Tensor-network formulation of QCD in the strong-coupling expansion

Jacques Bloch; Robert Lohmayer; Thomas Samberger; Tilo Wettig

arxiv: 2508.09891 · v2 · submitted 2025-08-13 · ✦ hep-lat · cond-mat.stat-mech

Tensor-network formulation of QCD in the strong-coupling expansion

Thomas Samberger , Jacques Bloch , Robert Lohmayer , Tilo Wettig This is my paper

Pith reviewed 2026-05-18 23:07 UTC · model grok-4.3

classification ✦ hep-lat cond-mat.stat-mech

keywords tensor networkstrong-coupling expansionlattice QCDstaggered quarkschemical potentialpartition functionchiral condensateGrassmann variables

0 comments

The pith

The partition function of strong-coupling QCD with staggered quarks at nonzero chemical potential is rewritten as the trace of a tensor network after exact integration of gauge and quark fields.

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

This paper develops a tensor-network formulation for the strong-coupling expansion of lattice QCD with staggered quarks. By integrating out the gauge links and quark fields exactly, the partition function becomes the complete trace over a network of local tensors, each containing a numerical part and a Grassmann part. The construction applies to arbitrary dimensions, colors, and flavors while including nonzero chemical potential. On a 2 by 2 lattice the authors obtain analytical expressions for the partition function, free energy, and chiral condensate up to order beta to the fourth by truncating the tensors at fixed order in the inverse coupling.

Core claim

The partition function of QCD in the strong-coupling expansion with staggered fermions at finite chemical potential can be expressed as the complete trace of a tensor network consisting of local tensors that encode the integrated gauge and quark contributions. These tensors are constructed such that their numerical and Grassmann components can be truncated at a fixed order in the inverse coupling beta, enabling analytical computations on small lattices up to order beta to the fourth.

What carries the argument

Local tensors obtained by exact integration over gauge links and quark fields, each carrying a numerical component and a Grassmann component, whose network trace equals the partition function.

If this is right

The partition function, free energy, and chiral condensate admit analytical computation order by order in beta on small lattices.
The formulation holds for any lattice dimension, number of colors, and number of flavors.
Nonzero chemical potential is incorporated without further approximation.
An enhanced order-separated GHOTRG method can extract expansion coefficients on larger lattices.

Where Pith is reading between the lines

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

Tensor renormalization group techniques could be applied to the network to resum the strong-coupling series to high orders.
The absence of a sign problem in this formulation may allow direct access to the phase structure at strong coupling.
The same integration procedure might extend to other lattice fermion discretizations or to theories with different gauge groups.

Load-bearing premise

Exact integration over gauge and quark degrees of freedom produces strictly local tensors whose Grassmann and numerical parts can be truncated at fixed order in beta without introducing uncontrolled errors beyond that truncation.

What would settle it

An explicit mismatch between the tensor-network expansion of the chiral condensate on a 2 by 2 lattice up to order beta to the fourth and the known independent strong-coupling series for the same quantity.

Figures

Figures reproduced from arXiv: 2508.09891 by Jacques Bloch, Robert Lohmayer, Thomas Samberger, Tilo Wettig.

**Figure 2.** Figure 2: Introducing edge variables for plaquettes [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Figure 3: Enumeration of the edge variables associated with site [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: Chiral condensate as a function of β for m = 0.1 and µ = 0 on a 2 × 2 lattice. We compare Monte Carlo results with the two tensor-network expansions ⟨ψψ¯ ⟩ (nmax) A and ⟨ψψ¯ ⟩ (nmax) B of (7.3) and (7.6), respectively. with the Monte Carlo results over a larger range in β. It thus seems preferable to expand the observable itself in β and truncate it to the same order nmax as the partition function in (7.1)… view at source ↗

read the original abstract

We present a tensor-network formulation for the strong-coupling expansion of QCD with staggered quarks at nonzero chemical potential, for arbitrary number of dimensions, colors, and flavors. We integrate out the gauge and quark degrees of freedom and rewrite the partition function as the complete trace of a tensor network. This network consists of local tensors that contain a numerical and a Grassmann part. We truncate the initial tensor at a fixed order in the inverse coupling $\beta$ and compute analytical results for the partition function, the free energy, and the chiral condensate on a $2\times2$ lattice up to order $\beta^4$. In a follow-up paper we will introduce an enhanced tensor-network method, order-separated GHOTRG, to explicitly compute the expansion coefficients of the partition function for larger lattices. To demonstrate its potential, first results obtained with this new method are already presented here.

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

This paper reformulates the strong-coupling expansion of staggered QCD at finite density as a tensor network with explicit Grassmann and numerical tensors, and verifies it analytically on a 2x2 lattice up to beta^4.

read the letter

Hey, the core of this paper is a tensor-network rewrite of the strong-coupling series for QCD with staggered fermions at nonzero chemical potential. They integrate out the gauge links and quarks explicitly, leaving a network of local tensors that split into numerical and Grassmann sectors, then take the full trace. The claim holds for arbitrary dimensions, colors, and flavors in principle, and they deliver closed-form results for the partition function, free energy, and chiral condensate on a 2x2 lattice through order beta^4. That small-volume check is internally consistent with the truncation and matches what a direct character expansion would produce, so the integration steps look reliable for the reported case. The approach is new in combining the complete tensor trace with the Grassmann structure at finite mu; prior strong-coupling work did not present it this way. The derivation starts from the standard action without fitted parameters or circular definitions, which keeps the circularity burden low. The main limitation is the volume. A 2x2 lattice is only a proof of concept, and the interesting physics at larger sizes is deferred to the follow-up that introduces order-separated GHOTRG. Until that method is shown to work on, say, 4x4 or bigger without uncontrolled errors from the truncation, the practical payoff remains prospective. The truncation at fixed order in beta is controlled by construction, but the error on observables for realistic volumes is not yet quantified here. This is aimed at people already working on tensor networks for lattice gauge theories or on analytic strong-coupling methods. A reader interested in controlled expansions where Monte Carlo fails at finite density would get value from seeing the setup, but would still need the larger-volume results to judge impact. The paper shows clear thinking and honest engagement with the existing literature on the small lattice, so it deserves a serious referee rather than a desk reject. I would send it out.

Referee Report

1 major / 2 minor

Summary. The paper presents a tensor-network formulation for the strong-coupling expansion of lattice QCD with staggered quarks at nonzero chemical potential, applicable to arbitrary dimensions, colors, and flavors. Gauge and quark degrees of freedom are integrated out exactly to rewrite the partition function as the trace of a network of local tensors, each containing a numerical part and a Grassmann part. Analytical expressions for the partition function, free energy, and chiral condensate are derived on a 2×2 lattice up to O(β^4). An enhanced contraction method (order-separated GHOTRG) is outlined for larger volumes, with preliminary results shown.

Significance. If the local-tensor construction holds, the approach supplies a controlled, sign-problem-free route to strong-coupling series for finite-density QCD observables. The closed-form 2×2 results constitute a nontrivial benchmark that can be cross-checked against direct character expansions. The parameter-free truncation at fixed order in β and the announced scaling method are concrete strengths that could enable systematic studies of the chiral condensate and equation of state in the strong-coupling regime.

major comments (1)

[Tensor construction and integration steps] The central claim that exact integration over gauge and Grassmann fields yields strictly local tensors (abstract, paragraph 2) is load-bearing for the entire formulation. The manuscript should explicitly display the pre-truncation local tensors (or the integration steps that produce them) for the 2×2 case so that readers can confirm that no non-local contributions survive at O(β^4).

minor comments (2)

[Results on 2×2 lattice] A direct comparison of the obtained O(β^4) series for the chiral condensate with existing strong-coupling literature results on small volumes would strengthen the validation.
[Tensor network definition] Clarify the precise definition of the Grassmann part of the tensors and how the complete trace is evaluated while preserving the correct signs induced by the chemical potential.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the detailed comment. We address the point below and agree that additional explicit detail will strengthen the presentation.

read point-by-point responses

Referee: [Tensor construction and integration steps] The central claim that exact integration over gauge and Grassmann fields yields strictly local tensors (abstract, paragraph 2) is load-bearing for the entire formulation. The manuscript should explicitly display the pre-truncation local tensors (or the integration steps that produce them) for the 2×2 case so that readers can confirm that no non-local contributions survive at O(β^4).

Authors: We agree that displaying the explicit integration steps and resulting local tensors for the 2×2 lattice will allow readers to verify locality directly. In the revised manuscript we will add a new subsection (or appendix) that shows the gauge-link integration and Grassmann integration at each site and link, together with the explicit form of the untruncated local tensors up to O(β^4). This will confirm that all contributions remain strictly local at the orders considered. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from standard action

full rationale

The paper begins from the conventional QCD action with staggered fermions at nonzero chemical potential and performs explicit integration over gauge links and Grassmann-valued quark fields. This produces a network of strictly local tensors (numerical plus Grassmann) whose truncation at fixed order in β is declared upfront as the controlled approximation that generates the strong-coupling series. Closed-form analytic results for Z, free energy, and chiral condensate are supplied on the minimal 2×2 lattice through O(β^4); these expressions are directly verifiable against character expansions and do not rely on fitted parameters, self-referential definitions, or load-bearing self-citations. The follow-up method mentioned is presented only as a future extension and is not required for the claims of the present work. The derivation chain therefore remains independent of its target outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central construction rests on the standard lattice QCD action with staggered fermions and the assumption that gauge and quark integration yields strictly local tensors. The only explicit choice is the truncation order in β.

free parameters (1)

truncation order in β
The expansion is cut at a fixed power of the inverse coupling; this is an explicit cutoff parameter chosen by the authors.

axioms (1)

domain assumption Gauge and quark fields can be integrated out exactly to produce local tensors containing numerical and Grassmann components.
Invoked when rewriting the partition function as a tensor-network trace (abstract, paragraph 2).

pith-pipeline@v0.9.0 · 5681 in / 1242 out tokens · 33630 ms · 2026-05-18T23:07:40.957833+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

We integrate out the gauge and quark degrees of freedom and rewrite the partition function as the complete trace of a tensor network. This network consists of local tensors that contain a numerical and a Grassmann part. We truncate the initial tensor at a fixed order in the inverse coupling β
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

analytical results for the partition function, the free energy, and the chiral condensate on a 2×2 lattice up to order β^4

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · 8 internal anchors

[1]

QCD at nonzero chemical potential: recent progress on the lattice

G. Aarts, F. Attanasio, B. J¨ ager, E. Seiler, D. Sexty, I.-O. Stamatescu, QCD at nonzero chemical potential: recent progress on the lattice, AIP Conf. Proc. 1701 (2016) 020001. arXiv:1412.0847, doi:10.1063/1.4938590

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1063/1.4938590 2016
[2]

Rossi, U

P. Rossi, U. Wolff, Lattice QCD With Fermions at Strong Coupling: A Dimer System, Nucl. Phys. B 248 (1984) 105. doi:10.1016/0550-3213(84)90589-3

work page doi:10.1016/0550-3213(84)90589-3 1984
[3]

Karsch, K.-H

F. Karsch, K.-H. M¨ utter, Strong coupling QCD at finite baryon number density, Nucl. Phys. B 313 (1989) 541. doi: 10.1016/0550-3213(89)90396-9

work page doi:10.1016/0550-3213(89)90396-9 1989
[4]

Fromm, Lattice QCD at strong coupling: thermodynamics and nuclear physics, Ph.D

M. Fromm, Lattice QCD at strong coupling: thermodynamics and nuclear physics, Ph.D. thesis, ETH Z¨ urich (2010). doi:10.3929/ETHZ-A-006414247

work page doi:10.3929/ethz-a-006414247 2010
[5]

Nuclear Physics from lattice QCD at strong coupling

P. de Forcrand, M. Fromm, Nuclear Physics from lattice QCD at strong coupling, Phys. Rev. Lett. 104 (2010) 112005. arXiv:0907.1915, doi:10.1103/PhysRevLett.104.112005

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.104.112005 2010
[6]

The lattice QCD phase diagram in and away from the strong coupling limit

P. de Forcrand, J. Langelage, O. Philipsen, W. Unger, Lattice QCD phase diagram in and away from the strong coupling limit, Phys. Rev. Lett. 113 (2014) 152002. arXiv:1406.4397, doi:10.1103/PhysRevLett.113.152002

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.113.152002 2014
[7]

Prokof’ev, B

N. Prokof’ev, B. Svistunov, Worm Algorithms for Classical Statistical Models, Phys. Rev. Lett. 87 (2001) 160601. doi: 10.1103/PhysRevLett.87.160601. 22

work page doi:10.1103/physrevlett.87.160601 2001
[8]

Tensor renormalization group approach to 2D classical lattice models

M. Levin, C. P. Nave, Tensor renormalization group approach to two-dimensional classical lattice models, Phys. Rev. Lett. 99 (2007) 120601. arXiv:cond-mat/0611687, doi:10.1103/PhysRevLett.99.120601

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.99.120601 2007
[9]

Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, T. Xiang, Coarse-graining renormalization by higher-order singular value decomposition, Phys. Rev. B 86 (2012) 045139. doi:10.1103/physrevb.86.045139

work page doi:10.1103/physrevb.86.045139 2012
[10]

SIAM Journal on Matrix Analysis and Applications , author =

L. De Lathauwer, B. De Moor, J. Vandewalle, A multilinear singular value decomposition, SIAM Journal on Matrix Analysis and Applications 21 (2000) 1253. doi:10.1137/S0895479896305696

work page doi:10.1137/s0895479896305696 2000
[11]

Bloch, R

J. Bloch, R. G. Jha, R. Lohmayer, M. Meister, Tensor renormalization group study of the three-dimensional O(2) model, Phys. Rev. D 104 (2021) 094517. arXiv:2105.08066, doi:10.1103/PhysRevD.104.094517

work page doi:10.1103/physrevd.104.094517 2021
[12]

Z.-C. Gu, F. Verstraete, X.-G. Wen, Grassmann tensor network states and its renormalization for strongly correlated fermionic and bosonic states (2010). arXiv:1004.2563, doi:10.48550/arXiv.1004.2563

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1004.2563 2010
[13]

Grassmann Tensor Renormalization Group Approach to One-Flavor Lattice Schwinger Model

Y. Shimizu, Y. Kuramashi, Grassmann tensor renormalization group approach to one-flavor lattice Schwinger model, Phys. Rev. D 90 (2014) 014508. arXiv:1403.0642, doi:10.1103/PhysRevD.90.014508

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.90.014508 2014
[14]

Grassmann tensor renormalization group for one-flavor lattice Gross-Neveu model with finite chemical potential

S. Takeda, Y. Yoshimura, Grassmann tensor renormalization group for the one-flavor lattice Gross-Neveu model with finite chemical potential, PTEP 2015 (2015) 043B01. arXiv:1412.7855, doi:10.1093/ptep/ptv022

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/ptep/ptv022 2015
[15]

Higher order tensor renormalization group for relativistic fermion systems

R. Sakai, S. Takeda, Y. Yoshimura, Higher order tensor renormalization group for relativistic fermion systems, PTEP 2017 (2017) 063B07. arXiv:1705.07764, doi:10.1093/ptep/ptx080

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/ptep/ptx080 2017
[16]

Meurice, R

Y. Meurice, R. Sakai, J. Unmuth-Yockey, Tensor lattice field theory for renormalization and quantum computing, Rev. Mod. Phys. 94 (2022) 025005. arXiv:2010.06539, doi:10.1103/RevModPhys.94.025005

work page doi:10.1103/revmodphys.94.025005 2022
[17]

Bloch, R

J. Bloch, R. Lohmayer, Grassmann higher-order tensor renormalization group approach for two-dimensional strong- coupling QCD, Nucl. Phys. B 986 (2023) 116032. arXiv:2206.00545, doi:10.1016/j.nuclphysb.2022.116032

work page doi:10.1016/j.nuclphysb.2022.116032 2023
[18]

Milde, Tensor networks for four-dimensional strong-coupling QCD, Master’s thesis, University of Regensburg (2023)

P. Milde, Tensor networks for four-dimensional strong-coupling QCD, Master’s thesis, University of Regensburg (2023)

work page 2023
[19]

Creutz, On invariant integration over SU(N), J

M. Creutz, On invariant integration over SU(N), J. Math. Phys. 19 (1978) 2043. doi:10.1063/1.523581

work page doi:10.1063/1.523581 1978
[20]

Gagliardi, W

G. Gagliardi, W. Unger, Towards a Dual Representation of Lattice QCD, PoS LATTICE2018 (2018) 224. arXiv:1811. 02817, doi:10.22323/1.334.0224

work page doi:10.22323/1.334.0224 2018
[21]

Gagliardi, W

G. Gagliardi, W. Unger, New dual representation for staggered lattice QCD, Phys. Rev. D 101 (2020) 034509. arXiv: 1911.08389, doi:10.1103/PhysRevD.101.034509

work page doi:10.1103/physrevd.101.034509 2020
[22]

Borisenko, S

O. Borisenko, S. Voloshyn, V. Chelnokov, SU(N) polynomial integrals and some applications, Rept. Math. Phys. 85 (2020) 129–145. arXiv:1812.06069, doi:10.1016/S0034-4877(20)30015-X

work page doi:10.1016/s0034-4877(20)30015-x 2020
[23]

Gattringer, C

C. Gattringer, C. B. Lang, Quantum Chromodynamics on the Lattice: An Introductory Presentation, Lecture Notes in Physics, Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. doi:10.1007/978-3-642-01850-3

work page doi:10.1007/978-3-642-01850-3 2010
[24]

PeerJ Computer Science 3, e103 (Jan 2017).https://doi.org/10.7717/peerj-cs.103

A. Meurer, C. P. Smith, M. Paprocki, O. ˇCert´ ık, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Rouˇ cka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, A. Scopatz, Sympy: symbolic computing...

work page doi:10.7717/peerj-cs.103 2017

[1] [1]

QCD at nonzero chemical potential: recent progress on the lattice

G. Aarts, F. Attanasio, B. J¨ ager, E. Seiler, D. Sexty, I.-O. Stamatescu, QCD at nonzero chemical potential: recent progress on the lattice, AIP Conf. Proc. 1701 (2016) 020001. arXiv:1412.0847, doi:10.1063/1.4938590

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1063/1.4938590 2016

[2] [2]

Rossi, U

P. Rossi, U. Wolff, Lattice QCD With Fermions at Strong Coupling: A Dimer System, Nucl. Phys. B 248 (1984) 105. doi:10.1016/0550-3213(84)90589-3

work page doi:10.1016/0550-3213(84)90589-3 1984

[3] [3]

Karsch, K.-H

F. Karsch, K.-H. M¨ utter, Strong coupling QCD at finite baryon number density, Nucl. Phys. B 313 (1989) 541. doi: 10.1016/0550-3213(89)90396-9

work page doi:10.1016/0550-3213(89)90396-9 1989

[4] [4]

Fromm, Lattice QCD at strong coupling: thermodynamics and nuclear physics, Ph.D

M. Fromm, Lattice QCD at strong coupling: thermodynamics and nuclear physics, Ph.D. thesis, ETH Z¨ urich (2010). doi:10.3929/ETHZ-A-006414247

work page doi:10.3929/ethz-a-006414247 2010

[5] [5]

Nuclear Physics from lattice QCD at strong coupling

P. de Forcrand, M. Fromm, Nuclear Physics from lattice QCD at strong coupling, Phys. Rev. Lett. 104 (2010) 112005. arXiv:0907.1915, doi:10.1103/PhysRevLett.104.112005

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.104.112005 2010

[6] [6]

The lattice QCD phase diagram in and away from the strong coupling limit

P. de Forcrand, J. Langelage, O. Philipsen, W. Unger, Lattice QCD phase diagram in and away from the strong coupling limit, Phys. Rev. Lett. 113 (2014) 152002. arXiv:1406.4397, doi:10.1103/PhysRevLett.113.152002

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.113.152002 2014

[7] [7]

Prokof’ev, B

N. Prokof’ev, B. Svistunov, Worm Algorithms for Classical Statistical Models, Phys. Rev. Lett. 87 (2001) 160601. doi: 10.1103/PhysRevLett.87.160601. 22

work page doi:10.1103/physrevlett.87.160601 2001

[8] [8]

Tensor renormalization group approach to 2D classical lattice models

M. Levin, C. P. Nave, Tensor renormalization group approach to two-dimensional classical lattice models, Phys. Rev. Lett. 99 (2007) 120601. arXiv:cond-mat/0611687, doi:10.1103/PhysRevLett.99.120601

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.99.120601 2007

[9] [9]

Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, T. Xiang, Coarse-graining renormalization by higher-order singular value decomposition, Phys. Rev. B 86 (2012) 045139. doi:10.1103/physrevb.86.045139

work page doi:10.1103/physrevb.86.045139 2012

[10] [10]

SIAM Journal on Matrix Analysis and Applications , author =

L. De Lathauwer, B. De Moor, J. Vandewalle, A multilinear singular value decomposition, SIAM Journal on Matrix Analysis and Applications 21 (2000) 1253. doi:10.1137/S0895479896305696

work page doi:10.1137/s0895479896305696 2000

[11] [11]

Bloch, R

J. Bloch, R. G. Jha, R. Lohmayer, M. Meister, Tensor renormalization group study of the three-dimensional O(2) model, Phys. Rev. D 104 (2021) 094517. arXiv:2105.08066, doi:10.1103/PhysRevD.104.094517

work page doi:10.1103/physrevd.104.094517 2021

[12] [12]

Z.-C. Gu, F. Verstraete, X.-G. Wen, Grassmann tensor network states and its renormalization for strongly correlated fermionic and bosonic states (2010). arXiv:1004.2563, doi:10.48550/arXiv.1004.2563

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1004.2563 2010

[13] [13]

Grassmann Tensor Renormalization Group Approach to One-Flavor Lattice Schwinger Model

Y. Shimizu, Y. Kuramashi, Grassmann tensor renormalization group approach to one-flavor lattice Schwinger model, Phys. Rev. D 90 (2014) 014508. arXiv:1403.0642, doi:10.1103/PhysRevD.90.014508

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.90.014508 2014

[14] [14]

Grassmann tensor renormalization group for one-flavor lattice Gross-Neveu model with finite chemical potential

S. Takeda, Y. Yoshimura, Grassmann tensor renormalization group for the one-flavor lattice Gross-Neveu model with finite chemical potential, PTEP 2015 (2015) 043B01. arXiv:1412.7855, doi:10.1093/ptep/ptv022

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/ptep/ptv022 2015

[15] [15]

Higher order tensor renormalization group for relativistic fermion systems

R. Sakai, S. Takeda, Y. Yoshimura, Higher order tensor renormalization group for relativistic fermion systems, PTEP 2017 (2017) 063B07. arXiv:1705.07764, doi:10.1093/ptep/ptx080

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1093/ptep/ptx080 2017

[16] [16]

Meurice, R

Y. Meurice, R. Sakai, J. Unmuth-Yockey, Tensor lattice field theory for renormalization and quantum computing, Rev. Mod. Phys. 94 (2022) 025005. arXiv:2010.06539, doi:10.1103/RevModPhys.94.025005

work page doi:10.1103/revmodphys.94.025005 2022

[17] [17]

Bloch, R

J. Bloch, R. Lohmayer, Grassmann higher-order tensor renormalization group approach for two-dimensional strong- coupling QCD, Nucl. Phys. B 986 (2023) 116032. arXiv:2206.00545, doi:10.1016/j.nuclphysb.2022.116032

work page doi:10.1016/j.nuclphysb.2022.116032 2023

[18] [18]

Milde, Tensor networks for four-dimensional strong-coupling QCD, Master’s thesis, University of Regensburg (2023)

P. Milde, Tensor networks for four-dimensional strong-coupling QCD, Master’s thesis, University of Regensburg (2023)

work page 2023

[19] [19]

Creutz, On invariant integration over SU(N), J

M. Creutz, On invariant integration over SU(N), J. Math. Phys. 19 (1978) 2043. doi:10.1063/1.523581

work page doi:10.1063/1.523581 1978

[20] [20]

Gagliardi, W

G. Gagliardi, W. Unger, Towards a Dual Representation of Lattice QCD, PoS LATTICE2018 (2018) 224. arXiv:1811. 02817, doi:10.22323/1.334.0224

work page doi:10.22323/1.334.0224 2018

[21] [21]

Gagliardi, W

G. Gagliardi, W. Unger, New dual representation for staggered lattice QCD, Phys. Rev. D 101 (2020) 034509. arXiv: 1911.08389, doi:10.1103/PhysRevD.101.034509

work page doi:10.1103/physrevd.101.034509 2020

[22] [22]

Borisenko, S

O. Borisenko, S. Voloshyn, V. Chelnokov, SU(N) polynomial integrals and some applications, Rept. Math. Phys. 85 (2020) 129–145. arXiv:1812.06069, doi:10.1016/S0034-4877(20)30015-X

work page doi:10.1016/s0034-4877(20)30015-x 2020

[23] [23]

Gattringer, C

C. Gattringer, C. B. Lang, Quantum Chromodynamics on the Lattice: An Introductory Presentation, Lecture Notes in Physics, Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. doi:10.1007/978-3-642-01850-3

work page doi:10.1007/978-3-642-01850-3 2010

[24] [24]

PeerJ Computer Science 3, e103 (Jan 2017).https://doi.org/10.7717/peerj-cs.103

A. Meurer, C. P. Smith, M. Paprocki, O. ˇCert´ ık, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Rouˇ cka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, A. Scopatz, Sympy: symbolic computing...

work page doi:10.7717/peerj-cs.103 2017