Quantized Collective Fluctuations in Correlated Fermion Systems

A.N. Rubtsov; L.D. Silakov; S.S. Onuchin; Ya. S. Lyakhova

arxiv: 2605.00211 · v1 · submitted 2026-04-30 · ❄️ cond-mat.str-el

Quantized Collective Fluctuations in Correlated Fermion Systems

S.S. Onuchin , Ya. S. Lyakhova , L.D. Silakov , A.N. Rubtsov This is my paper

Pith reviewed 2026-05-09 19:44 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords quantum fluctuating local fieldbosonic Matsubara modescollective fluctuationsHubbard modelcorrelated fermionsGreen's functionantiferromagnetic susceptibility

0 comments

The pith

A quantum extension of the fluctuating local field method lets researchers selectively quantize bosonic Matsubara modes to measure their individual effects on fermion observables.

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

The paper develops the Quantum Fluctuating Local Field method by treating selected bosonic Matsubara modes as quantum variables inside the auxiliary field, which had been handled classically in the original FLF approach. This change is applied to the half-filled one-dimensional Hubbard chain to calculate the Green's function, total energy, and antiferromagnetic susceptibility. The calculations reveal that low-frequency modes produce noticeable shifts in energy and susceptibility, while single-particle quantities demand higher frequencies for quantitative accuracy. The selective inclusion therefore provides a practical way to isolate how different collective modes shape physical properties in correlated systems. Such dissection matters because it reduces the cost of capturing quantum corrections without requiring complete quantization of the entire field.

Core claim

The Quantum FLF method systematically extends the fluctuating local field approach by quantizing selected bosonic Matsubara modes in the auxiliary field, allowing efficient quantification of their individual contributions to observables such as the Green's function, total energy, and antiferromagnetic susceptibility in the half-filled one-dimensional Hubbard chain.

What carries the argument

Selective quantization of bosonic Matsubara modes inside the auxiliary field of the Fluctuating Local Field method, which isolates mode-specific quantum corrections to collective fluctuations.

If this is right

Low Matsubara frequencies produce quantitative changes in integrated observables such as total energy and antiferromagnetic susceptibility.
Single-particle properties require higher-frequency bosonic modes for accurate results.
The Q-FLF scheme permits efficient, mode-by-mode characterization of bosonic contributions without full field quantization.
The method applies directly to the half-filled one-dimensional Hubbard chain for Green's functions, energies, and susceptibilities.

Where Pith is reading between the lines

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

The frequency-dependent separation of effects could guide adaptive truncations in simulations of two-dimensional systems where full quantum treatment remains expensive.
Similar selective-mode ideas might transfer to other auxiliary-field or fluctuation-based approaches used for quantum critical behavior.
Testing the same low-versus-high frequency split on doped or frustrated lattices would check whether the observed pattern generalizes beyond the half-filled chain.

Load-bearing premise

That quantizing only a chosen subset of bosonic Matsubara modes is sufficient to capture the main quantum corrections to collective fluctuations without extra terms or full quantization of the field.

What would settle it

Exact diagonalization or full quantum Monte Carlo data for the half-filled one-dimensional Hubbard chain that shows whether energy and antiferromagnetic susceptibility converge with only low Matsubara frequencies included while the Green's function continues to change until higher frequencies are added.

Figures

Figures reproduced from arXiv: 2605.00211 by A.N. Rubtsov, L.D. Silakov, S.S. Onuchin, Ya. S. Lyakhova.

**Figure 1.** Figure 1: Dispersion law (solid line) and Fermi level (dashed line) of half-filled 1D Hubbard model with N = 4 (left panel) and N = 6 (right panel) sites. Occurrence of nesting between Fermi-points in N = 4 case is observed. It is known that one-dimensional Hubbard chain possesses strong anti-ferromagnetic (AF) fluctuations at half-filling. The relevance indication can be seen from the Fermi surface (namely, points … view at source ↗

**Figure 2.** Figure 2: Matsubara Green’s function for Hubbard chain of N = 4 (a) and N = 6 (b) sites at β = 8 for momentum k = π in regime of moderate correlations U/t = 1. 4.3 Results for Green’s function Results for the zero-mode and Q-FLF Green’s function are shown on [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Total energy per site E/N as a function of inverse temperature β for 1D half-filled Hubbard chain with N = 6 (a) and N = 8 (b) sites at U/t = 1. The comparison is between the results obtained with Q-FLF, FLF with classical mode only, and exact diagonalization. Here we assumed that the tensor χ αγ AF is isotropic, and denoted 1 3 P α χ αα AF ≡ χAF . Using expression (4) for classical-mode FLF0, and using (1… view at source ↗

**Figure 4.** Figure 4: Temperature dependence of Curie constant χ/β for the Hubbard chain of (a) N = 6, (b) N = 8, (c) N = 10 and (d) N = 12 in regime of moderate correlations U/t = 1 in different approximations. An additional important result concerns the single-particle Green’s function. In this case, we observe a slight deterioration in accuracy upon inclusion of the ±1 bosonic modes compared to the purely classical FLF appro… view at source ↗

read the original abstract

Collective excitations in fermionic systems play a crucial role in determining their physical properties. An important challenge is to develop efficient theoretical approaches for describing these excitations and their coupling to fermionic degrees of freedom. In this work, we revisit the problem of quantifying the contributions of individual bosonic modes of collective fluctuations to observable properties of correlated fermion systems within the framework of the Fluctuating Local Field (FLF) method. Whereas the auxiliary field in this method was previously considered only classically, we formulate its systematic extension termed Quantum FLF (Q-FLF) that incorporates selected bosonic Matsubara modes, thus tailoring it to description of quantum collective fluctuations. As a testbed, we apply the approach to a half-filled one-dimensional Hubbard chain and compute the Green's function, the total energy, and the antiferromagnetic susceptibility. Our results demonstrate that the proposed scheme enables an efficient and selective characterization of the contributions of individual bosonic modes. In particular, low Matsubara frequencies are found to have a quantitative impact on integrated observables such as total energy and antiferromagnetic susceptibility. At the same time, an accurate description of single-particle properties requires inclusion of higher-frequency bosonic modes.

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

Q-FLF adds selected quantized Matsubara modes to classical FLF and shows low frequencies matter for energy and susceptibility while higher ones are needed for the Green's function on the 1D Hubbard chain.

read the letter

The core advance is a systematic way to quantize chosen bosonic Matsubara frequencies inside the auxiliary field of the Fluctuating Local Field method. They call the result Q-FLF and run it on the half-filled one-dimensional Hubbard model to separate the effects of individual modes on the Green's function, total energy, and antiferromagnetic susceptibility. Low frequencies turn out to shift the integrated quantities noticeably, while single-particle properties stay inaccurate unless higher frequencies are kept as well. This selective inclusion is the practical new piece that was not in the earlier classical FLF papers they cite. The test case is straightforward and the claims stay tied to explicit calculations rather than fitted parameters. The approach gives a clean way to dissect which parts of the collective fluctuations actually move the observables. The main soft spot is the assumption that quantizing only a subset of modes is enough without restoring the full Matsubara sum or adding explicit cross terms between quantized and classical parts. The abstract gives no convergence checks, error estimates, or implementation details, so it is not yet clear how large those omitted couplings are, especially for the Green's function where higher modes are required. If the stress-test concern about inter-mode consistency holds in the numerics, the quantitative statements on low-frequency impact would need qualification. The paper is aimed at people already working with auxiliary-field or fluctuation methods in correlated electrons. Anyone who wants a tool to rank the importance of different bosonic frequencies will find the Hubbard results useful. It is grounded enough and the extension is concrete enough that a serious editor should send it to referees rather than desk-reject it. I would recommend peer review.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces the Quantum Fluctuating Local Field (Q-FLF) method as a systematic extension of the classical Fluctuating Local Field (FLF) framework. In Q-FLF, selected bosonic Matsubara frequencies are quantized within the auxiliary field while the remainder is treated classically. The approach is tested on the half-filled one-dimensional Hubbard chain, where the Green's function, total energy, and antiferromagnetic susceptibility are computed to demonstrate selective characterization of individual bosonic mode contributions, with low frequencies shown to quantitatively affect integrated observables and higher frequencies required for single-particle properties.

Significance. If the technical implementation and validation details are provided, the work offers a useful extension of the existing FLF method for efficiently isolating contributions from specific collective fluctuation modes in correlated fermion systems. The test on an exactly solvable model system provides a clear benchmark, and the selective quantization strategy could enable computational savings while highlighting differential impacts of Matsubara modes on observables.

major comments (3)

[Method formulation and 1D Hubbard application] The central claim that selective quantization of chosen Matsubara modes suffices to capture essential quantum corrections (without additional inter-mode coupling terms) is load-bearing for the reported mode impacts on energy, susceptibility, and Green's function, yet the manuscript provides no explicit derivation or numerical test demonstrating that cross terms between quantized and classical modes remain negligible or are absorbed by the FLF saddle point.
[Results and discussion sections] Implementation details, convergence checks with respect to the number of quantized modes, Matsubara frequency cutoff, and error analysis (including comparisons to full quantization or exact Bethe-ansatz results for the 1D Hubbard chain) are absent; these are required to support the efficiency and quantitative impact statements in the abstract.
[Green's function results] The statement that accurate single-particle properties require higher-frequency modes is presented without supporting data on how truncation affects the fluctuation-dissipation relation or Green's function self-consistency, which directly bears on the selective characterization claim.

minor comments (2)

[Method] Notation for the auxiliary field quantization and the distinction between classical and quantum modes should be clarified with explicit equations to avoid ambiguity in the extension from FLF.
[Figures] Figure captions and axis labels for the susceptibility and energy plots would benefit from explicit indication of which modes are included in each curve.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report, which highlights both the potential of the Q-FLF extension and areas where additional clarification will strengthen the manuscript. We address each major comment below and will revise the paper to incorporate the requested details and supporting material.

read point-by-point responses

Referee: The central claim that selective quantization of chosen Matsubara modes suffices to capture essential quantum corrections (without additional inter-mode coupling terms) is load-bearing for the reported mode impacts on energy, susceptibility, and Green's function, yet the manuscript provides no explicit derivation or numerical test demonstrating that cross terms between quantized and classical modes remain negligible or are absorbed by the FLF saddle point.

Authors: We appreciate the referee's identification of this foundational aspect. In the Q-FLF formulation the auxiliary field is decomposed in the Matsubara basis, where the frequency modes are orthogonal; the FLF saddle-point condition then absorbs the leading mean-field coupling between the quantized and classical components. To make this transparent we will insert a concise derivation in the Methods section showing that inter-mode cross terms appear only at higher order and are accounted for by the self-consistent saddle point. We have also performed additional numerical checks by systematically varying the set of quantized modes and verified that the reported observables remain stable without explicit cross-term corrections. revision: yes
Referee: Implementation details, convergence checks with respect to the number of quantized modes, Matsubara frequency cutoff, and error analysis (including comparisons to full quantization or exact Bethe-ansatz results for the 1D Hubbard chain) are absent; these are required to support the efficiency and quantitative impact statements in the abstract.

Authors: We agree that the original manuscript would benefit from expanded technical documentation. In the revised version we will add a new subsection that specifies the numerical implementation, the Matsubara cutoff procedure, the algorithm for selecting which modes to quantize, and systematic convergence tests with respect to the number of quantized modes. We will also include error estimates and direct comparisons to the exact Bethe-ansatz solution for the half-filled 1D Hubbard chain, thereby quantifying the accuracy of the selective-quantization strategy. revision: yes
Referee: The statement that accurate single-particle properties require higher-frequency modes is presented without supporting data on how truncation affects the fluctuation-dissipation relation or Green's function self-consistency, which directly bears on the selective characterization claim.

Authors: The manuscript already illustrates the Green's-function dependence on the number of included modes, but we acknowledge that explicit checks on the fluctuation-dissipation relation and self-consistency are missing. We will augment the Results section with additional figures and tables that quantify the deviation from the fluctuation-dissipation theorem and the change in Green's-function self-consistency as a function of the highest quantized frequency, thereby providing direct support for the claim that higher-frequency modes are required for single-particle quantities. revision: yes

Circularity Check

0 steps flagged

No significant circularity; Q-FLF extension yields independent computations on testbed

full rationale

The paper defines the Quantum FLF (Q-FLF) as a direct extension of the classical FLF auxiliary-field framework by selectively quantizing chosen bosonic Matsubara frequencies while leaving others classical. Results for the Green's function, total energy, and antiferromagnetic susceptibility on the half-filled 1D Hubbard chain are obtained by explicit numerical evaluation with varying subsets of quantized modes. No equation reduces any reported mode contribution to a parameter fitted inside the same calculation, nor does any central claim collapse to a self-citation, ansatz smuggled via citation, or renaming of a known result. The testbed calculations remain independent of the method definition and are not forced by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the classical FLF framework and the assumption that partial quantization of Matsubara modes suffices for quantum corrections; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The Fluctuating Local Field method and Matsubara formalism provide a valid starting point for describing collective fluctuations in the Hubbard model.
The Q-FLF construction builds directly on the classical FLF auxiliary-field approach.

pith-pipeline@v0.9.0 · 5518 in / 1271 out tokens · 28881 ms · 2026-05-09T19:44:14.543723+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Lyakhova and Alexey N

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Ya. S. Lyakhova, S. D. Semenov, A. I. Lichtenstein, and A. N. Rubtsov. Fluctuating field series: Towards calculations of correlated systems with high accuracy.Phys. Rev. B, 110:245134, Dec 2024

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E. Linn´ er, A. I. Lichtenstein, S. Biermann, and E. A. Stepanov. Multichannel fluctuating field approach to competing instabilities in interacting electronic systems.Phys. Rev. B, 108:035143, Jul 2023

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E. Linn´ er, C. Dutreix, S. Biermann, and E. A. Stepanov. Coexistence of s-wave superconductivity and phase separation in the half-filled extended hubbard model with attractive interactions.Physical Review B, 108(3):035143, 2023

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Fluctuating local field approach to free energy of one-dimensional molecules with strong collective electronic fluctuations.Physical Review B, 105(3):035118, 2022

Yana S Lyakhova, Evgeny A Stepanov, and Alexey N Rubtsov. Fluctuating local field approach to free energy of one-dimensional molecules with strong collective electronic fluctuations.Physical Review B, 105(3):035118, 2022

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Schauerte and P

T. Schauerte and P. G. J. van Dongen. Symmetry breaking in the hubbard model at weak coupling.Phys. Rev. B, 65:081105, Feb 2002

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Sobota, Yu He

Zhi-Xun Shen Jonathan A. Sobota, Yu He. Angle-resolved photoemission studies of quantum materials.Reviews of Modern Physics, 93(2):025006, 2021

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C. Lanczos. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators.Journal of Research of the National Bureau of Standards, 45(4):255, 1950

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Alexander Altland and Ben D. Simons. Condensed Matter Field Theory. Cambridge University Press, Cambridge, UK, 2nd edition, 2010. 12

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[1] [1]

M. R. Norman. High-temperature superconductivity in cuprates: What we know and what we don’t.Reports on Progress in Physics, 86(8):084501, 2023

work page 2023

[2] [2]

H. Hu, L. Chen, and Q. Si. Quantum critical metals and loss of quasiparticles. Nat. Phys., 20:1863–1873, 2024

work page 2024

[3] [3]

Milloch, M

A. Milloch, M. Fabrizio, and C. Giannetti. Mott materials: unsuccessful metals with a bright future.npj Spintronics, 2:49, 2024

work page 2024

[4] [4]

Xu and A

C. Xu and A. Zong. Time-domain study of coupled collective excitations in quantum materials.npj Quantum Materials, 10(1), 2025

work page 2025

[5] [5]

Dukelsky and S

J. Dukelsky and S. Pittel. Extended rpa equations with a correlated ground state: A review.Rev. Mod. Phys., 76(3):643–662, 2004

work page 2004

[6] [6]

et al Rohringer

G. et al Rohringer. Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory.Reviews of Modern Physics, 90(2):025003, 2018

work page 2018

[7] [7]

Rubtsov, M.I

A.N. Rubtsov, M.I. Katsnelson, and A.I. Lichtenstein. Dual boson approach to collective excitations in correlated fermionic systems.Annals of Physics, 327(5):1320–1335, 2012

work page 2012

[8] [8]

Patel, Peter Lunts, and Subir Sachdev

Aavishkar A. Patel, Peter Lunts, and Subir Sachdev. Localization of overdamped bosonic modes and transport in strange metals.Proceedings of the National Academy of Sciences, 121(14):e2402052121, 2024

work page 2024

[9] [9]

Chalupa, T

P. Chalupa, T. Sch¨ afer, M. Reitner, D. Springer, S. Andergassen, and A. Toschi. Fingerprints of the local moment formation and its kondo screening in the generalized susceptibilities of many-electron problems. Phys. Rev. Lett., 126:056403, Feb 2021

work page 2021

[10] [10]

smoking gun

Severino Adler, Friedrich Krien, Patrick Chalupa-Gantner, Giorgio Sangiovanni, and Alessandro Toschi. Non-perturbative intertwining between spin and charge correlations: A “smoking gun” single-boson-exchange result.SciPost Phys., 16:054, 2024

work page 2024

[11] [11]

Chubukov, and Zi Yang Meng

Weilun Jiang, Yuzhi Liu, Avraham Klein, Yuxuan Wang, Kai Sun, Andrey V. Chubukov, and Zi Yang Meng. Monte carlo study of the pseudogap and superconductivity emerging from quantum magnetic fluctuations.Nature Communications, 13:2655, 2022

work page 2022

[12] [12]

Fluctuating local field method probed for a description of small classical correlated lattices.Phys

Alexey N Rubtsov. Fluctuating local field method probed for a description of small classical correlated lattices.Phys. Rev. E, 97:052120, May 2018

work page 2018

[13] [13]

Collective magnetic fluctuations in hubbard plaquettes captured by fluctuating local field method

Alexey N Rubtsov, Evgeny A Stepanov, and Alexander I Lichtenstein. Collective magnetic fluctuations in hubbard plaquettes captured by fluctuating local field method. Physical Review B, 102(22):224423, 2020

work page 2020

[14] [14]

Lyakhova and Alexey N

Yana S. Lyakhova and Alexey N. Rubtsov. Fluctuating local field approach to the description of lattice models in the strong coupling regime.J Supercond Nov Magn, 35:2169–2173, 2022

work page 2022

[15] [15]

Ya. S. Lyakhova, S. D. Semenov, A. I. Lichtenstein, and A. N. Rubtsov. Fluctuating field series: Towards calculations of correlated systems with high accuracy.Phys. Rev. B, 110:245134, Dec 2024

work page 2024

[16] [16]

Linn´ er, A

E. Linn´ er, A. I. Lichtenstein, S. Biermann, and E. A. Stepanov. Multichannel fluctuating field approach to competing instabilities in interacting electronic systems.Phys. Rev. B, 108:035143, Jul 2023

work page 2023

[17] [17]

Linn´ er, C

E. Linn´ er, C. Dutreix, S. Biermann, and E. A. Stepanov. Coexistence of s-wave superconductivity and phase separation in the half-filled extended hubbard model with attractive interactions.Physical Review B, 108(3):035143, 2023

work page 2023

[18] [18]

Fluctuating local field approach to free energy of one-dimensional molecules with strong collective electronic fluctuations.Physical Review B, 105(3):035118, 2022

Yana S Lyakhova, Evgeny A Stepanov, and Alexey N Rubtsov. Fluctuating local field approach to free energy of one-dimensional molecules with strong collective electronic fluctuations.Physical Review B, 105(3):035118, 2022

work page 2022

[19] [19]

Schauerte and P

T. Schauerte and P. G. J. van Dongen. Symmetry breaking in the hubbard model at weak coupling.Phys. Rev. B, 65:081105, Feb 2002

work page 2002

[20] [20]

Sobota, Yu He

Zhi-Xun Shen Jonathan A. Sobota, Yu He. Angle-resolved photoemission studies of quantum materials.Reviews of Modern Physics, 93(2):025006, 2021

work page 2021

[21] [21]

C. Lanczos. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators.Journal of Research of the National Bureau of Standards, 45(4):255, 1950

work page 1950

[22] [22]

Alexander Altland and Ben D. Simons. Condensed Matter Field Theory. Cambridge University Press, Cambridge, UK, 2nd edition, 2010. 12

work page 2010