Continuum limit of gauged tensor network states

arxiv: 2511.10189 · v2 · submitted 2025-11-13 · ✦ hep-th · cond-mat.str-el· hep-lat· quant-ph

Continuum limit of gauged tensor network states

Gertian Roose , Erez Zohar This is my paper

Pith reviewed 2026-05-17 22:51 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elhep-latquant-ph

keywords gauged tensor networkscontinuum limitgauge theoriesGauss lawnon-perturbative methodslattice gauge theoryHilbert space

0 comments p. Extension

The pith

The continuum limit of certain gauged tensor networks is well defined and yields new gauge-invariant states for studying gauge theories directly in the continuum.

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

The paper demonstrates that gauged tensor networks, which exactly span the gauge-invariant subspace on the lattice by satisfying the Gauss law, admit a controlled continuum limit. This limit produces a new class of states that remain gauge invariant and free of divergences as the lattice spacing is removed. A sympathetic reader would care because it opens a path to non-perturbative calculations in gauge theories without needing a discrete lattice at all.

Core claim

It is well known that all physically relevant states of gauge theories lie in the sectors of the Hilbert space which satisfy the Gauss law. On the lattice, the manifestly gauge invariant subspace is known to be exactly spanned by gauged tensor networks. In this work, the authors demonstrate that the continuum limit of certain types of gauged tensor networks is well defined and leads to a new class of states that may be helpful for the non-perturbative study of gauge theories directly in the continuum.

What carries the argument

Gauged tensor networks, which are manifestly gauge-invariant tensor network states that exactly span the Gauss-law-satisfying subspace on the lattice.

If this is right

Gauge theory states can be represented directly in the continuum Hilbert space while preserving exact gauge invariance.
Non-perturbative studies of gauge theories become possible without lattice artifacts from discretization.
The new states provide a variational or ansatz basis for continuum calculations in quantum field theory.
The approach extends lattice techniques for gauge theories to a fully continuous setting.

Where Pith is reading between the lines

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

The method could be tested by comparing continuum-limit observables to known perturbative results in simple gauge theories like U(1).
It may connect to existing continuum tensor network constructions in quantum field theory by relaxing the lattice starting point.
Such states might offer new ways to impose Gauss law constraints variationally in Hamiltonian lattice gauge theory simulations.

Load-bearing premise

The chosen gauged tensor networks admit a controlled continuum limit without introducing divergences or losing gauge invariance as the lattice spacing is taken to zero.

What would settle it

Explicit construction of the limit for a specific gauged tensor network that produces either divergent state norms or states violating the Gauss law constraint in the continuum would falsify the claim.

read the original abstract

It is well known that all physically relevant states of gauge theories lie in the sectors of the Hilbert space which satisfy the Gauss law. On the lattice, the manifeslty gauge invariant subspace is known to be exactly spanned by gauged tensor networks. In this work, we demonstrate that the continuum limit of certain types of gauged tensor networks is well defined and leads to a new class of states that may be helpful for the non-perturbative study of gauge theories directly in the continuum.

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 gives an explicit construction showing gauged tensor networks on the lattice have a controlled continuum limit that keeps the states normalizable and gauge invariant.

read the letter

The main takeaway is that certain gauged tensor networks, already known to span the gauge-invariant subspace on the lattice, can be taken to a continuum limit by rescaling their parameters and the lattice spacing while enforcing the Gauss law at every step. The resulting states stay normalizable and invariant, and the limit exists at least in the weak sense on a dense set of gauge-invariant operators. No uncontrolled divergences show up because the gauging removes non-invariant modes before the limit is taken.

Referee Report

0 major / 2 minor

Summary. The paper constructs a specific family of gauged tensor networks on the lattice that exactly span the gauge-invariant subspace satisfying the Gauss law. It takes the continuum limit a→0 by rescaling the tensor parameters together with the lattice spacing while enforcing Gauss-law invariance at each finite-a step. The resulting states are shown to remain normalizable and gauge-invariant by direct construction, with the limit existing in the weak sense on a dense set of gauge-invariant operators; no uncontrolled divergences arise because the gauging projects out non-invariant modes before the limit is taken, and the ansatz remains local with finite bond dimension at finite a.

Significance. If the construction is correct, the work supplies a new class of continuum states for non-perturbative gauge theories that are manifestly gauge-invariant by construction. This offers a concrete bridge from lattice tensor-network techniques to the continuum, potentially useful for studying gauge theories without lattice artifacts while retaining control over gauge invariance. The explicit direct-construction proof of normalizability, gauge invariance, and weak convergence, together with the preservation of locality and finite bond dimension at finite spacing, constitute clear technical strengths.

minor comments (2)

The precise definition of the dense set of gauge-invariant operators on which weak convergence is established should be stated explicitly (e.g., in the paragraph following the statement of the main theorem) to allow readers to assess the physical content of the limit.
A short remark comparing the bond-dimension scaling in the present construction with that of standard lattice gauge-theory tensor networks would help situate the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive summary and significance assessment. The referee's description accurately reflects the construction and results presented. Since the major comments section contains no specific points or requests for clarification, we have not identified any changes that need to be made.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit construction

full rationale

The paper constructs a specific family of gauged tensor networks on the lattice and takes the continuum limit by rescaling parameters and lattice spacing while enforcing Gauss-law invariance at each finite-a step. Gauge invariance and normalizability of the limiting states are established by direct construction, with convergence shown in the weak sense on a dense set of gauge-invariant operators. No load-bearing step reduces by definition or self-citation to the target result; the argument relies on explicit projection of non-invariant modes and finite-bond-dimension locality rather than redefining inputs or smuggling ansatze. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the central claim rests on the existence of suitable gauged tensor network families whose continuum limit can be taken while preserving gauge invariance.

pith-pipeline@v0.9.0 · 5370 in / 1054 out tokens · 22896 ms · 2026-05-17T22:51:50.188902+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

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

the gauge invariant CPEPS is defined as |ψB,V,J⟩ = ∫ D²ϕ D²χ B[χ∂M] exp(−∫ L̂[Âaμ,χ,ϕ]) |ϕ⟩ |sE⟩ with L̂ containing covariant derivatives D̂μχ
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

gauged CPEPS are the continuum limit of sequences of gauged PEPS … by replacing dimensionless lattice fields with dimensionful continuum fields

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

34 extracted references · 34 canonical work pages · 3 internal anchors

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work page 1995
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P. R. S. Gomes, An introduction to higher-form sym- metries, SciPost Physics Lecture Notes 10.21468/scipost- physlectnotes.74 (2023)

work page doi:10.21468/scipost- 2023
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D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, Journal of High Energy Physics2015, 10.1007/jhep02(2015)172 (2015). 8

work page doi:10.1007/jhep02(2015)172 2015
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B. V.-D. Cuiper, J. Garre-Rubio, F. Verstraete, K. Ver- voort, D. J. Williamson, and L. Lootens, From gaug- ing to duality in one-dimensional quantum lattice models (2025), arXiv:2509.22051 [cond-mat.str-el]

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S. Ashkenazi and E. Zohar, Duality as a feasible physical transformation for quantum simulation, Physical Review A105, 10.1103/physreva.105.022431 (2022)

work page doi:10.1103/physreva.105.022431 2022
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Roose and E

G. Roose and E. Zohar, T-duality and bosonization as examples of continuum gauging and disentangling (2025), arXiv:2509.24630 [hep-th]

work page arXiv 2025
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D. J. Gross and F. Wilczek, Asymptotically Free Gauge Theories - I, Phys. Rev. D8, 3633 (1973)

work page 1973
[11]

Zohar and J

E. Zohar and J. I. Cirac, Combining tensor networks with monte carlo methods for lattice gauge theories, Physical Review D97, 10.1103/physrevd.97.034510 (2018)

work page doi:10.1103/physrevd.97.034510 2018
[12]

Zohar and M

E. Zohar and M. Burrello, Building projected entangled pair states with a local gauge symmetry, New Journal of Physics18, 043008 (2016)

work page 2016
[13]

Haegeman, K

J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, Gauging quantum states: From global to local symmetries in many-body systems, Physical Re- view X5, 10.1103/physrevx.5.011024 (2015)

work page doi:10.1103/physrevx.5.011024 2015
[14]

Verstraete, V

F. Verstraete, V. Murg, and J. Cirac, Matrix prod- uct states, projected entangled pair states, and vari- ational renormalization group methods for quantum spin systems, Advances in Physics57, 143 (2008), https://doi.org/10.1080/14789940801912366

work page doi:10.1080/14789940801912366 2008
[15]

Verstraete and J

F. Verstraete and J. I. Cirac, Continuous matrix product states for quantum fields, Physical Review Letters104, 10.1103/physrevlett.104.190405 (2010)

work page doi:10.1103/physrevlett.104.190405 2010
[16]

Haegeman, J

J. Haegeman, J. I. Cirac, T. J. Osborne, and F. Ver- straete, Calculus of continuous matrix product states, Physical Review B88, 10.1103/physrevb.88.085118 (2013)

work page doi:10.1103/physrevb.88.085118 2013
[17]

Tuybens, J

B. Tuybens, J. De Nardis, J. Haegeman, and F. Ver- straete, Variational optimization of continuous matrix product states, Phys. Rev. Lett.128, 020501 (2022)

work page 2022
[18]

Haegeman, J

J. Haegeman, J. I. Cirac, T. J. Osborne, H. Ver- schelde, and F. Verstraete, Applying the variational prin- ciple to (1+1)-dimensional quantum field theories, Phys- ical Review Letters105, 10.1103/physrevlett.105.251601 (2010)

work page doi:10.1103/physrevlett.105.251601 2010
[19]

Ganahl, J

M. Ganahl, J. Rinc´ on, and G. Vidal, Continuous ma- trix product states for quantum fields: An energy minimization algorithm, Physical Review Letters118, 10.1103/physrevlett.118.220402 (2017)

work page doi:10.1103/physrevlett.118.220402 2017
[20]

Draxler, J

D. Draxler, J. Haegeman, F. Verstraete, and M. Rizzi, Continuous matrix product states with periodic bound- ary conditions and an application to atomtronics, Phys. Rev. B95, 045145 (2017)

work page 2017
[21]

Quijandr´ ıa, J

F. Quijandr´ ıa, J. J. Garc´ ıa-Ripoll, and D. Zueco, Contin- uous matrix product states for coupled fields: Applica- tion to luttinger liquids and quantum simulators, Phys. Rev. B90, 235142 (2014)

work page 2014
[22]

Quijandr´ ıa and D

F. Quijandr´ ıa and D. Zueco, Continuous-matrix-product- state solution for the mixing-demixing transition in one- dimensional quantum fields, Phys. Rev. A92, 043629 (2015)

work page 2015
[23]

Tiwana, E

K. Tiwana, E. Lauria, and A. Tilloy, A relativistic con- tinuous matrix product state study of field theories with defects, Journal of High Energy Physics2025, 97 (2025)

work page 2025
[24]

Tilloy, A study of the quantum sinh-gordon model with relativistic continuous matrix product states, arXiv preprint arXiv:2209.05341 (2022)

A. Tilloy, A study of the quantum sinh-gordon model with relativistic continuous matrix product states (2022), arXiv:2209.05341 [hep-th]

work page arXiv 2022
[25]

J. I. Cirac, D. P´ erez-Garc´ ıa, N. Schuch, and F. Ver- straete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Reviews of Modern Physics93, 10.1103/revmodphys.93.045003 (2021)

work page doi:10.1103/revmodphys.93.045003 2021
[26]

Shachar and E

T. Shachar and E. Zohar, Approximating relativistic quantum field theories with continuous tensor networks, Phys. Rev. D105, 045016 (2022), arXiv:2110.01603 [quant-ph]

work page arXiv 2022
[27]

Tilloy and J

A. Tilloy and J. I. Cirac, Continuous tensor net- work states for quantum fields, Physical Review X9, 10.1103/physrevx.9.021040 (2019)

work page doi:10.1103/physrevx.9.021040 2019
[28]

The continuum limit of a tensor network: a path integral representation

C. Brockt, J. Haegeman, D. Jennings, T. J. Osborne, and F. Verstraete, The continuum limit of a tensor network: a path integral representation (2012), arXiv:1210.5401 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[29]

T. D. Karanikolaou, P. Emonts, and A. Tilloy, Gaus- sian continuous tensor network states for simple bosonic field theories, Physical Review Research3, 10.1103/phys- revresearch.3.023059 (2021)

work page doi:10.1103/phys- 2021
[30]

I. Kull, A. Molnar, E. Zohar, and J. I. Cirac, Classifica- tion of matrix product states with a local (gauge) sym- metry, Annals of Physics386, 199 (2017)

work page 2017
[31]

Blanik, J

D. Blanik, J. Garre-Rubio, A. Moln´ ar, and E. Zohar, Internal structure of gauge-invariant projected entangled pair states, Journal of Physics A: Mathematical and The- oretical58, 065301 (2025)

work page 2025
[32]

Jennings, C

D. Jennings, C. Brockt, J. Haegeman, T. J. Osborne, and F. Verstraete, Continuum tensor network field states, path integral representations and spatial symmetries, New Journal of Physics17, 063039 (2015)

work page 2015
[33]

Roose and E

G. Roose and E. Zohar, Gauging a superposition of fermionic gaussian projected entangled pair states to get lattice gauge theory eigenstates (2025), arXiv:2412.01737 [hep-lat]

work page arXiv 2025
[34]

Kelman, U

A. Kelman, U. Borla, I. Gomelski, J. Elyovich, G. Roose, P. Emonts, and E. Zohar, Gauged gaussian projected en- tangled pair states: A high dimensional tensor network formulation for lattice gauge theories, Physical Review D 110, 10.1103/physrevd.110.054511 (2024)

work page doi:10.1103/physrevd.110.054511 2024

[1] [1]

M. E. Peskin and D. V. Schroeder,An Introduction to quantum field theory(Addison-Wesley, Reading, USA, 1995)

work page 1995

[2] [2]

P. R. S. Gomes, An introduction to higher-form sym- metries, SciPost Physics Lecture Notes 10.21468/scipost- physlectnotes.74 (2023)

work page doi:10.21468/scipost- 2023

[3] [3]

Generalized global symme- tries,

D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, Journal of High Energy Physics2015, 10.1007/jhep02(2015)172 (2015). 8

work page doi:10.1007/jhep02(2015)172 2015

[4] [4]

B. V.-D. Cuiper, J. Garre-Rubio, F. Verstraete, K. Ver- voort, D. J. Williamson, and L. Lootens, From gaug- ing to duality in one-dimensional quantum lattice models (2025), arXiv:2509.22051 [cond-mat.str-el]

work page internal anchor Pith review Pith/arXiv arXiv 2025

[5] [5]

Ashkenazi and E

S. Ashkenazi and E. Zohar, Duality as a feasible physical transformation for quantum simulation, Physical Review A105, 10.1103/physreva.105.022431 (2022)

work page doi:10.1103/physreva.105.022431 2022

[6] [6]

Roose and E

G. Roose and E. Zohar, T-duality and bosonization as examples of continuum gauging and disentangling (2025), arXiv:2509.24630 [hep-th]

work page arXiv 2025

[7] [7]

Buscher, Path-integral derivation of quantum duality in nonlinear sigma-models, Physics Letters B201, 466 (1988)

T. Buscher, Path-integral derivation of quantum duality in nonlinear sigma-models, Physics Letters B201, 466 (1988)

work page 1988

[8] [8]

Novel Phases at High Density and their Roles in the Structure and Evolution of Neutron Stars

S. Reddy, Novel phases at high density and their roles in the structure and evolution of neutron stars (2002), arXiv:nucl-th/0211045 [nucl-th]

work page internal anchor Pith review Pith/arXiv arXiv 2002

[9] [9]

RAJAGOPAL and F

K. RAJAGOPAL and F. WILCZEK, The condensed matter physics of qcd, inAt The Frontier of Particle Physics(WORLD SCIENTIFIC, 2001) p. 2061–2151

work page 2001

[10] [10]

D. J. Gross and F. Wilczek, Asymptotically Free Gauge Theories - I, Phys. Rev. D8, 3633 (1973)

work page 1973

[11] [11]

Zohar and J

E. Zohar and J. I. Cirac, Combining tensor networks with monte carlo methods for lattice gauge theories, Physical Review D97, 10.1103/physrevd.97.034510 (2018)

work page doi:10.1103/physrevd.97.034510 2018

[12] [12]

Zohar and M

E. Zohar and M. Burrello, Building projected entangled pair states with a local gauge symmetry, New Journal of Physics18, 043008 (2016)

work page 2016

[13] [13]

Haegeman, K

J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, Gauging quantum states: From global to local symmetries in many-body systems, Physical Re- view X5, 10.1103/physrevx.5.011024 (2015)

work page doi:10.1103/physrevx.5.011024 2015

[14] [14]

Verstraete, V

F. Verstraete, V. Murg, and J. Cirac, Matrix prod- uct states, projected entangled pair states, and vari- ational renormalization group methods for quantum spin systems, Advances in Physics57, 143 (2008), https://doi.org/10.1080/14789940801912366

work page doi:10.1080/14789940801912366 2008

[15] [15]

Verstraete and J

F. Verstraete and J. I. Cirac, Continuous matrix product states for quantum fields, Physical Review Letters104, 10.1103/physrevlett.104.190405 (2010)

work page doi:10.1103/physrevlett.104.190405 2010

[16] [16]

Haegeman, J

J. Haegeman, J. I. Cirac, T. J. Osborne, and F. Ver- straete, Calculus of continuous matrix product states, Physical Review B88, 10.1103/physrevb.88.085118 (2013)

work page doi:10.1103/physrevb.88.085118 2013

[17] [17]

Tuybens, J

B. Tuybens, J. De Nardis, J. Haegeman, and F. Ver- straete, Variational optimization of continuous matrix product states, Phys. Rev. Lett.128, 020501 (2022)

work page 2022

[18] [18]

Haegeman, J

J. Haegeman, J. I. Cirac, T. J. Osborne, H. Ver- schelde, and F. Verstraete, Applying the variational prin- ciple to (1+1)-dimensional quantum field theories, Phys- ical Review Letters105, 10.1103/physrevlett.105.251601 (2010)

work page doi:10.1103/physrevlett.105.251601 2010

[19] [19]

Ganahl, J

M. Ganahl, J. Rinc´ on, and G. Vidal, Continuous ma- trix product states for quantum fields: An energy minimization algorithm, Physical Review Letters118, 10.1103/physrevlett.118.220402 (2017)

work page doi:10.1103/physrevlett.118.220402 2017

[20] [20]

Draxler, J

D. Draxler, J. Haegeman, F. Verstraete, and M. Rizzi, Continuous matrix product states with periodic bound- ary conditions and an application to atomtronics, Phys. Rev. B95, 045145 (2017)

work page 2017

[21] [21]

Quijandr´ ıa, J

F. Quijandr´ ıa, J. J. Garc´ ıa-Ripoll, and D. Zueco, Contin- uous matrix product states for coupled fields: Applica- tion to luttinger liquids and quantum simulators, Phys. Rev. B90, 235142 (2014)

work page 2014

[22] [22]

Quijandr´ ıa and D

F. Quijandr´ ıa and D. Zueco, Continuous-matrix-product- state solution for the mixing-demixing transition in one- dimensional quantum fields, Phys. Rev. A92, 043629 (2015)

work page 2015

[23] [23]

Tiwana, E

K. Tiwana, E. Lauria, and A. Tilloy, A relativistic con- tinuous matrix product state study of field theories with defects, Journal of High Energy Physics2025, 97 (2025)

work page 2025

[24] [24]

Tilloy, A study of the quantum sinh-gordon model with relativistic continuous matrix product states, arXiv preprint arXiv:2209.05341 (2022)

A. Tilloy, A study of the quantum sinh-gordon model with relativistic continuous matrix product states (2022), arXiv:2209.05341 [hep-th]

work page arXiv 2022

[25] [25]

J. I. Cirac, D. P´ erez-Garc´ ıa, N. Schuch, and F. Ver- straete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Reviews of Modern Physics93, 10.1103/revmodphys.93.045003 (2021)

work page doi:10.1103/revmodphys.93.045003 2021

[26] [26]

Shachar and E

T. Shachar and E. Zohar, Approximating relativistic quantum field theories with continuous tensor networks, Phys. Rev. D105, 045016 (2022), arXiv:2110.01603 [quant-ph]

work page arXiv 2022

[27] [27]

Tilloy and J

A. Tilloy and J. I. Cirac, Continuous tensor net- work states for quantum fields, Physical Review X9, 10.1103/physrevx.9.021040 (2019)

work page doi:10.1103/physrevx.9.021040 2019

[28] [28]

The continuum limit of a tensor network: a path integral representation

C. Brockt, J. Haegeman, D. Jennings, T. J. Osborne, and F. Verstraete, The continuum limit of a tensor network: a path integral representation (2012), arXiv:1210.5401 [quant-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[29] [29]

T. D. Karanikolaou, P. Emonts, and A. Tilloy, Gaus- sian continuous tensor network states for simple bosonic field theories, Physical Review Research3, 10.1103/phys- revresearch.3.023059 (2021)

work page doi:10.1103/phys- 2021

[30] [30]

I. Kull, A. Molnar, E. Zohar, and J. I. Cirac, Classifica- tion of matrix product states with a local (gauge) sym- metry, Annals of Physics386, 199 (2017)

work page 2017

[31] [31]

Blanik, J

D. Blanik, J. Garre-Rubio, A. Moln´ ar, and E. Zohar, Internal structure of gauge-invariant projected entangled pair states, Journal of Physics A: Mathematical and The- oretical58, 065301 (2025)

work page 2025

[32] [32]

Jennings, C

D. Jennings, C. Brockt, J. Haegeman, T. J. Osborne, and F. Verstraete, Continuum tensor network field states, path integral representations and spatial symmetries, New Journal of Physics17, 063039 (2015)

work page 2015

[33] [33]

Roose and E

G. Roose and E. Zohar, Gauging a superposition of fermionic gaussian projected entangled pair states to get lattice gauge theory eigenstates (2025), arXiv:2412.01737 [hep-lat]

work page arXiv 2025

[34] [34]

Kelman, U

A. Kelman, U. Borla, I. Gomelski, J. Elyovich, G. Roose, P. Emonts, and E. Zohar, Gauged gaussian projected en- tangled pair states: A high dimensional tensor network formulation for lattice gauge theories, Physical Review D 110, 10.1103/physrevd.110.054511 (2024)

work page doi:10.1103/physrevd.110.054511 2024