Abstract
A lot of progress has been made recently in our understanding of the random-field Ising model thanks to large-scale numerical simulations. In particular, it has been shown that, contrary to previous statements: the critical exponents for different probability distributions of the random fields and for diluted antiferromagnets in a field are the same. Therefore, critical universality, which is a perturbative renormalization-group prediction, holds beyond the validity regime of perturbation theory. Most notably, dimensional reduction is restored at five dimensions, i.e., the exponents of the random-field Ising model at five dimensions and those of the pure Ising ferromagnet at three dimensions are the same.
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Aharony, A., Imry, Y., Ma, S.-K.: Lowering of dimensionality in phase transitions with random fields. Phys. Rev. Lett. 37, 1364 (1976)
Young, A.P.: On the lowering of dimensionality in phase transitions with random fields. J. Phys. A 10, L257 (1977)
Parisi, G., Sourlas, N.: Random magnetic fields, supersymmetry and negative dimensions. Phys. Rev. Lett. 43, 744 (1979)
Bricmont, J., Kupiainen, A.: Lower critical dimensions of the random-field Ising Model. Phys. Rev. Lett. 59, 1829 (1987)
Fytas, N.G., Martín-Mayor, V.: Universality in the three-dimensional random-field Ising model. Phys. Rev. Lett. 110, 227201 (2013)
Fytas, N.G., Martín-Mayor, V.: Efficient numerical methods for the random-field Ising model: finite-size scaling, reweighting extrapolation, and computation of response functions. Phys. Rev. E 93, 063308 (2016)
Picco, M., Sourlas, N.: Diluted antiferromagnetic 3D Ising model in a field. Europhys. Lett. 109, 37001 (2015)
Fytas, N.G., Martín-Mayor, V., Picco, M., Sourlas, N.: Phase transitions in disordered systems: the example of the random-field Ising model in four dimensions. Phys. Rev. Lett. 116, 227201 (2016)
Fytas, N.G., Martín-Mayor, V., Picco, M., Sourlas, N.: Specific-heat exponent and modified hyperscaling in the 4D random-field Ising model. J. Stat. Mech. 033302 (2017)
Fytas, N.G., Martín-Mayor, V., Picco, M., Sourlas, N.: Restoration of dimensional reduction in the random-field Ising model at five dimensions. Phys. Rev. E 95, 042117 (2017)
Sourlas, N.: The \(\epsilon \) expansion and universality in three dimensions (2017). arXiv:1706.07176
Landau, L.D., Lifshitz, E.M.: Quantum Mechanics. Pergamon Press, Oxford (1965)
Tissier, M., Tarjus, G.: Supersymmetry and its spontaneous breaking in the random field Ising model. Phys. Rev. Lett. 107, 041601 (2011)
Tissier, M., Tarjus, G.: Nonperturbative functional renormalization group for random field models and related disordered systems. IV. Supersymmetry and its spontaneous breaking. Phys. Rev. B 85, 104203 (2012)
Tarjus, G., Balog, I., Tissier, M.: Critical scaling in random-field systems: 2 or 3 independent exponents? Europhys. Lett. 103, 61001 (2013)
Parisi, G., Sourlas, N.: Scale invariance in disordered systems: the example of the random-field ising model. Phys. Rev. Lett. 89, 257204 (2002)
Kardar, M.: Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities. Nucl. Phys. B 290, 582 (1987)
Medina, E., Kardar, M., Shapir, M., Wang, X.R.: Interference of directed paths in disordered systems. Phys. Rev. Lett. 62, 941 (1989)
Brézin, E., De Dominicis, C.: New phenomena in the random field Ising model. Europhys. Lett. 44, 13 (1998)
Brézin, E., De Dominicis, C.: Interactions of several replicas in the random field Ising model. Eur. Phys. J. B 19, 467 (2001)
Parisi, G.: Order parameter for spin-glasses. Phys. Rev. Lett. 50, 1946 (1983)
Mandelstam, S.: Soliton operators for the quantized sine-Gordon equation. Phys. Rev. D 11, 3026 (1975)
Coleman, S.: Classical lumps and their quantum descendants. In: Aspects of Symmetry: Selected Erice Lectures, pp. 185–264. Cambridge University Press, Cambridge (1985).
Mandelstam, S.: Soliton operators for the quantized sine-Gordon equation. Phys. Rep. 23, 307 (1976)
Acknowledgements
We would like to thank Giorgio Parisi for his hospitality in Rome, where part of this work has been completed. V.M.-M. was partially supported by MINECO (Spain) through Grant No. FIS2015-65078- C2-1-P (this contract partially funded by FEDER). N.G.F. and M.P. acknowledge support by the Royal Society’s International Exchange Scheme 2016/R1.
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Fytas, N.G., Martín-Mayor, V., Picco, M. et al. Review of Recent Developments in the Random-Field Ising Model. J Stat Phys 172, 665–672 (2018). https://doi.org/10.1007/s10955-018-1955-7
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DOI: https://doi.org/10.1007/s10955-018-1955-7