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arxiv: 2605.30663 · v1 · pith:XLGGADFSnew · submitted 2026-05-28 · 🧮 math-ph · math.MP

Hubbard--Heisenberg Thermodynamic Comparison at Half Filling in a Fixed Staggered Field

Tadahiro Miyao This is my paper

Pith reviewed 2026-06-29 00:05 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords Hubbard modelHeisenberg modelstrong couplinghalf fillingstaggered fieldpressure estimatesmagnetizationcharge suppression
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The pith

The Hubbard model at half filling has pressures and magnetizations comparable to the Heisenberg model with errors uniform in system size for bounded staggered fields.

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

The paper proves that in the strong-coupling regime, the finite-volume pressures of the repulsive Hubbard model at half filling are close to those of the Heisenberg model, with the difference bounded independently of the lattice size, provided the staggered field is not too large and the temperature is low on the Heisenberg scale. These estimates carry over to the thermodynamic limit when the limits exist. Magnetization comparisons then follow from the convexity of the pressures for fixed positive field values. Charge fluctuations are also shown to be suppressed. This provides a rigorous justification for viewing the Hubbard model as an effective antiferromagnetic spin model at the relevant energy scale.

Core claim

Uniformly for |h| ≤ h0 and βJ0(U) ≥ ℓ0, the finite-volume Hubbard-Heisenberg pressure estimates hold with errors uniform in the system size. These pass to thermodynamic limits whenever the limiting pressures exist. For fixed positive staggered-field windows I ⊂ (0,h0], the magnetisation comparison is derived from the pressure comparison by convexity of finite-volume pressures. Charge-sector suppression estimates show that the density of empty or doubly occupied sites and the double-occupancy density are small, and the squared staggered charge divided by |Λ|^2 is small.

What carries the argument

A strong-coupling unitary transformation which separates the singly occupied spin sector from sectors containing empty or doubly occupied sites, with the remaining sectors controlled through a decomposition of the transformed partition function according to the set of empty or doubly occupied sites.

Load-bearing premise

The strong-coupling unitary transformation cleanly isolates the singly occupied spin sector so that the remaining sectors can be bounded using the decomposition by empty or doubly occupied sites.

What would settle it

For a specific finite lattice and parameters satisfying the conditions, numerically compute the pressure difference between the Hubbard and Heisenberg models and check if it exceeds the error bound claimed.

read the original abstract

We study the repulsive Hubbard model at half filling in the strong-coupling regime, with a staggered magnetic field of strength \(h\). The analysis is carried out in the canonical half-filled ensemble, with temperature measured on the Heisenberg scale \(J_0(U)=4t^2/U\). Uniformly for \(|h|\le h_0\) and \(\beta J_0(U)\ge \ell_0\), we prove finite-volume Hubbard--Heisenberg pressure estimates with errors uniform in the system size. These estimates pass to thermodynamic limits whenever the limiting pressures exist. The proof uses a strong-coupling unitary transformation which separates the singly occupied spin sector from sectors containing empty or doubly occupied sites. On the singly occupied sector, the effective Hamiltonian is compared with the Heisenberg reference Hamiltonian; the remaining sectors are controlled through a decomposition of the transformed partition function according to the set of empty or doubly occupied sites. For fixed positive staggered-field windows \(I\Subset(0,h_0]\), the magnetisation comparison is then derived from the pressure comparison by convexity of finite-volume pressures. We also prove charge-sector suppression estimates, uniformly for \(|h|\le h_0\): in the large positive-\(U\) Heisenberg-scale regime, the density of empty or doubly occupied sites and the double-occupancy density are small, and the squared staggered charge divided by \(|\Lambda|^2\) is small. Thus the results give a quantitative Gibbs-state formulation of the strong-coupling picture in which the half-filled repulsive Hubbard model is described, at the Heisenberg scale, by effective antiferromagnetic spin degrees of freedom, while charge fluctuations are suppressed.

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

0 major / 2 minor

Summary. The paper establishes finite-volume pressure estimates comparing the repulsive Hubbard model at half filling (with staggered field h) to the Heisenberg antiferromagnet in the strong-coupling regime (temperature scaled by J0(U)=4t²/U). Uniformly for |h|≤h0 and βJ0(U)≥ℓ0, the estimates have errors independent of system size and pass to thermodynamic limits when the limits exist. Magnetization comparison for fixed positive h-windows follows from convexity of the finite-volume pressures. The proof relies on a strong-coupling unitary transformation isolating the singly-occupied sector, followed by decomposition of the transformed partition function over charge-defect configurations; charge-suppression bounds are also obtained.

Significance. If the central estimates hold, the work supplies a rigorous, volume-uniform quantitative formulation of the strong-coupling mapping from half-filled Hubbard to Heisenberg antiferromagnet, including explicit control on charge fluctuations. The direct derivation from model definitions via unitary transformation and convexity, without fitted parameters, strengthens the result; conditional passage to the thermodynamic limit is a standard and appropriate formulation.

minor comments (2)
  1. The abstract states the uniformity conditions |h|≤h0 and βJ0(U)≥ℓ0 but does not indicate how h0 and ℓ0 are chosen or whether they depend on the lattice dimension; a brief remark in the introduction would clarify the range of applicability.
  2. Notation for the finite-volume pressure p_Λ and the reference Heisenberg pressure is introduced without an explicit equation number in the abstract; cross-referencing the first displayed equation in §2 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment, including the recommendation to accept. The referee's summary correctly identifies the key contributions: the uniform finite-volume pressure estimates comparing the half-filled repulsive Hubbard model (with staggered field) to the Heisenberg antiferromagnet in the strong-coupling regime, the derivation of magnetization comparisons via convexity, and the charge-sector suppression bounds.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation consists of direct mathematical estimates on the Hubbard model at half filling: a strong-coupling unitary transformation isolates the singly-occupied sector for comparison to the Heisenberg Hamiltonian, followed by decomposition of the partition function over charge sectors and application of convexity to obtain magnetization bounds from pressure bounds. These steps are constructed from the model definitions and standard analytic tools (unitary equivalence, decomposition, convexity) without any fitted parameters renamed as predictions, without load-bearing self-citations, and without ansatzes or uniqueness claims imported from prior author work. The results are conditional on existence of thermodynamic limits and are uniform in volume by explicit controls, rendering the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions of lattice statistical mechanics without introducing new free parameters or postulated entities.

axioms (2)
  • domain assumption Thermodynamic limits of the pressures exist whenever taken
    Explicitly invoked when stating that the finite-volume estimates pass to the thermodynamic limit.
  • standard math Convexity of finite-volume pressures with respect to the staggered field
    Used to obtain magnetization comparison from pressure comparison.

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discussion (0)

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Reference graph

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