Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2412.01902 · Physical review. D/Physical review. D.

Precision study of the massive Schwinger model near quantum criticality

Erick Arguello Cruz, Grigory Tarnopolsky, Yuan Xin

Published Jul 27, 2025• 7 citations
arXivHTMLPDF

Abstract

We perform a numerical analysis of the massive Schwinger model in the presence of a background electric field. Using the Density Matrix Renormalization Group approach, we efficiently compute the spectrum of the Schwinger model on a staggered lattice with up to 3000 qubits. As a result, we achieve a precise computation of the critical mass of the massive Schwinger model to five digits using four different “criticality criteria,” observing perfect agreement among them Additionally, we discuss the effect of a four-fermion operator deformation of the Schwinger model and compute the critical mass for various values of the deformation parameter.

DOI
10.1103/vf7m-kzk2

Related papers

Density matrix renormalization group approach to the massive Schwinger model

Tim Byrnes, P. Sriganesh, Robert J. Bursill +1 more

The massive Schwinger model is studied using a density matrix renormalization group approach to the staggered lattice Hamiltonian version of the model. Lattice sizes up to 256 sites are calculated, and the estimates in the continuum limit are almost two orders of magnitude more accurate than previous calculations. Coleman's picture of ``half-asymptotic'' particles at a background field $\ensuremath{\theta}=\ensuremath{\pi}$ is confirmed. The predicted phase transition at finite fermion mass $(m/g)$ is accurately located and demonstrated to belong in the 2D Ising universality class.

arXiv:hep-lat/0202014•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 1, 2002•178 citations

Quantum computation of phase transition in the massive Schwinger model

Shane Thompson, George Siopsis

Abstract As pointed out by Coleman, physical quantities in the Schwinger model depend on a parameter θ that determines the background electric field. There is a phase transition for θ = π only. We develop a momentum space formalism on a lattice and use it to perform a quantum computation of the critical point of this phase transition on the NISQ device IMB Q Lima. After error mitigation, our results give strong indication of the existence of a critical point at m / e ≃ 0.32, where m is the bare fermion mass and e is the coupling strength, in good agreement with the classical numerical result m / e ≃ 0.3335.

arXiv:2110.13046•Quantum Science and Technology•Mar 21, 2022•22 citations

Numerical investigations of the Schwinger model and selected quantum spin models

Marcin Szyniszewski

Numerical investigations of the XY model, the Heisenberg model and the J-J' Heisenberg model are conducted, using the exact diagonalisation, the numerical renormalisation and the density matrix renormalisation group approach. The low-lying energy levels are obtained and finite size scaling is performed to estimate the bulk limit values. The results are found to be consistent with the exact values. The DMRG results are found to be most promising. The Schwinger model is also studied using the exact diagonalisation and the strong coupling expansion. The massless, the massive model and the model with a background electric field are explored. Ground state energy, scalar and vector particle masses and order parameters are examined. The achieved values are observed to be consistent with previous results and theoretical predictions. Path to the future studies is outlined.

arXiv:1303.0441•Jan 1, 2012•0 citations

First-order phase transition of the Schwinger model with a quantum computer

Takis Angelides, Pranay Naredi, Arianna Crippa +4 more

Abstract We explore the first-order phase transition in the lattice Schwinger model in the presence of a topological θ-term by means of the variational quantum eigensolver (VQE). Using two different fermion discretizations, Wilson and staggered fermions, we develop parametric ansatz circuits suitable for both discretizations, and compare their performance by simulating classically an ideal VQE optimization in the absence of noise. The states obtained by the classical simulation are then prepared on the IBM's superconducting quantum hardware. Applying state-of-the art error-mitigation methods, we show that the electric field density and particle number, observables which reveal the phase structure of the model, can be reliably obtained from the quantum hardware. To investigate the minimum system sizes required for a continuum extrapolation, we study the continuum limit using matrix product states, and compare our results to continuum mass perturbation theory. We demonstrate that taking the additive mass renormalization into account is vital for enhancing the precision that can be obtained with smaller system sizes. Furthermore, for the observables we investigate we observe universality, and both fermion discretizations produce the same continuum limit.

npj Quantum Information•Jan 17, 2025•21 citations

Grassmann tensor renormalization group for the massive Schwinger model with a θ term using staggered fermions

Hayato Kanno, Shinichiro Akiyama, Kotaro Murakami +1 more

A bstract We use the Grassmann tensor renormalization group method to investigate the N f = 2 Schwinger model with the staggered fermions in the presence of a 2 π periodic θ term in a broad range of mass. The method allows us to deal with the massive staggered fermions straightforwardly and to study the θ dependence of the free energy and topological charge in the thermodynamic limit. Our calculation provides consistent results with not only the analytical solution in the large mass limit but also the previous Monte Carlo studies in the small mass regime. Our numerical results also suggest that the N f = 2 Schwinger model on a lattice has a different phase structure, than the model in the continuum limit.

arXiv:2412.08959•Journal of High Energy Physics•Nov 10, 2025•5 citations

Computing the mass shift of Wilson and staggered fermions in the lattice Schwinger model with matrix product states

Takis Angelides, Lena Funcke, Karl Jansen +1 more

Simulations of lattice gauge theories with tensor networks and quantum computing have so far mainly focused on staggered fermions. In this paper, we use matrix product states to study Wilson fermions in the Hamiltonian formulation and present a novel method to determine the additive mass renormalization. Focusing on the single-flavor Schwinger model as a benchmark model, we investigate the regime of a nonvanishing topological $\ensuremath{\theta}$-term, which is inaccessible to conventional Monte Carlo methods. We systematically explore the dependence of the mass shift on the volume, the lattice spacing, the $\ensuremath{\theta}$-parameter, and the Wilson parameter. This allows us to follow lines of constant renormalized mass, and therefore to substantially improve the continuum extrapolation of the mass gap and electric field density. For small values of the mass, our continuum results agree with the theoretical prediction from mass perturbation theory. Going beyond Wilson fermions, our technique can also be applied to staggered fermions, and we demonstrate that the results of our approach agree with a recent theoretical prediction for the mass shift at sufficiently large volumes.

Physical review. D/Physical review. D.•Jul 26, 2023•25 citations
We perform a numerical analysis of the massive Schwinger model in the presence of a background electric field. Using the Density Matrix Renormalization Group approach, we efficiently compute the spectrum of the Schwinger model on a staggered lattice with up to 3000 qubits. As a result, we achieve a precise computation of the critical mass of the massive Schwinger model to five digits using four different “criticality criteria,” observing perfect agreement among them Additionally, we discuss the effect of a four-fermion operator deformation of the Schwinger model and compute the critical mass for various values of the deformation parameter.
The massive Schwinger model is studied using a density matrix renormalization group approach to the staggered lattice Hamiltonian version of the model. Lattice sizes up to 256 sites are calculated, and the estimates in the continuum limit are almost two orders of magnitude more accurate than previous calculations. Coleman's picture of ``half-asymptotic'' particles at a background field $\ensuremath{\theta}=\ensuremath{\pi}$ is confirmed. The predicted phase transition at finite fermion mass (m/g)(m/g)(m/g) is accurately located and demonstrated to belong in the 2D Ising universality class.
Abstract As pointed out by Coleman, physical quantities in the Schwinger model depend on a parameter θ that determines the background electric field. There is a phase transition for θ = π only. We develop a momentum space formalism on a lattice and use it to perform a quantum computation of the critical point of this phase transition on the NISQ device IMB Q Lima. After error mitigation, our results give strong indication of the existence of a critical point at m / e ≃ 0.32, where m is the bare fermion mass and e is the coupling strength, in good agreement with the classical numerical result m / e ≃ 0.3335.
Numerical investigations of the XY model, the Heisenberg model and the J-J' Heisenberg model are conducted, using the exact diagonalisation, the numerical renormalisation and the density matrix renormalisation group approach. The low-lying energy levels are obtained and finite size scaling is performed to estimate the bulk limit values. The results are found to be consistent with the exact values. The DMRG results are found to be most promising. The Schwinger model is also studied using the exact diagonalisation and the strong coupling expansion. The massless, the massive model and the model with a background electric field are explored. Ground state energy, scalar and vector particle masses and order parameters are examined. The achieved values are observed to be consistent with previous results and theoretical predictions. Path to the future studies is outlined.
Abstract We explore the first-order phase transition in the lattice Schwinger model in the presence of a topological θ-term by means of the variational quantum eigensolver (VQE). Using two different fermion discretizations, Wilson and staggered fermions, we develop parametric ansatz circuits suitable for both discretizations, and compare their performance by simulating classically an ideal VQE optimization in the absence of noise. The states obtained by the classical simulation are then prepared on the IBM's superconducting quantum hardware. Applying state-of-the art error-mitigation methods, we show that the electric field density and particle number, observables which reveal the phase structure of the model, can be reliably obtained from the quantum hardware. To investigate the minimum system sizes required for a continuum extrapolation, we study the continuum limit using matrix product states, and compare our results to continuum mass perturbation theory. We demonstrate that taking the additive mass renormalization into account is vital for enhancing the precision that can be obtained with smaller system sizes. Furthermore, for the observables we investigate we observe universality, and both fermion discretizations produce the same continuum limit.
A bstract We use the Grassmann tensor renormalization group method to investigate the N f = 2 Schwinger model with the staggered fermions in the presence of a 2 π periodic θ term in a broad range of mass. The method allows us to deal with the massive staggered fermions straightforwardly and to study the θ dependence of the free energy and topological charge in the thermodynamic limit. Our calculation provides consistent results with not only the analytical solution in the large mass limit but also the previous Monte Carlo studies in the small mass regime. Our numerical results also suggest that the N f = 2 Schwinger model on a lattice has a different phase structure, than the model in the continuum limit.
Simulations of lattice gauge theories with tensor networks and quantum computing have so far mainly focused on staggered fermions. In this paper, we use matrix product states to study Wilson fermions in the Hamiltonian formulation and present a novel method to determine the additive mass renormalization. Focusing on the single-flavor Schwinger model as a benchmark model, we investigate the regime of a nonvanishing topological $\ensuremath{\theta}$-term, which is inaccessible to conventional Monte Carlo methods. We systematically explore the dependence of the mass shift on the volume, the lattice spacing, the $\ensuremath{\theta}$-parameter, and the Wilson parameter. This allows us to follow lines of constant renormalized mass, and therefore to substantially improve the continuum extrapolation of the mass gap and electric field density. For small values of the mass, our continuum results agree with the theoretical prediction from mass perturbation theory. Going beyond Wilson fermions, our technique can also be applied to staggered fermions, and we demonstrate that the results of our approach agree with a recent theoretical prediction for the mass shift at sufficiently large volumes.