- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Precision study of the massive Schwinger model near quantum criticality

Erick Arguello Cruz, Grigory Tarnopolsky, and Yuan Xin

PDF

Phys. Rev. D 112, 034023 – Published 22 August, 2025

DOI: https://doi.org/10.1103/vf7m-kzk2

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.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (64)

  1. J. S. Schwinger, Gauge invariance and mass. 2., Phys. Rev. 128, 2425 (1962).
  2. J. H. Lowenstein and J. A. Swieca, Quantum electrodynamics in two-dimensions, Ann. Phys. (N.Y.) 68, 172 (1971).
  3. A. Casher, J. Kogut, and L. Susskind, Vacuum polarization and the absence of free quarks, Phys. Rev. D 10, 732 (1974).
  4. S. R. Coleman, R. Jackiw, and L. Susskind, Charge shielding and quark confinement in the massive Schwinger model, Ann. Phys. (N.Y.) 93, 267 (1975).
  5. S. R. Coleman, More about the massive Schwinger model, Ann. Phys. (N.Y.) 101, 239 (1976).
  6. M. Creutz, One flavor QCD, Ann. Phys. (N.Y.) 322, 1518 (2007).
  7. 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, 202 (2002).
  8. T. Byrnes, Density Matrix Renormalization Group: A New Approach to Lattice Gauge Theory, Ph.D. thesis, University of New South Wales, 2003.
  9. 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, 094509 (2017).
  10. H. Ohata, Phase diagram near the quantum critical point in Schwinger model at θ=π: Analogy with quantum Ising chain, Prog. Theor. Exp. Phys. 2024, 013B02 (2024).
  11. 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, 031603 (2024).
  12. H. Fujii, K. Fujikura, Y. Kikukawa, T. Okuda, and J. W. Pedersen, Critical behavior of the Schwinger model via gauge-invariant variational uniform matrix product states, Phys. Rev. D 111, 094505 (2025).
  13. 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, 043133 (2022).
  14. J. Ranft and A. Schiller, Local Hamiltonian Monte Carlo study of the massive Schwinger model in an external background field, Phys. Lett. 122B, 403 (1983).
  15. A. Schiller and J. Ranft, The massive Schwinger model on the lattice studied via a local Hamiltonian Monte Carlo method, Nucl. Phys. B225, 204 (1983).
  16. H. Ohata, Monte Carlo study of Schwinger model without the sign problem, J. High Energy Phys. 12 (2023) 007.
  17. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  18. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  19. S. Östlund and S. Rommer, Thermodynamic limit of density matrix renormalization, Phys. Rev. Lett. 75, 3537 (1995).
  20. G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Phys. Rev. Lett. 98, 070201 (2007).
  21. I. P. McCulloch, Infinite size density matrix renormalization group, revisited, arXiv:0804.2509.
  22. Y. Zou, A. Milsted, and G. Vidal, Conformal data and renormalization group flow in critical quantum spin chains using periodic uniform matrix product states, Phys. Rev. Lett. 121, 230402 (2018).
  23. M. Ganahl, J. Rincón, and G. Vidal, Continuous matrix product states for quantum fields: An energy minimization algorithm, Phys. Rev. Lett. 118, 220402 (2017).
  24. M. C. Bañuls, K. Cichy, K. Jansen, and J. I. Cirac, The mass spectrum of the Schwinger model with matrix product states, J. High Energy Phys. 11 (2013) 158.
  25. M. C. Bañuls, K. Cichy, J. I. Cirac, K. Jansen, and H. Saito, Matrix product states for lattice field theories, Proc. Sci., LATTICE2013 (2014) 332.
  26. 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, 074503 (2014).
  27. B. Buyens, J. Haegeman, K. Van Acoleyen, H. Verschelde, and F. Verstraete, Matrix product states for gauge field theories, Phys. Rev. Lett. 113, 091601 (2014).
  28. M. C. Bañuls, K. Cichy, K. Jansen, and H. Saito, Chiral condensate in the Schwinger model with matrix product operators, Phys. Rev. D 93, 094512 (2016).
  29. B. Buyens, J. Haegeman, H. Verschelde, F. Verstraete, and K. Van Acoleyen, Confinement and string breaking for QED2 in the Hamiltonian picture, Phys. Rev. X 6, 041040 (2016).
  30. B. Buyens, F. Verstraete, and K. Van Acoleyen, Hamiltonian simulation of the Schwinger model at finite temperature, Phys. Rev. D 94, 085018 (2016).
  31. Y. Shimizu and Y. Kuramashi, Berezinskii-Kosterlitz-Thouless transition in lattice Schwinger model with one flavor of Wilson fermion, Phys. Rev. D 97, 034502 (2018).
  32. L. Funcke, K. Jansen, and S. Kühn, Topological vacuum structure of the Schwinger model with matrix product states, Phys. Rev. D 101, 054507 (2020).
  33. 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, 014504 (2022).
  34. E. Itou, A. Matsumoto, and Y. Tanizaki, DMRG study of the theta-dependent mass spectrum in the 2-flavor Schwinger model, J. High Energy Phys. 09 (2024) 155.
  35. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calculations, SciPost Phys. Codebases 4 (2022).10.21468/SciPostPhysCodeb.4
  36. M. Fishman, S. R. White, and E. M. Stoudenmire, Codebase release 0.3 for ITensor, SciPost Phys. Codebases 4-r0.3 (2022).10.21468/SciPostPhysCodeb.4-r0.3
  37. J. B. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11, 395 (1975).
  38. T. Banks, L. Susskind, and J. Kogut, Strong-coupling calculations of lattice gauge theories: (1+1)-dimensional exercises, Phys. Rev. D 13, 1043 (1976).
  39. C. J. Hamer, W.-h. Zheng, and J. Oitmaa, Series expansions for the massive Schwinger model in Hamiltonian lattice theory, Phys. Rev. D 56, 55 (1997).
  40. Z. Komargodski, K. Ohmori, K. Roumpedakis, and S. Seifnashri, Symmetries and strings of adjoint QCD2, J. High Energy Phys. 03 (2021) 103.
  41. A. Cherman, T. Jacobson, M. Shifman, M. Unsal, and A. Vainshtein, Four-fermion deformations of the massless Schwinger model and confinement, J. High Energy Phys. 01 (2023) 087.
  42. A. Bondi, G. Curci, G. Paffuti, and P. Rossi, Metric and central charge in the perturbative approach to two-dimensional fermionic models, Ann. Phys. (N.Y.) 199, 268 (1990).
  43. L. Susskind, Lattice fermions, Phys. Rev. D 16, 3031 (1977).
  44. M. C. Banuls, K. Cichy, Y.-J. Kao, C. J. D. Lin, Y.-P. Lin, and D. T. L. Tan, Investigation of the 1+1 dimensional Thirring model using the method of matrix product states, Proc. Sci., LATTICE2018 (2018) 229.
  45. C. Hamer, J. Kogut, D. Crewther, and M. Mazzolini, The massive Schwinger model on a lattice: Background field, chiral symmetry and the string tension, Nucl. Phys. B208, 413 (1982).
  46. C. Nagele, J. E. Cejudo, T. Byrnes, and M. Kleban, Flux unwinding in the lattice Schwinger model, Phys. Rev. D 99, 094501 (2019).
  47. E. Fradkin and L. Susskind, Order and disorder in gauge systems and magnets, Phys. Rev. D 17, 2637 (1978).
  48. M. E. Fisher and M. N. Barber, Scaling theory for finite-size effects in the critical region, Phys. Rev. Lett. 28, 1516 (1972).
  49. C. J. Hamer and M. N. Barber, Finite lattice methods in quantum Hamiltonian field theory. 1. The Ising model, J. Phys. A 14, 241 (1981).
  50. J. L. Cardy, Conformal invariance and universality in finite-size scaling, J. Phys. A 17, L385 (1984).
  51. A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241, 333 (1984).
  52. J. L. Cardy, Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories, Nucl. Phys. B275, 200 (1986).
  53. P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  54. P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009).
  55. I. Affleck and A. W. W. Ludwig, Universal noninteger ground-state degeneracy in critical quantum systems, Phys. Rev. Lett. 67, 161 (1991).
  56. M. Campostrini, A. Pelissetto, and E. Vicari, Finite-size scaling at quantum transitions, Phys. Rev. B 89, 094516 (2014).
  57. T. Okuda, Schwinger model on an interval: Analytic results and DMRG, Phys. Rev. D 107, 054506 (2023).
  58. M. Honda, E. Itou, and Y. Tanizaki, DMRG study of the higher-charge Schwinger model and its ’t Hooft anomaly, J. High Energy Phys. 11 (2022) 141.
  59. B.-X. Lao and S. Rychkov, 3D Ising CFT and exact diagonalization on icosahedron: The power of conformal perturbation theory, SciPost Phys. 15, 243 (2023).
  60. I. P. McCulloch, From density-matrix renormalization group to matrix product states, J. Stat. Mech. (2007) P10014.
  61. G. M. Crosswhite and D. Bacon, Finite automata for caching in matrix product algorithms, Phys. Rev. A 78, 012356 (2008).
  62. G. M. Crosswhite, A. C. Doherty, and G. Vidal, Applying matrix product operators to model systems with long-range interactions, Phys. Rev. B 78, 035116 (2008).
  63. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (Amsterdam) 326, 96 (2011), January 2011 Special Issue.
  64. S. R. White, Spin gaps in a frustrated Heisenberg model for CaV4O9, Phys. Rev. Lett. 77, 3633 (1996).

Outline

Information

Phys. Rev. D 112, 034023– Published 22 August, 2025
  • Received 10 June 2025
  • Accepted 27 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation