Publication: Dynamical quantum phase transitions from random matrix theory
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2022
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Abstract
We uncover a novel dynamical quantum phase transition, using random matrix theory and its associated notion of planar limit. We study it for the isotropic XY Heisenberg spin chain. For this, we probe its real-time dynamics through the Loschmidt echo. This leads to the study of a random matrix ensemble with a complex weight, whose analysis requires novel technical considerations, that we develop. We obtain three main results: 1) There is a third order phase transition at a rescaled critical time, that we determine. 2) The third order phase transitions persists away from the thermodynamic limit. 3) For times below the critical value, the difference between the thermodynamic limit and a finite chain decreases exponentially with the system size. All these results depend in a rich manner on the parity of the number of flipped spins of the quantum state conforming the fidelity.
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[S1] T. Gorin, T. Prosen, T. H. Seligman and M. Žnidarič, "Dynamics of Loschmidt echoes and fidelity decay," Phys. Rep. 435, 33-156 (2006), arXiv:quant-ph/0607050 [quant-ph].
[S2] N. M. Bogoliubov and C. Malyshev, "The correlation functions of the XXZ Heisenberg Chain for zero or infinite anisotropy and random walks of vicious walkers," St. Petersburg Math. J. 22, 359-377 (2011), arXiv:0912.1138 [cond-mat.stat-mech].
[S3] N. M. Bogoliubov, "XX0 Heisenberg chain and random walks," J. Math. Sci. 138, 5636-5643 (2006).
[S4] N. M. Bogoliubov, "Integrable models for vicious and friendly walkers," J. Math. Sci. 143, 2729-2737 (2007).
[S5] C. Andréief, "Note sur une relation entre les intégrales définies des produits des fonctions," Mém . Soc. Sci. Phys. Nat. Bordeaux 3, 2, 1 (1886).
[S6] P. J. Forrester, \Meet Andréief, Bordeaux 1886, and Andreev, Kharkov 1882|1883," Random Matrices: Theory and Applications 08, 1930001 (2019), arXiv:1806.10411 [math-ph].
[S7] D. Bump and P. Diaconis, "Toeplitz minors," J. Combin. Theory Ser. A 97, 252-271 (2002).
[S8] D. Pérez-García and M. Tierz, "Mapping between the Heisenberg XX spin chain and low-energy QCD," Phys. Rev. X 4, no.2, 021050 (2014), arXiv:1305.3877 [cond-mat.str-el].
[S9] P. J. Forrester, Log-gases and random matrices, London Mathematical Society Monographs Series, Vol. 34 (Princeton University Press, Princeton, NJ, 2010).
[S10] J.-M. Stéphan, "Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain," J. Stat. Mech. 2017, no.10, 103108 (2017), arXiv:1707.06625 [cond-mat.stat-mech].
[S11] D. J. Gross and E. Witten, "Possible third order phase transition in the large N lattice gauge theory," Phys. Rev. D 21, 446-453 (1980).
[S12] S. R. Wadia, "N = Infinity phase transition in a class of exactly soluble model lattice gauge theories," Phys. Lett. B 93, 403-410 (1980).
[S13] S. R. Wadia, "A study of U(N) lattice gauge theory in 2-dimensions," arXiv:1212.2906 [hep-th].
[S14] M. Heyl, "Dynamical quantum phase transitions: a review," Rept. Prog. Phys. 81, no.5, 054001 (2018), arXiv:1709.07461 [cond-mat.stat-mech].
[S15] P. L. Krapivsky, J. M. Luck and K. Mallick, "Quantum return probability of a system of N non-interacting lattice fermions," J. Stat. Mech. 1802, no.2, 023104 (2018), arXiv:1710.08178 [cond-mat.mes-hall].
[S16] T. Kimura and S. Purkayastha, "Classical group matrix models and universal criticality," arXiv:2205.01236 [hep-th].
[S17] P. Di Francesco, P. H. Ginsparg and J. Zinn-Justin, \2-D Gravity and random matrices," Phys. Rept. 254, 1-133 (1995), arXiv:hep-th/9306153 [hep-th].
[S18] M. Marino, "Les Houches lectures on matrix models and topological strings," arXiv:hep-th/0410165 [hep-th].
[S19] B. Eynard, T. Kimura and S. Ribault, \Random matrices," arXiv:1510.04430 [math-ph].
[S20] G. Mandal, "Phase structure of unitary matrix models," Mod. Phys. Lett. A 5, 1147-1158 (1990).
[S21] S. Jain, S. Minwalla, T. Sharma, T. Takimi, S. R. Wadia and S. Yokoyama, "Phases of large N vector Chern-Simons theories on S2 × S1," JHEP 09, 009 (2013), arXiv:1301.6169 [hep-th].
[S22] L. Santilli and M. Tierz, "Phase transition in complex-time Loschmidt echo of short and long range spin chain," J. Stat. Mech. 2006, 063102 (2020), arXiv:1902.06649 [cond-mat.stat-mech].
[S23] L. Santilli and M. Tierz, "Exact equivalences and phase discrepancies between random matrix ensembles," J. Stat. Mech. 2008, 083107 (2020) arXiv:2003.10475 [math-ph].
[S24] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, "Planar diagrams," Commun. Math. Phys. 59, 35 (1978).
[S25] G. 't Hooft, "A planar diagram theory for strong interactions," Nucl. Phys. B 72, 461 (1974).
[S26] P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, Vol.3 (New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999).
[S27] F. G. Tricomi, Integral equations, Pure and Applied Mathematics, Vol.15 (Courier Corporation, 1985).
[S28] K. Johansson, "On random matrices from the compact classical groups," Annals Math. 145, no.3, 519-545 (1997).
[S29] G. Szegő, "On certain Hermitian forms associated with the Fourier series of a positive function," Comm. Sém. Math. Univ. Lund Tome Supplémentaire, 228-238 (1952).
[S30] D. García-García and M. Tierz, "Matrix models for classical groups and Toeplitz±Hankel minors with applications to Chern-Simons theory and fermionic models," J. Phys. A 53, no.34, 345201 (2020), arXiv:1901.08922 [hep-th].
[S31] S. Garcia, Z. Guralnik and G. S. Guralnik, "Theta vacua and boundary conditions of the Schwinger-Dyson equations," arXiv:hep-th/9612079 [hep-th].
[S32] G. Guralnik and Z. Guralnik, "Complexified path integrals and the phases of quantum field theory," Annals Phys. 325, 2486-2498 (2010), arXiv:0710.1256 [hep-th].
[S33] D. D. Ferrante, G. S. Guralnik, Z. Guralnik and C. Pehlevan, "Complex path integrals and the space of theories," arXiv:1301.4233 [hep-th].
[S34] M. Marino, "Nonperturbative effects and nonperturbative definitions in matrix models and topological strings," JHEP 12, 114 (2008), arXiv:0805.3033 [hep-th].
[S35] M. Mariño, "Lectures on non-perturbative effects in large N gauge theories, matrix models and strings," Fortsch. Phys. 62, 455-540 (2014), arXiv:1206.6272 [hep-th].
[S36] G. Penington, S. H. Shenker, D. Stanford and Z. Yang, "Replica wormholes and the black hole interior," JHEP 03, 205 (2022), arXiv:1911.11977 [hep-th].
[S37] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, "ReplicaWormholes and the Entropy of Hawking Radiation," JHEP 05, 013 (2020), arXiv:1911.12333 [hep-th].
[S38] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, "The entropy of Hawking radiation," Rev. Mod. Phys. 93, no.3, 035002 (2021) arXiv:2006.06872 [hep-th].
[S39] F. David, "Phases of the large N matrix model and nonperturbative effects in 2-d gravity," Nucl. Phys. B 348, 507-524 (1991).
[S40] C. Copetti, A. Grassi, Z. Komargodski and L. Tizzano, "Delayed deconfinement and the Hawking-Page transition," JHEP 04, 132 (2022), arXiv:2008.04950 [hep-th].
[S41] F. D. Cunden, P. Facchi, M. Ligabò and P. Vivo, "Third-order phase transition: random matrices and screened Coulomb gas with hard walls," J. Stat. Phys. 175, no.6, 1262-1297 (2019), arXiv:1810.12593 [math-ph].
[S42] A. Deaño, "Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval," J. Approx. Theory 186, 33-63 (2014), arXiv:1402.2085 [math.CA].
[S43] A. F. Celsus, A. Deaño, D. Huybrechs and A. Iserles, "The kissing polynomials and their Hankel determinants," Trans. Math. Appl. 6 (2022), arXiv:1504.07297 [math.CA].
[S44] A. F. Celsus and G. L. F. Silva, "Supercritical regime for the kissing polynomial," J. Approx. Theory 255, 105408 (2020), arXiv:1903.00960 [math.CA].
[S45] L. Santilli and M. Tierz, "Multiple phases and meromorphic deformations of unitary matrix models," Nucl. Phys. B 976, 115694 (2022) arXiv:2102.11305 [hep-th].
[S46] J. Baik, P. Deift, and K. Johansson, "On the distribution of the length of the longest increasing subsequence of random permutations," J. Amer. Math. Soc. 12, 1119-1178 (1999), arXiv:math/9810105 [math.CO].
[S47] J. Baik, "Random vicious walks and random matrices," Comm. Pure Appl. Math. 53, 1385-1410 (2000), arXiv:math/0001022 [math.PR].
[S48] E. Brezin and V. A. Kazakov, "Exactly solvable field theories of closed strings," Phys. Lett. B 236, 144-150 (1990).
[S49] D. J. Gross and A. A. Migdal, "Nonperturbative Two-Dimensional Quantum Gravity," Phys. Rev. Lett. 64, 127 (1990).
[S50] M. R. Douglas and S. H. Shenker, "Strings in less than one-dimension," Nucl. Phys. B 335, 635 (1990).
[S51] D. Aasen, R. S. K. Mong and P. Fendley, "Topological defects on the lattice I: The Ising model," J. Phys. A 49, no.35, 354001 (2016), arXiv:1601.07185 [cond-mat.stat-mech].
[S52] D. Aasen, P. Fendley and R. S. K. Mong, "Topological defects on the lattice: Dualities and degeneracies," arXiv:2008.08598 [cond-mat.stat-mech].
[S53] A. Roy and H. Saleur, "Entanglement entropy in the Ising model with topological defects," Phys. Rev. Lett. 128, no.9, 090603 (2022), arXiv:2111.04534 [hep-th].
[S54] A. Roy and H. Saleur, "Entanglement entropy in critical quantum spin chains with boundaries and defects," arXiv:2111.07927 [quant-ph].
[S55] M. T. Tan, Y. Wang and A. Mitra, "Topological defects in Floquet circuits," arXiv:2206.06272 [cond-mat.str-el].
[S56] S. A. Hartnoll and S. P. Kumar, "Higher rank Wilson loops from a matrix model," JHEP 08, 026 (2006), arXiv:hepth/0605027 [hep-th].
[S57] J. G. Russo and K. Zarembo, "Wilson loops in antisymmetric representations from localization in supersymmetric gauge theories," Rev. Math. Phys. 30, no.07, 1840014 (2018), arXiv:1712.07186 [hep-th].
[S58] L. Santilli and M. Tierz, "Phase transitions and Wilson loops in antisymmetric representations in Chern-Simons-matter theory," J. Phys. A 52, no.38, 385401 (2019), arXiv:1808.02855 [hep-th].
[S59] L. Santilli, "Phases of five-dimensional supersymmetric gauge theories," JHEP 07, 088 (2021), arXiv:2103.14049 [hep-th].
[S60] M. R. Douglas and V. A. Kazakov, "Large N phase transition in continuum QCD in two-dimensions," Phys. Lett. B 319, 219-230 (1993), arXiv:hep-th/9305047 [hep-th].
[S61] J. Baik and Z. Liu, "Discrete Toeplitz/Hankel determinants and the width of non-intersecting processes," Int. Math. Research Not. 20, 5737-5768 (2014), arXiv:1212.4467 [math.PR].
[S62] B. Zhou, Y. Zeng and S. Chen, "Exact zeros of the Loschmidt echo and quantum speed limit time for the dynamical quantum phase transition in finite-size systems," Phys. Rev. B 104, 094311 (2021), arXiv:2107.02709 [quant-ph].
[S63] C. Lupo and M. Schiró, "Transient Loschmidt echo in quenched Ising chains," Phys. Rev. B 94, 014310 (2016), arXiv:1604.01312 [cond-mat.stat-mech].
[S64] T. Fogarty, S. Deffner, T. Busch and S. Campbell, "Orthogonality catastrophe as a consequence of the quantum speed limit," Phys. Rev. Lett. 124, 110601 (2020), arXiv:1910.10728 [quant-ph].
[S65] E. Basor, F. Ge and M. O. Rubinstein, "Some multidimensional integrals in number theory and connections with the Painlevé V equation," J. Math. Phys. 59, 091404 (2018), arXiv:1805.08811 [math.NT].
[S66] M. Adler, and P. van Moerbeke, "Integrals over classical groups, random permutations, Toda and Toeplitz lattices," Commun. Pure Appl. Math. 54, 153-205 (2001), arXiv:math/9912143 [math.CO].
[S67] M. Adler, and P. van Moerbeke, "Virasoro action on Schur function expansions, skew Young tableaux and random walks," Commun. Pure Appl. Math. 58, 362-408 (2005), arXiv:math/0309202 [math.PR].