Nonpertubative Many-Body Theory for the Two-Dimensional Hubbard Model at Low Temperature: From Weak to Strong Coupling Regimes

Dingping Li; Huaqing Huang; Hui Li; Junnian Xiong; Ruitao Xiao; Yingze Su

arxiv: 2503.12468 · v3 · submitted 2025-03-16 · ❄️ cond-mat.str-el · physics.atom-ph· physics.comp-ph

Nonpertubative Many-Body Theory for the Two-Dimensional Hubbard Model at Low Temperature: From Weak to Strong Coupling Regimes

Ruitao Xiao , Yingze Su , Junnian Xiong , Hui Li , Huaqing Huang , Dingping Li This is my paper

Pith reviewed 2026-05-23 00:27 UTC · model grok-4.3

classification ❄️ cond-mat.str-el physics.atom-phphysics.comp-ph

keywords 2D Hubbard modelMermin-Wagner theoremGW approximationsymmetrization schemecorrelation functionsmany-body theoryquantum Monte Carlostrong coupling

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The pith

A symmetrization scheme that averages over symmetry-breaking states preserves the Mermin-Wagner theorem in many-body calculations of the two-dimensional Hubbard model.

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

The paper introduces a symmetrization scheme to address the issue of pseudo phase transitions in theoretical studies of two-dimensional systems, where many-body methods often predict finite-temperature symmetry breaking forbidden by the Mermin-Wagner theorem. By averaging physical quantities over all symmetry-breaking states, the scheme ensures compliance with the theorem while allowing calculations in the intermediate-to-strong coupling regime at low temperatures. Applied to the GW-covariance method for the half-filled repulsive Hubbard model, it produces one-body Green's functions and spin-spin correlation functions that match Determinant Quantum Monte Carlo results. The approach maintains the fluctuation-dissipation relation and Ward-Takahashi identities, and suggests the chi-sum rule as a check for method reliability.

Core claim

The symmetrization scheme, defined as averaging physical quantities over all symmetry-breaking states, ensures preservation of the Mermin-Wagner theorem. When combined with the GW-covariance calculation for the 2D repulsive Hubbard model at half-filling, it yields accurate one-body Green's functions and spin-spin correlation functions in the intermediate-to-strong coupling and low-temperature regime, in good agreement with DQMC benchmarks, while satisfying the fluctuation-dissipation relation and Ward-Takahashi identities.

What carries the argument

The symmetrization scheme, which averages physical quantities over all symmetry-breaking states to prevent violation of the Mermin-Wagner theorem in approximate many-body calculations.

If this is right

The symmetrized GW-covariance method can be used to study the doped regime of the 2D Hubbard model.
The chi-sum rule serves as a probe for the reliability of many-body methods when FDR and WTI are satisfied.
The framework is applicable to investigating strong-coupling regimes relevant to high-Tc cuprate superconductors.
Spin-spin correlation functions calculated via covariance theory preserve the fundamental relations from the Pauli exclusion principle.

Where Pith is reading between the lines

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

This symmetrization approach could be adapted to other 2D lattice models with continuous symmetries to avoid artificial phase transitions.
The method might resolve inconsistencies in other approximation schemes for low-dimensional strongly correlated systems.
Extending the covariance theory with symmetrization could provide new insights into the applicability of mean-field approaches in 2D.

Load-bearing premise

Averaging physical quantities over all symmetry-breaking states preserves the Mermin-Wagner theorem without distorting the underlying physics of the Hubbard model.

What would settle it

A significant disagreement between the symmetrized GW-covariance results and DQMC data on the spin-spin correlation function at low temperatures, or the emergence of a spurious finite-temperature phase transition, would indicate the scheme does not work as claimed.

read the original abstract

In theoretical studies of two-dimensional (2D) systems, the Mermin-Wagner theorem prevents continuous symmetry breaking at any finite temperature, thus forbidding a Landau phase transition at a critical temperature $T_c$. The difficulty arises when many-body theoretical studies predict a Landau phase transition at finite temperatures, which contradicts the Mermin-Wagner theorem and is termed a pseudo phase transition. To tackle this problem, we systematically develop a symmetrization scheme, defined as averaging physical quantities over all symmetry-breaking states, thus ensuring that it preserves the Mermin-Wagner theorem. We apply the symmetrization scheme to the GW-covariance calculation for the 2D repulsive Hubbard model at half-filling in the intermediate-to-strong coupling regime and at low temperatures, obtaining the one-body Green's function and spin-spin correlation function, and benchmark them against Determinant Quantum Monte Carlo (DQMC) with good agreement.The spin-spin correlation functions are approached within the covariance theory, a general method for calculating two-body correlation functions from a one-particle starting point, such as the GW formalism used here, which ensures the preservation of the fundamental fluctuation-dissipation relation (FDR) and Ward-Takahashi identities (WTI). With the FDR and WTI satisfied, we conjecture that the $\chi$-sum rule, a fundamental relation from the Pauli exclusion principle, can be used to probe the reliability of many-body methods, and demonstrate this by comparing the GW-covariance and mean-field-covariance approaches. This work provides a novel framework to investigate the strong-coupling and doped regime of the 2D Hubbard model, which is believed to be applicable to real high-$T_c$ cuprate superconductors.

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

Symmetrization in GW-covariance gives decent DQMC matches for the 2D Hubbard model but the Mermin-Wagner enforcement step still needs verification.

read the letter

The main takeaway is that this work applies a symmetrization scheme—averaging physical quantities over symmetry-breaking states—to GW-covariance calculations on the half-filled 2D repulsive Hubbard model at low temperatures. The goal is to eliminate pseudo phase transitions while producing one-body Green's functions and spin-spin correlations that line up with DQMC data in the intermediate-to-strong coupling range. They also keep the fluctuation-dissipation relation and Ward-Takahashi identities through the covariance approach for two-body functions and float the chi-sum rule as a reliability diagnostic. That combination is the concrete new piece. The benchmarks against independent DQMC runs are the clearest strength; external checks reduce the risk that the results are just internal fitting. The covariance framework itself is a reasonable way to build two-body quantities from a one-particle starting point without obvious contradictions. The soft spot sits in the symmetrization step itself. The abstract defines it as averaging over all symmetry-breaking states to preserve Mermin-Wagner, but it is not obvious from the description whether the averaging is folded into the self-consistency loop or applied afterward to states that already selected a preferred direction. If the latter, residual effective ordering could survive and the DQMC agreement might be partly coincidental. The chi-sum rule is labeled a conjecture, so that diagnostic carries less weight until it is tested more broadly. This paper is for people already working on numerical many-body methods for the 2D Hubbard model or similar systems where symmetry constraints matter. A reader who needs practical fixes for common approximation failures in two dimensions will get something usable. It is worth sending to serious referees because the problem it targets is real, the external benchmarks are present, and the technical choices are traceable even if the central enforcement claim requires closer inspection of the equations.

Referee Report

2 major / 2 minor

Summary. The paper develops a symmetrization scheme that averages physical quantities over all symmetry-breaking states to ensure compliance with the Mermin-Wagner theorem and avoid pseudo phase transitions in 2D systems. It applies this scheme within a GW-covariance framework to compute the one-body Green's function and spin-spin correlation functions for the half-filled 2D repulsive Hubbard model in the intermediate-to-strong coupling regime at low temperatures. Results are benchmarked against DQMC data with reported good agreement. The approach invokes the fluctuation-dissipation relation and Ward-Takahashi identities, and conjectures that the χ-sum rule serves as a reliability diagnostic for many-body methods, illustrated by comparison to mean-field covariance.

Significance. If the symmetrization procedure is shown to be internally consistent with Mermin-Wagner compliance and the DQMC benchmarks are robust, the framework offers a route to controlled calculations of correlation functions in the strong-coupling, low-T regime of the 2D Hubbard model without artificial finite-T ordering. The explicit use of covariance theory to enforce FDR and WTI, together with the external DQMC benchmark, constitutes a concrete strength. The χ-sum-rule conjecture, if validated, could provide a useful internal consistency check for other diagrammatic methods.

major comments (2)

[symmetrization scheme description (abstract and § on method)] The central claim that the symmetrization scheme (averaging over symmetry-breaking states) rigorously preserves the Mermin-Wagner theorem is asserted in the abstract and method description but lacks an explicit derivation or numerical test showing that long-range order is eliminated at any T>0 rather than restored post hoc. If the underlying GW self-consistency selects a preferred broken-symmetry direction before averaging, or if the ensemble of states is incomplete, the resulting Green's function and spin correlations could still encode an effective finite-T transition, undermining both the theorem preservation and the interpretation of the DQMC benchmark agreement.
[discussion of χ-sum rule] The conjecture that the χ-sum rule can diagnose reliability once FDR and WTI are satisfied is introduced without a derivation linking it directly to the Pauli principle in the presence of the symmetrization procedure. No table or figure quantifies how the sum-rule violation correlates with deviation from DQMC data across coupling strengths or temperatures, leaving the diagnostic status of the rule unverified.

minor comments (2)

[title] The title contains the spelling 'Nonpertubative'; correct to 'Nonperturbative'.
[abstract] The abstract states 'good agreement' with DQMC but provides no quantitative metrics (e.g., maximum relative error on G(k,ω) or integrated spin correlation); a table of error measures would strengthen the benchmark claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the work and the detailed, constructive comments. We respond to each major comment below. Where the comments identify areas that would benefit from additional clarification or supporting material, we agree to revise the manuscript accordingly.

read point-by-point responses

Referee: [symmetrization scheme description (abstract and § on method)] The central claim that the symmetrization scheme (averaging over symmetry-breaking states) rigorously preserves the Mermin-Wagner theorem is asserted in the abstract and method description but lacks an explicit derivation or numerical test showing that long-range order is eliminated at any T>0 rather than restored post hoc. If the underlying GW self-consistency selects a preferred broken-symmetry direction before averaging, or if the ensemble of states is incomplete, the resulting Green's function and spin correlations could still encode an effective finite-T transition, undermining both the theorem preservation and the interpretation of the DQMC benchmark agreement.

Authors: The symmetrization is performed after obtaining the broken-symmetry solutions from GW self-consistency for each direction of the order parameter; the final observables are obtained by explicit averaging over a complete set of directions in the continuous symmetry group. This construction ensures that any preferred direction is eliminated and that the averaged quantities remain invariant, thereby satisfying the Mermin-Wagner theorem at any finite T. We acknowledge that the manuscript would be strengthened by an explicit step-by-step derivation of this property together with a supplementary numerical demonstration that the order parameter vanishes for T>0. We will add both to the methods section in the revised version. revision: yes
Referee: [discussion of χ-sum rule] The conjecture that the χ-sum rule can diagnose reliability once FDR and WTI are satisfied is introduced without a derivation linking it directly to the Pauli principle in the presence of the symmetrization procedure. No table or figure quantifies how the sum-rule violation correlates with deviation from DQMC data across coupling strengths or temperatures, leaving the diagnostic status of the rule unverified.

Authors: Because the covariance framework already enforces the fluctuation-dissipation relation and Ward-Takahashi identities, the χ-sum rule (which follows directly from the Pauli principle applied to the two-particle density matrix) provides an internal consistency diagnostic. Since symmetrization is a linear averaging operation performed after the covariance construction, it preserves the sum rule whenever the rule holds for the individual symmetry-broken states. We agree that an explicit derivation of this preservation and a quantitative comparison of sum-rule violations versus DQMC deviations would make the diagnostic more convincing. We will include both the derivation and a new figure (or table) showing the correlation across U and T in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central results benchmarked externally against DQMC.

full rationale

The paper defines a symmetrization scheme via averaging over symmetry-breaking states and asserts MW preservation from that construction, but the load-bearing outputs (Green's functions, spin correlations) are obtained via GW-covariance and directly compared to independent DQMC data with reported agreement. The FDR/WTI preservation and χ-sum-rule conjecture are presented as external checks rather than self-referential derivations that reduce the main numerical claims to tautologies. No load-bearing step reduces by the paper's own equations to a fitted input or self-citation chain; external benchmarks keep the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Mermin-Wagner theorem, the fluctuation-dissipation relation, and the Ward-Takahashi identities as background facts; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Mermin-Wagner theorem prevents continuous symmetry breaking at finite temperature in 2D
Invoked to define the pseudo-transition problem that the symmetrization must solve
standard math Covariance theory preserves fluctuation-dissipation relation and Ward-Takahashi identities
Stated as the reason the χ-sum rule can be used as a reliability probe

pith-pipeline@v0.9.0 · 5865 in / 1576 out tokens · 58676 ms · 2026-05-23T00:27:58.012200+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

we systematically develop a symmetrization scheme, defined as averaging physical quantities over all symmetry-breaking states, thus ensuring that it preserves the Mermin-Wagner theorem
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

GW-covariance equations... covariance theory... FDR and WTI

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