Hamiltonian-based dimensional reduction and spectral reconstruction with Wilson-Dirac fermions

B. Hind; P. V. Buividovich

arxiv: 2606.20090 · v1 · pith:ZDMAN5KBnew · submitted 2026-06-18 · ✦ hep-lat

Hamiltonian-based dimensional reduction and spectral reconstruction with Wilson-Dirac fermions

P. V. Buividovich , B. Hind This is my paper

Pith reviewed 2026-06-26 15:01 UTC · model grok-4.3

classification ✦ hep-lat

keywords Wilson-Dirac fermionsdimensional reductionspectral reconstructionanisotropic latticeslattice QCDfermionic determinantpropagatorEuclidean time evolution

0 comments

The pith

Four-dimensional clover-improved Wilson-Dirac fermions on anisotropic lattices reduce exactly to a three-dimensional Hamiltonian at finite temporal spacing.

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

This paper derives explicit formulas for the fermionic determinant and the propagator of four-dimensional clover-improved Wilson-Dirac fermions on anisotropic lattices using only the three-dimensional Wilson-Dirac Hamiltonian. It introduces an effective Hamiltonian that controls the evolution in Euclidean time even when the time spacing is not small. These results provide a way to measure and correct for lattice artifacts when reconstructing real-time spectral functions from Euclidean-time correlators in lattice QCD.

Core claim

We derive explicit expressions for the fermionic determinant and the propagator of the four-dimensional clover-improved Wilson-Dirac fermions on anisotropic lattices in terms of the three-dimensional Wilson-Dirac Hamiltonian operator. We derive an effective Hamiltonian that governs Euclidean time evolution at finite temporal lattice spacing, and demonstrate its hermiticity and particle-anti-particle symmetry. Our results allow to quantify lattice artifacts of the numerical spectral reconstruction based on Euclidean fermionic correlators at finite temporal lattice spacing.

What carries the argument

The effective three-dimensional Wilson-Dirac Hamiltonian operator that governs Euclidean time evolution of the fermion fields at finite temporal lattice spacing.

If this is right

The fermionic determinant and propagator become computable from three-dimensional operators alone.
The effective Hamiltonian remains Hermitian and respects particle-anti-particle symmetry at finite temporal spacing.
Lattice artifacts in spectral-function reconstruction can be quantified without taking the temporal spacing to zero.
Euclidean correlators at finite temporal spacing can be related directly to the spectrum via the effective Hamiltonian.

Where Pith is reading between the lines

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

The reduction may allow simulations that evolve fermions in a hybrid three-dimensional plus separate time-step manner, reducing memory requirements.
Similar exact reductions could be explored for other improved fermion actions or for gauge-field operators.
The method supplies a controlled way to extrapolate spectral functions to the continuum temporal spacing without additional model assumptions.
Testing the expressions on quenched configurations would provide a concrete check before full dynamical simulations.

Load-bearing premise

The four-dimensional clover-improved Wilson-Dirac operator on anisotropic lattices admits an exact dimensional reduction to expressions involving only the three-dimensional Hamiltonian at finite temporal lattice spacing, without further approximations.

What would settle it

Direct numerical evaluation of the fermionic determinant from both the full four-dimensional operator and the reduced three-dimensional expressions on the same small anisotropic lattice would falsify the claim if the two results disagree beyond floating-point precision.

read the original abstract

Motivated by the process of reconstructing real-time spectral functions from Euclidean correlators in lattice QCD, we derive explicit expressions for the fermionic determinant and the propagator of the four-dimensional clover-improved Wilson-Dirac fermions on anisotropic lattices in terms of the three-dimensional Wilson-Dirac Hamiltonian operator. We derive an effective Hamiltonian that governs Euclidean time evolution at finite temporal lattice spacing, and demonstrate its hermiticity and particle-anti-particle symmetry. Our results allow to quantify lattice artifacts of the numerical spectral reconstruction based on Euclidean fermionic correlators at finite temporal lattice spacing.

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

The paper supplies explicit algebraic 4D-to-3D reductions for the clover Wilson-Dirac determinant and propagator on anisotropic lattices plus a finite-spacing effective Hamiltonian with verified hermiticity and symmetry.

read the letter

The main takeaway is that this paper works out explicit formulas expressing the four-dimensional clover-improved Wilson-Dirac determinant and propagator on anisotropic lattices in terms of the three-dimensional Wilson-Dirac Hamiltonian, together with an effective operator that governs Euclidean time evolution at finite temporal spacing. They also verify that this effective Hamiltonian is Hermitian and respects particle-antiparticle symmetry.

What stands out is the algebraic exactness of the reduction at finite lattice spacing. No further approximations are introduced, which directly addresses the need to quantify discretization artifacts when reconstructing spectral functions from Euclidean fermionic correlators. The derivations start from the standard Wilson-Dirac operator and keep the clover term, so the results are concrete and checkable.

The limitation is that everything stays formal. The abstract and claims give no numerical tests, no comparison to known limits, and no example of how the formulas change an actual spectral reconstruction. The scope is also narrow: anisotropic lattices only, fermions only. Broader impact on isotropic or gauge-field cases would require extra work.

This is for lattice QCD people already working on real-time observables who need tighter control over temporal artifacts. A reader who knows the Wilson operator and Hamiltonian formulations will follow the steps without trouble.

It deserves peer review. The central results are derivations that can be verified line by line, and the motivation matches a real methodological gap even if the paper itself stops at the formulas.

Referee Report

0 major / 2 minor

Summary. The paper derives explicit expressions for the fermionic determinant and the propagator of four-dimensional clover-improved Wilson-Dirac fermions on anisotropic lattices in terms of the three-dimensional Wilson-Dirac Hamiltonian operator. It constructs an effective Hamiltonian governing Euclidean time evolution at finite temporal lattice spacing and demonstrates its hermiticity and particle-antiparticle symmetry. The results are motivated by and intended to quantify lattice artifacts in spectral reconstruction from Euclidean fermionic correlators.

Significance. If the algebraic derivations hold, the work supplies an exact (non-approximate) framework for analyzing finite-a_t effects in the fermionic sector of anisotropic lattice QCD. The explicit expressions and the proof of hermiticity plus particle-antiparticle symmetry constitute a concrete technical contribution that can be used directly in spectral-function studies; this is a strength for a derivation-style manuscript.

minor comments (2)

[Abstract] The abstract states that explicit expressions are derived but does not preview their functional form; adding one or two key equations (or their structure) in the abstract or introduction would improve immediate readability.
A short numerical illustration or consistency check of the effective Hamiltonian (e.g., a small-volume test of hermiticity) would help readers verify the algebraic results in practice.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation to accept. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraic and self-contained

full rationale

The paper's central claim is an explicit algebraic derivation of 4D fermionic determinant/propagator expressions and an effective Euclidean-time Hamiltonian directly from the clover-improved Wilson-Dirac operator on anisotropic lattices. No load-bearing steps reduce to fitted inputs, self-citations, or self-definitional loops; the reduction is presented as exact by construction from the standard lattice operator. The work is therefore self-contained against external benchmarks (the Wilson-Dirac operator itself) with no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; the contribution is presented as a re-expression of the standard Wilson-Dirac operator.

pith-pipeline@v0.9.1-grok · 5615 in / 1107 out tokens · 32249 ms · 2026-06-26T15:01:18.829511+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 6 canonical work pages · 5 internal anchors

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work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.82.094027 2010
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QCD at non-zero density and canonical partition functions with Wilson fermions

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work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.83.034502 2011
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P. V. Buividovich, B. Hind,Spectral reconstruction based on dimensional reduction in high- temperature gauge theories(2025), ArXiv:2512.23560

arXiv 2025
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M. A. Zubkov,Momentum space topology of QCD, Annals of Physics393(2018), 264–287, ArXiv:1610.08041

Pith/arXiv arXiv 2018
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M. N. Chernodub, M. Zubkov,Scale magnetic effect in quantum electrodynamics and the Wigner-Weyl formalism, Physical Review D96(2017) (5), 056006, ArXiv:1703.06516

Pith/arXiv arXiv 2017
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C.Zhang,M.Zubkov,InfluenceofinteractionsontheanomalousquantumHalleffect,Journal of Physics A: Mathematical and Theoretical53(2020) (19), 195002, ArXiv:1902.06545

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J.Kogut,L.Susskind,HamiltonianformulationofWilson’slatticegaugetheories,Phys.Rev.D 11(1975), 395.http://dx.doi.org/10.1103/PhysRevD.11.395

work page doi:10.1103/physrevd.11.395 1975
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P. V. Buividovich, M. I. Polikarpov,Monte-Carlo study of the electron transport properties of monolayer graphene within the tight-binding model, Phys. Rev. B86(2012), 245117, ArXiv:1206.0619.http://dx.doi.org/10.1103/PhysRevB.86.245117

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevb.86.245117 2012
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Wenger,Transfer matrices and temporal factorization of the Wilson fermion determinant, PoSLATTICE2022(2023), 042, ArXiv:2302.07385

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arXiv 2023
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Futamura, S

Y. Futamura, S. Hashimoto, A. Imakura, K. Nagata, T. Sakurai,A filtering technique for the temporally reduced matrix of the Wilson fermion determinant, PoSLATTICE2014(2014), 049, ArXiv:1411.4262

Pith/arXiv arXiv 2014
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Bilgici, J

E. Bilgici, J. Danzer, C. Gattringer, C. B. Lang, L. Liptak,Canonical fermion determinants in lattice QCD: Numerical evaluation and properties, Phys. Lett. B697(2011) (1), 85–89, ArXiv:0906.1088

Pith/arXiv arXiv 2011
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Heavy Quarks on Anisotropic Lattices: The Charmonium Spectrum

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work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.64 2001
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G.Aarts,C.Allton,A.Amato,P.Giudice,S.Hands,J.Skullerud,Electricalconductivityand charge diffusion in thermal QCD from the lattice, JHEP02(2015), 186, ArXiv:1412.6411. http://dx.doi.org/10.1007/JHEP02(2015)186

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep02(2015)186 2015
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Datta, S

S. Datta, S. Gupta, A. Lytle,Using Wilson flow to study the SU(3) deconfinement transition, Phys. Rev. D94(2016) (9), 094502, ArXiv:1512.04892

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M.Ulybyshev,S.Zafeiropoulos,C.Winterowd,F.Assaad,Bridgingthegapbetweennumerics and experiment in free standing graphene(2021), ArXiv:2104.09655

arXiv 2021
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F. A. Bresciani, M. Bruno, M. T. Hansen,Finite-volume effects on smeared spectral densities (2026), ArXiv:2606.14349. 11

arXiv 2026

[1] [1]

K.Nagata,A.Nakamura,WilsonfermiondeterminantinlatticeQCD,Phys.Rev.D82(2010), 094027, ArXiv:1009.2149.http://dx.doi.org/10.1103/PhysRevD.82.094027

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.82.094027 2010

[2] [2]

QCD at non-zero density and canonical partition functions with Wilson fermions

A. Alexandru, U. Wenger,QCD at non-zero density and canonical partition functions with Wilsonfermions,Phys.Rev.D83(2011),034502,ArXiv:1009.2197.http://dx.doi.org/ 10.1103/PhysRevD.83.034502 9 Hamiltonian dimensional reduction with Wilson-Dirac fermionsP. Buividovich

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.83.034502 2011

[3] [3]

P. V. Buividovich, B. Hind,Spectral reconstruction based on dimensional reduction in high- temperature gauge theories(2025), ArXiv:2512.23560

arXiv 2025

[4] [4]

M. A. Zubkov,Momentum space topology of QCD, Annals of Physics393(2018), 264–287, ArXiv:1610.08041

Pith/arXiv arXiv 2018

[5] [5]

M. N. Chernodub, M. Zubkov,Scale magnetic effect in quantum electrodynamics and the Wigner-Weyl formalism, Physical Review D96(2017) (5), 056006, ArXiv:1703.06516

Pith/arXiv arXiv 2017

[6] [6]

C.Zhang,M.Zubkov,InfluenceofinteractionsontheanomalousquantumHalleffect,Journal of Physics A: Mathematical and Theoretical53(2020) (19), 195002, ArXiv:1902.06545

arXiv 2020

[7] [7]

J.Kogut,L.Susskind,HamiltonianformulationofWilson’slatticegaugetheories,Phys.Rev.D 11(1975), 395.http://dx.doi.org/10.1103/PhysRevD.11.395

work page doi:10.1103/physrevd.11.395 1975

[8] [8]

Blankenbecler, D

R. Blankenbecler, D. J. Scalapino, R. L. Sugar,Monte Carlo calculations of coupled boson-fermion systems. I, Phys. Rev. D24(1981), 2278.http://dx.doi.org/10.1103/ PhysRevD.24.2278

1981

[9] [9]

P. V. Buividovich, M. I. Polikarpov,Monte-Carlo study of the electron transport properties of monolayer graphene within the tight-binding model, Phys. Rev. B86(2012), 245117, ArXiv:1206.0619.http://dx.doi.org/10.1103/PhysRevB.86.245117

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevb.86.245117 2012

[10] [10]

Wenger,Transfer matrices and temporal factorization of the Wilson fermion determinant, PoSLATTICE2022(2023), 042, ArXiv:2302.07385

U. Wenger,Transfer matrices and temporal factorization of the Wilson fermion determinant, PoSLATTICE2022(2023), 042, ArXiv:2302.07385

arXiv 2023

[11] [11]

Futamura, S

Y. Futamura, S. Hashimoto, A. Imakura, K. Nagata, T. Sakurai,A filtering technique for the temporally reduced matrix of the Wilson fermion determinant, PoSLATTICE2014(2014), 049, ArXiv:1411.4262

Pith/arXiv arXiv 2014

[12] [12]

K.Nagata,A.Nakamura,S.Hashimoto,Apropertyoffermionsatfinitedensitybyareduction formula of fermion determinant, PoSLATTICE2013(2014), 207

2014

[13] [13]

Bilgici, J

E. Bilgici, J. Danzer, C. Gattringer, C. B. Lang, L. Liptak,Canonical fermion determinants in lattice QCD: Numerical evaluation and properties, Phys. Lett. B697(2011) (1), 85–89, ArXiv:0906.1088

Pith/arXiv arXiv 2011

[14] [14]

Heavy Quarks on Anisotropic Lattices: The Charmonium Spectrum

P. Chen,Heavy quarks on anisotropic lattices: The Charmonium spectrum, Phys. Rev. D64 (2001), 034509, ArXiv:hep-lat/0006019.https://dx.doi.org/10.1103/PhysRevD.64. 034509

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.64 2001

[15] [15]

Electrical conductivity and charge diffusion in thermal QCD from the lattice

G.Aarts,C.Allton,A.Amato,P.Giudice,S.Hands,J.Skullerud,Electricalconductivityand charge diffusion in thermal QCD from the lattice, JHEP02(2015), 186, ArXiv:1412.6411. http://dx.doi.org/10.1007/JHEP02(2015)186

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep02(2015)186 2015

[16] [16]

Rylund Glesaaen, B

J. Rylund Glesaaen, B. Jäger,openQCD-fastsum code(2018).https://gitlab.com/ fastsum/openqcd-fastsum 10 Hamiltonian dimensional reduction with Wilson-Dirac fermionsP. Buividovich

2018

[17] [17]

Datta, S

S. Datta, S. Gupta, A. Lytle,Using Wilson flow to study the SU(3) deconfinement transition, Phys. Rev. D94(2016) (9), 094502, ArXiv:1512.04892

Pith/arXiv arXiv 2016

[18] [18]

M.Ulybyshev,S.Zafeiropoulos,C.Winterowd,F.Assaad,Bridgingthegapbetweennumerics and experiment in free standing graphene(2021), ArXiv:2104.09655

arXiv 2021

[19] [19]

F. A. Bresciani, M. Bruno, M. T. Hansen,Finite-volume effects on smeared spectral densities (2026), ArXiv:2606.14349. 11

arXiv 2026