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arxiv: 2604.02769 · v1 · submitted 2026-04-03 · ❄️ cond-mat.str-el

Semiclassical representation of the Hubbard model

Yuki Yamasaki , Hidemaro Suwa , Cristian D. Batista , Shintaro Hoshino This is my paper

Pith reviewed 2026-05-13 18:21 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Hubbard modelsemiclassical approximationcoherent statespath integralstrongly correlated electronsfinite temperatureintersite correlationsdouble occupancy
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The pith

Semiclassical coherent-state method qualitatively reproduces exact Hubbard model results for small clusters.

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

The paper introduces a semiclassical approximation to the Hubbard model by revisiting its path-integral formulation and using an unconventional coherent-state representation in which a subset of dynamical variables is held fixed. This yields a nonperturbative scheme that operates at finite temperature and includes intersite correlations while remaining extendable to multiorbital systems. Validation against exact solutions on one- and two-site clusters shows that the method reproduces the qualitative dependence of particle number, double occupancy, hopping amplitude, and spin correlations. Quantitative discrepancies are traced to the continuum character of the density of states employed. An exact transformation associated with the coherent-state construction is also derived.

Core claim

Treating a subset of the dynamical variables as static within the semiclassical coherent-state representation produces a nonperturbative approximation to the Hubbard model that qualitatively matches exact results for particle number, double occupancy, hopping amplitude, and spin correlations on one- and two-site systems at finite temperature.

What carries the argument

Semiclassical coherent-state representation in which a subset of dynamical variables is treated as static

If this is right

  • The scheme applies directly to finite-temperature calculations of the Hubbard model.
  • Intersite correlations are incorporated without perturbative expansion.
  • Natural extension to multiorbital systems is possible within the same framework.
  • The derived exact transformation clarifies the representation of the Hubbard interaction in coherent states.

Where Pith is reading between the lines

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

  • The method could serve as a starting point for larger-cluster or lattice calculations where exact diagonalization becomes intractable.
  • Systematic improvement might be obtained by restoring dynamics to the currently static variables in a controlled way.
  • The continuum-density-of-states limitation suggests testing the same approximation on discretized spectra to isolate its source of quantitative error.

Load-bearing premise

Treating a subset of dynamical variables as static remains valid when the underlying density of states is taken as continuum rather than discretized.

What would settle it

Exact diagonalization results for double occupancy or nearest-neighbor spin correlation on a three-site Hubbard cluster at finite temperature compared against the semiclassical approximation would reveal whether the qualitative agreement persists or degrades with increasing cluster size.

Figures

Figures reproduced from arXiv: 2604.02769 by Cristian D. Batista, Hidemaro Suwa, Shintaro Hoshino, Yuki Yamasaki.

Figure 1
Figure 1. Figure 1: FIG. 1. Overview on the represenations of the Hubbard model [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Top row) Filling versus chemcal potential [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Top row) Temperature dependence of the two-site spin correlation [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Temperature depnendece of the spin correlation for [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Top row) Filling dependence of the two-site spin correlation at several temperatures, obtained by (a,b) the semiclassical [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
read the original abstract

By revisiting the path-integral formulation of the Hubbard model, we propose a theoretical approach based on a semiclassical approximation employing an unconventional coherent-state representation. Within this framework, a subset of the dynamical variables is treated as static, yielding a nonperturbative scheme that is applicable at finite temperature, incorporates intersite correlations, and can be naturally extended to multiorbital systems. We assess the validity of the approximation by comparing its results with exact solutions for one- and two-site systems, focusing in particular on the particle number, double occupancy, hopping amplitude, and spin correlations, and find that the present approach qualitatively reproduces the exact behavior. Quantitatively, deviations arise, which is associated with the continuum (non-discretized) character of the underlying density of states. Furthermore, we derive the exact transformation associated with the coherent-state construction, thereby providing additional insight into the representation of the Hubbard model.

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

1 major / 1 minor

Summary. The manuscript proposes a semiclassical approximation to the Hubbard model derived from its path-integral formulation using an unconventional coherent-state representation. A subset of dynamical variables is treated as static, resulting in a nonperturbative scheme suitable for finite temperatures that incorporates intersite correlations and can be extended to multiorbital systems. The approach is validated through comparisons with exact solutions for one- and two-site clusters, demonstrating qualitative agreement in quantities such as particle number, double occupancy, hopping amplitude, and spin correlations, with quantitative deviations linked to the use of a continuum density of states. Additionally, the exact transformation associated with the coherent-state construction is derived.

Significance. If the static approximation remains controlled for extended systems under the continuum DOS, the method could provide a useful nonperturbative route to finite-temperature properties of the Hubbard model that includes intersite correlations without perturbative expansions. The extensibility to multiorbital cases and the exact transformation derivation are additional strengths that could facilitate broader applications in strongly correlated systems.

major comments (1)
  1. [comparison to exact solutions] The central validation (abstract and results on small clusters) shows qualitative agreement with exact diagonalization for 1- and 2-site systems, but attributes quantitative deviations explicitly to the continuum (non-discretized) DOS. No separate test or error analysis is provided to confirm that the static treatment of dynamical variables remains valid once the DOS is forced to continuum form, which is the regime required for the intended extension to larger lattices. This makes the static approximation the load-bearing step whose justification is incomplete for the target applications.
minor comments (1)
  1. [Introduction] The abstract refers to an 'unconventional coherent-state representation' without a concise statement of its difference from standard coherent states; adding one sentence in the introduction would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the method's potential, and the constructive major comment. We address the point below and will revise the manuscript to strengthen the justification of the static approximation.

read point-by-point responses

  1. Referee: The central validation (abstract and results on small clusters) shows qualitative agreement with exact diagonalization for 1- and 2-site systems, but attributes quantitative deviations explicitly to the continuum (non-discretized) DOS. No separate test or error analysis is provided to confirm that the static treatment of dynamical variables remains valid once the DOS is forced to continuum form, which is the regime required for the intended extension to larger lattices. This makes the static approximation the load-bearing step whose justification is incomplete for the target applications.

    Authors: We appreciate the referee's emphasis on this key aspect of the validation. The one- and two-site benchmarks are performed with the continuum DOS inside the semiclassical approximation, while the exact diagonalization uses the discrete spectrum; the qualitative agreement in particle number, double occupancy, hopping amplitude, and spin correlations is therefore obtained under the continuum-DOS conditions relevant to larger lattices. This already indicates that the static treatment does not produce uncontrolled qualitative errors. Quantitative deviations are, as stated in the manuscript, linked to the continuum DOS rather than the static approximation. To make this separation explicit, we will add a short paragraph in the revised discussion section that recalls the path-integral origin of the static approximation, notes that it becomes exact in the high-temperature and infinite-dimensional limits (where the DOS is continuum-like), and briefly quantifies the expected error using the exact coherent-state transformation already derived in the paper. These additions will clarify the justification without new numerical tests. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from path integral with external validation

full rationale

The paper starts from the standard path-integral formulation of the Hubbard model and introduces a semiclassical coherent-state representation by holding a subset of dynamical variables static. This approximation is assessed through direct comparison to exact diagonalization results on one- and two-site clusters for particle number, double occupancy, hopping, and spin correlations. Deviations are explicitly linked to the continuum DOS rather than any fitted parameters or self-referential definitions. An exact transformation for the coherent-state construction is also derived independently. No load-bearing step reduces by construction to inputs, self-citations, or renamed known results; the central claims rest on explicit benchmarks outside the approximation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, additional axioms, or invented entities are identified; the central addition is the semiclassical treatment itself.

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Reference graph

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