Abstract
After an introduction to quantum phase transitions, we study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one-dimension: the dynamical exponent is infinite and, at the critical point, the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point, there may be different exponents for the divergence of the average and typical correlation lengths, again as in one-dimension, but the evidence for this is less strong.
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References
R. B. Griffiths, Phys. Rev. Lett. 23. 17 (1969).
B. M. McCoy, Phys. Rev. Lett. 23, 383 (1969); Phys. Rev. 188, 1014 (1969).
P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
For a discussion of the localization problem in the presence of interactions see D. Belitz and T.R. Kirkpatrick Rev. Mod. Phys. 66, 261 (1994).
For a discussion of the localization problem in the presence of interactions see D. Belitz and T.R. Kirkpatrick Rev. Mod. Phys. 66, 261 (1994).
S.V. Kravchenko et al. Phys. Rev. Lett. 77, 4938 (1996).
S.L. Sondhi, J.P. Carini, S.M. Girvin and D. Shahar, Rev. Mod. Phys. 69, 315 (1997).
H.P. Wei, L.W. Engel, and D.C. Tsui, Phys. Rev. B. 50, 14609 (1994).
B. Huckstein, Rev. Mod. Phys. 67, 357 (1995).
D. Shahar (unpublished).
M.P.A. Fisher, P. Weichman, G. Grinstein and M.P.A. Fisher, Phys. Rev. B 40, 546, (1989).
W. Wu, B. Ellmann, T. F. Rosenbaum, G. Aeppli and D. H. Reich, Phys. Rev. Lett. 67, 2076 (1991); W. Wu, D. Bitko, T. F. Rosenbaum and G. Acppli; Phys. Rev. Lett. 71, 1919 (1993)
T. Rosenbaum (private communication).
A.H. Castro Neto, G. Castilla, and B. Jones, cond-mat/9710123.
M. C. de Andrade et al., cond-mat/9802081.
D. S. Fisher, Phys. Rev. Lett. 69, 534 (1992); Phys. Rev. B 51, 6411 (1995).
R. Shankar and G. Murthy, Phys. Rev. B, 36, 536(1987).
B. M. McCoy and T. T. Wu, Phys. Rev. 176, 631 (1968); 188, 982 (1969).
One can define the typical correlation function in different ways, which should all have the same critical behavior. One could take the exponential of the average of the log, exp[ln C(r)]av, but this is rather tricky for Monte Carlo simulations where statistical errors could make the estimated value of a very small correlation function go negative. Hence, here for our Monte Carlo simulations, we define the typical correlation function to be the median. Where data for both definitions are reliable, the results are very similar.
S. Katsura, Phys. Rev. 127, 1508 (1962); P. Pfeuty, Ann. Phys. (NY) 27, 79 (1970).
A. P. Young and H. Rieger, Phys. Rev. B, 53, 8486 (1996).
M.J. Thill and O.A. Huse, Physica 15, C321, (1995), 321 (1995).
A.P. Young, Phys. Rev. B. 56 11691, (1997).
H. Rieger and A. P. Young, Phys. Rev. Lett. 72, 4141 (1994); Phys. Rev. B54, 3328 (1996); M. Guo, R. N. Bhatt and D. A. Huse, Phys. Rev. Lett. 72, 4137 (1994); Phys. Rev. B54, 3336 (1996).
H. Rieger and N. Kawashima cond-mat/9802104.
However, T. Senthil and S. Sachdev, Phys. Rev. Lett 77, 5292 (1996), have shown, for a model with dilution, that activated dynamical scaling occurs for d ≥ 1 at the percolation concentration.
R. H. Swendsen and J. Wang, Phys. Rev. Lett.58,86(1987).
U. Wolff, Phys. Rev. Lett. 62, 361 (1989).
See e.g. H. Evertz cond-mat/9707221.
C. Pich and A. P. Young cond-mat/9802108.
See e.g. J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).
B.B. Beard and U.-J. Wiese Phys. Rev. Lett., 77, 5130 (1996).
By T dc we denoted the value of the effectiveclassicaltemperature at the transition. The temperature of the original quantum model is zero at the quantum critical point.
The followingL L Tpairs were used: 1-d (8,16), (12,32), (16,64), (24,128), (32,256), (48,1024), 2-d: (8,12), (12,22), (16,30), (24,62), (32,102), (48,228).
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Young, A.P., Pich, C. (1999). The Random Transverse Field Ising Ferromagnet: The Simplest Disordered Model with a Quantum Phase Transition. In: Landau, D.P., Schüttler, HB. (eds) Computer Simulation Studies in Condensed-Matter Physics XI. Springer Proceedings in Physics, vol 84. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60095-1_8
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DOI: https://doi.org/10.1007/978-3-642-60095-1_8
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