Revisiting the replica trick: competition between spin glass and conventional order. (English) Zbl 07723873
Summary: There is an ambiguity in how to apply the replica trick to spin glass models which have additional order parameters unrelated to spin glass order – with respect to which quantities does one minimize vs maximize the action, and in what sequence? Here we show that the correct procedure is to first maximize with respect to “replica” order parameters, and then minimize with respect to “conventional” order parameters. With this result, we further elucidate the relationship between quenched free energies, annealed free energies, and replica order – it is possible for the quenched and annealed free energies to differ even while all replica order parameters remain zero.
MSC:
| 82Bxx | Equilibrium statistical mechanics |
| 82Dxx | Applications of statistical mechanics to specific types of physical systems |
| 82-XX | Statistical mechanics, structure of matter |
References:
| [1] | Binder, K.; Young, AP, Spin glasses: experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys., 58, 801 (1986) · doi:10.1103/RevModPhys.58.801 |
| [2] | Mezard, M.; Parisi, G.; Virasoro, MA, Spin Glass Theory and Beyond (1987), Singapore: World Scientific, Singapore · Zbl 0992.82500 |
| [3] | Fischer, KH; Hertz, JA, Spin Glasses (1991), Cambridge: CUP, Cambridge · doi:10.1017/CBO9780511628771 |
| [4] | Mydosh, JA, Spin Glasses: An Experimental Introduction (1993), London: Taylor & Francis, London |
| [5] | Nishimori, H., Statistical Physics of Spin Glasses and Information Processing: An Introduction (2001), Oxford: OUP, Oxford · Zbl 1103.82002 · doi:10.1093/acprof:oso/9780198509417.001.0001 |
| [6] | Mezard, M.; Montanari, A., Information, Physics, and Computation (2009), Oxford: OUP, Oxford · Zbl 1163.94001 · doi:10.1093/acprof:oso/9780198570837.001.0001 |
| [7] | Maldacena, J.; Stanford, D., Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D, 94, 106002 (2016) · doi:10.1103/PhysRevD.94.106002 |
| [8] | Chowdhury, D.; Georges, A.; Parcollet, O.; Sachdev, S., Sachdev-Ye-Kitaev models and beyond: window into non-Fermi liquids, Rev. Mod. Phys., 94, 035004 (2022) · doi:10.1103/RevModPhys.94.035004 |
| [9] | Castellani, T.; Cavagna, A., Spin-glass theory for pedestrians, J. Stat. Mech.: Theory Exp., 2005, 5, P05012 (2005) · Zbl 1456.82490 · doi:10.1088/1742-5468/2005/05/P05012 |
| [10] | Charbonneau, P.: From the replica trick to the replica symmetry breaking technique. arXiv:2211.01802 (2022) |
| [11] | Zdeborová, L.; Krzakala, F., Statistical physics of inference: thresholds and algorithms, Adv. Phys., 65, 453 (2016) · doi:10.1080/00018732.2016.1211393 |
| [12] | Foini, L.; Kurchan, J., Annealed averages in spin and matrix models, SciPost Phys., 12, 080 (2022) · doi:10.21468/SciPostPhys.12.3.080 |
| [13] | Saad, P., Shenker, S.H., Stanford, D.: JT gravity as a matrix integral. arXiv:1903.11115 (2019) |
| [14] | Baldwin, CL; Swingle, B., Quenched vs annealed: glassiness from SK to SYK, Phys. Rev. X, 10, 031026 (2020) |
| [15] | Gur-Ari, G.; Mahajan, R.; Vaezi, A., Does the SYK model have a spin glass phase?, J. High Energy Phys., 2018, 11, 70 (2018) · Zbl 1404.81169 · doi:10.1007/JHEP11(2018)070 |
| [16] | Wang, H.; Bagrets, D.; Chudnovskiy, AL; Kamenev, A., On the replica structure of Sachdev-Ye-Kitaev model, J. High Energy Phys., 2019, 9, 57 (2019) · Zbl 1423.83075 · doi:10.1007/JHEP09(2019)057 |
| [17] | Tanaka, T., Moment problem in replica method, Interdiscip. Inf. Sci., 13, 1, 17 (2007) · Zbl 1146.82005 |
| [18] | Mourrat, J-C, Nonconvex interactions in mean-field spin glasses, Probab. Math. Phys., 2, 2, 281 (2021) · Zbl 1486.82044 · doi:10.2140/pmp.2021.2.281 |
| [19] | Sherrington, D.; Kirkpatrick, S., Solvable model of a spin-glass, Phys. Rev. Lett., 35, 1792 (1975) · doi:10.1103/PhysRevLett.35.1792 |
| [20] | Bovier, A., Statistical Mechanics of Disordered Systems: A Mathematical Perspective (2006), Cambridge: CUP, Cambridge · Zbl 1108.82002 · doi:10.1017/CBO9780511616808 |
| [21] | Talagrand, M., Mean Field Models for Spin Glasses (2011), New York: Springer, New York · Zbl 1214.82002 · doi:10.1007/978-3-642-22253-5 |
| [22] | Talagrand, M., Mean Field Models for Spin Glasses (2011), New York: Springer, New York · Zbl 1214.82002 · doi:10.1007/978-3-642-22253-5 |
| [23] | Panchenko, D., The Sherrington-Kirkpatrick Model (2013), New York: Springer, New York · Zbl 1266.82005 · doi:10.1007/978-1-4614-6289-7 |
| [24] | Goldschmidt, YY, Solvable model of the quantum spin glass in a transverse field, Phys. Rev. B, 41, 4858 (1990) · doi:10.1103/PhysRevB.41.4858 |
| [25] | Nieuwenhuizen, TM; Ritort, F., Quantum phase transition in spin glasses with multi-spin interactions, Phys. A: Stat. Mech. Appl., 250, 1-4, 8 (1998) · doi:10.1016/S0378-4371(97)00546-3 |
| [26] | Cugliandolo, LF; Grempel, DR; da Silva Santos, CA, From second to first order transitions in a disordered quantum magnet, Phys. Rev. Lett., 85, 2589 (2000) · doi:10.1103/PhysRevLett.85.2589 |
| [27] | Cugliandolo, LF; Grempel, DR; da Silva Santos, CA, Imaginary-time replica formalism study of a quantum spherical p-spin-glass model, Phys. Rev. B, 64, 014403 (2001) · doi:10.1103/PhysRevB.64.014403 |
| [28] | Mottishaw, P., First-order spin glass transitions: an exact solution, EPL, 1, 409 (1986) · doi:10.1209/0295-5075/1/8/007 |
| [29] | Panchenko, D., Free energy in the Potts spin glass, Ann. Probab., 46, 829 (2018) · Zbl 1430.60088 · doi:10.1214/17-AOP1193 |
| [30] | Panchenko, D., Free energy in the mixed p-spin models with vector spins, Ann. Probab., 46, 865 (2018) · Zbl 1430.60032 · doi:10.1214/17-AOP1194 |
| [31] | Camilli, F.; Contucci, P.; Mingione, E., An inference problem in a mismatched setting: a spin-glass model with Mattis interactions, SciPost Phys., 12, 125 (2022) · doi:10.21468/SciPostPhys.12.4.125 |
| [32] | Dobrosavljevic, V.; Thirumalai, D., 1/p expansion for a p-spin interaction spin-glass model in a transverse field, J. Phys. A: Math. Gen., 23, 15, L767 (1990) · doi:10.1088/0305-4470/23/15/013 |
| [33] | Baldwin, CL; Laumann, CR; Pal, A.; Scardicchio, A., Clustering of nonergodic eigenstates in quantum spin glasses, Phys. Rev. Lett., 118, 127201 (2017) · doi:10.1103/PhysRevLett.118.127201 |
| [34] | Biroli, G.; Facoetti, D.; Schiró, M.; Tarzia, M.; Vivo, P., Out-of-equilibrium phase diagram of the quantum random energy model, Phys. Rev. B, 103, 014204 (2021) · doi:10.1103/PhysRevB.103.014204 |
| [35] | Jörg, T.; Krzakala, F.; Kurchan, J.; Maggs, AC, Simple glass models and their quantum annealing, Phys. Rev. Lett., 101, 147204 (2008) · doi:10.1103/PhysRevLett.101.147204 |
| [36] | Baldwin, CL; Laumann, CR, Quantum algorithm for energy matching in hard optimization problems, Phys. Rev. B, 97, 224201 (2018) · doi:10.1103/PhysRevB.97.224201 |
| [37] | Smelyanskiy, VN; Kechedzhi, K.; Boixo, S.; Isakov, SV; Neven, H.; Altshuler, B., Nonergodic delocalized states for efficient population transfer within a narrow band of the energy landscape, Phys. Rev. X, 10, 011017 (2020) |
| [38] | Gardner, E., Spin glasses with p-spin interactions, Nucl. Phys. B, 257, 747 (1985) · doi:10.1016/0550-3213(85)90374-8 |
| [39] | Derrida, B., Random-energy model: limit of a family of disordered models, Phys. Rev. Lett., 45, 79 (1980) · doi:10.1103/PhysRevLett.45.79 |
| [40] | Derrida, B., Random-energy model: an exactly solvable model of disordered systems, Phys. Rev. B, 24, 2613 (1981) · Zbl 1323.60134 · doi:10.1103/PhysRevB.24.2613 |
| [41] | Gross, D.; Mezard, M., The simplest spin glass, Nucl. Phys. B, 240, 4, 431 (1984) · doi:10.1016/0550-3213(84)90237-2 |
| [42] | Derrida, B.; Gardner, E., Solution of the generalised random energy model, J. Phys. C: Solid State Phys., 19, 13, 2253 (1986) · doi:10.1088/0022-3719/19/13/015 |
| [43] | Mora, T.; Zdeborová, L., Random subcubes as a toy model for constraint satisfaction problems, J. Stat. Phys., 131, 6, 1121 (2008) · Zbl 1214.82045 · doi:10.1007/s10955-008-9543-x |
| [44] | Gross, DJ; Kanter, I.; Sompolinsky, H., Mean-field theory of the Potts glass, Phys. Rev. Lett., 55, 304 (1985) · doi:10.1103/PhysRevLett.55.304 |
| [45] | de Almeida, JRL; Thouless, DJ, Stability of the Sherrington-Kirkpatrick solution of a spin glass model, J. Phys. A: Math. Gen., 11, 5, 983 (1978) · doi:10.1088/0305-4470/11/5/028 |
| [46] | Aizenman, M.; Sims, R.; Starr, SL, Extended variational principle for the Sherrington-Kirkpatrick spin-glass model, Phys. Rev. B, 68, 214403 (2003) · doi:10.1103/PhysRevB.68.214403 |
| [47] | Crisanti, A.; Sommers, HJ, The spherical p-spin interaction spin glass model: the statics, Z. Phys. B, 87, 3, 341 (1992) · doi:10.1007/BF01309287 |
| [48] | Crisanti, A.; Horner, H.; Sommers, HJ, The spherical p-spin interaction spin-glass model, Z. Phys. B, 92, 2, 257 (1993) · doi:10.1007/BF01312184 |
| [49] | Auerbach, A., Interacting Electrons and Quantum Magnetism (1994), New York: Springer, New York · doi:10.1007/978-1-4612-0869-3 |
| [50] | den Hollander, F., Large Deviations (2000), Providence: AMS, Providence · Zbl 0949.60001 |
| [51] | Barbier, J.; Macris, N., The adaptive interpolation method for proving replica formulas. Applications to the Curie-Weiss and Wigner spike models, J. Phys. A: Math. Theor., 52, 29, 294002 (2019) · Zbl 1509.82018 · doi:10.1088/1751-8121/ab2735 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
