Path-integral representation for quantum spin models: Application to the quantum cavity method and Monte Carlo simulations

Physical Review B - Condensed Matter and Materials Physics

Volumn 78, Issue 13, 2008, Pages

  Krzakala, Florent ^a   Rosso, Alberto ^b   Semerjian, Guilhem ^c   Zamponi, Francesco ^c  

^a Centre National de la Recherche Scientifique   (France)

^b UNIVERSITÉ PARIS SUD   (France)

^c CNRS   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 55349147372     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.78.134428     Document Type: Article

References (71)

1. R. Abou-Chacra, D. J. Thouless, and P. W. Anderson, J. Phys. C 6, 1734 (1973). 10.1088/0022-3719/6/10/009
2. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 10.1103/RevModPhys.68.13
3. D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975). 10.1103/PhysRevLett.35.1792
4. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, New York, 1989).
5. L. Viana and A. J. Bray, J. Phys. C 18, 3037 (1985). 10.1088/0022-3719/ 18/15/013
6. M. Mézard and G. Parisi, Eur. Phys. J. B 20, 217 (2001). 10.1007/PL00011099
7. R. Monasson and R. Zecchina, Phys. Rev. E 56, 1357 (1997). 10.1103/PhysRevE.56.1357
8. M. Mézard, G. Parisi, and R. Zecchina, Science 297, 812 (2002). 10.1126/science.1073287
9. F. Krzakala, A. Montanari, F. Ricci-Tersenghi, G. Semerjian, and L. Zdeborová, Proc. Natl. Acad. Sci. U.S.A. 104, 10318 (2007). 10.1073/pnas.0703685104
10. M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
11. R. Monasson, J. Phys. A 31, 513 (1998). 10.1088/0305-4470/31/2/012
12. G. Biroli and L. F. Cugliandolo, Phys. Rev. B 64, 014206 (2001). 10.1103/PhysRevB.64.014206
13. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 1999).
14. A. J. Bray and M. A. Moore, J. Phys. C 13, L655 (1980). 10.1088/0022-3719/13/24/005
15. Y. Y. Goldschmidt, Phys. Rev. B 41, 4858 (1990). 10.1103/PhysRevB.41.4858
16. H. Nishimori and Y. Nonomura, J. Phys. Soc. Jpn. 65, 3780 (1996). 10.1143/JPSJ.65.3780
17. A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. Lett. 85, 840 (2000). 10.1103/PhysRevLett.85.840
18. L. F. Cugliandolo, D. R. Grempel, and C. A. da Silva Santos, Phys. Rev. B 64, 014403 (2001). 10.1103/PhysRevB.64.014403
19. M. P. Kennett, C. Chamon, and J. Ye, Phys. Rev. B 64, 224408 (2001). 10.1103/PhysRevB.64.224408
20. M. Müller and L. B. Ioffe, arXiv:0711.2668 (unpublished).
21. C. Laumann, A. Scardicchio, and S. L. Sondhi, Phys. Rev. B 78, 134424 (2008). 10.1103/PhysRevB.78.134424
22. M. Tarzia and G. Biroli, Europhys. Lett. 82, 67008 (2008). 10.1209/0295-5075/82/67008
23. S. Knysh and V. N. Smelyanskiy, arXiv:0803.0149 (unpublished).
24. D. Nagaj, E. Farhi, J. Goldstone, P. Shor, and I. Sylvester, Phys. Rev. B 77, 214431 (2008). 10.1103/PhysRevB.77.214431
25. B. Apolloni, N. Cesa-Bianchi, and D. de Falco, in Stochastic Processes, Physics and Geometry, Proceedings of the 1988 Ascona-Locarno Conference, Vol. 97, edited by, S. Albeverio,,, G. Casati,,, U. Cattaneo,,, D. Merlini,, and, R. Moresi, (World Scientific, Singapore, 1990).
26. T. Kadowaki and H. Nishimori, Phys. Rev. E 58, 5355 (1998). 10.1103/PhysRevE.58.5355
27. E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science 292, 472 (2001). 10.1126/science.1057726
28. T. Jörg, F. Krzakala, J. Kurchan, and A. C. Maggs, Phys. Rev. Lett. 101, 147204 (2008). 10.1103/PhysRevLett.101.147204
29. M. Suzuki, Prog. Theor. Phys. 56, 1454 (1976). 10.1143/PTP.56.1454
30. E. Farhi and S. Gutmann, Ann. Phys. (N.Y.) 213, 182 (1992). 10.1016/0003-4916(92)90288-W
31. R. P. Feynman, Phys. Rev. 91, 1291 (1953). 10.1103/PhysRev.91.1291
32. J. Ginibre, Commun. Math. Phys. 10, 140 (1968). 10.1007/BF01654238
33. G. Gallavotti, S. Miracle-Sole, and D. W. Robinson, Commun. Math. Phys. 10, 311 (1968).
34. M. Aizenman and B. Nachtergaele, Commun. Math. Phys. 164, 17 (1994). 10.1007/BF02108805
35. D. Ioffe, in Stochastic Geometry of Classical and Quantum Ising Models, Springer Lecture Notes in Mathematics (Springer, New York, 2008).
36. B. B. Beard and U.-J. Wiese, Phys. Rev. Lett. 77, 5130 (1996). 10.1103/PhysRevLett.77.5130
37. N. V. Prokof'ev, B. V. Svistunov, and I. S. Tupitsyn, JETP 87, 310 (1998). 10.1134/1.558661
38. H. Rieger and N. Kawashima, Eur. Phys. J. B 9, 233 (1999). 10.1007/s100510050761
39. H. G. Evertz, Adv. Phys. 52, 1 (2003). 10.1080/0001873021000049195
40. The weight w (σ) becomes λB/ Ns raised to the number of discontinuities in the spin trajectory and the factors λ/ Ns are absorbed in the change in variables from discrete to continuous time.
41. This can be done with a number of operations of order p. Let us define Zi (σ0,..., σi |h) = σi+1,..., σp Z (σ0,..., σp |h). First draw σ0 with probability Z0 (σ0 |h) /Z (h), then σ1 according to Z1 (σ0, σ1 |h) / Z0 (σ0 |h), and so on and so forth.
42. S. Janson, T. Luczak, and A. Rucinski, Random Graphs (Wiley, New York, 2000).
43. This is the graph theoretic distance between two vertices i and j, defined as the length of a shortest path of adjacent vertices joining i and j along the graph.
44. F. Kschischang, B. J. Frey, and H.-A. Loeliger, IEEE Trans. Inf. Theory 47, 498 (2001). 10.1109/18.910572
45. J. S. Yedidia, W. T. Freeman, and Y. Weiss, Adv. Neural Inf. Process. Syst. 13, 689 (2001).
46. A. Dembo and A. Montanari, arXiv:0804.4726 (unpublished).
47. F. Alet, P. Dayal, A. Grzesik, A. Honecker, M. Koerner, A. Laeuchli, S. R. Manmana, I. P. McCulloch, F. Michel, R. M. Noack, G. Schmid, U. Schollwoeck, F. Stoeckli, S. Todo, S. Trebst, M. Troyer, P. Werner, S. Wessel, and The ALPS Collaboration, J. Phys. Soc. Jpn. 74, 30 (2005).
48. R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 57, 2607 (1986). 10.1103/PhysRevLett.57.2607
49. O. Redner, J. Machta, and L. F. Chayes, Phys. Rev. E 58, 2749 (1998). 10.1103/PhysRevE.58.2749
50. J. Houdayer, Eur. Phys. J. B 22, 479 (2001). 10.1007/PL00011151
51. T. Jörg, Phys. Rev. B 73, 224431 (2006). 10.1103/PhysRevB.73.224431
52. E. Marinari, G. Parisi, and J. J. Ruiz-Lorenzo, in Spin Glasses and Random Fields, edited by, A. P. Young, (World Scientific, Singapore, 1997).
53. K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65, 1604 (1996). 10.1143/JPSJ.65.1604
54. M. Guo, R. N. Bhatt, and D. A. Huse, Phys. Rev. Lett. 72, 4137 (1994) 10.1103/PhysRevLett.72.4137
55. M. Guo, R. N. Bhatt, and D. A. Huse, Phys. Rev. B 54, 3336 (1996). 10.1103/PhysRevB.54.3336
56. H. Rieger and A. P. Young, Phys. Rev. Lett. 72, 4141 (1994). 10.1103/PhysRevLett.72.4141
57. H. Rieger, in Quantum Annealing and Other Optimization Methods, Lecture Notes in Physics (Springer, New York, 2005).
58. F. Alet, S. Wessel, and M. Troyer, Phys. Rev. E 71, 036706 (2005). 10.1103/PhysRevE.71.036706
59. R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987). 10.1103/PhysRevLett.58.86
60. D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 2nd ed. (Cambridge University Press, Cambridge, England, 2005).
61. Finite Size Scaling and Numerical Simulation of Statistical Systems, edited by, V. Privman, (World Scientific, Singapore, 1990).
62. E. Brezin, J. Phys. (France) 43, 15 (1982).
63. R. Botet and R. Jullien, Phys. Rev. B 28, 3955 (1983). 10.1103/PhysRevB.28.3955
64. J. L. Jones and A. P. Young, Phys. Rev. B 71, 174438 (2005). 10.1103/PhysRevB.71.174438
65. F. Liers, M. Palassini, A. K. Hartmann, and M. Jünger, Phys. Rev. B 68, 094406 (2003). 10.1103/PhysRevB.68.094406
66. T. Jörg, H. G. Katzgraber, and F. Krzakala, Phys. Rev. Lett. 100, 197202 (2008). 10.1103/PhysRevLett.100.197202
67. S. Franz, M. Mézard, F. Ricci-Tersenghi, M. Weigt, and R. Zecchina, Europhys. Lett. 55, 465 (2001). 10.1209/epl/i2001-00438-4
68. M. B. Hastings, Phys. Rev. B 76, 201102 (R) (2007). 10.1103/PhysRevB.76. 201102
69. R. Brout, K. A. Müller, and H. Thoma, Solid State Commun. 4, 507 (1966). 10.1016/0038-1098(66)90412-1
70. R. M. Stratt, Phys. Rev. B 33, 1921 (1986). 10.1103/PhysRevB.33.1921
71. L. Chayes, N. Crawford, D. Ioffe, and A. Levit, J. Stat. Phys. 133, 131 (2008). 10.1007/s10955-008-9608-x