AKLT models with quantum spin glass ground states

Physical Review B - Condensed Matter and Materials Physics

Volumn 81, Issue 17, 2010, Pages

  Laumann, C R ^a   Parameswaran, S A ^a   Sondhi, S L ^a   Zamponi, F ^b  

^a PRINCETON UNIVERSITY   (United States)

^b ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77955465334     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.81.174204     Document Type: Article

References (33)

1. I. Affleck, T. Kennedy, E. Lieb, and H. Tasaki, Commun. Math. Phys. 115, 477 (1988). 10.1007/BF01218021
2. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987). 10.1103/PhysRevLett.59.799
3. S. A. Parameswaran, S. L. Sondhi, and D. P. Arovas, Phys. Rev. B 79, 024408 (2009). 10.1103/PhysRevB.79.024408
4. M. Fannes, B. Nachtergaele, and R. Werner, J. Stat. Phys. 66, 939 (1992). 10.1007/BF01055710
5. C. R. Laumann, S. A. Parameswaran, and S. L. Sondhi, Phys. Rev. B 80, 144415 (2009). 10.1103/PhysRevB.80.144415
6. D. P. Arovas, A. Auerbach, and F. D. M. Haldane, Phys. Rev. Lett. 60, 531 (1988). 10.1103/PhysRevLett.60.531
7. A few comments on the nature of the classical statistical state space of a vector spin are in order, as we have so cavalierly complexified and tensored it into a much more quantum-mechanical looking system. Physical cavity distributions ψ (n) must be real, normalizable, and non-negative functions on the sphere. By its construction as a marginalization (summing out) procedure, Eq. must produce such a physical output given physical inputs, even though we have extended it over C. A more important subtlety arises in the normalization of states-the standard L 2 norm associated with the Dirac inner product is not necessarily 1 for a properly normalized probability distribution. Since the probabilistic L 1 norm is incompatible with the Hilbert space structure, it is much simpler to work with un-normalized vectors and keep in mind that a probabilistic interpretation only applies in the standard basis.
8. M. E. Fisher, Am. J. Phys. 32, 343 (1964). 10.1119/1.1970340
9. In principle, an exponential regulator with a sufficiently fast decay constant also works. The calculation proceeds in a similar fashion but the interpretation is more complicated and no more enlightening.
10. S. Hoory, N. Linial, and A. Wigderson, Bull. Am. Math. Soc. 43, 439 (2006). 10.1090/S0273-0979-06-01126-8
11. S. Janson, T. Tuczak, and A. Ruciński, Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley, Hoboken, New Jersey, 2000).
12. M. Mezard and G. Parisi, Eur. Phys. J. B 20, 217 (2001). 10.1007/PL00011099
13. See Ref. for a general discussion and Ref. for the explicit computation of the phase diagram of a classical Ising antiferromagnet.
14. M. Mézard and A. Montanari, Information, Physics, and Computation (Oxford University Press, New York, 2009). 10.1093/acprof:oso/9780198570837.001. 0001
15. K. Binder and A. Young, Rev. Mod. Phys. 58, 801 (1986). 10.1103/RevModPhys.58.801
16. Although there is a replica symmetric treatment of a related model with the additional complication of random interactions. See Ref..
17. F. Krzakala and L. Zdeborová, EPL 81, 57005 (2008). 10.1209/0295-5075/81/57005
18. M. Mézard, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
19. Other mean-field spin glass models, such as the p -spin model, can have a number of states scaling exponentially in system size. These models show a discontinuous spin glass transition. However, antiferromagnetic models with two-body interactions usually do not show this phenomenology, therefore we will not investigate this transition in this paper.
20. See Ref., and in particular the reprint on page 226, for a more detailed discussion of this delicate statement.
21. G. Biroli, C. Chamon, and F. Zamponi, Phys. Rev. B 78, 224306 (2008). 10.1103/PhysRevB.78.224306
22. Such observables are diagonal in the coherent-state basis and therefore their correlations follow from the classical measure.
23. B. I. Halperin and W. M. Saslow, Phys. Rev. B 16, 2154 (1977). 10.1103/PhysRevB.16.2154
24. V. Gurarie and J. T. Chalker, Phys. Rev. B 68, 134207 (2003). 10.1103/PhysRevB.68.134207
25. C. R. Laumann, A. Scardicchio, and S. L. Sondhi, Phys. Rev. B 78, 134424 (2008). 10.1103/PhysRevB.78.134424
26. T. K. Kopeć and K. D. Usadel, Phys. Status Solidi B 243, 502 (2006) 10.1002/pssb.200541282
27. T. K. Kopeć and K. D. Usadel, Phys. Rev. Lett. 78, 1988 (1997). 10.1103/PhysRevLett.78.1988
28. M. Tarzia and G. Biroli, Europhys. Lett. 82, 67008 (2008). 10.1209/0295-5075/82/67008
29. G. Carleo, M. Tarzia, and F. Zamponi, Phys. Rev. Lett. 103, 215302 (2009). 10.1103/PhysRevLett.103.215302
30. T. Jorg, F. Krzakala, G. Semerjian, and F. Zamponi, Phys. Rev. Lett. (to be published), arXiv:0911.3438 (unpublished).
31. P. Fendley and O. Tchernyshyov, Nucl. Phys. B 639, 411 (2002). 10.1016/S0550-3213(02)00481-9
32. L. Zdeborová and F. Krzakała, Phys. Rev. E 76, 031131 (2007). 10.1103/PhysRevE.76.031131
33. A. C. C. Coolen, N. S. Skantzos, I. P. Castillo, C. J. P. Vicente, J. P. L. Hatchett, B. Wemmenhove, and T. Nikoletopoulos, J. Phys. A 38, 8289 (2005). 10.1088/0305-4470/38/39/001