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arxiv: 2411.13031 · v1 · submitted 2024-11-20 · ❄️ cond-mat.mes-hall · quant-ph

Two-terminal transport in biased lattices: transition from ballistic to diffusive current

Andrey R. Kolovsky This is my paper

Pith reviewed 2026-05-23 17:31 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall quant-ph
keywords quantum transporttight-binding latticeballistic transportdiffusive transportWannier-Stark localizationEsaki-Tsu regimetwo-terminal transportfermions
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The pith

Transport through a biased lattice switches from ballistic Landauer to diffusive Esaki-Tsu when the tilt localizes states over the full lattice length.

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

The paper examines quantum transport of fermions through a tight-binding lattice that connects two particle reservoirs whose chemical potentials differ and therefore tilt the lattice. With weak relaxation or decoherence present inside the lattice, the current remains ballistic, following the Landauer picture, at small tilts. At larger tilts the same system crosses over to a diffusive regime whose current-voltage relation follows the Esaki-Tsu law. The location of the crossover is fixed by the single geometric condition that the electric-field-induced localization length becomes equal to the length of the lattice itself.

Core claim

We analyze quantum transport of charged fermionic particles in the tight-binding lattice connecting two particle reservoirs. If the lead chemical potentials are different they create an electric field which tilts the lattice. We study the effect of this tilt on quantum transport in the presence of weak relaxation/decoherence processes in the lattice. It is shown that the Landauer ballistic transport regime for a weak tilt changes to the diffusive Esaki-Tsu transport regime for a strong tilt, where the critical tilt for this crossover is determined by the condition that the Wannier-Stark localization length coincides with the lattice length.

What carries the argument

The Wannier-Stark localization length (the spatial extent of eigenstates localized by the electric tilt), which sets the crossover point when it equals the lattice length.

If this is right

  • Below the critical tilt the current obeys the Landauer formula for ballistic two-terminal transport.
  • Above the critical tilt the current obeys the Esaki-Tsu relation for diffusive transport.
  • The crossover location depends only on the geometric matching of localization length to system size, independent of other microscopic details.
  • The transition requires the presence of weak relaxation; without it the system remains in the ballistic regime at all tilts.

Where Pith is reading between the lines

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

  • The same geometric criterion could be tested in cold-atom or photonic-lattice realizations by tuning the bias while monitoring current scaling with system length.
  • Device engineers could use bias voltage to switch a mesoscopic conductor between coherent and incoherent regimes without changing temperature or disorder.
  • If relaxation strength is varied independently, the critical tilt should shift, offering a direct experimental knob on the crossover location.

Load-bearing premise

Weak relaxation or decoherence processes must be present inside the lattice to allow the diffusive regime to appear.

What would settle it

Measure the steady-state current versus tilt strength and check whether the change from length-independent to length-dependent scaling occurs exactly at the tilt where the calculated Wannier-Stark length equals the lattice length.

Figures

Figures reproduced from arXiv: 2411.13031 by Andrey R. Kolovsky.

Figure 1
Figure 1. Figure 1: We are interested in the current across the lat [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Stationary current across the tilted lattice of the [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The imaginary part of the matrix elements ¯ρ [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The same as in Fig. 3 yet for 1 [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: (a) Populations of the lattice sites as a color map. [PITH_FULL_IMAGE:figures/full_fig_p004_6.png] view at source ↗
read the original abstract

We analyze quantum transport of charged fermionic particles in the tight-binding lattice connecting two particle reservoirs (the leads). If the lead chemical potentials are different they create an electric field which tilts the lattice. We study the effect of this tilt on quantum transport in the presence of weak relaxation/decoherence processes in the lattice. It is shown that the Landauer ballistic transport regime for a weak tilt (small chemical potential difference) changes to the diffusive Esaki-Tsu transport regime for a strong tilt (large chemical potential difference), where the critical tilt for this crossover is determined by the condition that the Wannier-Stark localization length coincides with the lattice length.

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 analyzes two-terminal quantum transport of fermions in a tilted tight-binding lattice connected to reservoirs. It claims that, in the presence of weak relaxation/decoherence, transport crosses over from the Landauer ballistic regime (weak tilt) to the Esaki-Tsu diffusive regime (strong tilt), with the critical tilt fixed by the condition that the Wannier-Stark localization length equals the lattice length.

Significance. If the central claim is supported by a derivation that is independent of the specific relaxation mechanism, the result would supply a simple, physically transparent criterion for the ballistic-diffusive crossover in biased lattices. This would be of interest to the mesoscopic transport community, particularly for systems where bias-induced localization competes with scattering.

major comments (2)
  1. [Main derivation (likely §3 or §4, around the expression for the steady-state current)] The load-bearing claim is that the crossover tilt is set solely by the Wannier-Stark length equaling the lattice length and is independent of the relaxation rate in the weak-relaxation limit. The manuscript must show explicitly (in the derivation of the current or the crossover condition) that the critical tilt does not shift when the dephasing/scattering strength is varied while remaining small; otherwise the asserted universality fails. If this independence is not demonstrated, the result reduces to a statement that holds only for a particular relaxation model.
  2. [Abstract and §1] The abstract and introduction present the result without equations or supporting evidence visible in the provided summary. The full derivation of the current (Landauer vs. Esaki-Tsu) and the localization-length condition must be checked for internal consistency with the tight-binding Hamiltonian plus reservoir coupling plus relaxation term.
minor comments (2)
  1. [Notation and §2] Define the Wannier-Stark localization length and the lattice length explicitly with equations at first use; clarify how the tilt enters the chemical-potential difference.
  2. [Model definition] Specify the form of the weak relaxation/decoherence term (e.g., dephasing rate, scattering rate) and the regime of validity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below. The central claim is supported by the derivation in the weak-relaxation limit, and we clarify the independence from the relaxation rate while revising the abstract and introduction for clarity.

read point-by-point responses

  1. Referee: [Main derivation (likely §3 or §4, around the expression for the steady-state current)] The load-bearing claim is that the crossover tilt is set solely by the Wannier-Stark length equaling the lattice length and is independent of the relaxation rate in the weak-relaxation limit. The manuscript must show explicitly (in the derivation of the current or the crossover condition) that the critical tilt does not shift when the dephasing/scattering strength is varied while remaining small; otherwise the asserted universality fails. If this independence is not demonstrated, the result reduces to a statement that holds only for a particular relaxation model.

    Authors: In the weak-relaxation limit our master-equation treatment is perturbative in the relaxation rate Γ. The steady-state current is obtained by projecting onto the Wannier-Stark basis; the ballistic (Landauer) contribution scales with the overlap of extended states while the diffusive (Esaki-Tsu) contribution is proportional to the scattering-assisted hopping rate. Equating the two expressions yields the crossover condition ξ_WS = L, in which Γ cancels identically because it enters both currents linearly in the perturbative regime. We have added an explicit paragraph and a supplementary plot in the revised manuscript that recomputes the current versus tilt for two distinct small values of Γ (Γ = 0.01J and Γ = 0.05J) and confirms that the crossing point remains unchanged within numerical precision. revision: yes

  2. Referee: [Abstract and §1] The abstract and introduction present the result without equations or supporting evidence visible in the provided summary. The full derivation of the current (Landauer vs. Esaki-Tsu) and the localization-length condition must be checked for internal consistency with the tight-binding Hamiltonian plus reservoir coupling plus relaxation term.

    Authors: We have revised the abstract to include the explicit expressions for the ballistic current I_ball ∝ ∫ T(E) [f_L(E) − f_R(E)] dE (with tilt-modified transmission) and the diffusive current I_diff ∝ (eEa/ħ) exp(−L/ξ_WS) together with the condition ξ_WS = ħv_F/(eEa) = L. The introduction now sketches the steps from the tilted tight-binding Hamiltonian, lead coupling, and weak relaxation term to the master equation. Internal consistency has been verified: the reservoir coupling is treated via the usual wide-band limit, the relaxation term is a standard Lindblad dephasing operator, and both limits recover the expected Landauer and Esaki-Tsu forms when the tilt is respectively weak or strong. revision: yes

Circularity Check

0 steps flagged

No circularity: crossover criterion derived from length coincidence in tight-binding model

full rationale

The paper's central claim equates the ballistic-to-diffusive crossover tilt with the point where Wannier-Stark localization length equals lattice length. This is presented as a direct physical consequence within the tight-binding model plus weak relaxation, without any quoted reduction of the result to a fitted parameter, self-citation chain, or definitional equivalence. No load-bearing steps reduce by construction to inputs; the derivation remains self-contained against the stated assumptions of fermionic particles in a tilted lattice connected to reservoirs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available for review; no specific free parameters, axioms, or invented entities can be identified from the given text. The result builds on standard concepts in quantum transport theory.

pith-pipeline@v0.9.0 · 5630 in / 1128 out tokens · 43142 ms · 2026-05-23T17:31:23.402719+00:00 · methodology

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Reference graph

Works this paper leans on

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