- 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
  • Letter
  • Open Access

Parton distribution functions in the Schwinger model from tensor network states

Mari Carmen Bañuls1, Krzysztof Cichy2, C.-J. David Lin3,4,5, and Manuel Schneider3,4,*

  • *Contact author: manuel.schneider@nycu.edu.tw
PDF

Phys. Rev. D 113, L011502 – Published 29 January, 2026

DOI: https://doi.org/10.1103/lcyh-rxf3

Abstract

Parton distribution functions (PDFs) describe the inner, nonperturbative structure of hadrons. Their computation involves matrix elements with a Wilson line along a direction on the light cone, posing significant challenges in Euclidean lattice calculations, where the time direction is not directly accessible. We propose implementing the light-front Wilson line within the Hamiltonian formalism using tensor network techniques. The approach is demonstrated in the massive Schwinger model (quantum electrodynamics in 1+1 dimensions), a toy model that shares key features with quantum chromodynamics. We present accurate continuum results for the fermion PDF of the vector meson at varying fermion masses, obtained from first-principle calculations directly in Minkowski space. Our strategy also provides a useful path for quantum simulations and quantum computing.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (88)

  1. National Academies of Sciences, Engineering, and Medicine, An Assessment of U.S.-Based Electron-Ion Collider Science (The National Academies Press, Washington, DC, 2018), 10.17226/25171.
  2. K.-F. Liu, Parton degrees of freedom from the path integral formalism, Phys. Rev. D 62, 074501 (2000).
  3. W. Detmold and C. J. D. Lin, Deep-inelastic scattering and the operator product expansion in lattice QCD, Phys. Rev. D 73, 014501 (2006).
  4. V. Braun and D. Müller, Exclusive processes in position space and the pion distribution amplitude, Eur. Phys. J. C 55, 349 (2008).
  5. Z. Davoudi and M. J. Savage, Restoration of rotational symmetry in the continuum limit of lattice field theories, Phys. Rev. D 86, 054505 (2012).
  6. X. Ji, Parton physics on a Euclidean lattice, Phys. Rev. Lett. 110, 262002 (2013).
  7. Y.-Q. Ma and J.-W. Qiu, Extracting parton distribution functions from lattice QCD calculations, Phys. Rev. D 98, 074021 (2018).
  8. A. V. Radyushkin, Quasi-parton distribution functions, momentum distributions, and pseudo-parton distribution functions, Phys. Rev. D 96, 034025 (2017).
  9. A. J. Chambers, R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller, K. Somfleth, R. D. Young, and J. M. Zanotti, Nucleon structure functions from operator product expansion on the lattice, Phys. Rev. Lett. 118, 242001 (2017).
  10. W. Detmold, A. V. Grebe, I. Kanamori, C. J. D. Lin, R. J. Perry, and Y. Zhao (HOPE Collaboration), Parton physics from a heavy-quark operator product expansion: Formalism and wilson coefficients, Phys. Rev. D 104, 074511 (2021).
  11. A. Shindler, Moments of parton distribution functions of any order from lattice QCD, Phys. Rev. D 110, L051503 (2024).
  12. K. Cichy and M. Constantinou, A guide to light-cone pdfs from lattice QCD: An overview of approaches, techniques and results, Adv. High Energy Phys. 2019, 3036904 (2019).
  13. A. Radyushkin, Theory and applications of parton pseudodistributions, Int. J. Mod. Phys. A 35, 2030002 (2020).
  14. X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang, and Y. Zhao, Large-momentum effective theory, Rev. Mod. Phys. 93, 035005 (2021).
  15. M. Constantinou, The x-dependence of hadronic parton distributions: A review on the progress of lattice QCD, Eur. Phys. J. A 57, 77 (2021).
  16. K. Cichy, Progress in x-dependent partonic distributions from lattice QCD, Proc. Sci. LATTICE2021 (2022) 017 [arXiv:2110.07440].
  17. K. Cichy, Overview of lattice calculations of the x-dependence of PDFs, GPDs and TMDs, EPJ Web Conf. 258, 01005 (2022).
  18. X. Gao, Overview of hadron structure from lattice QCD, Proc. Sci. LATTICE2023 (2024) 128.
  19. Y. Mo and R. J. Perry, Basis function calculations for the massive Schwinger model in the light-front Tamm-Dancoff approximation, J. Comput. Phys. 108, 159 (1993).
  20. H. Bergknoff, Physical particles of the massive Schwinger model, Nucl. Phys. B122, 215 (1977).
  21. H. Kröger and N. Scheu, The massive Schwinger model—a Hamiltonian lattice study in a fast moving frame, Phys. Lett. B 429, 58 (1998).
  22. F. Verstraete, V. Murg, and J. I. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).
  23. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (Amsterdam) 326, 96 (2011).
  24. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (Amsterdam) 349, 117 (2014).
  25. J. C. Bridgeman and C. T. Chubb, Hand-waving and interpretive dance: An introductory course on tensor networks, J. Phys. A 50, 223001 (2017).
  26. P. Silvi, F. Tschirsich, M. Gerster, J. Jünemann, D. Jaschke, M. Rizzi, and S. Montangero, The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems, SciPost Phys. Lect. Notes 8, 1 (2019).
  27. J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  28. K. Okunishi, T. Nishino, and H. Ueda, Developments in the Tensor Network—from statistical mechanics to quantum entanglement, J. Phys. Soc. Japan 91, 062001 (2022).
  29. M. C. Bañuls, Tensor network algorithms: A route map, Annu. Rev. Condens. Matter Phys. 14, 173 (2023).
  30. M. C. Bañuls and K. Cichy, Review on novel methods for lattice gauge theories, Rep. Prog. Phys. 83, 024401 (2020).
  31. M. C. Bañuls et al., Simulating lattice gauge theories within quantum technologies, Eur. Phys. J. D 74, 165 (2020).
  32. Y. Meurice, R. Sakai, and J. Unmuth-Yockey, Tensor lattice field theory for renormalization and quantum computing, Rev. Mod. Phys. 94, 025005 (2022).
  33. D. Kadoh, Recent progress in the tensor renormalization group, Proc. Sci. LATTICE2021 (2022) 633.
  34. T. Felser, P. Silvi, M. Collura, and S. Montangero, Two-dimensional quantum-link lattice quantum electrodynamics at finite density, Phys. Rev. X 10, 041040 (2020).
  35. D. Robaina, M. C. Bañuls, and J. I. Cirac, Simulating 2+1D Z3 lattice gauge theory with an infinite projected entangled-pair state, Phys. Rev. Lett. 126, 050401 (2021).
  36. G. Magnifico, T. Felser, P. Silvi, and S. Montangero, Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks, Nat. Commun. 12, 3600 (2021).
  37. P. Emonts, A. Kelman, U. Borla, S. Moroz, S. Gazit, and E. Zohar, Finding the ground state of a lattice gauge theory with fermionic tensor networks: A 2+1D Z2 demonstration, Phys. Rev. D 107, 014505 (2023).
  38. A. Kelman, U. Borla, I. Gomelski, J. Elyovich, G. Roose, P. Emonts, and E. Zohar, Gauged Gaussian projected entangled pair states: A high dimensional tensor network formulation for lattice gauge theories, Phys. Rev. D 110, 054511 (2024).
  39. S. Akiyama, Y. Meurice, and R. Sakai, Tensor renormalization group for fermions, J. Phys. Condens. Matter 36, 343002 (2024).
  40. S. Paeckel, T. Kohler, A. Swoboda, S. R. Manmana, U. Schollwock, and C. Hubig, Time-evolution methods for matrix-product states, Ann. Phys. (Amsterdam) 411, 167998 (2019).
  41. Z. Davoudi, M. Hafezi, C. Monroe, G. Pagano, A. Seif, and A. Shaw, Towards analog quantum simulations of lattice gauge theories with trapped ions, Phys. Rev. Res. 2, 023015 (2020).
  42. L. Funcke, T. Hartung, K. Jansen, and S. Kühn, Review on quantum computing for lattice field theory, Proc. Sci. LATTICE2022 (2023) 228 [arXiv:2302.00467].
  43. C. W. Bauer et al., Quantum simulation for high-energy physics, PRX Quantum 4, 027001 (2023).
  44. J. C. Halimeh, M. Aidelsburger, F. Grusdt, P. Hauke, and B. Yang, Cold-atom quantum simulators of gauge theories, Nat. Phys. 21, 25 (2025).
  45. A. Di Meglio et al., Quantum computing for high-energy physics: State of the art and challenges, PRX Quantum 5, 037001 (2024).
  46. J. S. Pedernales, R. Di Candia, I. L. Egusquiza, J. Casanova, and E. Solano, Efficient quantum algorithm for computing n-time correlation functions, Phys. Rev. Lett. 113, 020505 (2014).
  47. H. Lamm, S. Lawrence, and Y. Yamauchi (NuQS Collaboration), General methods for digital quantum simulation of gauge theories, Phys. Rev. D 100, 034518 (2019).
  48. H. Lamm, S. Lawrence, and Y. Yamauchi (NuQS Collaboration), Parton physics on a quantum computer, Phys. Rev. Res. 2, 013272 (2020).
  49. M. G. Echevarria, I. L. Egusquiza, E. Rico, and G. Schnell, Quantum simulation of light-front parton correlators, Phys. Rev. D 104, 014512 (2021).
  50. R. D. Pisarski, Wilson loops in the Hamiltonian formalism, Phys. Rev. D 105, L111501 (2022).
  51. M. Kreshchuk, S. Jia, W. M. Kirby, G. Goldstein, J. P. Vary, and P. J. Love, Simulating hadronic physics on noisy intermediate-scale quantum devices using basis light-front quantization, Phys. Rev. A 103, 062601 (2021).
  52. M. Kreshchuk, W. M. Kirby, G. Goldstein, H. Beauchemin, and P. J. Love, Quantum simulation of quantum field theory in the light-front formulation, Phys. Rev. A 105, 032418 (2022).
  53. T. Li, X. Guo, W. K. Lai, X. Liu, E. Wang, H. Xing, D.-B. Zhang, and S.-L. Zhu (QuNu Collaboration), Partonic collinear structure by quantum computing, Phys. Rev. D 105, L111502 (2022).
  54. W. Qian, R. Basili, S. Pal, G. Luecke, and J. P. Vary, Solving hadron structures using the basis light-front quantization approach on quantum computers, Phys. Rev. Res. 4, 043193 (2022).
  55. S. Grieninger, K. Ikeda, and I. Zahed, Quasiparton distributions in massive QED2: Toward quantum computation, Phys. Rev. D 110, 076008 (2024).

  56. The method has been presented at the 41st International Symposium on Lattice Field Theory [57]. Similar ideas have since been applied to a purely fermionic model [58].

  57. M. Schneider, M. C. Bañuls, K. Cichy, and C.-J. D. Lin, Parton distribution functions in the Schwinger model with tensor networks, Proc. Sci. LATTICE2024 (2025) 024 [arXiv:2409.16996].
  58. Z.-B. Kang, N. Moran, P. Nguyen, and W. Qian, Partonic distribution functions and amplitudes using tensor network methods, J. High Energy Phys. 09 (2025) 176.
  59. J. Schwinger, Gauge invariance and mass. II, Phys. Rev. 128, 2425 (1962).
  60. S. Coleman, R. Jackiw, and L. Susskind, Charge shielding and quark confinement in the massive Schwinger model, Ann. Phys. (N.Y.) 93, 267 (1975).

  61. We use Einstein summation convention, the slashed notation A=γμAμ with γ0=σz=(100−1) and γ1=iσy=(0−110), Ψ¯=Ψ†γ0, and σ±=12(σx±iσy)=12(01∓11±10).

  62. J. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11, 395 (1975).
  63. M. C. Bañuls, K. Cichy, J. I. Cirac, and K. Jansen, The mass spectrum of the Schwinger model with matrix product states, J. High Energy Phys. 11 (2013) 158.
  64. C. J. Hamer, Z. Weihong, and J. Oitmaa, Series expansions for the massive Schwinger model in Hamiltonian lattice theory, Phys. Rev. D 56, 55 (1997).
  65. T. Banks, L. Susskind, and J. Kogut, Strong-coupling calculations of lattice gauge theories: (1+1)-dimensional exercises, Phys. Rev. D 13, 1043 (1976).
  66. C. Muschik, M. Heyl, E. Martinez, T. Monz, P. Schindler, B. Vogell, M. Dalmonte, P. Hauke, R. Blatt, and P. Zoller, U(1) Wilson lattice gauge theories in digital quantum simulators, New J. Phys. 19, 103020 (2017).
  67. See Supplemental Material at http://link.aps.org/supplemental/10.1103/lcyh-rxf3 for (A) the Hamiltonian formulation of the Schwinger model, (B) the implementation of the Wilson line in the physical subspace, (C) symmetry properties of the matrix elements, and (D) systematic errors and extrapolations. The Supplemental Material includes Refs. [68–70].
  68. D. Crewther and C. Hamer, Eigenvalues for the massive Schwinger model from a finite-lattice Hamiltonian approach, Nucl. Phys. B170, 353 (1980).
  69. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton paperbacks (Princeton University Press, Princeton, NJ, 1992), 10.2307/j.ctv10crg0r.
  70. E. Zohar and M. Burrello, Formulation of lattice gauge theories for quantum simulations, Phys. Rev. D 91, 054506 (2015).
  71. 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).
  72. T. Angelides, L. Funcke, K. Jansen, and S. Kühn, Computing the mass shift of Wilson and staggered fermions in the lattice Schwinger model with matrix product states, Phys. Rev. D 108, 014516 (2023).
  73. T. Angelides, L. Funcke, K. Jansen, and S. Kühn, Mass renormalization of the Schwinger model with Wilson and staggered fermions in the Hamiltonian lattice formulation, Proc. Sci. LATTICE2022 (2023) 046 [arXiv:2211.12169].
  74. J. Collins, Foundations of Perturbative QCD, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology (Cambridge University Press, Cambridge, England, 2011), 10.1017/CBO9780511975592.

  75. The rescaling factor ag22 from Appendix A [67] is included here.

  76. A. Irving and A. Thomas, Finite lattice fermions and gauge fields in 1+1 dimensions, Nucl. Phys. B215, 23 (1983).

  77. Notice that, at the initial time, a single (positive) static charge sits on the same vertex as the initial fermionic operator, thus canceling the change in electric flux introduced by the latter.

  78. From here on, we use a notation where site 0 is an even site in the middle of the lattice.

  79. H. F. Trotter, On the product of semi-groups of operators, Proc. Am. Math. Soc. 10, 545 (1959).
  80. M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations, Phys. Lett. A 146, 319 (1990).
  81. M. C. Bañuls, K. Cichy, J. I. Cirac, K. Jansen, and H. Saito, Thermal evolution of the Schwinger model with matrix product operators, Proc. Sci. LATTICE2015 (2016) 283 [arXiv:1511.00794].
  82. J. Dai, J. Hughes, and J. Liu, Perturbative analysis of the massless Schwinger model, Phys. Rev. D 51, 5209 (1995).
  83. J. Dai, J. Hughes, and J. Liu, Next-to-leading order parton model calculations in the massless Schwinger model, Phys. Lett. B 327, 347 (1994).
  84. G. Korchemsky and A. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B283, 342 (1987).
  85. Y. Guo, M. Schneider, T. Angelides, K. Jansen, C.-J. D. Lin, and Y. T. Su, Preparation of initial states with open and periodic boundary conditions on quantum devices using matrix product states, arXiv:2510.07125.
  86. Z. Davoudi, A. F. Shaw, and J. R. Stryker, General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory, Quantum 7, 1213 (2023).
  87. 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, 114510 (2024).
  88. S. Hariprakash, N. S. Modi, M. Kreshchuk, C. F. Kane, and C. W. Bauer, Strategies for simulating the time evolution of Hamiltonian lattice field theories, Phys. Rev. A 111, 022419 (2025).

Outline

Information

Phys. Rev. D 113, L011502– Published 29 January, 2026
  • Received 19 April 2025
  • Accepted 13 January 2026
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