Critical behavior of the Schwinger model via gauge-invariant VUMPS

Hirotsugu Fujii; Juan W. Pedersen; Kohei Fujikura; Takuya Okuda; Yoshio Kikukawa

REVIEW 3 major objections 5 minor 2 cited by

Critical behavior of the Schwinger model via gauge-invariant VUMPS

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims the continuum Schwinger model at $\theta = \pi$ has its critical endpoint at $(m/g)_c = 0.333556(5)$, with Ising universality, established by gauge-invariant VUMPS with double extrapolation.

desk verdict A precise and probably correct critical mass for the Schwinger model, from a genuinely new gauge-invariant VUMPS pipeline; the quoted uncertainty leans on an extrapolation that is unchecked in the critical region. read the letter →

arxiv 2412.03569 v3 pith:XL7W5X6S submitted 2024-12-04 hep-lat cond-mat.str-elhep-thquant-ph

Hirotsugu Fujii , Kohei Fujikura , Yoshio Kikukawa , Takuya Okuda , Juan W. Pedersen This is my paper · ORCID

classification hep-latcond-mat.str-elhep-thquant-ph

keywords SchwingermodelthetaanglelatticegaugetheorymatrixproductstatesVUMPSGausslawconstraintIsinguniversalityclasscriticalmass

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to pin down the critical endpoint of the massive Schwinger model at $\theta=\pi$: the value of the fermion mass (in units of the gauge coupling) at which the theory's first-order transition turns second-order. Combining the variational uniform matrix product state (VUMPS) algorithm with a gauge-invariant ansatz that enforces Gauss's law locally, the authors obtain the continuum critical mass $(m/g)_c = 0.333556(5)$, roughly an order of magnitude more precise than earlier numerical estimates. They further show that the correlation length, a local order parameter, and the entanglement entropy collapse onto a single curve in the simultaneous continuum and infinite-bond-dimension limits. The collapse uses Ising exponents in the infrared and a conformal contribution with central charge $c=1$ in the ultraviolet, giving independent support to the quoted critical mass.

What carries the argument

The central object is the gauge-invariant uniform matrix product state (uMPS) ansatz, in which each variational matrix carries a virtual index structure that enforces the Gauss law locally while keeping the electric field explicit. The VUMPS algorithm optimizes this ansatz in the infinite-volume limit, and the resulting transfer matrix yields the correlation length $1/\epsilon_1$ and the gap parameter $\delta = \epsilon_2 - \epsilon_1$. The argument is carried by the linear extrapolation $\epsilon_1(D) = \epsilon_{1,\infty} + c_1\,\delta(D)$, which treats finite bond dimension as a finite-size effect, followed by a polynomial extrapolation of the lattice critical mass in the lattice spacing $ga$; a double data collapse of scale-invariant combinations of observables serves as a cross-check.

What would settle it

Repeat the extraction of $\epsilon_{1,\infty}$ at $ga = 0.1$ and $m/g = 0.3335$ with bond dimensions above $D = 500$ (or including the CT-broken points omitted from Fig. 2) and test whether the linear relation $\epsilon_1(D) = \epsilon_{1,\infty} + c_1\,\delta(D)$ still holds and whether the resulting $(m/g)_c$ leaves $0.333556(5)$; alternatively, add a cubic term to the $ga$-fit and check whether the intercept moves by more than the quoted uncertainty.

Watch

Extended reading notes

Core claim

The central claim is that the continuum Schwinger model at $\theta=\pi$ has its critical endpoint at $(m/g)_c = 0.333556(5)$ and that its critical behavior belongs to the Ising universality class. The authors compute ground states of the lattice-regularized Hamiltonian with gauge-invariant uniform matrix product states, extract the correlation length from the leading gap of the MPS transfer matrix, extrapolate the bond dimension to infinity, and then extrapolate the lattice spacing to zero. A finite-size-scaling double collapse of the correlation length, the local order parameter (the CT-odd electric field), and the entanglement entropy onto universal curves confirms the critical mass, fixes the infrared Ising exponents, and exposes a UV conformal contribution with central charge $c=1$.

Load-bearing premise

The result rests on the assumption that finite bond dimension acts like a finite system size, so the correlation length extrapolates linearly in the transfer-matrix gap $\delta(D)$; if that linear law fails or CT-breaking states contaminate the data at the bond dimensions used, the inferred critical mass shifts.

Editorial extensions

If this is right

The continuum Schwinger model at $\theta=\pi$ has its transition point at $(m/g)_c = 0.333556(5)$, refining the earlier best estimate $0.3335(2)$ by an order of magnitude in uncertainty.
The infrared critical behavior of the lattice model near the continuum limit is governed by Ising universality-class exponents, $\Delta_t = 1$, $\Delta_\phi = 1/8$, and central charge $c_{\rm IR} = 1/2$.
The ultraviolet scaling of the entanglement entropy carries a conformal contribution with central charge $c_{\rm UV} = 1$, so the finite-lattice-spacing cutoff behaves like a free-boson conformal field theory.
The double-collapse analysis yields $(m/g)_c = 0.333560$, $0.333560$, and $0.333559$ from the correlation length, order parameter, and entanglement entropy, all consistent with the main extrapolation.
The gauge-invariant VUMPS approach locates the $\theta=\pi$ critical point without a sign problem, making precise continuum extrapolations practical for this lattice gauge theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the same double-extrapolation template should transfer to multi-flavor Schwinger models and two-dimensional adjoint QCD, whose richer phase structures are mentioned in the paper's outlook, but the linear-in-$\delta$ ansatz would need to be revalidated in each theory.
Inference: because the paper's note records an overlapping independent estimate of $0.333561(4)$, the agreement suggests the systematic error from the extrapolation ansatz is no larger than the quoted uncertainty; that comparison is a consistency check, not part of the paper's own argument.
Inference: combining several bond dimensions in a single simultaneous fit of the $\delta$-extrapolation and the $ga$-extrapolation could reduce the scatter among the different data-subset estimates seen in the paper's Fig. 4.
Inference: if the finite-bond-dimension scaling is truly a finite-size effect, the same $\delta$-based extrapolation could be applied to the local order parameter and entanglement entropy directly, yielding independent continuum estimates beyond the data-collapse cross-check.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read of 2412.03569. The genuinely new thing is the pipeline: VUMPS with the gauge-invariant uMPS ansatz, which locally enforces Gauss law. That combination lets them push to large bond dimensions and extract a precise continuum critical mass, (m/g)c = 0.333556(5). The number agrees with Byrnes et al.'s DMRG value 0.3335(2) and with the overlapping preprint [66] at 0.333561(4), which is a strong signal the central value is right. The double data collapse for the correlation length, order parameter, and entanglement entropy is a nice demonstration, and it is consistent with Ising IR exponents plus a c=1 UV contribution.

The soft spots are real but not fatal. The entire error budget rests on Eq. (25), the linear relation between epsilon1 and delta(D), justified as a finite-size-like effect. That ansatz is plausible, but footnote 7 removes exactly the near-critical points where D up to 500 cannot reach the CT-invariant sector. The left branch of the two-line fit in Fig. 2 is then anchored away from m* and extrapolated across the omitted interval. Nothing the paper does rules out a nonlinear c2*delta^2 term or a crossover contribution near criticality, and a 1e-5 bias would move the quoted central value outside the stated uncertainty. The double collapse does not cure this: it uses the same delta-to-inverse-size identification and the same (m/g)* ansatz (37), and it fixes the Ising exponents rather than testing them. So the collapse is a consistency check, not an independent universality test. Also, no code or data is provided, which limits how much a referee can verify.

In proportion: the paper's main claim is almost certainly correct, given the agreement with two other methods. But the precision claim (5e-6) is probably too aggressive for an extrapolation that is unchecked in precisely the region that matters. A serious referee should ask for the omitted points at larger D, a demonstration that Eq. (25) is stable under adding a quadratic term, and ideally release of the code and data.

This paper deserves peer review, not desk rejection. The method is a real contribution, the result is a useful benchmark, and the concerns are fixable with additional analysis rather than conceptual flaws.

Referee Report

3 major / 5 minor

Summary. The paper studies the lattice Schwinger model at theta=pi using a gauge-invariant uniform matrix product state (VUMPS) ansatz that enforces the Gauss law locally. The authors extract the inverse correlation length from the MPS transfer matrix, extrapolate it to infinite bond dimension using Eq. (25) and then to zero lattice spacing, obtaining a continuum critical mass (m/g)c = 0.333556(5). They additionally perform a double collapse of the correlation length, local order parameter, and entanglement entropy in the simultaneous critical and continuum limits, concluding that the data are consistent with the Ising universality class. The main numerical claim agrees with previous DMRG results and with the overlapping preprint [66].

Significance. If the central claim holds, the paper provides a high-precision determination of the critical endpoint of the theta=pi Schwinger model and demonstrates that gauge-invariant VUMPS is an effective tool for studying critical phenomena in lattice gauge theories without a sign problem. Strengths of the work include the use of a manifestly gauge-invariant ansatz, a clearly described two-step extrapolation procedure, an explicit sensitivity analysis over random data subsets, and a comparison with independent results. The paper also makes a specific, falsifiable prediction for the continuum critical mass that is consistent with previous determinations. The main caveat is that the quoted precision rests on an extrapolation ansatz whose validity is not demonstrated in the very region where it matters most.

major comments (3)

[Section IV A, Eq. (25) and footnote 7] The linear ansatz epsilon1(D) = epsilon1,infty + c1*delta(D) is the sole bridge from finite-D VUMPS data to the exact correlation length, yet for ga=0.1 the two mass values closest to the fitted critical point (m/g = 0.3335 and 0.3336) are omitted because D up to 500 does not reach the linear regime and produces CT-breaking states. The left branch of the two-line fit (26) is therefore anchored at m/g <= 0.3334 and must be extrapolated across the interval where corrections to Eq. (25) are expected to be largest. Since the final uncertainty is quoted as 5e-6, a bias at the 1e-5 level would change the central claim. Please validate Eq. (25) in the critical region (e.g., with larger bond dimensions or a different extrapolation variable) and quantify the systematic error introduced by the footnote-7 omission.
[Section IV B, Eqs. (35)-(39)] The double collapse is presented as confirmation of Eq. (28) and of Ising universality, but it fixes the IR exponents (Delta_t = 1, Delta_phi = 1/8, c_IR = 1/2) as inputs, uses the same identification of delta as an inverse system size, and uses the same ansatz (37) for (m/g)*. The optimized values in (39) are therefore consistency checks under the assumed universality class, not independent tests of it. The abstract's wording 'confirm that the data collapse aligns with the Ising universality class' should be softened to 'consistent with'. In addition, because the cost function (38) is minimized on the same data that define the collapsed curve, a goodness-of-fit measure or cross-validation is needed to quantify the quality of the collapse.
[Section IV A, uncertainty estimation] The uncertainty in epsilon1,infty is obtained by choosing the uncertainty in epsilon1 so that the reduced chi-squared of the linear fit (25) equals 1. This procedure propagates only statistical scatter under the assumed model and does not include model error in Eq. (25) or the omission of the footnote-7 points. The final error bar (5e-6) is thus best interpreted as a statistical error conditional on the extrapolation ansatz, not as a total systematic error. Please state this limitation explicitly and add a systematic component, for example from the spread of the fits in Fig. 4 or from an alternate extrapolation form.

minor comments (5)

[Section II A] The sentence 'This property motives the definition' contains a typo; it should read 'motivates'.
[Section III] The statement that the exponent eta takes the value 1/2 with a small correction for one spatial dimension in a deep gapped phase is vague; please clarify what correction is meant and provide a precise reference.
[Figure 2 caption] The caption says the data are fitted separately in the two regions m/g < 0.3336 and m/g > 0.3336, but the split point should be identified with the fitted m*/g rather than a fixed abscissa, especially since the two points nearest the crossing are omitted.
[Footnote 7] The content of footnote 7 describes an essential limitation of the extrapolation procedure and should at least be summarized in the main text of Section IV A, not relegated to a figure caption footnote.
[Section II B] The phrase 'We also find the exponential suppression on the Schmidt coefficient of the large electric charge in our simulation' should be rephrased, for example as 'We also find exponential suppression of the Schmidt coefficients for large electric charge'.

Circularity Check

1 steps flagged · score 3.0 of 10

The critical mass determination is a genuine extrapolation from VUMPS data, not a fit to the answer; the only circular element is the claim that the double collapse 'confirms' Ising universality while the Ising exponents are fixed inputs to the collapse construction.

self definitional [Sec. IV B (Eq. (35) and following paragraph); Sec. V Conclusion]
""We assume the Ising universality class, ∆IR t = 1, ∆ϕ = 1/8, cIR = 1/2, for the IR scale transformation" (Sec. IV B) and "confirmed that the IR critical exponents are consistent with those of the Ising universality class" (Sec. V)."

The collapse variables in Eq. (35) are constructed with the Ising IR exponents as fixed inputs; the cost function in Eq. (38) only optimizes (m/g)c, b1, b2, and l1, never the exponents. Therefore the successful collapse demonstrates only that the data are compatible with the assumed Ising values, not that the data independently determine those exponents. The conclusion that the IR critical exponents are consistent with the Ising universality class restates the input of the construction, so it cannot serve as an independent confirmation of Ising universality.

full rationale

The central determination of (m/g)c = 0.333556(5) is not circular. Section IV A takes raw VUMPS outputs {ε1, δ}, extrapolates to D → ∞ through the linear ansatz (25), extracts ε1,∞, then fits the two linear branches in m/g (26) and extrapolates m*/g to ga → 0 by polynomial fits; the final value is a numerical extrapolation, not an input. No load-bearing result is imported from the authors' own prior papers: the gauge-invariant MPS construction [14], VUMPS [55], the δ parametrization [74,78], and the scaling-collapse method [64,65] are all external references. The double-collapse analysis re-extracts (m/g)c from the same VUMPS data with the ansatz (37); while this is a consistency check rather than an independent confirmation, it is not forced by construction to equal (28), so it is not circular in the strict sense. The genuine circular element is the Ising-universality confirmation: the exponents are fixed to Ising values in Eq. (35), so the collapse cannot independently establish those exponents. Footnote 7's omission of m/g = 0.3335 and 0.3336 points is a possible systematic bias in the ε1,∞ extrapolation, but that is a robustness concern, not a circularity, because the omitted points are not used to define the fitted result. Overall the paper is largely self-contained; the main critical-mass result stands on its own, with score 3 reflecting the one assumption-laden 'confirmation'.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on several fit parameters and on the assumed scaling forms. The free parameters appear at three stages: extrapolation in the transfer-matrix gap delta, extrapolation in lattice spacing ga, and the double-collapse ansatz. The axioms are mostly domain assumptions about MPS representability, the scaling hypothesis, and the Ising/CFT exponents. No new physical entities are introduced.

free parameters (6)

c1 in Eq. (25) = not reported
Slope of linear fit between epsilon1 and delta for each (ga, m/g); used to extract intercept epsilon1,infty.
epsilon1,infty intercept in Eq. (25) = values in Fig. 2, not tabulated
Extrapolated inverse correlation length at infinite bond dimension for each lattice spacing and mass; central input to critical-mass estimate.
c- and c+ slopes in Eq. (26) = not reported
Slopes of linear fits of epsilon1,infty versus m/g on either side of the critical point, per ga; define the intersection m*/g.
Continuum fit coefficients C0, C1, C2 = C0 = 0.333556(5) median; C1 small; not separately reported
Polynomial coefficients in ga used to extrapolate (m/g)* to the continuum limit.
l1 coefficient in Eq. (37) = not reported
Coefficient of delta/ga in the ansatz for the lattice critical point used in the double collapse; shifted by optimization.
Noise scale for epsilon1 = chosen so reduced chi-squared = 1
The paper assigns uncertainties to epsilon1 by the common procedure of rescaling errors until reduced chi-squared is 1; this affects error bars in Fig. 2 and the final uncertainty.

assumptions (5)

ad hoc to paper Finite-bond-dimension correction is controlled by delta = epsilon2 - epsilon1 through a linear relation (Eq. 25).
Assumed in Sec. IV A to extrapolate epsilon1 to infinite D; not derived from the model. The paper acknowledges it as an assumption in footnote 6.
ad hoc to paper The lattice critical point has the analytic form (m/g)* = (m/g)c + b1*ga + b2*ga^2 + l1*delta/ga (Eq. 37).
Used in the double collapse; higher-order terms neglected. The paper states it works well in practice but does not derive it.
domain assumption IR critical behavior is described by the Ising universality class with Delta_t = 1, Delta_phi = 1/8, c_IR = 1/2, and UV by c_UV = 1.
These exponents and central charges are fixed inputs to the collapse in Sec. IV B, not extracted from the data.
domain assumption The VUMPS-optimized gauge-invariant uMPS approximates the true ground state in the CT-symmetric or CT-broken phase, and the transfer matrix has a spectral decomposition.
The whole analysis rests on the uMPS ansatz and the eigendecomposition (19); phase selection is nontrivial, as footnote 7 shows CT-breaking states can be found.
domain assumption The entanglement entropy scaling (34) with c/6 log terms applies.
Used to define the scale-invariant entropy in the collapse; from CFT, but its applicability to the lattice data is assumed.

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Cite this review

Pith. "Pith review of Critical behavior of the Schwinger model via gauge-invariant VUMPS." pith.science (2026). https://pith.science/paper/XL7W5X6S

@misc{pith2026241203569,
  author       = {Pith},
  title        = {Pith review of: Critical behavior of the Schwinger model via gauge-invariant VUMPS},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XL7W5X6S}},
  note         = {Machine review of arXiv:2412.03569}
}

abstract

We study the lattice Schwinger model by combining the variational uniform matrix product state (VUMPS) algorithm with a gauge-invariant matrix product ansatz that locally enforces the Gauss law constraint. Both the continuum and lattice versions of the Schwinger model with $\theta=\pi$ are known to exhibit first-order phase transitions for the values of the fermion mass above a critical value, where a second-order phase transition occurs. Our algorithm enables a precise determination of the critical endpoint in the continuum theory. We further analyze the scaling in the simultaneous critical and continuum limits and confirm that the data collapse aligns with the Ising universality class to remarkable precision.

Figures

Figures reproduced from arXiv: 2412.03569 by the authors.

**Figure 3.** FIG. 3. Plot of the data for ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 5.** FIG. 5. Color online. Generated data points used in the double collapse on ( [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Reference graph

Works this paper leans on

85 extracted references · 24 canonical work pages · cited by 2 Pith papers

[66]

Vanhecke, F

B. Vanhecke, F. Verstraete and K. Van Acoleyen, Entanglement scaling for λϕ24, Phys. Rev. D 106 (2022) L071501 [ 2104.10564]

arXiv 2022
[1]

good fits

As reported in [20], the Schmidt coefficients σqαq for large |q| are exponentially suppressed. Indeed, a state with a larger |q| leads to a larger energy cost when we restrict to the region 0 ≤ θ ≤ π. We also find the expo- nential suppression on the Schmidt coefficient of the large electric charge in our simulation. In order to systemat- ically control t...

2024
[2]

J. S. Schwinger, Gauge Invariance and Mass , Phys. Rev. 125 (1962) 397

1962
[3]

J. S. Schwinger, Gauge Invariance and Mass. 2. , Phys. Rev. 128 (1962) 2425

1962
[4]

Duncan and M

A. Duncan and M. Furman, Monte Carlo Calculations With Fermions: The Schwinger Model , Nucl. Phys. B 190 (1981) 767

1981
[5]

Ranft and A

J. Ranft and A. Schiller, Local Hamiltonian Monte Carlo Study of the Massive Schwinger Model in an External Background Field , Phys. Lett. B 122 (1983) 403

1983
[6]

Schiller and J

A. Schiller and J. Ranft, The Massive Schwinger Model on the Lattice Studied via a Local Hamiltonian Monte Carlo Method, Nucl. Phys. B 225 (1983) 204

work page 1983
[7]

Bender, H

I. Bender, H. J. Rothe and K. D. Rothe, Monte Carlo Study of Screening Versus Confinement in the Massless and Massive Schwinger Model , Nucl. Phys. B 251 (1985) 745

work page 1985

Show all 85 references

[8]

Potvin, A nonperturbative study of hadronization with heavy sources

J. Potvin, A nonperturbative study of hadronization with heavy sources. 1. The screening length as a function of the quark mass in the Schwinger model , Phys. Rev. D 32 (1985) 2070

1985
[9]

W. A. Bardeen, A. Duncan, E. Eichten and H. Thacker, Quenched approximation artifacts: A Study in two-dimensional QED , Phys. Rev. D 57 (1998) 3890

1998
[10]

Ohata, Monte Carlo study of Schwinger model without the sign problem , JHEP 12 (2023) 007 [2303.05481]

H. Ohata, Monte Carlo study of Schwinger model without the sign problem , JHEP 12 (2023) 007 [2303.05481]. 9

2023 arXiv
[11]

Ohata, Phase diagram near the quantum critical point in Schwinger model at θ = π: analogy with quantum Ising chain , PTEP 2024 (2024) 013B02 [2311.04738]

H. Ohata, Phase diagram near the quantum critical point in Schwinger model at θ = π: analogy with quantum Ising chain , PTEP 2024 (2024) 013B02 [2311.04738]

2024 arXiv
[12]

Byrnes, P

T. Byrnes, P. Sriganesh, R. J. Bursill and C. J. Hamer, Density matrix renormalization group approach to the massive Schwinger model , Nucl. Phys. B Proc. Suppl. 109 (2002) 202 [ hep-lat/0201007]

2002 arXiv
[13]

Byrnes, P

T. Byrnes, P. Sriganesh, R. J. Bursill and C. J. Hamer, Density matrix renormalization group approach to the massive Schwinger model , Phys. Rev. D 66 (2002) 013002 [hep-lat/0202014]

2002 arXiv
[14]

M. C. Ba˜ nuls, K. Cichy, K. Jansen and J. I. Cirac, The mass spectrum of the Schwinger model with Matrix Product States, JHEP 11 (2013) 158 [ 1305.3765]

2013 arXiv
[15]

Buyens, J

B. Buyens, J. Haegeman, K. Van Acoleyen, H. Verschelde and F. Verstraete, Matrix product states for gauge field theories , Phys. Rev. Lett. 113 (2014) 091601 [1312.6654]

2014 arXiv
[16]

Shimizu and Y

Y. Shimizu and Y. Kuramashi, Grassmann tensor renormalization group approach to one-flavor lattice Schwinger model, Phys. Rev. D 90 (2014) 014508 [1403.0642]

2014 arXiv
[17]

Shimizu and Y

Y. Shimizu and Y. Kuramashi, Critical behavior of the lattice Schwinger model with a topological term at θ = π using the Grassmann tensor renormalization group , Phys. Rev. D 90 (2014) 074503 [ 1408.0897]

2014 arXiv
[18]

M. C. Ba˜ nuls, K. Cichy, J. I. Cirac, K. Jansen and H. Saito, Thermal evolution of the Schwinger model with Matrix Product Operators , Phys. Rev. D 92 (2015) 034519 [1505.00279]

2015 arXiv
[19]

Buyens, J

B. Buyens, J. Haegeman, H. Verschelde, F. Verstraete and K. Van Acoleyen, Confinement and string breaking for QED 2 in the Hamiltonian picture , Phys. Rev. X 6 (2016) 041040 [ 1509.00246]

2016 arXiv
[20]

M. C. Ba˜ nuls, K. Cichy, K. Jansen and H. Saito,Chiral condensate in the Schwinger model with Matrix Product Operators, Phys. Rev. D 93 (2016) 094512 [1603.05002]

2016 arXiv
[21]

Buyens, S

B. Buyens, S. Montangero, J. Haegeman, F. Verstraete and K. Van Acoleyen, Finite-representation approximation of lattice gauge theories at the continuum limit with tensor networks , Phys. Rev. D 95 (2017) 094509 [1702.08838]

2017 arXiv
[22]

Ercolessi, P

E. Ercolessi, P. Facchi, G. Magnifico, S. Pascazio and F. V. Pepe, Phase Transitions in Zn Gauge Models: Towards Quantum Simulations of the Schwinger-Weyl QED, Phys. Rev. D 98 (2018) 074503 [ 1705.11047]

2018 arXiv
[23]

Funcke, K

L. Funcke, K. Jansen and S. K¨ uhn,Topological vacuum structure of the Schwinger model with matrix product states, Phys. Rev. D 101 (2020) 054507 [ 1908.00551]

2020 arXiv
[24]

Magnifico, M

G. Magnifico, M. Dalmonte, P. Facchi, S. Pascazio, F. V. Pepe and E. Ercolessi, Real Time Dynamics and Confinement in the Zn Schwinger-Weyl lattice model for 1+1 QED , Quantum 4 (2020) 281 [ 1909.04821]

2020 arXiv
[25]

Okuda, Schwinger model on an interval: Analytic results and DMRG , Phys

T. Okuda, Schwinger model on an interval: Analytic results and DMRG , Phys. Rev. D 107 (2023) 054506 [2210.00297]

2023 arXiv
[26]

Honda, E

M. Honda, E. Itou and Y. Tanizaki, DMRG study of the higher-charge Schwinger model and its ’t Hooft anomaly, JHEP 11 (2022) 141 [ 2210.04237]

2022 arXiv
[27]

Angelides, L

T. Angelides, L. Funcke, K. Jansen and S. K¨ uhn, Computing the mass shift of Wilson and staggered fermions in the lattice Schwinger model with matrix product states, Phys. Rev. D 108 (2023) 014516 [2303.11016]

2023 arXiv
[28]

E. A. Martinez et al., Real-time dynamics of lattice gauge theories with a few-qubit quantum computer , Nature 534 (2016) 516 [ 1605.04570]

2016 arXiv
[29]

Muschik, M

C. Muschik, M. Heyl, E. Martinez, T. Monz, P. Schindler, B. Vogell et al., U(1) Wilson lattice gauge theories in digital quantum simulators , New J. Phys. 19 (2017) 103020 [ 1612.08653]

2017 arXiv
[30]

N. Klco, E. F. Dumitrescu, A. J. McCaskey, T. D. Morris, R. C. Pooser, M. Sanz et al., Quantum-classical computation of Schwinger model dynamics using quantum computers, Phys. Rev. A 98 (2018) 032331 [1803.03326]

2018 arXiv
[31]

Kokail et al., Self-verifying variational quantum simulation of lattice models , Nature 569 (2019) 355 [1810.03421]

C. Kokail et al., Self-verifying variational quantum simulation of lattice models , Nature 569 (2019) 355 [1810.03421]

2019 arXiv
[32]

F. M. Surace, P. P. Mazza, G. Giudici, A. Lerose, A. Gambassi and M. Dalmonte, Lattice gauge theories and string dynamics in Rydberg atom quantum simulators, Phys. Rev. X 10 (2020) 021041 [1902.09551]

2020 arXiv
[33]

Chakraborty, M

B. Chakraborty, M. Honda, T. Izubuchi, Y. Kikuchi and A. Tomiya, Classically emulated digital quantum simulation of the Schwinger model with a topological term via adiabatic state preparation , Phys. Rev. D 105 (2022) 094503 [ 2001.00485]

2022 arXiv
[34]

D. E. Kharzeev and Y. Kikuchi, Real-time chiral dynamics from a digital quantum simulation , Phys. Rev. Res. 2 (2020) 023342 [ 2001.00698]

2020 arXiv
[35]

Rajput, A

A. Rajput, A. Roggero and N. Wiebe, Hybridized Methods for Quantum Simulation in the Interaction Picture, Quantum 6 (2022) 780 [ 2109.03308]

2022 arXiv
[36]

Thompson and G

S. Thompson and G. Siopsis, Quantum computation of phase transition in the massive Schwinger model , Quantum Sci. Technol. 7 (2022) 035001 [ 2110.13046]

2022 arXiv
[37]

Yamamoto, Quantum variational approach to lattice gauge theory at nonzero density , Phys

A. Yamamoto, Quantum variational approach to lattice gauge theory at nonzero density , Phys. Rev. D 104 (2021) 014506 [ 2104.10669]

2021 arXiv
[38]

Honda, E

M. Honda, E. Itou, Y. Kikuchi and Y. Tanizaki, Negative string tension of a higher-charge Schwinger model via digital quantum simulation , PTEP 2022 (2022) 033B01 [ 2110.14105]

2022 arXiv
[39]

W. A. de Jong, K. Lee, J. Mulligan, M. P losko´ n, F. Ringer and X. Yao, Quantum simulation of nonequilibrium dynamics and thermalization in the Schwinger model, Phys. Rev. D 106 (2022) 054508 [2106.08394]

2022 arXiv
[40]

Honda, E

M. Honda, E. Itou, Y. Kikuchi, L. Nagano and T. Okuda, Classically emulated digital quantum simulation for screening and confinement in the Schwinger model with a topological term , Phys. Rev. D 105 (2022) 014504 [ 2105.03276]

2022 arXiv
[41]

Rajput, A

A. Rajput, A. Roggero and N. Wiebe, Quantum error correction with gauge symmetries , npj Quantum Inf. 9 (2023) 41 [ 2112.05186]

2023 arXiv
[42]

N. H. Nguyen, M. C. Tran, Y. Zhu, A. M. Green, C. H. Alderete, Z. Davoudi et al., Digital Quantum Simulation of the Schwinger Model and Symmetry Protection with Trapped Ions, PRX Quantum 3 (2022) 020324 [2112.14262]

2022 arXiv
[43]

Cheng, S

Y. Cheng, S. Liu, W. Zheng, P. Zhang and H. Zhai, Tunable Confinement-Deconfinement Transition in an Ultracold-Atom Quantum Simulator, PRX Quantum 3 10 (2022) 040317 [ 2204.06586]

2022 arXiv
[44]

Tomiya, Schwinger model at finite temperature and density with beta VQE , 2205.08860

A. Tomiya, Schwinger model at finite temperature and density with beta VQE , 2205.08860

arXiv
[45]

Nagano, A

L. Nagano, A. Bapat and C. W. Bauer, Quench dynamics of the Schwinger model via variational quantum algorithms , Phys. Rev. D 108 (2023) 034501 [2302.10933]

2023 arXiv
[46]

Ikeda, D

K. Ikeda, D. E. Kharzeev, R. Meyer and S. Shi, Detecting the critical point through entanglement in the Schwinger model, Phys. Rev. D 108 (2023) L091501 [2305.00996]

2023 arXiv
[47]

Sakamoto, H

K. Sakamoto, H. Morisaki, J. Haruna, E. Itou, K. Fujii and K. Mitarai, End-to-end complexity for simulating the Schwinger model on quantum computers , Quantum 8 (2024) 1474 [ 2311.17388]

2024 arXiv
[48]

R. C. Farrell, M. Illa, A. N. Ciavarella and M. J. Savage, Scalable Circuits for Preparing Ground States on Digital Quantum Computers: The Schwinger Model Vacuum on 100 Qubits , PRX Quantum 5 (2024) 020315 [2308.04481]

2024 arXiv
[49]

R. C. Farrell, M. Illa, A. N. Ciavarella and M. J. Savage, Quantum simulations of hadron dynamics in the Schwinger model using 112 qubits , Phys. Rev. D 109 (2024) 114510 [ 2401.08044]

2024 arXiv
[50]

Ghim and M

D. Ghim and M. Honda, Digital Quantum Simulation for Spectroscopy of Schwinger Model , PoS LA TTICE2023(2024) 213 [ 2404.14788]

2024 arXiv
[51]

Kaikov, T

O. Kaikov, T. Saporiti, V. Sazonov and M. Tamaazousti, Phase Diagram of the Schwinger Model by Adiabatic Preparation of States on a Quantum Simulator, 2407.09224

arXiv
[52]

Y. Guo, T. Angelides, K. Jansen and S. K¨ uhn, Concurrent VQE for Simulating Excited States of the Schwinger Model, 2407.15629

arXiv
[53]

J. Y. Araz, S. Bhowmick, M. Grau, T. J. McEntire and F. Ringer, State preparation of lattice field theories using quantum optimal control , 2407.17556

arXiv
[54]

X.-W. Li, F. Li, J. Zhuang and M.-H. Yung, Simulating the Schwinger Model with a Regularized Variational Quantum Imaginary Time Evolution , 2409.13510

arXiv
[55]

Schollw¨ ock,The density-matrix renormalization group in the age of matrix product states , Annals of Physics 326 (2011) 96 [ 1008.3477]

U. Schollw¨ ock,The density-matrix renormalization group in the age of matrix product states , Annals of Physics 326 (2011) 96 [ 1008.3477]

2011 arXiv
[56]

Zauner-Stauber, L

V. Zauner-Stauber, L. Vanderstraeten, M. T. Fishman, F. Verstraete and J. Haegeman, Variational optimization algorithms for uniform matrix product states, Phys. Rev. B 97 (2018) 045145 [ 1701.07035]

2018 arXiv
[57]

J. B. Kogut and L. Susskind, Hamiltonian Formulation of Wilson ’s Lattice Gauge Theories, Phys. Rev. D 11 (1975) 395

1975
[58]

S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69 (1992) 2863

1992
[59]

I. P. McCulloch, Infinite size density matrix renormalization group, revisited, 0804.2509

arXiv
[60]

Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension , Phys

G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension , Phys. Rev. Lett. 98 (2007) 070201 [ cond-mat/0605597]

2007 arXiv
[61]

Haegeman, J

J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pizorn, H. Verschelde and F. Verstraete, Time-Dependent Variational Principle for Quantum Lattices , Phys. Rev. Lett. 107 (2011) 070601 [ 1103.0936]

2011 arXiv
[62]

Haegeman, C

J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken and F. Verstraete, Unifying time evolution and optimization with matrix product states , arXiv e-prints (2014) arXiv:1408.5056 [ 1408.5056]

2014 arXiv
[63]

S. R. Coleman, More About the Massive Schwinger Model, Annals Phys. 101 (1976) 239

1976
[64]

C. J. Hamer, J. B. Kogut, D. P. Crewther and M. M. Mazzolini, The Massive Schwinger Model on a Lattice: Background Field, Chiral Symmetry and the String Tension, Nucl. Phys. B 208 (1982) 413

1982
[65]

Vanhecke, J

B. Vanhecke, J. Haegeman, K. Van Acoleyen, L. Vanderstraeten and F. Verstraete, Scaling Hypothesis for Matrix Product States , Phys. Rev. Lett. 123 (2019) 250604 [1907.08603]

2019 arXiv
[67]

Arguello Cruz, G

E. Arguello Cruz, G. Tarnopolsky and Y. Xin, Precision study of the massive Schwinger model near quantum criticality, 2412.01902

arXiv
[68]

Banks, L

T. Banks, L. Susskind and J. B. Kogut, Strong Coupling Calculations of Lattice Gauge Theories: (1+1)-Dimensional Exercises, Phys. Rev. D 13 (1976) 1043

1976
[69]

Jordan and E

P. Jordan and E. P. Wigner, About the Pauli exclusion principle, Z. Phys. 47 (1928) 631

1928
[70]

Dempsey, I

R. Dempsey, I. R. Klebanov, S. S. Pufu and B. Zan, Discrete chiral symmetry and mass shift in the lattice Hamiltonian approach to the Schwinger model , Phys. Rev. Res. 4 (2022) 043133 [ 2206.05308]

2022 arXiv
[71]

Berruto, G

F. Berruto, G. Grignani, G. W. Semenoff and P. Sodano, On the correspondence between the strongly coupled two flavor lattice Schwinger model and the Heisenberg antiferromagnetic chain, Annals Phys. 275 (1999) 254 [ hep-th/9901142]

1999 arXiv
[72]

Van Acoleyen, B

K. Van Acoleyen, B. Buyens, J. Haegeman and F. Verstraete, Matrix product states for Hamiltonian lattice gauge theories , PoS LA TTICE2014(2014) 308 [1411.0020]

2014 arXiv
[73]

Buyens, F

B. Buyens, F. Verstraete and K. Van Acoleyen, Hamiltonian simulation of the Schwinger model at finite temperature, Phys. Rev. D 94 (2016) 085018 [1606.03385]

2016 arXiv
[74]

Buyens, J

B. Buyens, J. Haegeman, F. Hebenstreit, F. Verstraete and K. Van Acoleyen, Real-time simulation of the Schwinger effect with Matrix Product States , Phys. Rev. D 96 (2017) 114501 [ 1612.00739]

2017 arXiv
[75]

Zauner, D

V. Zauner, D. Draxler, L. Vanderstraeten, M. Degroote, J. Haegeman, M. M. Rams et al., Transfer Matrices and Excitations with Matrix Product States , New J. Phys. 17 (2015) 053002 [ 1408.5140]

2015 arXiv
[76]

L. S. Ornstein and F. Zernike, Accidental deviations of density and opalescence at the critical point of a single substance, Proc. Akad. Sci. 17 (1914) 793

1914
[77]

Kennedy, Ornstein-zernike decay in the ground state of the quantum ising model in a strong transverse field , Communications in Mathematical Physics 137 (1991) 599

T. Kennedy, Ornstein-zernike decay in the ground state of the quantum ising model in a strong transverse field , Communications in Mathematical Physics 137 (1991) 599

1991
[78]

J. L. Cardy, Scaling and renormalization in statistical physics. Cambridge University Press, Cambridge, England, 1996

1996
[79]

M. M. Rams, P. Czarnik and L. Cincio, Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models , Physical Review X 8 (2018) 041033 [ 1801.08554]. 11

2018 arXiv
[80]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes 3rd Edition: The Art of Scientific Computing . Cambridge University Press, USA, 3 ed., 2007

2007
[81]

Calabrese and J

P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory , J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152]

2004 arXiv
[82]

Gepner, Nonabelian Bosonization and Multiflavor QED and QCD in Two-dimensions , Nucl

D. Gepner, Nonabelian Bosonization and Multiflavor QED and QCD in Two-dimensions , Nucl. Phys. B 252 (1985) 481

1985
[83]

Affleck, On the Realization of Chiral Symmetry in (1+1)-dimensions, Nucl

I. Affleck, On the Realization of Chiral Symmetry in (1+1)-dimensions, Nucl. Phys. B 265 (1986) 448

1986
[84]

Dempsey, I

R. Dempsey, I. R. Klebanov, S. S. Pufu, B. T. Søgaard and B. Zan, Phase Diagram of the Two-Flavor Schwinger Model at Zero Temperature, Phys. Rev. Lett. 132 (2024) 031603 [ 2305.04437]

2024 arXiv
[85]

Dempsey, I

R. Dempsey, I. R. Klebanov, S. S. Pufu and B. T. Søgaard, Lattice Hamiltonian for adjoint QCD 2, JHEP 08 (2024) 009 [ 2311.09334]

2024 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [66]

Vanhecke, F

B. Vanhecke, F. Verstraete and K. Van Acoleyen, Entanglement scaling for λϕ24, Phys. Rev. D 106 (2022) L071501 [ 2104.10564]

arXiv 2022

[2] [1]

good fits

As reported in [20], the Schmidt coefficients σqαq for large |q| are exponentially suppressed. Indeed, a state with a larger |q| leads to a larger energy cost when we restrict to the region 0 ≤ θ ≤ π. We also find the expo- nential suppression on the Schmidt coefficient of the large electric charge in our simulation. In order to systemat- ically control t...

2024

[3] [2]

J. S. Schwinger, Gauge Invariance and Mass , Phys. Rev. 125 (1962) 397

1962

[4] [3]

J. S. Schwinger, Gauge Invariance and Mass. 2. , Phys. Rev. 128 (1962) 2425

1962

[5] [4]

Duncan and M

A. Duncan and M. Furman, Monte Carlo Calculations With Fermions: The Schwinger Model , Nucl. Phys. B 190 (1981) 767

1981

[6] [5]

Ranft and A

J. Ranft and A. Schiller, Local Hamiltonian Monte Carlo Study of the Massive Schwinger Model in an External Background Field , Phys. Lett. B 122 (1983) 403

1983

[7] [6]

Schiller and J

A. Schiller and J. Ranft, The Massive Schwinger Model on the Lattice Studied via a Local Hamiltonian Monte Carlo Method, Nucl. Phys. B 225 (1983) 204

work page 1983

[8] [7]

Bender, H

I. Bender, H. J. Rothe and K. D. Rothe, Monte Carlo Study of Screening Versus Confinement in the Massless and Massive Schwinger Model , Nucl. Phys. B 251 (1985) 745

work page 1985