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Exact density profile in a tight-binding chain with dephasing noise

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Pith's one-line read Exact solution: any dephasing makes a quantum chain diffuse

desk verdict Genuinely exact density profiles for a dephasing tight-binding chain, with a derivation that checks out; a strong paper that only needs minor cleanup. read the letter →

arxiv 2501.07095 v1 pith:5N4IV3I3 submitted 2025-01-13 cond-mat.stat-mech cond-mat.quant-gasmath-phmath.MPquant-ph

Taiki Ishiyama , Kazuya Fujimoto , Tomohiro Sasamoto This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.quant-gasmath-phmath.MPquant-ph MSC 82C1082C23
keywords tight-bindingchaindephasingnoiseLindbladequationBetheansatzaveragedensityprofiledomainwallinitialconditionalternatingsymmetricsimpleexclusionprocess
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

This paper solves, exactly, the time-dependent average particle density of a tight-binding (hopping) fermion chain whose sites are subjected to dephasing noise, for two many-body initial states: a filled half-line (domain wall) and particles on alternating sites. The main claim is that for any positive dephasing strength $\gamma$, the domain-wall density spreads diffusively at long times, with an explicit scaling form, while for the alternating state the density relaxes either with oscillations (weak dephasing) or with overdamped exponential decay (strong dephasing), with a polynomial-times-exponential decay exactly at the transition. The exact formulas also confirm that the long-time behaviors match those of the symmetric simple exclusion process and give the leading corrections away from it. A sympathetic reader would care because exact dynamical results for interacting, non-free open quantum systems are rare, and these formulas turn a previously numerical observation into a proven statement.

What carries the argument

The carrying object is the two-particle wavefunction of the Fermi-Hubbard model with pure-imaginary interaction strength (the Bethe wavefunction, Eq. (6)), which solves the equation of motion for the two-point correlation function after a phase twist. The key step is the double contour-integral representation of the Green's function (Eq. (10)), which builds in the two-particle eigenstates without solving Bethe equations: working directly on the infinite lattice lets every eigenstate contribute through contour integrals, and summing contours against the initial state produces Bessel-function integrals that are then expanded by saddle-point analysis. For the alternating initial condition, the divergent sum over initial positions is handled by a contour deformation that keeps only the pole at $z_1=-1/z_2$, and the coalescence of poles at $\gamma=1$ produces the polynomial factor in the decay.

What would settle it

Directly integrate the equation of motion, Eq. (2), for the two-point correlation function on a large finite chain and extrapolate to the infinite-size limit: at $\gamma=1$, the alternating-state quantity $(\langle n_x\rangle_t-1/2)(-1)^x e^{4t}$ should grow linearly in $t$ according to Eq. (23), and any other growth or saturation would falsify the claimed transition polynomial; for the domain wall, the rescaled profile $\sqrt{\tau}\,\langle n_x\rangle_t$ plotted against $X=x/\sqrt{\tau}$ should collapse to the curve of Eq. (19) at large $\tau$.

Watch

Extended reading notes

Core claim

The paper's central discovery is an exact, closed-form Green's function for the two-point correlation function of the Lindblad-evolved dephasing tight-binding chain, written as a double contour integral built from the two-particle Fermi-Hubbard wavefunction with imaginary interaction (Eq. (10)). Summing this Green's function over the domain-wall and alternating initial conditions yields the exact density profiles of Eqs. (12) and (22), and their large-time asymptotics, Eqs. (19), (20), (23), (29), and (32). The density exhibits diffusive scaling for any $\gamma>0$ in the domain-wall case; in the alternating case, the relaxation changes from oscillatory to overdamped as $\gamma$ increases, with a polynomial-times-exponential decay at $\gamma=1$. These results provide an analytic confirmation of the previously numerical transition, and identify exactly how the density differs from the symmetric simple exclusion process.

Load-bearing premise

Everything rests on the two-particle Bethe wavefunction, Eq. (6), being exactly right at coincident sites, and on the contour deformation in Appendix D being legitimate for all times; if either fails, the density formulas cease to be exact.

Editorial extensions

If this is right

  • For any positive dephasing strength, no matter how small, the long-time domain-wall density obeys the diffusive scaling form Eq. (19), so coherent hopping alone cannot prevent diffusion once dephasing is present.
  • The average integrated current converges to the diffusive result with the explicit next-order correction Eq. (20).
  • In the alternating initial state, changing $\gamma$ across 1 switches the relaxation from oscillatory to overdamped; at $\gamma=1$, the deviation of $\langle n_x\rangle_t$ from $1/2$ relaxes as $4(-1)^x t e^{-4t}$.
  • The exact density reproduces the symmetric simple exclusion process in the leading long-time term for both initial states; corrections are of order $1/\tau$ for the domain wall, while for the alternating state the difference is already visible at the subleading exponential level, where the transition is absent in SEP.
  • The zero-dephasing limit of Eq. (12) reduces to the known XX-chain result, providing a consistency check on the exact formula.

Reading between the lines

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

  • Editorial inference: the same Green's function, Eq. (10), can likely be summed against other translation-invariant or periodic initial conditions, so exact density profiles should be available for a wider family of step-like or periodic states than the two studied here.
  • Editorial inference: because the exact density is explicit for all $t$ and $\gamma$, one could extract full current-fluctuation statistics beyond the average, a direction the paper flags but does not pursue.
  • Editorial inference: the analogy between the $\gamma=1$ pole coalescence and exceptional points suggests that the polynomial factor is a genuine non-Hermitian degeneracy effect; testing the off-diagonal correlation at $\gamma=1$ would show whether the same coalescence controls all correlations.
  • Editorial inference: the domain-wall correction to SEP is of relative order $1/\tau$, while the alternating-state correction is visible in the leading exponential; this suggests the SEP correspondence is quantitatively stronger for initial states that break two-site translation symmetry, a pattern that other periodicities could test.
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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

0 major / 5 minor

Summary. The paper studies the GKSL dynamics of an infinite tight-binding chain with dephasing noise. The authors close the equation of motion for the two-point correlation function, map it to a two-particle Fermi-Hubbard model with imaginary interaction, and use the nested Bethe ansatz to obtain a double-contour-integral Green's function, Eq. (10). From this they derive exact average densities for the domain-wall and alternating initial conditions, Eqs. (12) and (22), together with long-time asymptotics: diffusive scaling for the domain wall, Eqs. (19) and (20), and a dynamical transition from oscillatory to overdamped decay for the alternating initial condition, Eq. (23), including the exceptional-point polynomial decay at the transition. The paper then compares these results with the symmetric simple exclusion process and identifies corrections, Eqs. (28), (29), (32), and (34).

Significance. The result is significant. It provides one of the first exact time-dependent many-body correlation functions for an interacting, non-free Lindblad system, obtained by applying the Bethe ansatz directly on the infinite lattice and thereby avoiding the Bethe equations. The derivation is explicit and parameter-free: Eq. (10) is verified against the equation of motion and the initial condition, and the exact formulas reduce to the known free-fermion limit at gamma=0 and to SEP in the strong-dephasing limit. The asymptotic expressions are concrete falsifiable predictions, and the SEP corrections in Eqs. (29) and (32) are a useful benchmark. I found no circularity: the SEP comparison is an external check, not an input to the derivation. I also checked the two points that carry the most risk, namely the validity of the Bethe wavefunction at coincident sites and the contour regularization in Appendix D, and found both sound. The paper is honest about the scope: it treats the average density, not fluctuations, and it explicitly leaves open the relation between the pole coalescence and the exceptional-point spectral picture.

minor comments (5)
  1. [§5.2, Eq. (30)] The expression for Q in Eq. (30) has a typo: the second modified Bessel function should be I1(2τ), not I1(τ), to agree with the contour representation (26) and with the asymptotic expansion (31).
  2. [§5.3] The sentence 'We next consider the large τ (with finite τ ) case' should read 'large τ (with finite γ)', otherwise the comparison with the strong-dephasing expansion is misstated.
  3. [§4.2, §5.2, Fig. 2, Conclusion] Several typographical errors should be corrected: 'Appnedix E' in §4.2, 'nemely' in §5.2, 'dased line' in the Fig. 2 caption, and 'depahsing' in the Conclusion.
  4. [Appendix A, Eq. (38)] The displayed identity is algebraically correct, but only if the second exponential term is read as a boundary contribution outside the s-integral; writing the identity as an integral of e^{As}/(2A) from 0 to t minus e^{At}/(2A) would remove the ambiguity.
  5. [Fig. 3 caption] The plotted quantity appears as (⟨n_{2x}⟩_t − 1/2)e^{4γt}; if this notation is meant to denote the density on even sites, it should be defined explicitly in the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exact density formulas are derived from an independently verifiable Bethe-ansatz input and an explicit contour integral, with no fitted parameters or prediction-by-construction steps.

full rationale

The paper's central results, Eqs. (12), (22), and their asymptotics (19), (20), (23), are obtained by summing the explicitly constructed Green's function (10) over the domain-wall and alternating initial conditions. The integral representation (10) is proved directly by verifying the equation of motion and the initial condition, using the standard two-particle Fermi-Hubbard Bethe wavefunction (6) from an established textbook [84]; the dephasing strength γ is a physical input, not a fitted parameter. No subset of data is used to determine a parameter that is later 'predicted.' The SEP comparison in Sec. 5 is an external benchmark applied after the exact formulas are derived, not an ingredient in their derivation. The papers cited for the Bethe ansatz and for the earlier numerical observation of the transition [85] are outside the present authorship, and the authors' own prior works are used only for related methods or Bessel-function identities, not to force the central claim. The analytic-continuation step in Appendix D, while delicate, is a mathematical derivation rather than an appeal to the desired result. Thus no circular step can be exhibited; the derivation chain is self-contained given the stated Bethe-ansatz input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the dephasing strength is the model's physical input. The derivation leans on prior Bethe ansatz results for the imaginary-interaction Hubbard model and on standard saddle point analysis, none of which introduces new entities. The only notable internal issue is a formula typo in Eq. (30), not an input assumption.

assumptions (4)
  • domain assumption The two-point correlation function equation of motion, Eq. (2), is closed and exact.
    Invoked in Sec. 2; it follows from the quadratic nature of the Liouvillian per Ref. [88], and the paper relies on it to avoid higher-order correlations.
  • standard math The two-particle Bethe ansatz wavefunction, Eq. (6), with pure-imaginary interaction solves the Schrodinger equation, Eq. (4), including the coincident-site delta term.
    Taken from Refs. [83,84] and used in Eq. (10); the paper does not rederive the Bethe ansatz solution itself, only the contour representation built on top of it.
  • domain assumption Infinite-lattice contour integrals can be interchanged with summation over initial sites and deformed without missing singular contributions.
    Used in Sec. 4 and Appendices A and D when summing the Green's function over initial positions; the alternating case explicitly splits contours to make the geometric series converge.
  • standard math The saddle point and pole expansions in Appendices C and E give the true large-time asymptotics with the stated error orders.
    The paper provides error bounds for some steps, such as Eq. (50), but the overall uniform validity of the expansions is asserted rather than fully proven.

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Pith. "Pith review of Exact density profile in a tight-binding chain with dephasing noise." pith.science (2026). https://pith.science/paper/5N4IV3I3

@misc{pith2026250107095,
  author       = {Pith},
  title        = {Pith review of: Exact density profile in a tight-binding chain with dephasing noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5N4IV3I3}},
  note         = {Machine review of arXiv:2501.07095}
}
read the original abstract

We theoretically investigate the many-body dynamics of a tight-binding chain with dephasing noise on the infinite interval. We obtain the exact solution of an average particle-density profile for the domain wall and the alternating initial conditions via the Bethe ansatz, analytically deriving the asymptotic expressions for the long time dynamics. For the domain wall initial condition, we obtain the scaling form of the average density, elucidating that the diffusive transport always emerges in the long time dynamics if the strength of the dephasing, no matter how small, is positive. For the alternating initial condition, our exact solution leads to the fact that the average density displays oscillatory decay or over-damped decay depending on the strength of the dissipation. Furthermore, we demonstrate that the asymptotic forms approach those of the symmetric simple exclusion process, identifying corrections from it.

Figures

Figures reproduced from arXiv: 2501.07095 by the authors.

Figure 1
Figure 1. Schematic illustrations of the initial states. The blue circles represent fermions [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Density profile ⟨nx⟩t with rescaled coordinate X = x/√ τ . The colored markers show the exact solution (12) for γ = 1, while the dased line does the leading term of the asymptotic expansion (19). 4.2 Alternating initial condition As in the case of the domain wall initial condition, we have the exact solution of GAlt x1,x2 (t) as follows, G Alt x1,x2 (t) = 1 + (−1)x1 2 δx1,x2 + (−1)xQ(1) i x2−x1 Z t 0 du α(u) e −4γt(… view at source ↗
Figure 3
Figure 3. Time evolution of ⟨nx⟩ for the alternating initial condition. The colored lines show the exact solution, Eq. (22). The dashed lines show the asymptotic form, Eq. (23). was previously identified through the numerical analysis in Ref. [85]. The equation (23) gives the analytical demonstration for the presence of the transition. Moreover, in Appnedix E, we show that the off-diagonal element of GAlt x1,x2 (t) also exhib… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: , ∆⟨nx⟩t becomes larger as γ decreases, and this deviation is attributed to the effect of the coherent hopping, which becomes more significant compared to the dissipation. 0 1 2 3 4 5 X −0.04 −0.03 −0.02 −0.01 0.00 τ × ∆hn xit γ = 0.5 γ = 1 γ = 1.5 [PITH_FULL_IMAGE:fi…
Figure 5
Figure 5. Figure 5: Schematic illustration of the contour in the asymptotic analysis of Eq. (59). The [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Schematic illustrations of the contours. The arrowed lines represent the contours. [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]

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