- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

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

Phase Diagram of 1+1D Abelian-Higgs Model and Its Critical Point

Titas Chanda1,2,*, Maciej Lewenstein3,4, Jakub Zakrzewski2,5, and Luca Tagliacozzo6,7

  • 1The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
  • 2Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, Łojasiewicza 11, 30-348 Kraków, Poland
  • 3ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Avenue Carl Friedrich Gauss 3, 08860 Barcelona, Spain
  • 4ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain
  • 5Mark Kac Complex Systems Research Center, Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków, Poland
  • 6Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
  • 7Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Catalonia, Spain

  • *Corresponding author. tchanda@ictp.it
PDF

Phys. Rev. Lett. 128, 090601 – Published 28 February, 2022

DOI: https://doi.org/10.1103/PhysRevLett.128.090601

Abstract

We determine the phase diagram of the Abelian-Higgs model in one spatial dimension and time (1+1D) on a lattice. We identify a line of first order phase transitions separating the Higgs region from the confined one. This line terminates in a quantum critical point above which the two regions are connected by a smooth crossover. We analyze the critical point and find compelling evidence for its description as the product of two noninteracting systems: a massless free fermion and a massless free boson. However, we find also some surprising results that cannot be explained by our simple picture, suggesting this newly discovered critical point is an unusual one.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (76)

  1. C. Borgs and F. Nill, J. Stat. Phys. 47, 877 (1987).
  2. D. Brydges, J. Fröhlich, and E. Seiler, Nucl. Phys. B152, 521 (1979).
  3. T. Chanda, J. Zakrzewski, M. Lewenstein, and L. Tagliacozzo, Phys. Rev. Lett. 124, 180602 (2020).
  4. J. Schwinger, Phys. Rev. 82, 664 (1951).
  5. J. Schwinger, Phys. Rev. 125, 397 (1962).
  6. J. Schwinger, Phys. Rev. 128, 2425 (1962).
  7. S. Coleman, Ann. Phys. (N.Y.) 101, 239 (1976).
  8. P. W. Anderson, Phys. Rev. 130, 439 (1963).
  9. F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964).
  10. P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964).
  11. G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1964).
  12. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley Pub. Co, Reading, Mass, 1995).
  13. S. Coleman, Aspects of Symmetry: Selected Erice Lectures (Cambridge University Press, Cambridge, England, 1985).
  14. Z. Komargodski, A. Sharon, R. Thorngren, and X. Zhou, SciPost Phys. 6, 3 (2019).
  15. D. Tong, Lectures on gauge theory, https://www.damtp.cam.ac.uk/user/tong/gaugetheory.html.
  16. E. Fradkin and S. H. Shenker, Phys. Rev. D 19, 3682 (1979).
  17. D. J. E. Callaway and L. J. Carson, Phys. Rev. D 25, 531 (1982).
  18. O. Dutta, L. Tagliacozzo, M. Lewenstein, and J. Zakrzewski, Phys. Rev. A 95, 053608 (2017).
  19. C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero, E. Demler, N. Goldman, I. Bloch, and M. Aidelsburger, Nat. Phys. 15, 1168 (2019).
  20. F. Görg, K. Sandholzer, J. Minguzzi, R. Desbuquois, M. Messer, and T. Esslinger, Nat. Phys. 15, 1161 (2019).
  21. A. Mil, T. V. Zache, A. Hegde, A. Xia, R. P. Bhatt, M. K. Oberthaler, P. Hauke, J. Berges, and F. Jendrzejewski, Science 367, 1128 (2020).
  22. M. C. Bañuls, R. Blatt, J. Catani, A. Celi, J. I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero et al., Eur. Phys. J. D 74, 165 (2020).
  23. M. Aidelsburger et al., Phil. Trans. R. Soc. A 380, 20210064 (2022).
  24. K. Kasamatsu, I. Ichinose, and T. Matsui, Phys. Rev. Lett. 111, 115303 (2013).
  25. Y. Kuno, S. Sakane, K. Kasamatsu, I. Ichinose, and T. Matsui, Phys. Rev. D 95, 094507 (2017).
  26. D. González-Cuadra, E. Zohar, and J. I. Cirac, New J. Phys. 19, 063038 (2017).
  27. J. Zhang, J. Unmuth-Yockey, J. Zeiher, A. Bazavov, S.-W. Tsai, and Y. Meurice, Phys. Rev. Lett. 121, 223201 (2018).
  28. J. Unmuth-Yockey, J. Zhang, A. Bazavov, Y. Meurice, and S.-W. Tsai, Phys. Rev. D 98, 094511 (2018).
  29. J. Park, Y. Kuno, and I. Ichinose, Phys. Rev. A 100, 013629 (2019).
  30. Y. Meurice, Phys. Rev. D 104, 094513 (2021).
  31. D. Pekker and C. Varma, Annu. Rev. Condens. Matter Phys. 6, 269 (2015).
  32. C. Schori, T. Stöferle, H. Moritz, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 93, 240402 (2004).
  33. T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 92, 130403 (2004).
  34. M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P. Schauß, C. Gross, E. Demler, S. Kuhr, and I. Bloch, Nature (London) 487, 454 (2012).
  35. T. D. Kühner and H. Monien, Phys. Rev. B 58, R14741 (1998).
  36. I. Danshita and A. Polkovnikov, Phys. Rev. A 84, 063637 (2011).
  37. M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, Rev. Mod. Phys. 83, 1405 (2011).
  38. O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Lühmann, B. A. Malomed, T. Sowiński, and J. Zakrzewski, Rep. Prog. Phys. 78, 066001 (2015).
  39. U. Schollwöck, Ann. Phys. (Amsterdam) 326, 96 (2011).
  40. R. Orús, Ann. Phys. (Amsterdam) 349, 117 (2014).
  41. See Supplementary Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.128.090601, which includes Refs. [42–54], for the derivation of the lattice Hamiltonian, some extra results not reported in the main text, specifically the convincing evidence towards Lorentz invariance of the critical point by spectral analysis, and for the details about numerical simulations; For details about the density matrix renormalization group simulations.
  42. H. W. J. Blöte, J. L. Cardy, and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986).
  43. I. Affleck, Phys. Rev. Lett. 56, 746 (1986).
  44. I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman, J. Phys. A 22, 511 (1989).
  45. K. Hallberg, X. Q. G. Wang, P. Horsch, and A. Moreo, Phys. Rev. Lett. 76, 4955 (1996).
  46. J. C. Xavier, Phys. Rev. B 81, 224404 (2010).
  47. M. Dalmonte, E. Ercolessi, and L. Taddia, Phys. Rev. B 85, 165112 (2012).
  48. N. Chepiga and F. Mila, Phys. Rev. B 96, 054425 (2017).
  49. L. Tagliacozzo, A. Celi, and M. Lewenstein, Phys. Rev. X 4, 041024 (2014).
  50. B. Buyens, J. Haegeman, K. Van Acoleyen, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 113, 091601 (2014).
  51. P. Silvi, E. Rico, T. Calarco, and S. Montangero, New J. Phys. 16, 103015 (2014).
  52. I. Kull, A. Molnar, E. Zohar, and J. I. Cirac, Ann. Phys. (Amsterdam) 386, 199 (2017).
  53. S. Singh, R. N. C. Pfeifer, and G. Vidal, Phys. Rev. A 82, 050301(R) (2010).
  54. S. Singh, R. N. C. Pfeifer, and G. Vidal, Phys. Rev. B 83, 115125 (2011).

  55. The operators are defined as ϕ^j=12(a^j+b^j†),Π^j=i2(a^j†−b^j),ϕ^j†=12(a^j†+b^j),Π^j†=i2(b^j†−a^j),as discussed in, e.g., [3].

  56. S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
  57. S. R. White, Phys. Rev. B 48, 10345 (1993).
  58. S. R. White, Phys. Rev. B 72, 180403(R) (2005).
  59. U. Schollwöck, Rev. Mod. Phys. 77, 259 (2005).
  60. C. Hubig, I. P. McCulloch, U. Schollwöck, and F. A. Wolf, Phys. Rev. B 91, 155115 (2015).
  61. C. Callan and F. Wilczek, Phys. Lett. B 333, 55 (1994).
  62. G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003).
  63. P. Calabrese and J. Cardy, J. Stat. Mech. (2004) P06002.
  64. M. P. Nightingale, Physica (Amsterdam) 83A, 561 (1975).
  65. T. Koffel, M. Lewenstein, and L. Tagliacozzo, Phys. Rev. Lett. 109, 267203 (2012).
  66. A. S. Buyskikh, L. Tagliacozzo, D. Schuricht, C. A. Hooley, D. Pekker, and A. J. Daley, Phys. Rev. Lett. 123, 090401 (2019).

  67. Notice that our analysis is based on the full scaling form of the entropy in Eq. (3), which holds for conformally invariant systems only. The logarithmic divergence of the half chain entropy, on the other hand, can be observed also for systems that only possess scale invariance [65, 68, 69].

  68. G. Refael and J. E. Moore, Phys. Rev. Lett. 93, 260602 (2004).
  69. J. I. Latorre, R. Orús, E. Rico, and J. Vidal, Phys. Rev. A 71, 064101 (2005).
  70. A. B. Zomolodchikov, JETP Lett. 43, 730 (1986), http://www.jetpletters.ru/ps/1413/article_21504.pdf.
  71. J. Cardy and E. Tonni, J. Stat. Mech. (2016) 123103.
  72. in Finite-Size Scaling, Current Physics-Sources and Comments, edited by J. L. Cardy (Elsevier, New York, 1988), Vol. 2, pp. 1–7.
  73. J. Latorre, E. Rico, and G. Vidal, Quantum Inf. Comput. 4, 48 (2004).
  74. M. Campostrini and E. Vicari, Phys. Rev. A 81, 063614 (2010).
  75. A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosenbaum, and D. Sen, Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge University Press, Cambridge, England, 2015).
  76. T. Chanda, M. Dalmonte, M. Lewenstein, J. Zakrzewski, and L. Tagliacozzo (unpublished).

Outline

Information

Phys. Rev. Lett. 128, 090601– Published 28 February, 2022
  • Received 6 August 2021
  • Accepted 8 February 2022
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