pith. sign in

arxiv: 2507.04041 · v3 · pith:BRMFFROXnew · submitted 2025-07-05 · ❄️ cond-mat.str-el

Specific heat and density anomaly in the Hubbard model

M. A. Habitzreuter , Willdauany C. de Freitas Silva , Eduardo O. Rizzatti , Thereza Paiva , Marcia C. Barbosa This is my paper

Pith reviewed 2026-05-19 06:07 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Hubbard modelspecific heatdensity anomalythermal expansionquantum Monte Carlostrong correlationsfilling dependenceSeebeck coefficient
0
0 comments X

The pith

Strong correlations in the Hubbard model produce three peaks in specific heat versus electron filling.

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

The paper investigates the specific heat of the Hubbard model as a function of filling using quantum Monte Carlo simulations on lattices interpolating between square and triangular geometries. It finds that at strong interactions the specific heat develops three maxima separated by local minima. These features arise from distinct kinetic and potential energy contributions to the specific heat. The work also identifies a related density anomaly visible in the thermal expansion coefficient, which connects to the sign change in the Seebeck coefficient. Such quantities are measurable in cold-atom experiments, offering new ways to probe correlations away from half filling.

Core claim

With strong correlations the specific heat as a function of filling shows a three-maxima structure with local minima between them. This structure is explained by separating the kinetic and potential contributions to the specific heat. Connected to this behavior is a density anomaly detected through the thermal expansion coefficient, which can be linked to the change of sign in the Seebeck coefficient.

What carries the argument

Determinant Quantum Monte Carlo simulations that decompose the specific heat into kinetic and potential parts while interpolating between square and triangular lattice geometries.

If this is right

  • The kinetic contribution in momentum space reveals the density anomaly.
  • The thermal expansion anomaly connects directly to the Seebeck coefficient sign change.
  • These effects occur away from half-filling and are accessible via cold-atom measurements at multiple temperatures.
  • The location of the phenomena can be mapped in the parameter space of filling, interaction, and temperature.

Where Pith is reading between the lines

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

  • Similar three-peak structures might appear in other strongly correlated models beyond the Hubbard Hamiltonian.
  • Experimental verification in cold atoms could guide searches for related anomalies in real materials.
  • The connection to thermoelectric properties suggests applications in understanding transport in correlated systems.

Load-bearing premise

Finite-lattice Determinant Quantum Monte Carlo simulations with the chosen lattice interpolation accurately reflect the thermodynamic limit behavior without dominant finite-size or sign-problem artifacts.

What would settle it

A direct measurement or simulation showing the absence of three distinct maxima in specific heat as a function of filling at strong coupling would falsify the central claim.

Figures

Figures reproduced from arXiv: 2507.04041 by Eduardo O. Rizzatti, M. A. Habitzreuter, Marcia C. Barbosa, Thereza Paiva, Willdauany C. de Freitas Silva.

Figure 1
Figure 1. Figure 1: FIG. 1. a) Specific heat at constant density [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Specific heat and kinetic/potential contributions for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Specific heat as a function of lattice filling [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The momentum distribution of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The gradient of the momentum distribution of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. a) Thermal expansion [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Entropy [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Location of [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Heat map of [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
read the original abstract

Understanding thermal properties of materials is fundamental to technological applications and to discovering new phenomena. In particular, advances in experimental techniques such as cold-atom measurements allow the simulation of paradigmatic Hamiltonians with great control over model parameters, such as the Hubbard model. One aspect of this model which is not much explored is the behavior of the specific heat as a function of density. In this work, we perform Determinant Quantum Monte Carlo simulations of the Hubbard model interpolating between the square and triangular lattices to analyze the specific heat as the filling, interaction, and temperature of the system are changed. We found that, with strong correlations, the specific heat presents a three-maxima structure as a function of filling, with local minima between them. This effect can be explained by a decomposition of kinetic and potential contributions to the specific heat, demonstrating interesting phenomena away from the commonly studied half-filling regime. Moreover, by analyzing the kinetic contribution in momentum space we show that, connected to this specific heat behavior, there is a density anomaly detected through the thermal expansion coefficient. These momentum-space quantities are accessible using cold-atom experiments measurements at multiple temperatures. Finally, we map the location of these phenomena and connect the thermal expansion anomaly with the well-known Seebeck coefficient change of sign. Our results provide a new perspective to analyze this change of sign.

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 performs Determinant Quantum Monte Carlo simulations of the Hubbard model on lattices that interpolate between square and triangular geometries. It reports that, for strong on-site repulsion U, the specific heat C as a function of filling n exhibits a three-maxima structure separated by local minima; this structure is decomposed into kinetic and potential contributions. The work further identifies a density anomaly through the thermal expansion coefficient and connects it to the sign change of the Seebeck coefficient.

Significance. If the numerical results are robust, the identification of a filling-dependent three-maxima specific-heat structure and the associated density anomaly would provide a useful perspective on correlation effects away from half-filling, with direct implications for cold-atom experiments that can access momentum-resolved quantities at multiple temperatures.

major comments (3)
  1. [Methods] § Methods (DQMC implementation): the manuscript does not report the average sign of the Monte Carlo weights as a function of filling n or temperature T for the interpolated geometries. Because the sign problem is expected to be severe away from n=1 once geometric frustration is introduced, it is unclear whether the reported three-maxima structure in C(n) and the thermal-expansion anomaly remain free of uncontrolled sampling errors.
  2. [Results] Results section on specific heat (figures showing C versus n): no error bars, no finite-size scaling, and no explicit thermodynamic-limit extrapolation are provided for the local minima between the three maxima. Specific heat extracted from energy fluctuations is particularly sensitive to both statistical noise and finite-size effects; without these checks the claimed structure cannot be considered established.
  3. [Results] Section discussing the thermal expansion coefficient: the coefficient is obtained from derivatives of density with respect to temperature at fixed pressure-like quantities. The manuscript should demonstrate that these derivatives remain numerically stable when the average sign is appreciably less than unity, as exponential cancellations can amplify errors in fluctuation-derived observables.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a brief statement of the lattice interpolation parameter and the range of fillings and temperatures explored.
  2. [Figures] Figure captions should explicitly state the system sizes used and whether the data are for a single size or extrapolated.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below and will incorporate revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Methods] § Methods (DQMC implementation): the manuscript does not report the average sign of the Monte Carlo weights as a function of filling n or temperature T for the interpolated geometries. Because the sign problem is expected to be severe away from n=1 once geometric frustration is introduced, it is unclear whether the reported three-maxima structure in C(n) and the thermal-expansion anomaly remain free of uncontrolled sampling errors.

    Authors: We agree that reporting the average sign is essential for evaluating the reliability of DQMC results, particularly away from half filling. In the revised manuscript we will add explicit data (plots or tabulated values) for the average sign versus filling n and temperature T across the interpolated geometries. For the parameter regime in which the three-maxima structure and density anomaly appear, the sign remains sufficiently close to unity that the reported features are not affected by uncontrolled sampling errors; we will state this explicitly and discuss the range of fillings and temperatures where the sign problem remains manageable. revision: yes

  2. Referee: [Results] Results section on specific heat (figures showing C versus n): no error bars, no finite-size scaling, and no explicit thermodynamic-limit extrapolation are provided for the local minima between the three maxima. Specific heat extracted from energy fluctuations is particularly sensitive to both statistical noise and finite-size effects; without these checks the claimed structure cannot be considered established.

    Authors: The referee is correct that error bars and finite-size analysis are necessary to establish the local minima. We will revise the figures to include statistical error bars on all specific-heat data. In addition, we have performed additional simulations on larger lattices and will include a finite-size scaling analysis showing that the locations and depths of the minima converge with increasing system size, thereby confirming that the three-maxima structure survives in the thermodynamic limit. revision: yes

  3. Referee: [Results] Section discussing the thermal expansion coefficient: the coefficient is obtained from derivatives of density with respect to temperature at fixed pressure-like quantities. The manuscript should demonstrate that these derivatives remain numerically stable when the average sign is appreciably less than unity, as exponential cancellations can amplify errors in fluctuation-derived observables.

    Authors: We acknowledge the importance of verifying numerical stability of the thermal-expansion coefficient when the average sign is reduced. In the revised manuscript we will add a supplementary analysis that examines the stability of the relevant derivatives by comparing results obtained from independent runs of different lengths and by monitoring the anomaly as the sign decreases. This will demonstrate that the reported density anomaly is robust and not an artifact of amplified statistical errors. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct outputs of standard DQMC simulations

full rationale

The paper obtains its central claims—the three-maxima specific-heat structure versus filling and the linked density anomaly via thermal expansion—by direct numerical simulation of the Hubbard Hamiltonian using Determinant Quantum Monte Carlo on finite lattices. No parameters are fitted to a subset of data and then relabeled as predictions, no quantities are defined in terms of each other, and no load-bearing steps reduce to self-citations or imported ansatzes. The reported features emerge from the computed energies, fluctuations, and derivatives without any self-referential construction that would make the outputs equivalent to the inputs by definition.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The work rests on the standard Hubbard Hamiltonian and the validity of the DQMC algorithm; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond conventional model parameters (U, t, filling, temperature) that are scanned rather than fitted to produce the reported features.

free parameters (2)
  • interaction strength U
    Scanned as a control parameter; the three-maxima feature appears only above a threshold value of U.
  • filling n
    Varied continuously; the reported structure is a function of n.
axioms (1)
  • domain assumption Determinant Quantum Monte Carlo provides unbiased estimates of thermodynamic quantities on finite lattices for the Hubbard model away from half-filling
    Invoked implicitly when interpreting the specific-heat maxima and thermal-expansion anomaly as physical.

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