- Physical Review Journals
All JournalsPhysics Magazine

Physical Review C

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

Massive Thirring / sine-Gordon model with nonzero current density

Eric Oevermann*

Thomas D. Cohen†

  • *Contact author: eric.oevermann@uni-jena.de
  • †Contact author: cohen@umd.edu
PDF

Phys. Rev. C 114, 045202 – Published 1 October, 2026

DOI: https://doi.org/10.1103/vyq7-6hxx

Abstract

This paper determines the zero-temperature equation of state for the massive Thirring / sine-Gordon model. This demonstrates recently derived model-independent upper and lower bounds on the zero-temperature equation of state with fixed number density from systems with a nonzero current density. That approach is potentially valuable as Monte Carlo calculations with a current density avoid the sign problem in the Euclidean formulation. An advantage to illustrating these bounds in the massive Thirring/sine-Gordon model is that the relevant calculations with both a number density and a current density can be done using a Bethe ansatz. For this model, optimal bounds constrain the energy density as a function of number density by a factor of two from above and below at high densities for all choices of couplings. The lower bound becomes exact at low densities, while the upper bound approaches the worst constraint of a factor of 4.90.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (41)

  1. O. Philipsen, The QCD equation of state from the lattice, Prog. Part. Nucl. Phys. 70, 55 (2013).
  2. H.-T. Ding, F. Karsch, and S. Mukherjee, Thermodynamics of strong-interaction matter from lattice QCD, Int. J. Mod. Phys. E 24, 1530007 (2015).
  3. G. Aarts, Introductory lectures on lattice QCD at nonzero baryon number, J. Phys.: Conf. Ser. 706, 022004 (2016).
  4. A. Monnai, B. Schenke, and C. Shen, QCD equation of state at finite chemical potentials for relativistic nuclear collisions, Int. J. Mod. Phys. A 36, 2130007 (2021).
  5. G. F. Burgio, H. J. Schulze, I. Vidana, and J. B. Wei, Neutron stars and the nuclear equation of state, Prog. Part. Nucl. Phys. 120, 103879 (2021).
  6. K. Chatziioannou, H. T. Cromartie, S. Gandolfi, I. Tews, D. Radice, A. W. Steiner, and A. L. Watts, Neutron stars and the dense matter equation of state, Rev. Mod. Phys. 97, 045007 (2025).
  7. T. D. Cohen and E. S. Werbos, Magnetization of the QCD vacuum at large fields, Phys. Rev. C 80, 015203 (2009).
  8. J. O. Andersen, W. R. Naylor, and A. Tranberg, Phase diagram of QCD in a magnetic field: A review, Rev. Mod. Phys. 88, 025001 (2016).
  9. G. S. Bali, F. Bruckmann, G. Endrödi, S. D. Katz, and A. Schäfer, The QCD equation of state in background magnetic fields, J. High Energy Phys. 08 (2014) 177.
  10. P. Adhikari et al., Strongly interacting matter in extreme magnetic fields, Prog. Part. Nucl. Phys. 146, 104199 (2026).
  11. S. Borsanyi and P. Parotto, The QCD phase diagram, arXiv:2512.08843.
  12. C. Ratti, Equation of state for QCD from lattice simulations, Prog. Part. Nucl. Phys. 129, 104007 (2023).
  13. J. O. Andersen and M. Strickland, The equation of state for dense QCD and quark stars, Phys. Rev. D 66, 105001 (2002).
  14. E. S. Fraga, A. Kurkela, and A. Vuorinen, Interacting quark matter equation of state for compact stars, Astrophys. J. Lett. 781, L25 (2014).
  15. E. Annala, T. Gorda, A. Kurkela, J. Nättilä, and A. Vuorinen, Evidence for quark-matter cores in massive neutron stars, Nat. Phys. 16, 907 (2020).
  16. Y. Fujimoto and K. Fukushima, Equation of state of cold and dense QCD matter in resummed perturbation theory, Phys. Rev. D 105, 014025 (2022).
  17. Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, and K. K. Szabo, The order of the quantum chromodynamics transition predicted by the standard model of particle physics, Nature (London) 443, 675 (2006).
  18. C. Ratti, Lattice QCD and heavy ion collisions: A review of recent progress, Rept. Prog. Phys. 81, 084301 (2018).
  19. J.-L. Jiang, S.-P. Tang, Y.-Z. Wang, Y.-Z. Fan, and D.-M. Wei, PSR J0030+0451, GW170817 and the nuclear data: Joint constraints on equation of state and bulk properties of neutron stars, Astrophys. J. 892, 55 (2020).
  20. M. Oertel, M. Hempel, T. Klähn, and S. Typel, Equations of state for supernovae and compact stars, Rev. Mod. Phys. 89, 015007 (2017).
  21. T. D. Cohen, Method to obtain bounds on the equation of state of cold nuclear matter from imaginary chemical potentials, Phys. Rev. Lett. 136, 181901 (2026).
  22. M. Troyer and U.-J. Wiese, Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations, Phys. Rev. Lett. 94, 170201 (2005).
  23. E. Y. Loh, J. E. Gubernatis, R. T. Scalettar, S. R. White, D. J. Scalapino, and R. L. Sugar, Sign problem in the numerical simulation of many-electron systems, Phys. Rev. B 41, 9301 (1990).
  24. G. Aarts and D. Sexty, Lattice field theories with a sign problem, arXiv:2604.24290.
  25. P. de Forcrand, Simulating QCD at finite density, PoS LAT2009, 010 (2010).
  26. C. Gattringer and K. Langfeld, Approaches to the sign problem in lattice field theory, Int. J. Mod. Phys. A 31, 1643007 (2016).
  27. C. E. Berger, L. Rammelmüller, A. C. Loheac, F. Ehmann, J. Braun, and J. E. Drut, Complex Langevin and other approaches to the sign problem in quantum many-body physics, Phys. Rep. 892, 1 (2021).
  28. M. D'Elia and M.-P. Lombardo, Finite density QCD via imaginary chemical potential, Phys. Rev. D 67, 014505 (2003).
  29. T. D. Cohen, Functional integrals for QCD at nonzero chemical potential and zero density, Phys. Rev. Lett. 91, 222001 (2003).
  30. T. D. Cohen, The Silver Blaze problem in QCD arXiv:2601.21053.
  31. F. Gao and J. M. Pawlowski, QCD phase structure from functional methods, Phys. Rev. D 102, 034027 (2020).
  32. F. D. M. Haldane, Quantum field ground state of the sine-Gordon model with finite soliton density: Exact results, J. Phys. A: Math. Gen. 15, 507 (1982).
  33. S. Coleman, The quantum sine-Gordon equation as the massive Thirring model, Phys. Rev. D 11, 2088 (1975).
  34. S. Mandelstam, Soliton operators for the quantized sine-Gordon equation, Phys. Rev. D 11, 3026 (1975).
  35. R. Rajaraman, Solitons and Instantons. An Introduction to Solitons and Instantons in Quantum Field Theory (North-Holland, Amsterdam, 1982).
  36. T. D. Cohen and M. Li, Exact effective potential for a scalar source coupled to the sine-Gordon model: Test of effective potentials for composite nucleons, Phys. Rev. C 42, 970 (1990).
  37. T. D. Cohen, Mean field theory and solitonic matter, Nucl. Phys. A 495, 545 (1989).
  38. H. Bergknoff and H. B. Thacker, Structure and solution of the massive Thirring model, Phys. Rev. D 19, 3666 (1979).
  39. H. B. Thacker, Exact integrability in quantum field theory and statistical systems, Rev. Mod. Phys. 53, 253 (1981).
  40. M. G. Alford, A. Kapustin, and F. Wilczek, Imaginary chemical potential and finite fermion density on the lattice, Phys. Rev. D 59, 054502 (1999).
  41. E. S. Allhands, A numerical method for solving Fredholm integral equations of the second kind (2022).

Outline

Information

Phys. Rev. C 114, 045202– Published 1 October, 2026
  • Received 8 June 2026
  • Accepted 3 September 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation