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Observations outside the light cone: Algorithms for nonequilibrium and thermal states

M. B. Hastings

  • Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
  • Kavli Institute of Theoretical Physics, University of California, Santa Barbara, California 93106, USA

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Phys. Rev. B 77, 144302 – Published 9 April, 2008

DOI: https://doi.org/10.1103/PhysRevB.77.144302

Abstract

We apply algorithms based on Lieb-Robinson bounds to simulate time-dependent and thermal quantities in quantum systems. For time-dependent systems, we modify a previous mapping to quantum circuits to significantly reduce the computer resources required. This modification is based on a principle of “observing” the system outside the light cone. We apply this method to study spin relaxation in systems started out of equilibrium with initial conditions that give rise to a very rapid entanglement growth. We also show that it is possible to approximate time evolution under a local Hamiltonian by a quantum circuit whose light cone naturally matches the Lieb-Robinson velocity. Asymptotically, these modified methods allow a doubling of the system size that one can obtain as compared to a direct simulation. We then consider a different problem of thermal properties of disordered spin chains and use quantum belief propagation to average over different configurations. We test this algorithm on one-dimensional systems with mixed ferromagnetic and antiferromagnetic bonds, where we can compare to quantum Monte Carlo, and then we apply it to the study of disordered, frustrated spin systems.

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References (38)

  1. U. Schollwoeck, Rev. Mod. Phys. 77, 259 (2005).
  2. F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Phys. Rev. Lett. 93, 207204 (2004); M. Zwolak and G. Vidal, ibid. 93, 207205 (2004).
  3. G. Vidal, Phys. Rev. Lett. 91, 147902 (2003); 93, 040502 (2004).
  4. M. B. Hastings, J. Stat. Mech.: Theory Exp. (2007) P08024.
  5. M. B. Hastings, Phys. Rev. B 73, 085115 (2006).
  6. M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Phys. Rev. Lett. 100, 070502 (2008).
  7. T. J. Osborne, Phys. Rev. Lett. 97, 157202 (2006).
  8. E. H. Lieb and D. W. Robinson, Commun. Math. Phys. 28, 251 (1972).
  9. M. B. Hastings and T. Koma, Commun. Math. Phys. 265, 781 (2006).
  10. B. Nachtergaele and R. Sims, Commun. Math. Phys. 265, 119 (2006).
  11. M. B. Hastings, Phys. Rev. B 69, 104431 (2004).
  12. G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003).
  13. V. Eisler and I. Peschel, J. Stat. Mech.: Theory Exp. (2007) P06005.
  14. P. Calabrese and J. Cardy, J. Stat. Mech.: Theory Exp. (2007) P10004.
  15. D. Gobert, C. Kollath, U. Schollwock, and G. Schutz, Phys. Rev. E 71, 036102 (2005).
  16. S. Bravyi, M. B. Hastings, and F. Verstraete, Phys. Rev. Lett. 97, 050401 (2006); J. Eisert and T. J. Osborne, ibid. 97, 150404 (2006).
  17. G. De Chiara, S. Montangero, P. Calabrese, and R. Fazio, J. Stat. Mech.: Theory Exp. (2006) P03001.
  18. P. Calabrese and J. Cardy, J. Stat. Mech.: Theory Exp. (2005) P04010.
  19. S. Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995); M. Fannes, B. Nachtergaele, and R. F. Werner, Commun. Math. Phys. 144, 443 (1992); F. Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett. 93, 227205 (2004).

  20. To understand why 2N∕2 states suffice, consider how much entanglement is possible across each bond. If we divide the chain of N sites into a subchain of n sites and another subchain of N−n sites, with n⩽N−n, then there are only 2n states available on the subchain and so it is possible to perform a Schmidt decomposition and find a matrix product state representation with 2n states across that bond. The worst case is midway across the chain, when n=N∕2. If we wish to write matrix product operators instead of states, then these estimates are increased to 4N∕2.

  21. M. B. Hastings, Phys. Rev. B 76, 201102(R) (2007); M. Leifer and D. Poulin, arXiv:0708.1337 (unpublished).
  22. X. Wang and T. Xiang, Phys. Rev. B 56, 5061 (1997).

  23. The Krylov space methods may lead to some improvement in the time required as compared to the series methods. However, since we are considering a situation in which the entropy rapidly increases, and hence the state exp(iHLt)ΨL has very little overlap with ΨL, it seems likely that it would be necessary to take a fairly large k to obtain accurate results for large t. Therefore, we are not sure how much advantage would actually be obtained by using Krylov methods here.

  24. V. E. Korepin, N. M. Boloiubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions (Cambridge University Press, Cambridge, England, 1993).
  25. M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, arXiv:0703314 (unpublished).
  26. P. Zhang, Zh. Xu, H. Ying, and J. Dai, Proceedings of the Conference on Non-Perturbative Quantum Field Theory: Lattice and Beyond, Guangzhou, P. R. China, November 2004 (unpublished).
  27. E. Yusuf and Kun Yang, Phys. Rev. B 68, 024425 (2003).

  28. The quantum belief propagation method, as currently implemented, requires manipulating operators, rather than pure states, which increases the computational effort.

  29. D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. Phys. (N.Y.) 321, 1126 (2006); D. M. Basko, I. L. Aleiner, and B. L. Altshuler, in Problems of Condensed Matter Physics, edited by Alexei L. Ivanov and Sergei G. Tikhodeev (Oxford University Press, Oxford, 2007).
  30. C. K. Burrell and T. J. Osborne, Phys. Rev. Lett. 99, 167201 (2007).
  31. H. Manaka, I. Yamada, and H. Aruga Katori, Phys. Rev. B 63, 104408 (2001).
  32. H. Manaka, I. Yamada, and K. Yamaguchi, J. Phys. Soc. Jpn. 66, 564 (1997).
  33. H. Manaka and I. Yamada, J. Phys. Soc. Jpn. 66, 1908 (1997).
  34. H. Hida, J. Phys. Soc. Jpn. 72, 688 (2003).
  35. T. Nakamura, J. Phys. Soc. Jpn. 72, 789 (2003).
  36. J. E. Lorenzo, K. Hirota, G. Shirane, J. M. Tranquada, M. Hase, K. Uchinokura, H. Kojima, I. Tanaka, and Y. Shibuya, Phys. Rev. B 50, 1278 (1994).
  37. M. Hase, K. Ozawa, and N. Shinya, Phys. Rev. B 68, 214421 (2003).
  38. T. Barthel and U. Schollwöck, arXiv:0711.4896 (unpublished).

Outline

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Phys. Rev. B 77, 144302– Published 9 April, 2008
  • Received 16 January 2008

© 2008 American Physical Society

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