arxiv: 2604.12475 · v1 · submitted 2026-04-14 · ❄️ cond-mat.quant-gas

Recognition: unknown

Explicit proof of Anderson's orthogonality catastrophe for the one-dimensional Fermi polaron with attractive interaction

Giuliano Orso

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:27 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas

keywords Anderson orthogonality catastropheFermi polaronone-dimensionalBethe ansatzquasi-particle residueCauchy matricesphase shiftthermodynamic limit

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The pith

The quasi-particle residue of the one-dimensional Fermi polaron decays algebraically as Z = W N^{-θ} with θ = 2δ_F²/π².

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

The paper gives an explicit analytical proof that the overlap between the interacting ground state and the non-interacting Fermi sea vanishes in the thermodynamic limit for the attractive one-dimensional Fermi polaron. It starts from the Bethe-ansatz determinant expressions for the wave-function norm and the overlap, then uses the asymptotic properties of Cauchy matrices to extract the leading large-N behavior at fixed density. This produces an algebraic decay of the quasi-particle residue with an exponent fixed by the scattering phase shift at the Fermi edge. The result matters because it supplies the first fully rigorous confirmation of Anderson's orthogonality catastrophe inside an exactly solvable model without further approximations.

Core claim

In the thermodynamic limit the two determinants yield the quasi-particle residue Z = W N^{-θ}, where the Anderson exponent θ equals 2δ_F²/π² and δ_F is the Bethe-ansatz phase shift evaluated at the Fermi momentum. The prefactor W is computed numerically as a function of the interaction strength.

What carries the argument

Determinant representations of the norm and overlap obtained from the Bethe-ansatz solution, whose leading asymptotics are found using the special properties of Cauchy matrices.

If this is right

The interacting ground state becomes orthogonal to the non-interacting Fermi sea in the infinite-volume limit.
The power-law exponent is locked to the two-body phase shift at the Fermi edge.
Finite-size corrections to the quasi-particle weight follow the same algebraic scaling for any interaction strength.
The prefactor W supplies a concrete interaction-dependent amplitude that can be compared with other calculations.

Where Pith is reading between the lines

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

The same determinant-plus-Cauchy-matrix route may give exact decay rates in other one-dimensional integrable models whose wave functions are known explicitly.
Spectroscopic measurements on finite one-dimensional polaron systems will need to include this power-law suppression when extracting residues.
The result provides a benchmark for numerical methods that simulate the same model at large but finite N.

Load-bearing premise

The determinant expressions coming from the Bethe-ansatz solution share the same leading asymptotic behavior as Cauchy matrices when the system size is taken to infinity at fixed density.

What would settle it

Direct numerical evaluation of the overlap determinant for increasing particle numbers N at fixed density, to check whether log(Z) versus log(N) approaches a slope equal to the predicted −θ.

Figures

Figures reproduced from arXiv: 2604.12475 by Giuliano Orso.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Main panel [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Prefactor [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

read the original abstract

We provide a fully analytical derivation of Anderson's orthogonality catastrophe for the one dimensional Fermi polaron integrable model, describing a system of $N$ spin-up fermions, with fixed density $n=N/L$, interacting with a single spin-down fermion via an attractive contact potential. The proof combines the determinant representations of the norm of the many-body wave function and of its scalar product with the noninteracting ground state, obtained from the Bethe ansatz solution, with the special properties of Cauchy matrices. We derive the leading asymptotics of the two determinants in the thermodynamic limit and show that the quasi-particle residue $Z$ decays algebraically, $Z=W N^{-\theta}$. We confirm that the Anderson exponent $\theta$ is equal to $2\delta_F^2/\pi^2$, where $\delta_F$ is the Bethe-ansatz phase shift at the Fermi edge. The prefactor $W$ is obtained numerically as a function of the interaction parameter.

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.

Desk Editor's Note private letter to a colleague

The paper gives a clean analytical derivation of the algebraic decay of the quasiparticle residue in the attractive 1D Fermi polaron, confirming the expected Anderson exponent from the Bethe-ansatz phase shift while extracting a numerical prefactor.

read the letter

The central result is an explicit proof that the overlap Z between the interacting ground state and the non-interacting Fermi sea decays as W N to the minus theta, where theta equals 2 delta_F squared over pi squared and delta_F comes directly from the Bethe-ansatz equations. They start from the known determinant expressions for the norm and the overlap that follow from the Bethe-ansatz wave functions, then use the large-N asymptotics of Cauchy matrices to extract the leading power-law behavior in the thermodynamic limit at fixed density. The prefactor W is evaluated numerically as a function of the interaction parameter. This is new: prior work on this model had numerical evidence or indirect arguments, but not this closed-form treatment of both the exponent and the prefactor from the determinants. The approach stays within standard integrable-model machinery and the phase shift is taken from the equations rather than adjusted to fit the target quantity, so the logic is straightforward and the circularity is low. The derivation is technically involved but the steps are laid out clearly enough that a reader familiar with Bethe-ansatz determinants can follow the logic. The main soft spot is the interchange of limits. The Bethe roots are not uniformly spaced, and the matrix entries deviate from the pure Cauchy form because of the density-dependent distribution and the finite-size cutoff. The paper asserts that these corrections do not affect the power theta and only enter W, but without explicit bounds on the error terms from the root density or the Jacobian it is hard to be fully confident that nothing leaks into the exponent. If those bounds are present and tight, the claim holds; if they are only sketched, that section would benefit from more detail. This paper is aimed at people working on exact solutions in one-dimensional quantum gases and on polaron problems in cold atoms. A reader who needs a benchmark result or wants to see how Cauchy-matrix techniques extend to Bethe-ansatz overlaps will get concrete value from it. It is worth sending to peer review; the result is specific, the method is reproducible, and the potential gap on error control is fixable with modest additional work.

Referee Report

1 major / 2 minor

Summary. The manuscript presents an explicit analytical proof of Anderson's orthogonality catastrophe for the one-dimensional Fermi polaron with attractive contact interaction. Using Bethe-ansatz determinant representations for the wave-function norm and the overlap with the non-interacting ground state, the authors apply asymptotic properties of Cauchy matrices in the thermodynamic limit at fixed density to show that the quasiparticle residue Z decays algebraically as Z = W N^{-θ}, where the exponent θ equals 2 δ_F² / π² with δ_F the Bethe-ansatz phase shift at the Fermi edge, and W is a numerically determined prefactor.

Significance. If the central derivation is correct, this work provides a valuable explicit verification of the Anderson exponent in an exactly solvable model, linking it directly to the phase shift without fitting parameters. The combination of integrable-model techniques with Cauchy-matrix asymptotics offers a template for similar analyses in other one-dimensional systems, and the numerical evaluation of the prefactor adds practical utility for comparisons with experiments or numerics in quantum gases.

major comments (1)

[Thermodynamic limit analysis of the determinants] The application of known large-N asymptotics for Cauchy matrices to the Bethe-ansatz determinants requires justification that corrections arising from the non-uniform distribution of rapidities (due to the transcendental BA equations with δ(k) = -arctan(k/c)), the Jacobian of the root map, and finite-size effects are either exponentially suppressed or affect only the prefactor W and not the power-law exponent θ. Explicit error bounds or a detailed analysis of these corrections in the thermodynamic limit at fixed density n = N/L should be provided to confirm that θ = 2δ_F²/π² holds rigorously.

minor comments (2)

[Abstract and Introduction] The abstract claims a 'fully analytical derivation' while noting that the prefactor W is obtained numerically; this distinction should be stated clearly in the introduction and conclusion to avoid any misinterpretation of the scope of the analytic result.
[Bethe-ansatz solution section] Ensure that the definition of the Fermi-edge phase shift δ_F is explicitly tied to the specific Bethe-ansatz quantization condition used in the thermodynamic limit, including any reference to the two-body phase shift function.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive overall assessment. We address the single major comment below, providing additional clarification on the thermodynamic-limit analysis while agreeing that further explicit bounds would strengthen the presentation.

read point-by-point responses

Referee: [Thermodynamic limit analysis of the determinants] The application of known large-N asymptotics for Cauchy matrices to the Bethe-ansatz determinants requires justification that corrections arising from the non-uniform distribution of rapidities (due to the transcendental BA equations with δ(k) = -arctan(k/c)), the Jacobian of the root map, and finite-size effects are either exponentially suppressed or affect only the prefactor W and not the power-law exponent θ. Explicit error bounds or a detailed analysis of these corrections in the thermodynamic limit at fixed density n = N/L should be provided to confirm that θ = 2δ_F²/π² holds rigorously.

Authors: We agree that a more explicit discussion of error terms is desirable. In the thermodynamic limit at fixed density the Bethe-ansatz rapidities become dense according to the smooth density ρ(k) solving the integral equation obtained from the BA equations; the discrete sum over roots is replaced by an integral weighted by this density. The Cauchy-matrix determinants that appear in the norm and overlap admit a continuum representation whose logarithm yields an integral whose singular part at the Fermi points is controlled solely by the phase shift δ_F. The Jacobian of the root map and the global shape of ρ(k) contribute only multiplicative constants to the prefactor W. Finite-size corrections to the rapidity distribution are O(1/N) and produce corrections to log Z that are either exponentially small or O(1/N), leaving the algebraic exponent θ unaffected. In the revised manuscript we will add an appendix that (i) recalls the relevant large-N asymptotics for Cauchy determinants with points sampled from a smooth density, (ii) derives the leading singular contribution from the Fermi-edge phase shift, and (iii) supplies explicit bounds showing that all other corrections are o(1) in the exponent as N→∞ at fixed n. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external Cauchy-matrix asymptotics to BA-derived determinants without reduction to fitted inputs or self-citations

full rationale

The paper obtains norm and overlap determinants from the standard Bethe-ansatz solution for the integrable 1D Fermi polaron, then invokes known large-N asymptotics of Cauchy matrices to extract the algebraic decay of the overlap Z ~ N^{-θ} with θ expressed via the BA phase shift δ_F. No step defines the target exponent in terms of itself, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose validity is internal to the present work. The phase shift δ_F is an output of the BA equations rather than an adjustable parameter tuned to the orthogonality result, and the thermodynamic-limit analysis at fixed density is presented as a direct application of external mathematical results on Cauchy determinants. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The proof rests on the integrability of the model (Bethe ansatz) and on standard mathematical properties of Cauchy matrices; no new physical entities are postulated and the only numerical element is the prefactor W.

free parameters (1)

interaction strength
W is computed numerically as a function of the attraction parameter; the exponent itself contains no free parameters.

axioms (2)

domain assumption Bethe-ansatz solution supplies exact determinant representations for the norm and overlap
Invoked to obtain the starting expressions whose asymptotics are then analyzed.
standard math Cauchy matrices possess known leading asymptotics in the thermodynamic limit
Used to extract the algebraic decay from the determinants.

pith-pipeline@v0.9.0 · 5464 in / 1485 out tokens · 31037 ms · 2026-05-10T14:27:50.313609+00:00 · methodology

discussion (0)

Reference graph

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3 as a function ofy= nj/nN , together with the approximating functionln(1 +y)− ln(1−y)appearing in Eq

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