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Heat current in a dissipative quantum Hall edge

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Dissipative neutral modes carry a fixed fraction of the quantum Hall heat flux quantum.

desk verdict Plausible but unverified: the dissipative AG mode's heat fraction is a real idea, but both key equations are asserted and cutoff-sensitive. read the letter →

arxiv 1908.01213 v1 pith:6VI5VT6B submitted 2019-08-03 cond-mat.mes-hall

Anna Goremykina , Artem Borin , Eugene Sukhorukov This is my paper · ORCID
classification cond-mat.mes-hall
keywords quantumHalledgeheatcurrentfluxdissipativecompressiblestriphydrodynamicneutralmodesfluctuation-dissipationtheoremintegereffect
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

At integer quantum Hall fillings, a clean edge transports heat in exact quanta even when additional neutral modes are present. This paper argues that if the edge hosts a compressible strip—a dissipative region with a finite diagonal conductivity $\sigma_{xx}$—the lowest hydrodynamic neutral mode becomes overdamped and carries only the fraction $\sigma_{xy}/(2\pi\sigma_{xx})$ of one heat flux quantum $J_q=\pi T^2/12$. Because the lowest mode carries the same fraction at both $\nu=1$ and $\nu=2$, the total heat current becomes $(1+\sigma_{xy}/2\pi\sigma_{xx})J_q$ at $\nu=1$ and $(2+\sigma_{xy}/2\pi\sigma_{xx})J_q$ at $\nu=2$. This offers a concrete explanation for the missing heat flux seen in the $\nu=2$ heat transport experiment reported in Ref. [13], and it makes heat transport a probe of the dissipative properties of the edge. The authors stress that a microscopic model of dissipation, beyond their low-energy effective theory, is still needed to confirm the prediction.

What carries the argument

The load-bearing object is a simplified electrostatic model of the dissipative compressible strip: the Hall-conductivity profile $\sigma_{xy}(y)$ has a half-jump of width $\xi$, and the strip carries a constant diagonal conductivity $\sigma_{xx}$. Charged and neutral charge amplitudes live on the two boundaries of the strip; the neutral mode acquires a dominant dissipative part $-2i(\sigma_{xx}/\sigma_{xy})\varepsilon_0$ with $\varepsilon_0=2v_\sigma/\xi$ acting as the high-energy cutoff. The heat current is computed as an integral of density-density correlation functions obtained from the equations of motion through the fluctuation-dissipation theorem; the same machinery is repeated for the three amplitudes at $\nu=2$, where the overdamped mode mixes with the two underdamped ones and contributes the same fractional heat current.

What would settle it

Measure the total heat current of a clean integer quantum Hall edge at $\nu=1$ or $\nu=2$ while tuning the compressible strip, for example by gate voltage or magnetic field to move the local $\sigma_{xx}$ peak. If the heat current remains exactly $J_q$ (at $\nu=1$) or $2J_q$ (at $\nu=2$) even when a dissipative strip is present, or if the deficit does not scale with $\sigma_{xy}/\sigma_{xx}$ as predicted, the dissipative-neutral-mode explanation is ruled out. A $\nu=1$ measurement is the cleanest test, since the model predicts only a single fractional correction $(\sigma_{xy}/2\pi\sigma_{xx})J_q$ on top of the quantized charged mode.

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Extended reading notes

Core claim

The paper's central claim is that dissipation inside the compressible edge strip removes the exact quantization of heat current at integer quantum Hall fillings. Modeling the strip as a symmetric jump in the Hall-conductivity profile with a large diagonal conductivity $\sigma_{xx}$, the authors find two modes: a charged edge mode that is insensitive to dissipation and carries exactly one heat flux quantum, and a hydrodynamic neutral mode with spectrum $\omega_\sigma = k v_\sigma - 2i(\sigma_{xx}/\sigma_{xy})\varepsilon_0$, i.e. an overdamped mode. Its contribution to the heat current, calculated from fluctuation-dissipation correlations, equals $(\sigma_{xy}/2\pi\sigma_{xx})J_q$ at $\nu=1$; repeating the calculation for $\nu=2$, where the overdamped mode couples to the two underdamped modes, gives the same fractional contribution on top of the two quanta carried by those modes. All auxiliary parameters—velocities, interaction details, and the cutoff—drop out of the leading-order answer, leaving only the conductivity ratio $\sigma_{xy}/\sigma_{xx}$.

Load-bearing premise

The central result collapses if the compressible strip does not satisfy $\sigma_{xx}\gg\sigma_{xy}$, and the non-quantized value also requires the finite high-energy cutoff $1/\xi$; if the wave-vector integral is extended to infinity, quantization is restored.

Editorial extensions

If this is right

  • At $\nu=1$ the total heat current is predicted to be $(1+\sigma_{xy}/2\pi\sigma_{xx})J_q$ rather than exactly $J_q$.
  • At $\nu=2$ the total becomes $(2+\sigma_{xy}/2\pi\sigma_{xx})J_q$, reproducing the qualitative 13% deficit seen in the experiment reported in Ref. [13].
  • The fractional correction is universal in the model: velocities, interaction details, and the cutoff scale $1/\xi$ all cancel, so the same ratio $\sigma_{xy}/(2\pi\sigma_{xx})$ controls both fillings.
  • The overdamped neutral mode is invisible to charge transport but contributes to heat transport, so heat measurements become a direct probe of the strip's diagonal conductivity.
  • For large $\sigma_{xx}/\sigma_{xy}$ the correction is small and quantization is approximately restored; the exact value of the measured deficit sets the conductivity ratio.

Reading between the lines

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

  • If a microscopic derivation supplied $\sigma_{xx}/\sigma_{xy}$ as a function of gate voltage and magnetic field, the heat-deficit measurement would become a quantitative estimator of that ratio; the model itself does not predict the ratio.
  • The same dissipative-strip mechanism could in principle affect heat transport at other integer fillings where a compressible strip forms, with the fractional correction set by the local conductivity ratio; the paper only works out $\nu=1$ and $\nu=2$.
  • Because the correction comes from wavevectors all the way up to the cutoff, a fully microscopic treatment that changes the high-energy behaviour could alter the numerical prefactor; the paper explicitly leaves this open.
  • A clean $\nu=1$ heat-current measurement would give the sharpest test: there the prediction is a single fractional contribution on top of the quantized charged mode, with no other modes to disentangle.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a low-energy effective model of a dissipative compressible strip at the edge of an integer quantum Hall system. Dissipation is modeled through a finite diagonal conductivity sigma_xx in the strip, and the lowest hydrodynamic (Aleiner-Glazman) neutral mode acquires an overdamped spectrum. The authors compute the heat current carried by this mode using response functions and the fluctuation-dissipation theorem. At nu=1 they claim the dissipative mode carries J_sigma = (sigma_xy/(2 pi sigma_xx)) J_q, and at nu=2 they claim the total heat current is J_E = [2 + sigma_xy/(2 pi sigma_xx)] J_q. The paper suggests these dissipative modes explain the missing heat current observed by le Sueur et al. (PRL 105, 056803 (2010)).

Significance. If the central claim is correct, the paper provides a concrete mechanism for the observed breakdown of heat-current quantization in integer quantum Hall edges and offers a potential explanation for the elusiveness of hydrodynamic neutral modes. The model is explicit, the FDT-based response-function formalism is appropriate, and the authors honestly flag the need for a microscopic model of dissipation. However, the significance is conditional: the claimed correction is dominated by the high-energy cutoff region rather than being a protected low-energy property, and the paper's own discussion concedes that the exact numerical prefactor depends on interaction details. The experimental consistency check is also not a parameter-free test, since the ratio sigma_xy/sigma_xx is an input parameter. These issues must be resolved before the universality claim can be accepted.

major comments (4)
  1. [Eq. (18)-(19), Sec. IV] The evaluation leading to Eq. (19) is not shown in sufficient detail, and the claim that all auxiliary parameters cancel is not demonstrated. A straightforward evaluation under the stated wide-Lorentzian approximation and a sharp cutoff at |k|=1/xi gives a prefactor of order 3/(2 pi^2), not exactly 1/(2 pi), so the exact coefficient depends on how the cutoff and the Lorentzian tails are treated. This is not a purely cosmetic issue, because the discussion section states that "the exact numerical prefactor depends on the interaction details," which appears to contradict the universality claimed in Eq. (19). The authors should provide the full step-by-step evaluation of the integral and clarify the range of validity of the stated cancellation.
  2. [Appendix A, Eq. (27)] The nu=2 heat current result is asserted rather than derived. After writing the response functions and the determinant, the appendix ends with "Following this strategy one can discover the correction sigma_xy/(2 pi sigma_xx) J_q." This is the key result used for the experimental comparison, and the relevant algebra is not shown. The authors should either display the full computation, including how the cross-correlators combine to produce exactly the same correction as at nu=1, or provide a reproducible symbolic calculation.
  3. [Introduction and Sec. VI] The claimed consistency with the 13% heat deficit in Ref. [13] appears numerically inconsistent with the central expansion assumption. If the deficit is interpreted as sigma_xy/(2 pi sigma_xx) = 0.13, then sigma_xy/sigma_xx is about 0.8, which violates the assumption sigma_xx >> sigma_xy used throughout to expand in sigma_xy/sigma_xx and to treat the neutral mode as overdamped. The authors should state explicitly how the measured 13% is normalized and clarify whether the inferred ratio lies within the regime where their leading-order result is controlled; otherwise the comparison to experiment is not a meaningful consistency check.
  4. [Sec. IV, renormalization of quantization] The paper correctly notes that extending the k integral in Eq. (18) to infinity restores full quantization, which means the non-quantized correction is dominated by the boundary of the hydrodynamic regime. Since the coefficient is set by physics at k ~ 1/xi, the prediction is not protected from microscopic details above the cutoff. This limitation is acknowledged in the discussion, but it directly affects the headline claim of universality; the paper should quantify the sensitivity of Eq. (19) to the cutoff profile and state explicitly that the result is not a universal low-energy prediction in the usual sense.
minor comments (4)
  1. [General] There are several typos, e.g., "seams reasonable" near Eqs. (3)-(7) should read "seems reasonable."
  2. [Sec. II, Fig. 2] The parameter delta is introduced as the effective width of the charge density in the edge channels, but its precise definition and relation to the strip width xi are not clear; a short clarifying sentence would help.
  3. [Eq. (18)] The upper limit of the frequency integral is written as T, but the integrand does not show the Bose factor or the vacuum subtraction that appears in Eq. (17); the authors should align the notation with the FDT formula in Eq. (15).
  4. [Sec. VI] The phrase "the exact numerical prefactor depends on the interaction details" should be reconciled with Eq. (19) and with the abstract's claim that the carried portion is the same at nu=1 and nu=2; as written, the two statements require qualification.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the heat-current result is a nontrivial computed function of the model's conductivity ratio, and the self-citations are methodological rather than load-bearing.

full rationale

The paper's central claim J_σσ = (σxy/2πσxx)J_q is obtained by evaluating the FDT correlator (16) in the integral (18) and then taking the σxx ≫ σxy limit. This is not a self-definitional reduction: the heat current is not the input dissipative rate or the input conductivity ratio by construction; it is a computed spectral integral whose value happens to depend only on the ratio after velocities and cutoff cancel. Equation (19) is not a renaming of Eq. (8), and no parameter is fitted to the 13% deficit: the text explicitly states that a microscopic model of dissipation is needed to confirm the prediction (abstract and Sec. VI), so the experimental consistency is conditional rather than a fitted prediction. The self-citations (Refs. 39 and 42) provide a hydrodynamics/FDT evaluation technique and an energy-flux identity, but the FDT itself is cited to Callen and Welton and to Kubo; these citations do not supply the target result or forbid alternatives. The ν=2 result Eq. (27) is supported by the response-function calculation in Appendix A, although the final integration is summarized as 'following this strategy one can discover the correction'; this is an omitted verification step, not circularity. The admitted sensitivity to the UV cutoff (Sec. IV) is a model limitation explicitly acknowledged by the authors, and it concerns correctness risk rather than circularity. No load-bearing step reduces by construction to its own inputs, so the appropriate score is 0.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The model introduces no new particles or fields. The overdamped neutral mode is the previously predicted Aleiner-Glazman mode with dissipation added; the dissipative strip is a modeling device, not a new entity. The central calculation depends on the free conductivity ratio and on the effective low-energy assumptions listed as axioms.

free parameters (3)
  • σxx/σxy ratio in the compressible strip
    The neutral mode heat current is (σxy/2πσxx)Jq, so the predicted deficit is set directly by this ratio. The paper does not derive it from a microscopic model or extract it from experiment, leaving it as a free input of the effective theory.
  • High-energy cut-off 1/ξ (or ε0 = 2vσ/ξ)
    The non-quantized heat current exists only because the k-integral is cut off at 1/ξ; the value cancels in the leading-order result but the existence of the cut-off is essential to the prediction.
  • Velocities v_c, v_σ and interaction parameters U(δ/D), U(ξ/D)
    These enter the model but cancel out of the leading heat current result; they are part of the effective theory's input.
assumptions (7)
  • standard math Fluctuation-dissipation theorem relates density correlators to linear response functions.
    Used in Sec. IV to compute S_ii(k,ω) from the response functions G_ij derived from the equations of motion.
  • domain assumption The compressible strip can be modeled as a symmetric half-jump in σxy with a uniform σxx inside the strip.
    Introduced in Sec. II (Fig. 2 and Eqs. (5)-(6)); this replaces the full density profile with a two-boundary model hosting one neutral mode.
  • domain assumption σxx ≫ σxy in the compressible strip, with expansion in σxy/σxx.
    Stated in the introduction and used throughout to treat the neutral mode as overdamped and to approximate integrals (Sec. IV, Eq. (18)). Not microscopically derived.
  • domain assumption Constant electric field inside the strip: U'(|ky|)|_{y=δ} = U'(|ky|)|_{y=ξ}.
    Assumed in Sec. II just before Eq. (7) as a low-energy approximation.
  • domain assumption Neglect of the k^2 (diffusion) term in the continuity equation.
    Stated in Sec. II; justified by chiral transport and the long-wavelength limit.
  • domain assumption Coulomb interaction is screened at distance D much larger than the strip width, making velocities dispersionless.
    Assumed in Sec. II after Eq. (9); motivated by metallic gates in the experiment.
  • domain assumption The existence of a high-energy cut-off 1/ξ for the hydrodynamic description.
    The non-quantized heat current in Eq. (19) relies on a finite integration range for k; the authors note quantization is restored if the cut-off is removed (Sec. IV).

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Pith. "Pith review of Heat current in a dissipative quantum Hall edge." pith.science (2026). https://pith.science/paper/6VI5VT6B

@misc{pith2026190801213,
  author       = {Pith},
  title        = {Pith review of: Heat current in a dissipative quantum Hall edge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6VI5VT6B}},
  note         = {Machine review of arXiv:1908.01213}
}
abstract

We explore the heat current in the quantum Hall edge at filling factors $\nu = 1$ and $\nu = 2$ in the presence of dissipation. Dissipation arises in the compressible strip forming at the edge in presence of a smooth confining potential. Such strip was predicted to host an infinite number of hydrodynamic neutral modes, which however were never observed. A possible explanation may be in their dissipative nature, which was not fully considered before. Heat transport measurements are capable of detecting neutral modes and experiment [H. le Sueur et al., Phys. Rev. Lett. 105, 056803 (2010)] at $\nu = 2$ captured additional degrees of freedom transferring heat at the edge. Surprisingly, the breakdown of heat current quantization has been found. We conjecture that the aforementioned dissipative modes might be responsible for this behavior. We build a low-energy effective model and show that the lowest hydrodynamic mode carries a portion of the heat flux quantum which is the same both at $\nu = 1$ and $\nu = 2$. Although our results are consistent with the experiment, a microscopic model of dissipation is needed to confirm the prediction of the low-energy approximation.

Figures

Figures reproduced from arXiv: 1908.01213 by the authors.

Figure 1
Figure 1. A scheme of the experimental set-up [13]. An outer edge channel at ν = 2 is driven out-of-equilibrium by the quantum point contact (triangles) and the energy distribu￾tion of the created excitation is extracted from probing the tunneling density of states ∝ ∂fD(ε)/∂ε by the quantum dot placed downstream. ture is extracted from the calculation of the energy flux JE = R dε 2π ε h f(ε)−θ(µ−ε) i , where µ is the electro… view at source ↗
Figure 2
Figure 2. A simplified Hall conductivity profile of the edge [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. A Hall conductivity profile of the edge cross-section [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Forward citations

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.