(x1,x2) = psi2(x2) psi1(x1).
standard math Standard imaginary-time path integral and Matsubara-sum machinery, Wick rotation, Fermi distribution integrals
SM S3, Eqs. (S39)-(S53); coefficients pi T^2/12 and 7 pi^3 T^4/60 follow from standard fermionic Matsubara sums with a cutoff, dropping temperature-independent divergent terms.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Broken Lorentz symmetry and violation of the Wiedemann-Franz law in topological insulators." pith.science (2026). https://pith.science/paper/7JOK4KIH

@misc{pith2026241113649,
  author       = {Pith},
  title        = {Pith review of: Broken Lorentz symmetry and violation of the Wiedemann-Franz law in topological insulators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7JOK4KIH}},
  note         = {Machine review of arXiv:2411.13649}
}

read the original abstract

We study a two-dimensional topological insulator in the presence of a static non-uniform gravitational field, which mimics the variations in the temperature distribution. We derive an effective boundary free energy functional for the gravitational field and show that, in contrast to the case of massive Dirac fermions, the addition of a Newtonian mass term significantly modifies the quantum anomalous behavior of the system. A non-zero bulk thermal current appears, which violates the Wiedemann-Franz law. The systematic approach we develop to calculate the contribution of edge states to thermal Hall conductivity and energy magnetization can easily be extended to other models.

Figures

Figures reproduced from arXiv: 2411.13649 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 29 canonical work pages

[32]

Zhang, Y

Y. Zhang, Y. Gao, and D. Xiao, Thermodynamics of energy magnetization, Physical Review B 102, 235161 (2020)

work page 2020
[1]

Golan and A

O. Golan and A. Stern, Probing topological supercon- ductors with emergent gravity, Physical Review B 98, 064503 (2018)

work page 2018
[2]

Grissonnanche, S

G. Grissonnanche, S. Th´ eriault, A. Gourgout, M.-E. Boulanger, E. Lefran¸ cois, A. Ataei, F. Lalibert´ e, M. Dion, J.-S. Zhou, S. Pyon, et al., Chiral phonons in the pseudo- gap phase of cuprates, Nature Physics 16, 1108 (2020)

2020
[3]

Kane and M

C. Kane and M. P. Fisher, Quantized thermal transport in the fractional quantum hall effect, Physical Review B 55, 15832 (1997)

work page 1997
[4]

Ideue, T

T. Ideue, T. Kurumaji, S. Ishiwata, and Y. Tokura, Giant thermal hall effect in multiferroics, Nature materials 16, 797 (2017)

work page 2017
[5]

Shimizu, A

Y. Shimizu, A. Yamakage, and K. Nomura, Quantum thermal hall effect of majorana fermions on the surface of superconducting topological insulators, Physical Review B 91, 195139 (2015)

work page 2015
[6]

Zhang, Y

X.-T. Zhang, Y. H. Gao, and G. Chen, Thermal hall effects in quantum magnets, Physics Reports 1070, 1 (2024)

work page 2024
[7]

Grissonnanche, A

G. Grissonnanche, A. Legros, S. Badoux, E. Lefran¸ cois, V. Zatko, M. Lizaire, F. Lalibert´ e, A. Gourgout, J.-S. Zhou, S. Pyon, et al., Giant thermal hall conductivity in the pseudogap phase of cuprate superconductors, Nature 571, 376 (2019)

work page 2019

Show all 35 references

[8]

Samajdar, M

R. Samajdar, M. S. Scheurer, S. Chatterjee, H. Guo, C. Xu, and S. Sachdev, Enhanced thermal hall effect in the square-lattice n´ eel state, Nature Physics15, 1290 (2019)

2019
[9]

Fulga, Y

I. Fulga, Y. Oreg, A. D. Mirlin, A. Stern, and D. F. Mross, Temperature enhancement of thermal hall con- ductance quantization, Physical review letters 125, 236802 (2020)

2020
[10]

Cooper, B

N. Cooper, B. Halperin, and I. Ruzin, Thermoelectric response of an interacting two-dimensional electron gas in a quantizing magnetic field, Physical Review B 55, 2344 (1997)

1997
[11]

T. Qin, Q. Niu, and J. Shi, Energy magnetization and the thermal hall effect, Physical review letters 107, 236601 (2011)

2011
[12]

Luttinger, Theory of thermal transport coefficients, Physical Review 135, A1505 (1964)

J. Luttinger, Theory of thermal transport coefficients, Physical Review 135, A1505 (1964)

1964
[13]

Shitade, Heat transport as torsional responses and keldysh formalism in a curved spacetime, Progress of Theoretical and Experimental Physics 2014, 123I01 (2014)

A. Shitade, Heat transport as torsional responses and keldysh formalism in a curved spacetime, Progress of Theoretical and Experimental Physics 2014, 123I01 (2014)

2014
[14]

Paul and G

I. Paul and G. Kotliar, Thermal transport for many- body tight-binding models, Physical review B 67, 115131 (2003)

2003
[15]

Cappelli, M

A. Cappelli, M. Huerta, and G. R. Zemba, Thermal transport in chiral conformal theories and hierarchical quantum hall states, Nuclear Physics B 636, 568 (2002)

2002
[16]

Bradlyn and N

B. Bradlyn and N. Read, Low-energy effective theory in the bulk for transport in a topological phase, Physical Review B 91, 125303 (2015)

2015
[17]

Stone, Gravitational anomalies and thermal hall effect in topological insulators, Physical Review B—Condensed Matter and Materials Physics 85, 184503 (2012)

M. Stone, Gravitational anomalies and thermal hall effect in topological insulators, Physical Review B—Condensed Matter and Materials Physics 85, 184503 (2012)

2012
[18]

Nomura, S

K. Nomura, S. Ryu, A. Furusaki, and N. Nagaosa, Cross- correlated responses of topological superconductors and superfluids, Physical review letters 108, 026802 (2012)

2012
[19]

Read and D

N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum hall effect, Phys- ical Review B 61, 10267 (2000)

2000
[20]

Sumiyoshi and S

H. Sumiyoshi and S. Fujimoto, Quantum thermal hall ef- fect in a time-reversal-symmetry-broken topological su- perconductor in two dimensions: approach from bulk cal- culations, Journal of the Physical Society of Japan 82, 023602 (2013)

2013
[21]

Nakai, S

R. Nakai, S. Ryu, and K. Nomura, Finite-temperature effective boundary theory of the quantized thermal hall effect, New Journal of Physics 18, 023038 (2016)

2016
[22]

F. Liu, A. D. Dumitriu-I, and A. Principi, No bulk ther- 5 mal currents in massive dirac fermions, Physical Review B 110, L081404 (2024)

2024
[23]

Qi and S.-C

X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Reviews of modern physics 83, 1057 (2011)

2011
[24]

Czajka, Exotic Thermal Transport in a Kitaev Magnet, Ph.D

P. Czajka, Exotic Thermal Transport in a Kitaev Magnet, Ph.D. thesis, Princeton University (2022)

2022
[25]

Vinkler-Aviv and A

Y. Vinkler-Aviv and A. Rosch, Approximately quantized thermal hall effect of chiral liquids coupled to phonons, Physical Review X 8, 031032 (2018)

2018
[26]

Gromov and A

A. Gromov and A. G. Abanov, Thermal hall effect and geometry with torsion, Physical review letters 114, 016802 (2015)

2015
[27]

S. M. Carroll, Spacetime and geometry (Cambridge Uni- versity Press, 2019)

2019
[28]

See supplemental material for details of the derivations
[29]

H. T. Stoof, K. B. Gubbels, and D. Dickerscheid, Ultracold quantum fields (Springer, 2009)

2009
[30]

The variable χ should not be confused with the value χ(x) that it is substituted into the expression for the energy current after the functional derivative of the free energy has been taken
[31]

Since we neglected temperature-independent terms in our derivation, this form of the boundary free energy is only valid up to a constant which is independent of tempera- ture
[33]

H. Guo, R. Samajdar, M. S. Scheurer, and S. Sachdev, Gauge theories for the thermal hall effect, Physical Re- view B 101, 195126 (2020). 1 Supplemental material: Broken Lorentz symmetry and violation of the Wiedemann-Franz law in topological insulators S1. ENERGY CURRENT OPERA...

2020
[35]

− (−i← −∂ 2γ ˆ2 − ← −∂ 2 2)− →∂ 2|s⟩ = ie2 ∗ˆ0ϕg − ← −∂ 2 − →∂ 2 2 + ← −∂ 2 2 − →∂ 2 II src = i⟨ψ1|ei ∗ˆ0 ← −∂ i −i 2 (∂jϕg)γj + i 2 (∂jϕg)γj− →∂ i |ψ1⟩ = i⟨ψ1|ei ∗ˆ0 ← −∂ i −i 2 (∂2ϕg)γ ˆ2 + i 2 (∂2ϕg)γ ˆ2− →∂ i |ψ1⟩ = 0 III src = i⟨ψ1|ei ∗ˆ0 − ← −∂ i(∂jϕg)− →∂ j + ← −∂ j(∂jϕ...
[253]

acknowledges support from the Engineer- ing and Physical Sciences Research Council, Grant No

A.D.D.-I. acknowledges support from the Engineer- ing and Physical Sciences Research Council, Grant No. EP/T517823/1. ∗ feng.liu-6@postgrad.manchester.ac.uk † alexandra-daria.dumitriu-iovanescu@outlook.com ‡ alessandro.principi@manchester.ac.uk

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [32]

Zhang, Y

Y. Zhang, Y. Gao, and D. Xiao, Thermodynamics of energy magnetization, Physical Review B 102, 235161 (2020)

work page 2020

[2] [1]

Golan and A

O. Golan and A. Stern, Probing topological supercon- ductors with emergent gravity, Physical Review B 98, 064503 (2018)

work page 2018

[3] [2]

Grissonnanche, S

G. Grissonnanche, S. Th´ eriault, A. Gourgout, M.-E. Boulanger, E. Lefran¸ cois, A. Ataei, F. Lalibert´ e, M. Dion, J.-S. Zhou, S. Pyon, et al., Chiral phonons in the pseudo- gap phase of cuprates, Nature Physics 16, 1108 (2020)

2020

[4] [3]

Kane and M

C. Kane and M. P. Fisher, Quantized thermal transport in the fractional quantum hall effect, Physical Review B 55, 15832 (1997)

work page 1997

[5] [4]

Ideue, T

T. Ideue, T. Kurumaji, S. Ishiwata, and Y. Tokura, Giant thermal hall effect in multiferroics, Nature materials 16, 797 (2017)

work page 2017

[6] [5]

Shimizu, A

Y. Shimizu, A. Yamakage, and K. Nomura, Quantum thermal hall effect of majorana fermions on the surface of superconducting topological insulators, Physical Review B 91, 195139 (2015)

work page 2015

[7] [6]

Zhang, Y

X.-T. Zhang, Y. H. Gao, and G. Chen, Thermal hall effects in quantum magnets, Physics Reports 1070, 1 (2024)

work page 2024

[8] [7]

Grissonnanche, A

G. Grissonnanche, A. Legros, S. Badoux, E. Lefran¸ cois, V. Zatko, M. Lizaire, F. Lalibert´ e, A. Gourgout, J.-S. Zhou, S. Pyon, et al., Giant thermal hall conductivity in the pseudogap phase of cuprate superconductors, Nature 571, 376 (2019)

work page 2019