Publication: Phase transition in three-dimensional Heisenberg spin glasses with strong random anisotropies through a multi-GPU parallelization
Full text at PDC
Advisors (or tutors)
American Physical Society
We characterize the phase diagram of anisotropic Heisenberg spin glasses, finding both the spin and the chiral glass transition. We remark on the presence of strong finite-size effects in the chiral sector. On the spin glass sector, we find that the universality class is that of Ising spin glasses. Our data are compatible with a unique phase transition for the chiral and spin glass sector. We focus on keeping finite-size effects under control, and we stress that they are important to understand experiments. Thanks to large GPU clusters we have been able to thermalize cubic lattices with up to 643 spins, over a vast range of temperatures (hence, of relaxation times).
© 2014 American Physical Society. We thank J. J. Ruiz Lorenzo and E. Marinari for useful discussions. We were partly supported by MINECO, Spain, through the Research Contract No. FIS2012-35719-C02. M.B.J. was supported by the FPU program (MECD, Spain). The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement No. . The computations were carried out in the GPU-accelerated clusters Tianh-1A (Tianjin, China) and Minotauro (Barcelona, Spain). The total amount of time devoted to this project was 2.2 × 105 GPU hours in Tianhe 1A and 2.0 × 105 GPU hours in Minotauro. Access to Tianhe 1A was granted through Research Contract No. 287746 by the EU-FP7. The authors thankfully acknowledge the computer resources, technical expertise, and assistance provided by the staff at the National Supercomputing Center-Tianjin and at the Red Española de Supercomputación–Barcelona Supercomputing Center.
 M. Mézard, G. Parisi, M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).  J. A. Mydosh, Spin Glasses: An Experimental Introduction (Taylor and Francis, London 1993).  A. P. Young, Spin Glasses and Random Fields(World Scientific, Singapore, 1997).  M. A. Ruderman, C. Kittel, Phys. Rev., 96, 99 (1954).  T. Kasuya, Prog. Theor. Phys., 16, 45 (1956).  K. Yosida, Phys. Rev., 106, 893 (1957).  I. Dzyaloshinsky, J. Phys. Chem. Solids, 4, 241 (1958).  T. Moriya, Phys. Rev. Lett., 4, 228 (1960).  S. F. Edwards, P. W. Anderson, J. Phys. F, 5, 965 (1975).  D. Petit, L. Fruchter, I. A. Campbell, Phys. Rev. Lett., 88, 207206 (2002).  F. Bert, V. Dupuis, E. Vincent, J. Hammann, J.-P. Bouchaud, Phys. Rev. Lett., 92, 167203 (2004).  O. E. Peil, A. V. Ruban, B. Johansson, Phys. Rev. B, 79, 024428 (2009).  J. W. Cable, S. A. Werner, G. P. Felcher, N. Wakabayashi, Phys. Rev. Lett., 49, 829 (1982).  J. W. Cable, S. A. Werner, G. P. Felcher, N. Wakabayashi, Phys. Rev. B 29, 1268 (1984).  F. J. Lamelas, S. A. Werner, S. M. Shapiro, J. A. Mydosh, Phys. Rev. B, 51, 621 (1995).  H. Bouchiat, J. Phys., (Paris) 47, 71 (1986).  L. P. Levy, ´ Phys. Rev. B, 38, 4963 (1988).  K. Gunnarsson, P. Svedlindh, P. Nordblad, L. Lundgren, H. Aruga, A. Ito, Phys. Rev. B, 43, 8199 (1991).  S. Franz, G. Parisi, M. A. Virasoro, J. Phys. I France, 4, 1657 (1994).  M. Palassini, S. Caracciolo, Phys. Rev. Lett., 82, 5128 (1999).  H. G. Ballesteros, A. Cruz, L. A. Fernández, V. Martín-Mayor, J. Pech, J. J. Ruiz-Lorenzo, A. Tarancón, P. Téllez, C. L. Ullod, C. Ungil, Phys. Rev. B, 62, 14237 (2000).  W. L. McMillan, Phys. Rev. B, 31, 342 (1985).  J. A. Olive, A. P. Young, D. Sherrington, Phys. Rev B, 34, 6341 (1986).  B. M. Morris, et al., J. Phys. C, 19, 1157 (1986).  F. Matsubara, T. Iyota, S. Inawashiro, Phys. Rev. Lett., 67, 1458 (1991).  M. J. P. Gingras, Phys. Rev. Lett., 71, 1637 (1993).  B. Coluzzi, J. of Phys. A: Math. Gen., 28, 747 (1995).  A. Mauger, J. Villaín, Y. Zhou, C. Rigaux, N. Bontemps, J. Ferré, Phys. Rev. B, 41, 4587 (1990).  H. Kawamura, Phys. Rev. Lett., 68, 3785 (1992) --- ibid., 80, 5421 (1998).  L. W. Lee, A. P. Young, Phys. Rev. Lett., 90, 227203 (2003).  L. A. Fernández, V. Martín-Mayor, S. Pérez-Gaviro, A. Tarancón, and A. P. Young, Phys. Rev. B, 80, 024422 (2009).  I. Campos, M. Cotallo-Aban, V. Martín-Mayor, S. Pérez-Gaviro, A. Tarancón, Phys. Rev. Lett., 97, 217204 (2006).  D. X. Viet, H. Kawamura, Phys. Rev. Lett., 102, 027202 (2009).  H. Kawamura, Phys. Rev. Lett., 90, 047202 (2003).  T. Taniguchi, J. Phys.: Condens. Matter, 19, 145213 (2007).  I. A. Campbell, D. C. M. C. Petit, J. Phys. Soc. Jpn., 79, 011006 (2010).  D. Amit, V. Martín-Mayor, Field Theory: the Renormalization Group and Critical Phenomena, 3rd ed.(World Scientific, Singapore, 2005).  F. Belletti, et al., Phys. Rev. Lett., 101, 157201 (2008) --- ibid., J. Stat. Phys., 135, 1121 (2009).  Y. G. Joh, R. Orbach, G. G. Wood, J. Hammann, E. Vincent, Phys. Rev. Lett., 82, 438 (1999).  A. J. Bray, M. A. Moore, J. Phys. C: Solid State Phys., 15, 3897 (1982).  V. Martín-Mayor, S. Pérez-Gaviro, Phys. Rev. B, 84, 024419 (2011).  Recall that γ_(CG) = ν(2 − η_(CG)), where γ_(CG) is the critical index for the CG susceptibility, while ν is the correlation-length exponent.  Of course the limiting factor is in the wide range of relaxation times, rather than temperatures. However, relaxation times depend on a variety of implementation dependent factors (such as the temperature spacing in the parallel tempering, or the number of over-relaxation sweeps). Hence, comparison with other work will be easier in terms of temperatures.  National Super-Computing Center, Tianjin, China, http://www.nscc-tj.gov.cn/en/  Barcelona Supercomputing Center, Barcelona, Spain http://www.bsc.es  F. Parisen Toldin, A. Pelissetto, E. Vicari, J. Stat. Mech., (2006) P06002.  F. Liers, J. Lukic, E. Marinari, A. Pelissetto, E. Vicari, Phys. Rev. B, 76, 174423 (2007).  Independence from microscopic details such as the disorder distribution has been found for spin glasses [49-52], as well as for other disordered systems like the random field Ising model , or disordered ferromagnets (either site  or bond [55, 56] diluted).  M. Hasenbusch, A. Pelissetto, E. Vicari, Phys. Rev. B, 78, 214205 (2008).  T. Jorg, Phys. Rev. B, 73, 224431 (2006).  H. G. Katzgraber, M. Korner, A. P. Young, Phys. Rev. B, 73, 224432 (2006).  T. Jorg, H. G. Katzgraber, Phys. Rev. Lett., 101, 197205 (2008).  N. G. Fytas, V. Martín-Mayor, Phys. Rev. Lett., 110, 227201 (2013).  H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, J. J. Ruiz-Lorenzo, Phys. Rev. B, 58, 2740 (1998).  P.-E. Berche, C. Chatelain, B. Berche, W. Janke, Eur. Phys. J. B, 38, 463 (2004).  A. Malakis, A. N. Berker, N. G. Fytas, T. Papakonstantinou, Phys. Rev. E, 85, 061106 (2012).  M. Baity-Jesi, et al., Eur. Phys. J. Special Topics, 210, 33 (2012).  L. W. Lee, A. P. Young, Phys. Rev. B, 76, 024405 (2007).  F. R. Brown, T. J. Woch, Phys. Rev. Lett., 58, 2394 (1987).  K. Hukushima, K. Nemoto, J. Phys. Soc. Jpn, 65, 1604 (1996).  E. Marinari in Advances in Computer Simulations, edited by J. Kerstész and I. Kondor (Springer, Berlin, 1995).  J. L. Alonso, A. Tarancón, H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, Phys. Rev. B, 53, 2537 (1996).  E. Marinari, V. Martín-Mayor, A. Pagnani, Phys. Rev. B, 62, 4999 (2000).  It is enough to define the local field as →hx = ∑_(||x-y||=1)[J_(xy)→S_(y)+D_(xy)→S_(y)].  L. A. Fernández, A. Maiorano, E. Marinari, V. Martín-Mayor, D. Navarro, D. Sciretti, A. Tarancón, J. L. Velasco, Phys. Rev. B, 77, 104432 (2008).  M. P. Nightingale, Physica A, 83, 561 (1975).  H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz-Sudupe, Phys. Lett. B, 378, 207 (1996).  A more recent article from the Janus Collaboration (Ref. ) gives a more precise determination of the critical exponents. Using one or the other does not change significantly our results and conclusions.  Janus Collaboration: M. Baity-Jesi, R. A. Baños, A. Cruz, L. A. Fernández, J. M. Gil-Narvión, A. Gordillo Guerrero, D. Íñiguez, A. Maiorano, F. Mantovani, E. Marinari, V. Martín-Mayor, J. Monforte-García, A. Muñoz Sudupe, D. Navarro, G. Parisi, S. Pérez-Gaviro, M. Pivanti, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancón, R. Tripiccione, D. Yllanes, Phys. Rev. B, 88, 224416 (2013).  Some of the points we used for those extrapolations shared some of the data. For example, the crossing of ξ for L = 8,16, had in common the points from size L = 16 with the pair L = 16,32. Hence, in the fits we have taken into account the covariance matrix that gave a measure of the anticorrelation between measures that share data.  In Ref.  the D^(αβ)_(ij) are uniformly distributed between ±D = 0.05, while in our work we use binary couplings. If we want to compare them, we have to use D = 1/√1200 ≈ 0.03.  In the phase diagram we show the D = 0 point comes from Ref. , where chiral and spin glass transition are assumed to be coupled. There is disagreement on whether T_(SG) = T_(CG) also in the isotropic case. Yet, we do plot it as a single transition because although T_(SG) might be lower than T_(CG), their best estimates are compatible (and not distinguishable in the plot).  In a typical system N = L^(3) ~ 6 x 10^(23) → L ≈ 10^(8).  G. F. Rodríguez, G. G. Kenning, R. Orbach, Phys. Rev. B, 88, 054302 (2013).  J. H. Pixley, A. P. Young, Phys. Rev. B, 78, 014419 (2008).  NVIDIA Corporation, CUDA C Programming Guide, docs.nvidia.com/cuda/cuda-c-programming-guide/index.html  M. Baity-Jesi, Ph.D. thesis, Universidad Complutense de Madrid (work in progress).  M. Bernaschi, G. Parisi, L. Parisi, Comput. Phys. Commun., 182, 1265 (2011).  T. Yavors’kii, M. Weigel, Eur. Phys. J. Special Topics, 210, 159 (2012).  D. E. Knuth, The Art of Computer Programming, 2nd ed. (Addison-Wesley, Reading, MA, 1981), Vol. 2.  L. A. Fernández, V. Martín-Mayor, D. Yllanes, Nucl. Phys. B, 807, 424 (2009).  V. Parisi, cited in G. Parisi and F. Rapuano, Phys. Lett. B, 157, 301 (1985).  H. G. Ballesteros, V. Martín-Mayor, Phys. Rev. E, 58, 6787 (1998).  L. A. Fernández, V. Martín-Mayor, D. Sciretti, A. Tarancón, J. L. Velasco, Phys. Lett. B, 628, 281 (2005).  P. L’Ecuyer, Math. Comp., 68, 249 (1999).  G. Ossola, A. D. Sokal, Nucl. Phys. B, 691, 259 (2004).  M. Luescher, Comput. Phys. Commun., 79, 100 (1994).  G. Marsaglia, Diehard Battery of Tests of Randomness, http://www.stat.fsu.edu/pub/diehard.  R. J. Rivers, Path Integral Methods in Quantum Field Theories (Cambridge University Press, Cambridge, 1990).