- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Conformally invariant free-parafermionic quantum chains with multispin interactions

Francisco C. Alcaraz* and Lucas M. Ramos†

  • Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-970, São Carlos, SP, Brazil

  • *alcaraz@ifsc.usp.br
  • †lucas.morais@ifsc.usp.br
PDF

Phys. Rev. E 109, 044138 – Published 16 April, 2024

DOI: https://doi.org/10.1103/PhysRevE.109.044138

Abstract

We calculate the spectral properties of two related families of non-Hermitian free-particle quantum chains with N-multispin interactions (N=2,3,...). The first family have a Z(N) symmetry and are described by free parafermions. The second have a U(1) symmetry and are generalizations of XX quantum chains described by free fermions. The eigenspectra of both free-particle families are formed by the combination of the same pseudo-energies. The models have a multicritical point with dynamical critical exponent z=1. The finite-size behavior of their eigenspectra, as well as the entanglement properties of their ground-state wave function, indicate the models are conformally invariant. The models with open and periodic boundary conditions show quite distinct physics due to their non-Hermiticity. The models defined with open boundaries have a single conformal invariant phase, while the XX multispin models show multiple phases with distinct conformal central charges in the periodic case.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (41)

  1. T. Schultz, D. Mattis and E. H. Lieb, Two-dimensional Ising model as a soluble problem of many fermions, Rev. Mod. Phys. 36, 856 (1964).
  2. P. Pfeuty, The one-dimensional Ising model with a transverse field, Ann. Phys. 57, 79 (1970).
  3. P. Fendley, Free fermions in disguise, J. Phys. A: Math. Theor. 52, 335002 (2019).
  4. R. J. Baxter, A simple solvable ZN Hamiltonian, Phys. Lett. A 140, 155 (1989).
  5. R. J. Baxter, Superintegrable chiral Potts model: Thermodynamic properties, an inverse model and a simple associated Hamiltonian, J. Stat. Phys. 57, 1 (1989).
  6. P. Fendley, Free parafermions, J. Phys. A: Math. Theor. 47, 075001 (2014).
  7. R. J. Baxter, The τ2 model and parafermions, J. Phys. A: Math. Theor. 47, 315001 (2014).
  8. H. Au-Yang and J. H. H. Perk, Parafermions in the τ2 model, J. Phys. A: Math. Theor. 47, 315002 (2014).
  9. H. Au-Yang and J. H. H. Perk, Parafermions in the tau-2 model II, arXiv:1606.06319.
  10. F. C. Alcaraz, M. T. Batchelor, and Z.-Z. Liu, Energy spectrum and critical exponents of the free parafermion ZN spin chain, J. Phys. A: Math. Theor. 50, 16LT03 (2017).
  11. F. C. Alcaraz and M. T. Batchelor, Anomalous bulk behavior in the free parafermion Z(N) spin chain, Phys. Rev. E 97, 062118 (2018).
  12. F. C. Alcaraz and R. A. Pimenta, Free fermionic and parafermionic quantum spin chains with multispin interactions, Phys. Rev. B 102, 121101(R) (2020).
  13. F. C. Alcaraz and R. A. Pimenta, Integrable quantum spin chains with free fermionic and parafermionic spectrum, Phys. Rev. B 102, 235170 (2020).
  14. S. J. Elman, A. Chapman, and S. T. Flammia, Free Fermions behind the disguise, Commun. Math. Phys. 388, 969 (2021); A. Chapman, S. T. Flammia, and A. J. Kollár, Free-Fermion subsystem codes, PRX Quantum 3, 030321 (2022); A. Chapman, S. J. Elman, and R. L. Mann, A unified graph-theoretic framework for Free-Fermion solvability, arXiv:2305.15625.
  15. B. Pozsgay, Quantum circuits with free fermions in disguise, arXiv:2402.02984.
  16. P. Fendley and B. Pozsgay, Free fermions with no Jordan-Wigner transformation, arXiv:2310.19897.
  17. P. D. Mannheim, PT symmetry as a necessary and sufficient condition for unitary time evolution, Phil. Tram. R. Soc. A 371, 2120060 (2012).
  18. F. C. Alcaraz and R. A. Pimenta, Free-parafermionic Z(N) and free-fermionic XY quantum chains, Phys. Rev. E 104, 054121 (2021).
  19. F. C. Alcaraz, R. A. Pimenta and J. Sirker, Ising analogs of quantum spin chains with multispin interactions, Phys. Rev. B 107, 235136 (2023).
  20. F. C. Alcaraz, J. A. Hoyos, and R. A. Pimenta, Powerful method to evaluate the gaps of free-particle quantum critical chains, Phys. Rev. B 104, 174206 (2021).
  21. R. A. Henry and M. T. Batchelor, Exceptional points in the Baxter-Fendley free parafermion model, Sci. Post 15, 016 (2023).
  22. F. C. Alcaraz, M. N. Barber, and M. T. Batchelor, Conformal invariance, the XXZ chain and the operator content of two-dimensional critical systems, Ann. Phys. 182, 280 (1988).
  23. J. L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270, 186 (1986).
  24. Q. I. Rahman and G. Schmeisser, in Analytic Theory of Polynomials, edited by H. G. Dales and P. M. Neumanni, London Mathematical Society Monographs Vol. 26 (Clarendon Press, Oxford, 2002; Academic Press, New York, 1983).
  25. F. C. Alcaraz, J. A. Hoyos and R. A. Pimenta, Random free-fermion quantum spin chain with multispin interactions, Phys. Rev. B 108, 214413 (2023).
  26. H. W. J. Blöte, J. L. Cardy and M. P. Nightingale, Conformal invariance, the central charge, and universal finite-size amplitudes at criticality, Phys. Rev. Lett. 56, 742 (1986).
  27. L. Titvinidze and G. Japaridze, Phase diagram of the spin extended model, Eur. Phys. J. B 32, 383 (2003).
  28. I. Affleck, Universal term in the free energy at a critical point and conformal anomaly, Phys. Rev. Lett. 56, 746 (1986).
  29. L. P. Kadanoff, Multicritical behavior at the Kosterlitz-Thouless critical point, Ann. Phys. 120, 39 (1979).
  30. L. P. Kadanoff and A. Brown, Correlation functions on the critical lines of the Baxter and Ashkin-Teller models, Ann. Phys. 121, 318 (1979).
  31. F. C. Alcaraz, M. Baake and V. Rittenberg, Operator content of the XXZ chain, J. Phys. A: Math. Gen. 21, L117 (1988).
  32. F. C. Alcaraz, U. Grimm and V. Rittenberg, The XXZ Heisenberg chain, conformal invariance and the operator content of c<1 systems, Nucl. Phys. B 316, 735 (1989).
  33. V. Alba, M. Fagotti, and P. Calabrese, Entanglement entropy of excited states, J. Stat. Mech. (2009) P10020.
  34. Y.-B. Guo, Y.-C. Yu, R.-Z. Huang, L.-P. Yang, R.-Z. Chi, H.-J. Liao, and T. Xiang, Entanglement entropy of non-Hermitian free fermions, J. Phys.: Condens. Matter 33, 475502 (2021).
  35. C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424, 443 (1994).
  36. P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  37. P. Calabrese and J. L. Cardy, Entanglement entropy and conformal field theory, J. Phys. A: Math. Theor. 42, 504005 (2009).
  38. V. Korepin, Universality of entropy scaling in one dimensional gapless models, Phys. Rev. Lett. 92, 096402 (2004).
  39. G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90, 227902 (2003).
  40. I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A: Math. Gen. 36, L205 (2003).
  41. P.-Y. Chang, J-S. You, X. Wen, and S. Ryu, Entanglement spectrum and entropy in topological non-Hermitian systems and nonunitary conformal field theory, Phys. Rev. Res. 2, 033069 (2020).

Outline

Information

Phys. Rev. E 109, 044138– Published 16 April, 2024
  • Received 16 February 2024
  • Accepted 21 March 2024
Crossmark - Check for updates button

©2024 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation