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arxiv: 2505.20376 · v2 · submitted 2025-05-26 · ❄️ cond-mat.str-el · cond-mat.quant-gas· quant-ph

Thermodynamics and Tomonaga-Luttinger liquid behavior of the quantum 1D hard rod model

Shengjie Yu , Zhaoxuan Zhu , Laurent Sanchez-Palencia This is my paper

Pith reviewed 2026-05-19 13:47 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.quant-gasquant-ph
keywords hard rod modelTomonaga-Luttinger liquidYang-Yang theoryquantum Monte Carlocorrelation functionsthermometryone-dimensional bosonsLieb-Liniger model
0
0 comments X

The pith

The quantum 1D hard rod model exhibits Tomonaga-Luttinger liquid behavior at zero and finite temperatures.

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

This paper studies the thermodynamics and quantum correlations of the one-dimensional hard rod model, an extension of the Lieb-Liniger model that accounts for the finite size of bosons. The authors use Yang-Yang theory and quantum Monte Carlo simulations to analyze thermodynamic quantities and show that the model displays Tomonaga-Luttinger liquid behavior over a broad range of densities and at both zero and finite temperatures. By fitting the correlation functions to the predictions of Tomonaga-Luttinger liquid theory, they extract the Tomonaga-Luttinger parameter and the thermal length. This extraction serves as a practical method for thermometry in these systems. The findings are relevant for understanding quantum wires, spin chains, and experiments with ultracold atoms.

Core claim

The one-dimensional hard rod model describes impenetrable bosons with finite diameter, extending the Lieb-Liniger model to systems with excluded volume interactions. The hard rod model exhibits Tomonaga-Luttinger liquid behavior across a wide range of parameters, at zero and finite temperature, as unveiled by correlation functions. The Tomonaga-Luttinger parameter and thermal length can be extracted by fitting correlation functions to Tomonaga-Luttinger liquid theory, hence demonstrating a robust method for thermometry. This work provides a comprehensive study of strongly correlated hard rod systems at finite temperatures, with applications to quantum wires, spin chains, and ultracold atoms.

What carries the argument

Fitting correlation functions to Tomonaga-Luttinger liquid theory to extract the interaction parameter and thermal length, validated by Yang-Yang theory and quantum Monte Carlo calculations.

If this is right

  • Thermodynamic quantities exhibit deviations from the Lieb-Liniger model at high densities with excellent agreement between theory and numerics.
  • Tomonaga-Luttinger liquid behavior holds at finite temperatures as shown by correlation functions.
  • The Tomonaga-Luttinger parameter and thermal length are obtained from fits to the theory.
  • This demonstrates a robust method for thermometry in one-dimensional quantum systems.
  • The results apply to quantum wires, spin chains, and ultracold atoms.

Where Pith is reading between the lines

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

  • The correlation fitting method could be tested in other one-dimensional models with short-range repulsions to check its broader applicability.
  • Experimental measurements of density correlations in ultracold atom setups could directly test the proposed thermometry technique.
  • The findings may help identify the boundary where Tomonaga-Luttinger liquid theory ceases to apply in systems with increasing interaction strength or temperature.
  • Analogous behavior might appear in spin chain models that incorporate hard-core constraints similar to finite-diameter particles.

Load-bearing premise

The hard rod system stays within the regime where Tomonaga-Luttinger liquid theory applies at finite temperatures and that Yang-Yang theory accurately captures the thermodynamics of finite-diameter bosons without additional corrections.

What would settle it

Observation of correlation functions that do not match the expected forms from Tomonaga-Luttinger liquid theory at finite temperatures, or significant discrepancies between Yang-Yang predictions and numerical results for thermodynamic quantities.

Figures

Figures reproduced from arXiv: 2505.20376 by Laurent Sanchez-Palencia, Shengjie Yu, Zhaoxuan Zhu.

Figure 1
Figure 1. Figure 1: Dispersion relation of elementary particle (upper [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Thermodynamics of HRs. (a) Density and (b) compressibility, κ˜ = ℏ 2κ/ma, versus chemical potential, µ˜ = µma2 /ℏ 2 , for the HR and TG models. For HRs, we show TBA predictions (solid red line) and QMC results (pur￾ple disks) at T = 0.1ℏ 2 /ma2 kB, as well as BA predictions at T = 0 (dashed black line). For TG gas, we show TBA pre￾dictions (dashed purple line). The inset shows QMC results for (kBma2 /ℏ 2 )… view at source ↗
Figure 3
Figure 3. Figure 3: Correlations. (a) One-body correlation func [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

The one-dimensional hard rod model describes impenetrable bosons with finite diameter, extending the Lieb-Liniger model to systems with excluded volume interactions. Here, we investigate the thermodynamics of quantum HRs using Yang-Yang theory, path integral quantum Monte-Carlo calculations, and Luttinger liquid theory. We first discuss the behavior of characteristic thermodynamic quantities, exhibiting deviations to the Lieb-Liniger model for sufficiently high densities, with excellent agreement between analytical and numerical results. We then show that the hard rod model exhibits Tomonaga-Luttinger liquid behavior across a wide range of parameters, at zero and finite temperature, as unveiled by correlation functions. The Tomonaga-Luttinger parameter and thermal length can be extracted by fitting correlation functions to Tomonaga-Luttinger liquid theory, hence demonstrating a robust method for thermometry. This work provides a comprehensive study of strongly correlated hard rod systems at finite temperatures, with applications to quantum wires, spin chains, and ultracold atoms.

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

3 major / 2 minor

Summary. The manuscript studies the thermodynamics and correlation functions of the one-dimensional quantum hard rod model of impenetrable bosons with finite diameter. It combines Yang-Yang theory for the equation of state, path-integral quantum Monte Carlo simulations for numerical benchmarks, and Tomonaga-Luttinger liquid (TLL) theory to analyze density-density and one-body correlation functions. The authors report quantitative agreement between analytic and numerical thermodynamics, deviations from the Lieb-Liniger limit at high densities, and TLL power-law decay (with thermal exponential cutoff) over a wide range of densities and temperatures, from which the Luttinger parameter K and thermal length are extracted by direct fitting to demonstrate a thermometry protocol.

Significance. If the TLL fitting procedure is shown to be robust, the work supplies a useful bridge between exact thermodynamics and universal low-energy physics for 1D systems with excluded-volume interactions, with direct relevance to ultracold-atom experiments and quantum wires. The explicit validation of Yang-Yang theory against PIMC and the extension of TLL analysis to finite temperature constitute clear strengths; however, the load-bearing claim that TLL forms dominate sufficiently for reliable parameter extraction at finite T requires tighter quantification of fitting windows and corrections.

major comments (3)
  1. [§4] §4 (finite-T correlation functions): the central claim that TLL behavior persists across a wide parameter range at finite temperature rests on fits of the correlation functions to the TLL form (power-law times exp(-r/ξ_T)). The manuscript does not specify the fitting interval in units of the rod diameter a or mean interparticle spacing, nor does it report χ² values or sensitivity to the cutoff; without this, it is unclear whether non-universal corrections from the hard-core diameter bias the extracted K and ξ_T.
  2. [§3] §3 (thermodynamic comparison): while excellent agreement between Yang-Yang and PIMC is stated, the text provides neither statistical error bars on the Monte Carlo data nor the precise density-temperature window in which deviations from Lieb-Liniger become statistically significant. This information is necessary to assess whether the reported deviations are robust or merely within numerical uncertainty.
  3. [§5] §5 (thermometry application): the proposal that K and ξ_T extracted from TLL fits constitute a robust thermometry method assumes that the thermal length remains the dominant scale and that finite-size or irrelevant-operator corrections are negligible. A quantitative test (e.g., comparison of fitted ξ_T against the independently computed thermal length from Yang-Yang thermodynamics) is missing and would directly test the assumption.
minor comments (2)
  1. [Figures 4-6] Figure captions should explicitly state the fitting ranges and the value of the rod diameter a used in each panel.
  2. [Introduction] Notation for the hard-rod diameter (a) versus the Lieb-Liniger interaction strength should be introduced once and used consistently.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the major comments below and have incorporated revisions to enhance the manuscript's clarity and rigor.

read point-by-point responses

  1. Referee: [§4] §4 (finite-T correlation functions): the central claim that TLL behavior persists across a wide parameter range at finite temperature rests on fits of the correlation functions to the TLL form (power-law times exp(-r/ξ_T)). The manuscript does not specify the fitting interval in units of the rod diameter a or mean interparticle spacing, nor does it report χ² values or sensitivity to the cutoff; without this, it is unclear whether non-universal corrections from the hard-core diameter bias the extracted K and ξ_T.

    Authors: We agree that additional details on the fitting procedure will strengthen the presentation. In the revised manuscript we will explicitly report the fitting intervals in units of both the rod diameter a and the mean interparticle spacing, include the corresponding χ² values, and discuss the sensitivity of the extracted K and ξ_T to the choice of cutoff. We will also add a brief analysis showing that the fits are performed in the regime r ≫ a, where universal TLL behavior is expected to dominate over non-universal corrections arising from the finite rod diameter. revision: yes

  2. Referee: [§3] §3 (thermodynamic comparison): while excellent agreement between Yang-Yang and PIMC is stated, the text provides neither statistical error bars on the Monte Carlo data nor the precise density-temperature window in which deviations from Lieb-Liniger become statistically significant. This information is necessary to assess whether the reported deviations are robust or merely within numerical uncertainty.

    Authors: We thank the referee for this observation. The revised manuscript will include statistical error bars on all PIMC data points for the thermodynamic quantities. We will also delineate the density-temperature window in which the deviations from the Lieb-Liniger equation of state exceed the Monte Carlo uncertainties, thereby demonstrating that the reported differences are statistically significant rather than numerical artifacts. revision: yes

  3. Referee: [§5] §5 (thermometry application): the proposal that K and ξ_T extracted from TLL fits constitute a robust thermometry method assumes that the thermal length remains the dominant scale and that finite-size or irrelevant-operator corrections are negligible. A quantitative test (e.g., comparison of fitted ξ_T against the independently computed thermal length from Yang-Yang thermodynamics) is missing and would directly test the assumption.

    Authors: We agree that a direct quantitative test would provide stronger support for the proposed thermometry protocol. In the revised manuscript we will add a comparison of the thermal length ξ_T obtained from TLL fits to the correlation functions against the independently computed thermal length derived from the Yang-Yang thermodynamic equations, performed across a range of densities and temperatures. This will directly test the dominance of the thermal length scale and the negligibility of finite-size or irrelevant-operator corrections within the fitting windows. revision: yes

Circularity Check

1 steps flagged

Fitting correlation functions to TLL theory to extract K and thermal length reduces the thermometry claim to the assumed functional form

specific steps
  1. fitted input called prediction [Abstract (and corresponding results section on correlation functions)]
    "The Tomonaga-Luttinger parameter and thermal length can be extracted by fitting correlation functions to Tomonaga-Luttinger liquid theory, hence demonstrating a robust method for thermometry."

    K and the thermal length are parameters internal to TLL theory; fitting the computed correlators to the TLL functional form (power-law times exp(-r/ξ_T)) directly yields those parameters by construction. The 'demonstration' that the hard-rod model exhibits TLL behavior and that this enables thermometry is therefore equivalent to the success of the fit rather than an independent prediction or first-principles derivation.

full rationale

The paper's demonstration of TLL behavior at finite temperature and the proposed thermometry method rest on fitting density or one-body correlation functions to the TLL power-law decay with exponential thermal cutoff. Because the TLL parameter K and thermal length are defined inside the very functional form being fitted, the extraction step is statistically forced once a fitting window is chosen; the claim that this constitutes independent evidence for TLL behavior or a robust thermometer therefore reduces to the quality of the fit itself. Yang-Yang thermodynamics for the equation of state is an external input and does not participate in this loop, but the central finite-T correlation analysis does. Monte Carlo data supply the raw correlators, yet the mapping from those data to TLL parameters and the subsequent interpretation remain internal to the TLL ansatz. This produces partial circularity (score 6) without rendering the entire thermodynamics section circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review limits visibility into specific free parameters or invented entities; the work relies on standard assumptions of Yang-Yang Bethe-ansatz thermodynamics and Luttinger liquid phenomenology applied to hard rods.

axioms (2)
  • domain assumption Yang-Yang thermodynamic Bethe ansatz applies directly to the quantum hard rod model with finite diameter
    Used to obtain analytical thermodynamic quantities
  • domain assumption The system remains in the Tomonaga-Luttinger liquid regime at finite temperature and varying densities
    Required for fitting correlation functions and extracting parameters

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