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arxiv: 2605.21774 · v3 · pith:4ILSRYKVnew · submitted 2026-05-20 · ❄️ cond-mat.str-el

Conductivity of a Non-Galilean--Invariant Fermi Liquid: Exact Solution of the Kinetic Equation

Tatia Kiliptari , Vladimir I. Yudson , Dmitrii L. Maslov This is my paper

Pith reviewed 2026-05-22 08:22 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Fermi liquid conductivitykinetic equationhydrodynamic regimequasiparticle scattering timePomeranchuk quantum critical pointnon-Galilean invariantoptical conductivityEliashberg approximation
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The pith

An exact solution of the kinetic equation shows that electron-electron interactions affect the conductivity of a non-Galilean-invariant Fermi liquid only through the quasiparticle scattering time.

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

The paper derives an exact expression for the conductivity of a disordered non-Galilean-invariant Fermi liquid by solving the kinetic equation that includes both screened Coulomb interactions and z=3 Pomeranchuk critical interactions. This exact solution demonstrates that interactions enter the conductivity solely via the quasiparticle scattering time τ_ee, which sets the scale for crossovers between collisionless and hydrodynamic regimes when compared to the impurity scattering rate or the probe frequency Ω. The approach also yields the optical conductivity deep in the hydrodynamic regime Ω much less than 1/τ_ee, a limit that perturbation theory cannot access. A sympathetic reader would care because the result supplies a unified description of transport across regimes in interacting electron systems, including near quantum critical points.

Core claim

By solving the kinetic equation exactly for a disordered non-Galilean-invariant Fermi liquid with screened Coulomb and z=3 Pomeranchuk interactions, the authors obtain an expression showing that electron-electron interactions influence conductivity exclusively through the quasiparticle scattering time τ_ee. Crossovers between collisionless and hydrodynamic regimes therefore occur when 1/τ_ee becomes comparable to the larger of the impurity scattering rate and the frequency Ω. The solution further provides the optical response in the hydrodynamic regime Ω ≪ 1/τ_ee. Near a z=3 Pomeranchuk quantum critical point, consistency with the Kubo formula requires inclusion of mass renormalization in an

What carries the argument

Exact solution of the kinetic equation that incorporates both impurity scattering and electron-electron scattering while accounting for the absence of Galilean invariance.

If this is right

  • The conductivity depends on electron-electron interactions only through τ_ee, independent of other details of the interaction.
  • Regime crossovers are controlled by the single scale 1/τ_ee relative to the impurity rate or Ω.
  • The hydrodynamic optical conductivity is obtained exactly, beyond the reach of perturbative expansions.
  • Near the z=3 Pomeranchuk critical point the crossover occurs at the Planckian scale Ω ∼ T once mass renormalization is accounted for.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that breaking Galilean invariance is essential for interactions to affect conductivity beyond a simple relaxation-time picture.
  • Similar exact solutions might be constructed for other critical interaction channels or in two-dimensional systems.
  • Transport measurements at frequencies around the Planckian scale near quantum critical points could directly test the predicted crossover location.
  • The approach may connect to hydrodynamic descriptions of other non-Galilean-invariant liquids such as certain quantum fluids.

Load-bearing premise

Consistency between the kinetic-equation and Kubo approaches holds only when mass renormalization is properly included within the Eliashberg approximation.

What would settle it

Observation of an optical conductivity crossover near the z=3 Pomeranchuk point at a scale distinctly different from the Planckian Ω ∼ T would contradict the necessity of the mass-renormalization step.

Figures

Figures reproduced from arXiv: 2605.21774 by Dmitrii L. Maslov, Tatia Kiliptari, Vladimir I. Yudson.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

We obtain an exact expression for the conductivity of a disordered, non-Galilean-invariant Fermi liquid by solving the kinetic equation with both screened Coulomb and $z=3$ Pomeranchuk critical interactions. While consistent with previous asymptotic results, our solution shows that electron-electron interactions enter the conductivity solely via the quasiparticle scattering time, $\tau_\mathrm{ee}$. Accordingly, the crossovers between the collisionless and hydrodynamic regimes occur when $1/\tau_\mathrm{ee}$ becomes comparable to the larger of the impurity scattering rate and the probe frequency, $\Omega$. In addition, the exact solution yields the optical response in the hydrodynamic regime, $\Omega\ll 1/\tau_\mathrm{ee}$, which is inaccessible within perturbation theory. Near a $z=3$ Pomeranchuk quantum critical point, consistency between the kinetic-equation and Kubo approaches requires proper inclusion of mass renormalization within the Eliashberg approximation, which also ensures that the crossover between the collisionless and hydrodynamic regimes in the optical conductivity occurs at the Planckian scale $\Omega\sim T$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives an exact solution of the Boltzmann kinetic equation for the dc and optical conductivity of a disordered, non-Galilean-invariant Fermi liquid that includes both screened Coulomb interactions and z=3 Pomeranchuk critical fluctuations. The central result is that electron-electron scattering enters the conductivity exclusively through the quasiparticle lifetime τ_ee; regime crossovers occur when 1/τ_ee becomes comparable to the larger of the impurity scattering rate and the probe frequency Ω. The solution also supplies an explicit expression for the optical conductivity in the hydrodynamic limit Ω ≪ 1/τ_ee. Near a z=3 Pomeranchuk quantum critical point, consistency between the kinetic-equation and Kubo formalisms is asserted to require inclusion of mass renormalization within the Eliashberg approximation, which simultaneously produces a Planckian crossover scale Ω ∼ T.

Significance. If the exact solution and the stated consistency condition hold, the work supplies a useful non-perturbative bridge between kinetic theory and linear-response formalism in systems lacking Galilean invariance. The isolation of all ee effects to a single τ_ee and the explicit hydrodynamic optical response are concrete advances that are inaccessible to perturbative treatments. The discussion of Planckian scaling near a z=3 QCP is timely. The derivation contains no free parameters and reproduces known asymptotic limits, which strengthens its internal coherence.

major comments (2)
  1. [final section / z=3 QCP consistency paragraph] Discussion of kinetic-Kubo consistency near the z=3 QCP (final section): the requirement that mass renormalization within the Eliashberg approximation be included to recover the Ω ∼ T crossover is load-bearing for the Planckian claim, yet the manuscript does not address possible O(1) renormalizations of the current vertex or current relaxation rate that arise once vertex corrections are restored. Such corrections are known to appear in Eliashberg treatments of critical fermions and could shift the hydrodynamic onset scale by a numerical factor without altering the functional form.
  2. [Abstract and exact-solution section] Abstract and § on the exact solution: the statement that the kinetic equation is solved exactly is correct within the stated approximations, but the subsequent claim that this yields a parameter-free result for the hydrodynamic optical conductivity rests on the same Eliashberg mass renormalization that is invoked for Kubo consistency. Any uncontrolled approximation in that step propagates directly into the hydrodynamic expression.
minor comments (2)
  1. [interaction Hamiltonian section] Notation for the screened Coulomb and critical interaction vertices should be defined once in a single equation rather than reintroduced in multiple places.
  2. [figures] Figure captions for the crossover plots should explicitly state the values of the impurity scattering rate and Ω used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed and insightful report. The comments highlight important points regarding the consistency between kinetic and Kubo approaches near the quantum critical point and the nature of the exact solution. We address each major comment below and propose revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: Discussion of kinetic-Kubo consistency near the z=3 QCP (final section): the requirement that mass renormalization within the Eliashberg approximation be included to recover the Ω ∼ T crossover is load-bearing for the Planckian claim, yet the manuscript does not address possible O(1) renormalizations of the current vertex or current relaxation rate that arise once vertex corrections are restored. Such corrections are known to appear in Eliashberg treatments of critical fermions and could shift the hydrodynamic onset scale by a numerical factor without altering the functional form.

    Authors: We thank the referee for pointing this out. In full Eliashberg theory, vertex corrections can indeed introduce O(1) renormalizations to the current vertex and relaxation rates. Our kinetic-equation solution captures the leading quasiparticle scattering effects, with mass renormalization providing the dominant contribution to the Planckian scale. The functional form Ω ∼ T is robust, although numerical prefactors of order unity may appear. We will revise the final section to note explicitly that the crossover occurs at Ω ∼ T up to O(1) coefficients and to discuss the limitations of neglecting higher-order vertex corrections. revision: partial

  2. Referee: Abstract and § on the exact solution: the statement that the kinetic equation is solved exactly is correct within the stated approximations, but the subsequent claim that this yields a parameter-free result for the hydrodynamic optical conductivity rests on the same Eliashberg mass renormalization that is invoked for Kubo consistency. Any uncontrolled approximation in that step propagates directly into the hydrodynamic expression.

    Authors: We agree that the hydrodynamic optical conductivity is obtained exactly from the kinetic equation once the quasiparticle lifetime τ_ee is given, but determining τ_ee near the QCP relies on the Eliashberg approximation. The parameter-free character refers to the conductivity formula itself containing no additional adjustable parameters beyond the input scattering rates. We will update the abstract and the exact-solution section to clarify that the solution is exact within the relaxation-time approximation for the collision integral and the Eliashberg treatment of the self-energy, with the hydrodynamic expression being parameter-free given those inputs. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper obtains an exact solution of the kinetic equation for conductivity in a disordered non-Galilean-invariant Fermi liquid with screened Coulomb and z=3 Pomeranchuk interactions. It reports that interactions enter conductivity only through the quasiparticle scattering time τ_ee, with crossovers set by 1/τ_ee relative to impurity rate or frequency Ω, plus an explicit hydrodynamic optical response. The additional statement that kinetic-Kubo consistency near the z=3 QCP requires mass renormalization in the Eliashberg approximation is presented as an external consistency condition rather than a self-definition, fitted parameter, or reduction of the main result to its inputs. No quoted equations or steps reduce by construction to prior fits, self-citations, or ansatzes; the central derivation remains self-contained within the kinetic-equation framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the claim rests on standard Fermi liquid assumptions and the kinetic equation framework plus the Eliashberg approximation for mass renormalization near the critical point. No free parameters or new entities are introduced in the summary.

axioms (2)
  • domain assumption Validity of the kinetic equation description for quasiparticles in a disordered non-Galilean-invariant Fermi liquid
    Invoked as the starting point for obtaining the exact conductivity expression.
  • domain assumption Proper inclusion of mass renormalization within the Eliashberg approximation for consistency near z=3 Pomeranchuk QCP
    Required for matching kinetic-equation and Kubo results and for Planckian crossover.

pith-pipeline@v0.9.0 · 5735 in / 1522 out tokens · 42644 ms · 2026-05-22T08:22:24.336701+00:00 · methodology

discussion (0)

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