Abstract
We review the theoretical description of the random field Ising and O(N) models obtained from the functional renormalization group, either in its nonperturbative implementation or, in some limits, in perturbative implementations. The approach solves questions concerning the critical behavior of random-field systems that have stayed pending for many years: What is the mechanism for the breakdown of dimensional reduction and the breaking of the underlying supersymmetry below d = 6? Can one provide a theoretical computation of the critical exponents, including the exponent ψ characterizing the activated dynamic scaling? Is it possible to theoretically describe collective phenomena such as avalanches and droplets? Is the critical scaling described by 2 or 3 independent exponents? What is the phase behavior of the random-field O(N) model in the whole (N, d) plane and what is the lower critical dimension of quasi-long range order for N = 2? Are the equilibrium and out-of-equilibrium critical points of the RFIM in the same universality class?
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References
M. Schechter, N. Laflorencie, Phys. Rev. Lett. 97, 137204 (2006)
Y.G. Pollack, M. Schechter, Phys. Rev. B 89, 064414 (2014)
D.M. Silevitch, D. Bitko, J. Brooke, S. Ghosh, G. Aeppli, T.F. Rosenbaum, Nature 448, 567 (2007)
S. Fishman, A. Aharony, J. Phys. C 12, L729 (1979)
J.L. Cardy, Phys. Rev. B 29, 505 (1984)
For a review, see D.P. Belanger, inSpin glasses and random fields, edited by A.P. Young (World scientific, Singapore, 1998), p. 251
F. Brochard, P.G. de Gennes, J. Phys. (France) Lett. 44, L785 (1983)
P.G. de Gennes, J. Phys. Chem. 88, 6469 (1984)
A.P.Y. Wong, M.H.W. Chan, Phys. Rev. Lett. 65, 2567 (1990)
E. Pitard, M.L. Rosinberg, G. Stell, G. Tarjus, Phys. Rev. Lett. 74, 4361 (1995)
R.L.C. Vink, K. Binder, H. Lšwen, J. Phys.: Condens. Matter 20, 404222 (2008)
T. Giamarchi, P. Le Doussal, Phys. Rev. Lett. 72, 1530 (1994)
T. Giamarchi, P. Le Doussal, Phys. Rev. B 52, 1242 (1995)
G. Blatter, M.V. Feigel’man, V.B. Geshkenbern, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66, 1125 (1994)
T. Giamarchi, P. Le Doussal, inSpin glasses and random fields, edited by A.P. Young (World scientific, Singapore, 1998), p. 321
L. Nie, G. Tarjus, S.A. Kivelson, Proc. Natl. Acad. Sci. USA 111, 7980 (2014)
L. Nie, L.E. Hayward Sierens, R.G. Melko, S. Sachdev, S.A. Kivelson, Phys. Rev. B 92, 174505 (2015)
B. Phillabaum, E.W. Carlson, K.A. Dahmen, Nat. Commun. 3, 915 (2012)
S. Liu, B. Phillabaum, E.W. Carlson, K.A. Dahmen, N.S. Vidhyadhiraja, M.M. Qazilbash, D.N. Basov, Phys. Rev. Lett. 116, 036401 (2016)
J.D. Stevenson, A.M. Walczak, R.W. Hall, P.G. Wolynes, J. Chem. Phys. 129, 194505 (2008)
S. Franz, G. Parisi, F. Ricci-Tersenghi, T. Rizzo, Eur. Phys. J. E 34, 102 (2011)
S. Franz, G. Parisi, J. Stat. Mech. 2013, P11012 (2013)
G. Biroli, C. Cammarota, M. Tarzia, G. Tarjus, Phys. Rev. Lett. 112, 175701 (2014)
G. Biroli, C. Cammarota, G. Tarjus, M. Tarzia, Phys. Rev. B 98, 174205 and 174206 (2018)
J.P. Sethna, K.A. Dahmen, C.R. Myers, Nature 410, 242 (2001)
J.P. Sethna, K. Dahmen, S. Kartha, J.A. Krumhansi, B.W. Roberts, J.D. Shore, Phys. Rev. Lett. 70, 3347 (1993)
K. Dahmen, J.P. Sethna, Phys. Rev. B 53, 14872 (1996)
O. Perkovic, K.A. Dahmen, J.P. Sethna, Phys. Rev. B 59, 6106 (1999)
O. Perkovic, K.A. Dahmen, J.P. Sethna, https://arXiv:cond-mat/9609072 (1996)
F.J. Perez-Reche, E. Vives, Phys. Rev. B 67, 134421 (2003)
F.J. Perez-Reche, E. Vives, Phys. Rev. B 70, 214422 (2004)
G. Bertotti,Hysteresis in Magnetism (Academic Press, New York, 1998)
J.P. Sethna, K.A. Dahmen, O. Perkovic, inThe Science of Hysteresis, edited by G. Bertotti, I. Mayergoyz (Elsevier, Amsterdam, 2005), p. 107
F.J. Pérez-Reche, E. Vives, L. Manosa, A. Planes, Phys. Rev. Lett. 87, 195701 (2001)
E. Kierlik, P.A. Monson, M.L. Rosinberg, L. Sarkisov, G. Tarjus, Phys. Rev. Lett. 87, 055701 (2001)
F. Detcheverry, E. Kierlik, M.L. Rosinberg, G. Tarjus, Langmuir 20, 8006 (2004)
F. Bonnet, T. Lambert, B. Cross, L. Guyon, F. Despetis, L. Puech, P.E. Wolf, EPL 82, 56003 (2008)
H. Borda da Rocha, L. Truskinovsky, Phys. Rev. Lett. 122, 088103 (2019)
M. Ozawa, L. Berthier, G. Biroli, A. Rosso, G. Tarjus, Proc. Natl. Acad. Sci. USA 115, 6656 (2018)
J.-P. Bouchaud, J. Stat. Phys. 151, 567 (2013)
G. Tarjus, M. Tissier, Phys. Rev. Lett 93, 267008 (2004)
J. Zinn-Justin,Quantum Field Theory and Critical Phenomena, 3rd edn. (Oxford University Press, New York, 1989)
G. Parisi, N. Sourlas, Phys. Rev. Lett. 43, 744 (1979)
H. Ji, M.O. Robbins, Phys. Rev. A 44, 2538 (1991)
H. Ji, M.O. Robbins, Phys. Rev. B 46, 14519 (1992)
T. Giamarchi, P. Le Doussal, Phys. Rev. Lett. 76, 3408 (1996)
T. Haga, Phys. Rev. B 96, 184202 (2017)
Y. Imry, S.K. Ma, Phys. Rev. Lett. 35, 1399 (1975)
A. Aharony, Y. Imry, S.K. Ma, Phys. Rev. Lett. 37, 1364 (1976)
G. Grinstein, Phys. Rev. Lett. 37, 944 (1976)
A.P. Young, J. Phys. C 10, L257 (1977)
M. Aizenman, J. Wehr, Phys. Rev. Lett. 62, 2503 (1989)
M. Aizenman, J. Wehr, Commun. Math. Phys. 130, 489 (1990)
J.Z. Imbrie, Phys. Rev. Lett. 53, 1747 (1984)
J. Bricmont, A. Kupiainen, Phys. Rev. Lett. 59, 1829 (1987)
T. Nattermann, inSpin glasses and random fields, edited by A.P. Young (World Scientific, Singapore, 1998), p. 277
D.E. Feldman, Phys. Rev. B 61, 382 (2000)
M.J.P. Gingras, D.A. Huse, Phys. Rev. B 53, 15193 (1996)
J. Villain, Phys. Rev. Lett. 52, 1543 (1984)
J. Villain, J. Phys. 46, 1843 (1985)
D.S. Fisher, Phys. Rev. Lett. 56, 416 (1986)
A.J. Bray, M.A. Moore, J. Phys. C: Solid State Phys. 18, L927 (1985)
See for instance H. Rieger, inAdvances in Computer Simulation, Lecture Notes in Physics edited by J. Kertesz, I. Kondor (Springer, Heidelberg, 1998), Vol. 501
A.A. Middleton, D.S. Fisher, Phys. Rev. B 65, 134411 (2002)
G. Parisi, V. Dotsenko, J. Phys. A: Math. Gen. 25, 3143 (1992)
C. De dominicis, H. Orland, T. Temesvari, J. Phys. I (Paris) 5, 987 (1995)
E. Brézin, C. De Dominicis, Eur. Phys. J. B 19, 467 (2001)
G. Parisi, N. Sourlas, Phys. Rev. Lett. 89, 257204 (2002)
M. Mézard, A.P. Young, Europhys. Lett. 18, 653 (1992)
D.S. Fisher, Phys. Rev. Lett. 56, 1964 (1986)
O. Narayan, D.S. Fisher, Phys. Rev. B 46, 11520 (1992)
O. Narayan, D.S. Fisher, Phys. Rev. B 46, 11520 (1993)
T. Nattermann, S. Stepanow, L.-H. Tang, H. Leschhorn, J. Phys. II 2, 1483 (1992)
P. Le Doussal, K.J. Wiese, P. Chauve, Phys. Rev. B 66, 174201 (2002)
P. Le Doussal, K.J. Wiese, P. Chauve, Phys. Rev. E 69, 026112 (2004)
P. Chauve, P. Le Doussal, K.J. Wiese, Phys. Rev. Lett. 86, 1785 (2001)
G. Parisi, inProceedings of Les Houches 1982, Session XXXIX, edited by J.B. Zuber, R. Stora (North Holland, Amsterdam, 1984), p. 473
M. Guagnelli, E. Marinari, G. Parisi, J. Phys. A 26, 5675 (1993)
F.J. Perez-Reche, M.L. Rosinberg, G. Tarjus, Phys. Rev. B 77, 064422 (2008)
C. Frontera, J. Goicoechea, J. Ortín, E. Vives, J. Comput. Phys. 160, 117 (2000)
C. Frontera, E. Vives, Comput. Phys. Commun. 147, 455 (2002)
Y. Wu, J. Machta, Phys. Rev. Lett. 95, 137208 (2005)
Y. Wu, J. Machta, Phys. Rev. B 74, 064418 (2006)
Y. Liu, K.A. Dahmen, Phys. Rev. E 76, 031106 (2007)
See, e.g., P. Le Doussal, A.A. Middleton, K.J. Wiese, Phys. Rev. E 79, 050101 (2009)
L. Balents, J.P. Bouchaud, M. Mézard, J. Phys. I 6, 1007 (1996)
See, e.g., A. Rosso, P. Le Doussal, K.J. Wiese, Phys. Rev. B 80, 144204 (2009)
A.J. Bray, M.A. Moore, J. Phys. C 17, L463 (1984)
D.S. Fisher, D.A. Huse, Phys. Rev. B 38, 373 (1988)
D.S. Fisher, D.A. Huse, Phys. Rev. B 38, 386 (1988)
M. Zumsande, A.K. Hartmann, Eur. Phys. J. B 72, 619 (2009)
L. Balents, P. Le Doussal, Europhys. Lett. 65, 685 (2004)
P. Chauve, T. Giamarchi, P. Le Doussal, Phys. Rev. B 62, 6241 (2000)
L. Balents, P. Le Doussal, Ann. Phys. (NY) 315, 213 (2005)
P. Le Doussal, Ann. Phys. (NY) 325, 49 (2010)
M. Mézard, G. Parisi, M.A. Virasoro,Spin glass theory and beyond (World Scientific, Singapore, Singapore, 1987)
P. Le Doussal, K.J. Wiese, Nucl. Phys. B 701, 409 (2004)
G. Tarjus, M. Tissier, Phys. Rev. B 78, 024203 (2008)
M. Tissier, G. Tarjus, Phys. Rev. B 85, 104202 (2012)
M. Tissier, G. Tarjus, Phys. Rev. B 85, 104203 (2012)
D. Mouhanna, G. Tarjus, Phys. Rev. E 81, 051101 (2010)
M. Tissier, G. Tarjus, Phys. Rev. Lett 96, 087202 (2006)
M. Tissier, G. Tarjus, Phys. Rev. B 78, 024204 (2008)
M. Tissier, G. Tarjus, Phys. Rev. Lett. 107, 041601 (2011)
M. Tissier, G. Tarjus, Phys. Rev. B 74, 214419 (2006)
P. Le Doussal, K.J. Wiese, Phys. Rev. Lett. 96, 197202 (2006)
G. Tarjus, M. Tissier, J. Stat. Mech. 2016, 023207 (2016)
P. Le Doussal, K.J. Wiese, Phys. Rev. E 79, 051106 (2009)
M. Baczyk, G. Tarjus, M. Tissier, I. Balog, J. Stat. Mech. 2014, P06010 (2014)
I. Balog, G. Tarjus, M. Tissier, https://arXiv:1906.10058 (2019)
F. Krzakala, F. Ricci-Tersenghi, L. Zdeborova, Phys. Rev. Lett. 104, 207208 (2010)
F. Krzakala, F. Ricci-Tersenghi, D. Sherrington, L. Zdeborova, J. Phys. A: Math. Theor. 44, 042003 (2011)
S. Chatterjee, Commun. Math. Phys. 337, 93 (2015)
M. Tissier, G. Tarjus, unpublished.
N.G. Fytas, V. Martin-Mayor, M. Picco, N. Sourlas, Phys. Rev. E 95, 042117 (2017)
N.G. Fytas, V. Martin-Mayor, G. Parisi, M. Picco, N. Sourlas, Phys. Rev. Lett. 122, 240603 (2019)
G. Tarjus, M. Baczyk, M. Tissier, Phys. Rev. Lett. 110, 135703 (2013)
G. Parisi, N. Sourlas, Phys. Rev. Lett. 46, 871 (1981)
D.C. Brydges, J.Z. Imbrie, J. Stat. Phys. 110, 503 (2003)
D.C. Brydges, J.Z. Imbrie, Ann. Math. 158, 1019 (2003)
I. Balog, G. Tarjus, Phys. Rev. B 91, 214201 (2015)
Y. Liu, K.A. Dahmen, Phys. Rev. E 79, 061124 (2009)
Y. Liu, K.A. Dahmen, Europhys. Lett. 86, 56003 (2009)
M. Schwartz, J. Phys. C: Solid State Phys. 18, 135 (1985)
M. Schwartz, A. Soffer, Phys. Rev. B 33, 2059 (1986)
M. Schwartz, M. Gofman, T. Natterman, Physica A 178, 6 (1991)
G. Tarjus, I. Balog, M. Tissier, Europhys. Lett. 103, 61001 (2013)
N.G. Fytas, V. Martin-Mayor, Phys. Rev. Lett. 110, 227201 (2013)
N.G. Fytas, V. Martin-Mayor, M. Picco, N. Sourlas, Phys. Rev. Lett. 116, 227201 (2016)
A.K. Hartmann, A.P. Young, Phys. Rev. B 64, 214419 (2001)
A.A. Middleton, https://arXiv:cond-mat/0208182 (2002)
A.K. Hartmann, Phys. Rev. B 65, 134411 (2002)
P.C. Martin, E.D. Siggia, H.A. Rose, Phys. Rev. A 8, 423 (1973)
H.K. Janssen, Z. Phys. B 23, 377 (1976)
C. de Dominicis, J. Phys. (Paris) Colloq. 37, 247 (1976)
I. Balog, G. Tarjus, M. Tissier, Phys. Rev. B 97, 094204 (2018)
K.G. Wilson J. Kogut, Phys. Rep. C 12, 77 (1974)
J. Polchinski, Nucl. Phys. B 231, 269 (1984)
C. Wetterich, Phys. Lett. B 301, 90 (1993)
J. Berges, N. Tetradis, C. Wetterich, Phys. Rep. 363, 223 (2002)
I. Balog, H. Chaté, B. Delamotte, M. Marohnić, N. Wschebor, Phys. Rev. Lett. 123, 240604 (2019)
G. Tarjus, M.-L. Rosinberg, E. Kierlik, M. Tissier, Mol. Phys. 109, 2863 (2011)
M.C. Angelini, C. Lucibello, G. Parisi, F. Ricci-Tersenghi, T. Rizzo, Proc. Natl. Acad. Sci. USA, to appear (2019)
M. Schwartz, A. Soffer, Phys. Rev. Lett. 55, 2499 (1985)
P. Le Doussal, K.J. Wiese, Phys. Rev. E 79, 051106 (2009)
P. Le Doussal, K.J. Wiese, Phys. Rev. E 85, 061102 (2012)
G. Tarjus, unpublished.
D.S. Fisher, Phys. Rev. B 31, 7233 (1985)
D.E. Feldman, Phys. Rev. Lett. 88, 177202 (2002)
Y. Sakamoto, H. Mukaida, C. Itoi, Phys. Rev. B 74, 064402 (2006)
Y. Sakamoto, H. Mukaida, C. Itoi, J. Phys.: Condens. Matter 19, 145219 (2007)
Y. Sakamoto, H. Mukaida, C. Itoi, Phys. Rev. Lett. 98, 269703 (2007)
I. Balog, M. Tissier, G. Tarjus, J. Stat. Mech. 2014, P10017 (2014)
M. Baczyk, M. Tissier, G. Tarjus, Y. Sakamoto, Phys. Rev. B 88, 014204 (2013)
A.J. Bray, J. Phys. C: Solid State Phys. 19, 6225 (1986)
A. Maritan, M. Cieplak, M.R. Swift, J.R. Banavar, Phys. Rev. Lett. 72, 946 (1994)
R. Harris, M. Plischke, M.J. Zuckermann, Phys. Rev. Lett. 31, 160 (1973)
R.W. Cochrane, R. Harris, M.J. Zuckermann, Phys. Rep. 48, 1 (1978)
D.E. Feldman, Int. J. Mod. Phys. B 15, 2945 (2001)
D. Mouhanna, G. Tarjus, Phys. Rev. B 94, 214205 (2016)
N. Dupuis, Phys. Rev. E 100, 030102 (2019)
I. Balog, G. Tarjus, M. Tissier, J. Stat. Mech.: Theory and Experiment 2019, 103301 (2019)
T. Machado, N. Dupuis, Phys. Rev. E 82, 041128 (2010)
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Contribution to the Topical Issue “Recent Advances in the Theory of Disordered Systems”, edited by Ferenc Iglói and Heiko Rieger.
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Tarjus, G., Tissier, M. Random-field Ising and O(N) models: theoretical description through the functional renormalization group. Eur. Phys. J. B 93, 50 (2020). https://doi.org/10.1140/epjb/e2020-100489-1
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DOI: https://doi.org/10.1140/epjb/e2020-100489-1