1Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
3Departamento de Matemática, Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
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 transition 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.
Featured image: Dynamical free energy. N = 10 and L ≫ 2N
Popular summary
Currently, dynamical quantum phase transitions are attracting an enormous amount of effort from both the theoretical and experimental communities. These transitions cause certain measurable physical quantities in a spin chain to be discontinuous in time. We present a new example of a dynamical phase transition that exhibits several exotic features, distinguishing it from previously observed transitions. Our results are obtained from the Heisenberg XY model, a well-known and extensively studied spin chain. Two strengths of our study are its mathematical soundness and experimental verifiability. We develop tailor-made tools inspired by the discipline of random matrix theory and argue quantitatively that the transition should be detectable in a quantum device of modest size.
This work opens up two clear avenues: on the one hand, setting up an experiment to observe the dynamical phase transition, and on the other hand, extending our techniques to predict new dynamical phase transitions.
► BibTeX data
► References
[1] M. Srednicki, Chaos and Quantum Thermalization, Phys. Rev. E 50 (1994) 888 [ cond-mat/9403051].
https://doi.org/10.1103/PhysRevE.50.888
arXiv:cond-mat/9403051
[2] J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81 (2018) 082001 [1805.01616].
https://doi.org/10.1088/1361-6633/aac9f1
arXiv:1805.01616
[3] N. Shiraishi and T. Mori, Systematic construction of counterexamples to the eigenstate thermalization hypothesis, Phys. Rev. Lett. 119 (2017) 030601 [1702.08227].
https://doi.org/10.1103/PhysRevLett.119.030601
arXiv:1702.08227
[4] T. Mori, T. Ikeda, E. Kaminishi and M. Ueda, Thermalization and prethermalization in isolated quantum systems: a theoretical overview, J. Phys. B 51 (2018) 112001 [ 1712.08790].
https://doi.org/10.1088/1361-6455/aabcdf
arXiv:1712.08790
[5] R. Nandkishore and D. A. Huse, Many body localization and thermalization in quantum statistical mechanics, Ann. Rev. Condensed Matter Phys. 6 (2015) 15 [1404.0686].
https://doi.org/10.1146/annurev-conmatphys-031214-014726
arXiv:1404.0686
[6] R. Vasseur and J. E. Moore, Nonequilibrium quantum dynamics and transport: from integrability to many-body localization, J. Stat. Mech. 1606 (2016) 064010 [1603.06618].
https://doi.org/10.1088/1742-5468/2016/06/064010
arXiv:1603.06618
[7] J. Z. Imbrie, On many-body localization for quantum spin chains, J. Stat. Phys. 163 (2016) 998 [1403.7837].
https://doi.org/10.1007/s10955-016-1508-x
arXiv:1403.7837
[8] J. Z. Imbrie, V. Ros and A. Scardicchio, Local integrals of motion in many-body localized systems, Annalen der Physik 529 (2017) 1600278 [1609.08076].
https://doi.org/10.1002/andp.201600278
arXiv:1609.08076
[9] S. A. Parameswaran and R. Vasseur, Many-body localization, symmetry, and topology, Rept. Prog. Phys. 81 (2018) 082501 [1801.07731].
https://doi.org/10.1088/1361-6633/aac9ed
arXiv:1801.07731
[10] D. A. Abanin, E. Altman, I. Bloch and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91 (2019) 021001.
https://doi.org/10.1103/RevModPhys.91.021001
[11] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner et al., Probing many-body dynamics on a 51-atom quantum simulator, Nature 551 (2017) 579 [ 1707.04344].
https://doi.org/10.1038/nature24622
arXiv:1707.04344
[12] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nature Phys. 14 (2018) 745 [1711.03528].
https://doi.org/10.1038/s41567-018-0137-5
arXiv:1711.03528
[13] M. Serbyn, D. A. Abanin and Z. Papić, Quantum many-body scars and weak breaking of ergodicity, Nature Phys. 17 (2021) 675 [2011.09486].
https://doi.org/10.1038/s41567-021-01230-2
arXiv:2011.09486
[14] P. Sala, T. Rakovszky, R. Verresen, M. Knap and F. Pollmann, Ergodicity breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians, Phys. Rev. X 10 (2020) 011047 [1904.04266].
https://doi.org/10.1103/PhysRevX.10.011047
arXiv:1904.04266
[15] M. Heyl, A. Polkovnikov and S. Kehrein, Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model, Phys. Rev. Lett. 110 (2013) 135704 [1206.2505].
https://doi.org/10.1103/PhysRevLett.110.135704
arXiv:1206.2505
[16] C. Karrasch and D. Schuricht, Dynamical phase transitions after quenches in nonintegrable models, Phys. Rev. B 87 (2013) 195104 [1302.3893].
https://doi.org/10.1103/PhysRevB.87.195104
arXiv:1302.3893
[17] J. M. Hickey, S. Genway and J. P. Garrahan, Dynamical phase transitions, time-integrated observables, and geometry of states, Phys. Rev. B 89 (2014) 054301 [1309.1673].
https://doi.org/10.1103/PhysRevB.89.054301
arXiv:1309.1673
[18] S. Vajna and B. Dóra, Disentangling dynamical phase transitions from equilibrium phase transitions, Phys. Rev. B 89 (2014) 161105 [1401.2865].
https://doi.org/10.1103/PhysRevB.89.161105
arXiv:1401.2865
[19] M. Heyl, Dynamical quantum phase transitions in systems with broken-symmetry phases, Phys. Rev. Lett. 113 (2014) 205701 [1403.4570].
https://doi.org/10.1103/PhysRevLett.113.205701
arXiv:1403.4570
[20] J. N. Kriel, C. Karrasch and S. Kehrein, Dynamical quantum phase transitions in the axial next-nearest-neighbor Ising chain, Phys. Rev. B 90 (2014) 125106 [1407.4036].
https://doi.org/10.1103/PhysRevB.90.125106
arXiv:1407.4036
[21] S. Vajna and B. Dóra, Topological classification of dynamical phase transitions, Phys. Rev. B 91 (2015) 155127 [1409.7019].
https://doi.org/10.1103/PhysRevB.91.155127
arXiv:1409.7019
[22] J. C. Budich and M. Heyl, Dynamical topological order parameters far from equilibrium, Phys. Rev. B 93 (2016) 085416 [1504.05599].
https://doi.org/10.1103/PhysRevB.93.085416
arXiv:1504.05599
[23] M. Schmitt and S. Kehrein, Dynamical quantum phase transitions in the kitaev honeycomb model, Phys. Rev. B 92 (2015) 075114 [1505.03401].
https://doi.org/10.1103/PhysRevB.92.075114
arXiv:1505.03401
[24] M. Heyl, Scaling and universality at dynamical quantum phase transitions, Phys. Rev. Lett. 115 (2015) 140602 [1505.02352].
https://doi.org/10.1103/PhysRevLett.115.140602
arXiv:1505.02352
[25] S. Sharma, S. Suzuki and A. Dutta, Quenches and dynamical phase transitions in a nonintegrable quantum Ising model, Phys. Rev. B 92 (2015) 104306 [1506.00477].
https://doi.org/10.1103/PhysRevB.92.104306
arXiv:1506.00477
[26] J. M. Zhang and H.-T. Yang, Cusps in the quench dynamics of a Bloch state, EPL 114 (2016) 60001 [1601.03569].
https://doi.org/10.1209/0295-5075/114/60001
arXiv:1601.03569
[27] S. Sharma, U. Divakaran, A. Polkovnikov and A. Dutta, Slow quenches in a quantum Ising chain: Dynamical phase transitions and topology, Phys. Rev. B 93 (2016) 144306 [1601.01637].
https://doi.org/10.1103/PhysRevB.93.144306
arXiv:1601.01637
[28] T. Puskarov and D. Schuricht, Time evolution during and after finite-time quantum quenches in the transverse-field Ising chain, SciPost Phys. 1 (2016) 003 [ 1608.05584].
https://doi.org/10.21468/SciPostPhys.1.1.003
arXiv:1608.05584
[29] B. Zunkovic, M. Heyl, M. Knap and A. Silva, Dynamical quantum phase transitions in spin chains with long-range interactions: Merging different concepts of nonequilibrium criticality, Phys. Rev. Lett. 120 (2018) 130601 [1609.08482].
https://doi.org/10.1103/PhysRevLett.120.130601
arXiv:1609.08482
[30] J. C. Halimeh and V. Zauner-Stauber, Dynamical phase diagram of quantum spin chains with long-range interactions, Phys. Rev. B 96 (2017) 134427 [1610.02019].
https://doi.org/10.1103/PhysRevB.96.134427
arXiv:1610.02019
[31] S. Banerjee and E. Altman, Solvable model for a dynamical quantum phase transition from fast to slow scrambling, Phys. Rev. B 95 (2017) 134302 [1610.04619].
https://doi.org/10.1103/PhysRevB.95.134302
arXiv:1610.04619
[32] C. Karrasch and D. Schuricht, Dynamical quantum phase transitions in the quantum Potts chain, Phys. Rev. B 95 (2017) 075143 [1701.04214].
https://doi.org/10.1103/PhysRevB.95.075143
arXiv:1701.04214
[33] L. Zhou, Q.-h. Wang, H. Wang and J. Gong, Dynamical quantum phase transitions in non-hermitian lattices, Phys. Rev. A 98 (2018) 022129 [1711.10741].
https://doi.org/10.1103/PhysRevA.98.022129
arXiv:1711.10741
[34] E. Guardado-Sanchez, P. T. Brown, D. Mitra, T. Devakul, D. A. Huse, P. Schauss and W. S. Bakr, Probing the quench dynamics of antiferromagnetic correlations in a 2D quantum Ising spin system, Phys. Rev. X 8 (2018) 021069 [1711.00887].
https://doi.org/10.1103/PhysRevX.8.021069
arXiv:1711.00887
[35] M. Heyl, F. Pollmann and B. Dóra, Detecting Equilibrium and Dynamical Quantum Phase Transitions in Ising Chains via Out-of-Time-Ordered Correlators, Phys. Rev. Lett. 121 (2018) 016801 [1801.01684].
https://doi.org/10.1103/PhysRevLett.121.016801
arXiv:1801.01684
[36] S. Bandyopadhyay, S. Laha, U. Bhattacharya and A. Dutta, Exploring the possibilities of dynamical quantum phase transitions in the presence of a Markovian bath, Sci. Rep. 8 (2018) 11921 [ 1804.03865].
https://doi.org/10.1038/s41598-018-30377-x
arXiv:1804.03865
[37] J. Lang, B. Frank and J. C. Halimeh, Dynamical quantum phase transitions: A geometric picture, Phys. Rev. Lett. 121 (2018) 130603 [1804.09179].
https://doi.org/10.1103/PhysRevLett.121.130603
arXiv:1804.09179
[38] U. Mishra, R. Jafari and A. Akbari, Disordered Kitaev chain with long-range pairing: Loschmidt echo revivals and dynamical phase transitions, J. Phys. A 53 (2020) 375301 [1810.06236].
https://doi.org/10.1088/1751-8121/ab97de
arXiv:1810.06236
[39] T. Hashizume, I. P. McCulloch and J. C. Halimeh, Dynamical phase transitions in the two-dimensional transverse-field ising model, Phys. Rev. Res. 4 (2022) 013250 [1811.09275].
https://doi.org/10.1103/PhysRevResearch.4.013250
arXiv:1811.09275
[40] A. Khatun and S. M. Bhattacharjee, Boundaries and unphysical fixed points in dynamical quantum phase transitions, Phys. Rev. Lett. 123 (2019) 160603 [1907.03735].
https://doi.org/10.1103/PhysRevLett.123.160603
arXiv:1907.03735
[41] S. P. Pedersen and N. T. Zinner, Lattice gauge theory and dynamical quantum phase transitions using noisy intermediate scale quantum devices, Phys. Rev. B 103 (2021) 235103 [2008.08980].
https://doi.org/10.1103/PhysRevB.103.235103
arXiv:2008.08980
[42] S. De Nicola, A. A. Michailidis and M. Serbyn, Entanglement View of Dynamical Quantum Phase Transitions, Phys. Rev. Lett. 126 (2021) 040602 [2008.04894].
https://doi.org/10.1103/PhysRevLett.126.040602
arXiv:2008.04894
[43] S. Zamani, R. Jafari and A. Langari, Floquet dynamical quantum phase transition in the extended xy model: Nonadiabatic to adiabatic topological transition, Phys. Rev. B 102 (2020) 144306 [2009.09008].
https://doi.org/10.1103/PhysRevB.102.144306
arXiv:2009.09008
[44] S. Peotta, F. Brange, A. Deger, T. Ojanen and C. Flindt, Determination of dynamical quantum phase transitions in strongly correlated many-body systems using Loschmidt cumulants, Phys. Rev. X 11 (2021) 041018 [2011.13612].
https://doi.org/10.1103/PhysRevX.11.041018
arXiv:2011.13612
[45] Y. Bao, S. Choi and E. Altman, Symmetry enriched phases of quantum circuits, Annals Phys. 435 (2021) 168618 [2102.09164].
https://doi.org/10.1016/j.aop.2021.168618
arXiv:2102.09164
[46] H. Cheraghi and S. Mahdavifar, Dynamical Quantum Phase Transitions in the 1D Nonintegrable Spin-1/2 Transverse Field XZZ Model, Annalen Phys. 533 (2021) 2000542.
https://doi.org/10.1002/andp.202000542
[47] R. Okugawa, H. Oshiyama and M. Ohzeki, Mirror-symmetry-protected dynamical quantum phase transitions in topological crystalline insulators, Phys. Rev. Res. 3 (2021) 043064 [2105.12768].
https://doi.org/10.1103/PhysRevResearch.3.043064
arXiv:2105.12768
[48] J. C. Halimeh, M. Van Damme, L. Guo, J. Lang and P. Hauke, Dynamical phase transitions in quantum spin models with antiferromagnetic long-range interactions, Phys. Rev. B 104 (2021) 115133 [2106.05282].
https://doi.org/10.1103/PhysRevB.104.115133
arXiv:2106.05282
[49] J. Naji, M. Jafari, R. Jafari and A. Akbari, Dissipative Floquet dynamical quantum phase transition, Phys. Rev. A 105 (2022) 022220 [2111.06131].
https://doi.org/10.1103/PhysRevA.105.022220
arXiv:2111.06131
[50] R. Jafari, A. Akbari, U. Mishra and H. Johannesson, Floquet dynamical quantum phase transitions under synchronized periodic driving, Phys. Rev. B 105 (2022) 094311 [2111.09926].
https://doi.org/10.1103/PhysRevB.105.094311
arXiv:2111.09926
[51] F. J. González, A. Norambuena and R. Coto, Dynamical quantum phase transition in diamond: Applications in quantum metrology, Phys. Rev. B 106 (2022) 014313 [2202.05216].
https://doi.org/10.1103/PhysRevB.106.014313
arXiv:2202.05216
[52] M. Van Damme, T. V. Zache, D. Banerjee, P. Hauke and J. C. Halimeh, Dynamical quantum phase transitions in spin-S U(1) quantum link models, Phys. Rev. B 106 (2022) 245110 [2203.01337].
https://doi.org/10.1103/PhysRevB.106.245110
arXiv:2203.01337
[53] Y. Qin and S.-C. Li, Quantum phase transition of a modified spin-boson model, J. Phys. A 55 (2022) 145301.
https://doi.org/10.1088/1751-8121/ac5507
[54] A. L. Corps and A. Relaño, Dynamical and excited-state quantum phase transitions in collective systems, Phys. Rev. B 106 (2022) 024311 [2205.11199].
https://doi.org/10.1103/PhysRevB.106.024311
arXiv:2205.11199
[55] D. Mondal and T. Nag, Anomaly in the dynamical quantum phase transition in a non-Hermitian system with extended gapless phases, Phys. Rev. B 106 (2022) 054308 [2205.12859].
https://doi.org/10.1103/PhysRevB.106.054308
arXiv:2205.12859
[56] M. Heyl, Dynamical quantum phase transitions: a review, Rept. Prog. Phys. 81 (2018) 054001 [1709.07461].
https://doi.org/10.1088/1361-6633/aaaf9a
arXiv:1709.07461
[57] A. Zvyagin, Dynamical quantum phase transitions, Low Temperature Physics 42 (2016) 971 [1701.08851].
https://doi.org/10.1063/1.4969869
arXiv:1701.08851
[58] M. Heyl, Dynamical quantum phase transitions: a brief survey, EPL 125 (2019) 26001 [ 1811.02575].
https://doi.org/10.1209/0295-5075/125/26001
arXiv:1811.02575
[59] J. Marino, M. Eckstein, M. S. Foster and A. M. Rey, Dynamical phase transitions in the collisionless pre-thermal states of isolated quantum systems: theory and experiments, Rept. Prog. Phys. 85 (2022) 116001 [2201.09894].
https://doi.org/10.1088/1361-6633/ac906c
arXiv:2201.09894
[60] I. Bloch, Ultracold Bosonic Atoms in Optical Lattices, in Understanding Quantum Phase Transitions (L. Carr, ed.), Series in Condensed Matter Physics, ch. 19, p. 469. CRC Press, 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742, 2010.
[61] N. Fläschner, D. Vogel, M. Tarnowski, B. S. Rem, D. S. Lühmann, M. Heyl, J. C. Budich, L. Mathey, K. Sengstock and C. Weitenberg, Observation of dynamical vortices after quenches in a system with topology, Nature Phys. 14 (2018) 265 [1608.05616].
https://doi.org/10.1038/s41567-017-0013-8
arXiv:1608.05616
[62] P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt and C. F. Roos, Direct observation of dynamical quantum phase transitions in an interacting many-body system, Phys. Rev. Lett. 119 (2017) 080501 [1612.06902].
https://doi.org/10.1103/PhysRevLett.119.080501
arXiv:1612.06902
[63] J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong and C. Monroe, Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, Nature 551 (2017) 601 [ 1708.01044].
https://doi.org/10.1038/nature24654
arXiv:1708.01044
[64] X.-Y. Guo, C. Yang, Y. Zeng, Y. Peng, H.-K. Li, H. Deng, Y.-R. Jin, S. Chen, D. Zheng and H. Fan, Observation of a dynamical quantum phase transition by a superconducting qubit simulation, Phys. Rev. Applied 11 (2019) 044080 [1806.09269].
https://doi.org/10.1103/PhysRevApplied.11.044080
arXiv:1806.09269
[65] K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, W. Yi and P. Xue, Simulating dynamic quantum phase transitions in photonic quantum walks, Phys. Rev. Lett. 122 (2019) 020501 [1806.10871].
https://doi.org/10.1103/PhysRevLett.122.020501
arXiv:1806.10871
[66] T. Tian, Y. Ke, L. Zhang, S. Lin, Z. Shi, P. Huang, C. Lee and J. Du, Observation of dynamical phase transitions in a topological nanomechanical system, Phys. Rev. B 100 (2019) 024310 [1807.04483].
https://doi.org/10.1103/PhysRevB.100.024310
arXiv:1807.04483
[67] X. Nie et al., Experimental Observation of Equilibrium and Dynamical Quantum Phase Transitions via Out-of-Time-Ordered Correlators, Phys. Rev. Lett. 124 (2020) 250601 [1912.12038].
https://doi.org/10.1103/PhysRevLett.124.250601
arXiv:1912.12038
[68] R. A. Jalabert and H. M. Pastawski, Environment-independent decoherence rate in classically chaotic systems, Phys. Rev. Lett. 86 (2001) 2490 [ cond-mat/0010094].
https://doi.org/10.1103/PhysRevLett.86.2490
arXiv:cond-mat/0010094
[69] E. L. Hahn, Spin echoes, Phys. Rev. 80 (1950) 580.
https://doi.org/10.1103/PhysRev.80.580
[70] T. Gorin, T. Prosen, T. H. Seligman and M. Žnidarič, Dynamics of Loschmidt echoes and fidelity decay, Phys. Rep. 435 (2006) 33 [ quant-ph/0607050].
https://doi.org/10.1016/j.physrep.2006.09.003
arXiv:quant-ph/0607050
[71] D. J. Gross and E. Witten, Possible Third Order Phase Transition in the Large N Lattice Gauge Theory, Phys. Rev. D 21 (1980) 446.
https://doi.org/10.1103/PhysRevD.21.446
[72] S. R. Wadia, $N$ = Infinity Phase Transition in a Class of Exactly Soluble Model Lattice Gauge Theories, Phys. Lett. B 93 (1980) 403.
https://doi.org/10.1016/0370-2693(80)90353-6
[73] S. R. Wadia, A Study of U(N) Lattice Gauge Theory in 2-dimensions, [1212.2906].
arXiv:1212.2906
[74] A. LeClair, G. Mussardo, H. Saleur and S. Skorik, Boundary energy and boundary states in integrable quantum field theories, Nucl. Phys. B 453 (1995) 581 [hep-th/9503227].
https://doi.org/10.1016/0550-3213(95)00435-u
arXiv:hep-th/9503227
[75] D. Pérez-García and M. Tierz, Mapping between the Heisenberg XX Spin Chain and Low-Energy QCD, Phys. Rev. X 4 (2014) 021050 [1305.3877].
https://doi.org/10.1103/PhysRevX.4.021050
arXiv:1305.3877
[76] J.-M. Stéphan, Emptiness formation probability, Toeplitz determinants, and conformal field theory, J. Stat. Mech. 2014 (2014) P05010 [1303.5499].
https://doi.org/10.1088/1742-5468/2014/05/p05010
arXiv:1303.5499
[77] B. Pozsgay, The dynamical free energy and the Loschmidt echo for a class of quantum quenches in the Heisenberg spin chain, J. Stat. Mech. 2013 (2013) P10028 [1308.3087].
https://doi.org/10.1088/1742-5468/2013/10/p10028
arXiv:1308.3087
[78] D. Pérez-García and M. Tierz, Chern-Simons theory encoded on a spin chain, J. Stat. Mech. 1601 (2016) 013103 [1403.6780].
https://doi.org/10.1088/1742-5468/2016/01/013103
arXiv:1403.6780
[79] 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 (2017) 103108 [1707.06625].
https://doi.org/10.1088/1742-5468/aa8c19
arXiv:1707.06625
[80] L. Santilli and M. Tierz, Phase transition in complex-time Loschmidt echo of short and long range spin chain, J. Stat. Mech. 2006 (2020) 063102 [1902.06649].
https://doi.org/10.1088/1742-5468/ab837b
arXiv:1902.06649
[81] 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 (2018) 023104 [1710.08178].
https://doi.org/10.1088/1742-5468/aaa79a
arXiv:1710.08178
[82] J. Viti, J.-M. Stéphan, J. Dubail and M. Haque, Inhomogeneous quenches in a free fermionic chain: Exact results, EPL 115 (2016) 40011 [ 1507.08132].
https://doi.org/10.1209/0295-5075/115/40011
arXiv:1507.08132
[83] J.-M. Stéphan, Exact time evolution formulae in the XXZ spin chain with domain wall initial state, J. Phys. A 55 (2022) 204003 [ 2112.12092].
https://doi.org/10.1088/1751-8121/ac5fe8
arXiv:2112.12092
[84] L. Piroli, B. Pozsgay and E. Vernier, From the quantum transfer matrix to the quench action: the Loschmidt echo in XXZ Heisenberg spin chains, J. Stat. Mech. 1702 (2017) 023106 [1611.06126].
https://doi.org/10.1088/1742-5468/aa5d1e
arXiv:1611.06126
[85] L. Piroli, B. Pozsgay and E. Vernier, Non-analytic behavior of the Loschmidt echo in XXZ spin chains: Exact results, Nucl. Phys. B 933 (2018) 454 [1803.04380].
https://doi.org/10.1016/j.nuclphysb.2018.06.015
arXiv:1803.04380
[86] E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, Planar Diagrams, Commun. Math. Phys. 59 (1978) 35.
https://doi.org/10.1007/BF01614153
[87] S. Sachdev, Quantum Phase Transitions. Cambridge University Press, 2 ed., 2011, 10.1017/CBO9780511973765.
https://doi.org/10.1017/CBO9780511973765
[88] E. Canovi, P. Werner and M. Eckstein, First-order dynamical phase transitions, Phys. Rev. Lett. 113 (2014) 265702 [1408.1795].
https://doi.org/10.1103/PhysRevLett.113.265702
arXiv:1408.1795
[89] R. Hamazaki, Exceptional dynamical quantum phase transitions in periodically driven systems, Nature Commun. 12 (2021) 1 [ 2012.11822].
https://doi.org/10.1038/s41467-021-25355-3
arXiv:2012.11822
[90] S. M. A. Rombouts, J. Dukelsky and G. Ortiz, Quantum phase diagram of the integrable $p_x + ip_y$ fermionic superfluid, Phys. Rev. B 82 (2010) 224510.
https://doi.org/10.1103/PhysRevB.82.224510
[91] H. S. Lerma, S. M. A. Rombouts, J. Dukelsky and G. Ortiz, Integrable two-channel $p_x + ip_y$-wave superfluid model, Phys. Rev. B 84 (2011) 100503 [1104.3766].
https://doi.org/10.1103/PhysRevB.84.100503
arXiv:1104.3766
[92] T. Eisele, On a third-order phase transition, Commun. Math. Phys. 90 (1983) 125.
https://doi.org/10.1007/BF01209390
[93] J.-O. Choi and U. Yu, Phase transition in the diffusion and bootstrap percolation models on regular random and Erdős-Rényi networks, J. Comput. Phys. 446 (2021) 110670 [2108.12082].
https://doi.org/10.1016/j.jcp.2021.110670
arXiv:2108.12082
[94] J. Chakravarty and D. Jain, Critical exponents for higher order phase transitions: Landau theory and RG flow, J. Stat. Mech. 2021 (2021) 093204 [2102.08398].
https://doi.org/10.1088/1742-5468/ac1f11
arXiv:2102.08398
[95] S. N. Majumdar and G. Schehr, Top eigenvalue of a random matrix: large deviations and third order phase transition, J. Stat. Mech. 2014 (2014) P01012 [1311.0580].
https://doi.org/10.1088/1742-5468/2014/01/P01012
arXiv:1311.0580
[96] I. Bars and F. Green, Complete Integration of U ($N$) Lattice Gauge Theory in a Large $N$ Limit, Phys. Rev. D 20 (1979) 3311.
https://doi.org/10.1103/PhysRevD.20.3311
[97] K. Johansson, The longest increasing subsequence in a random permutation and a unitary random matrix model, Math. Res. Lett. 5 (1998) 63.
https://doi.org/10.4310/MRL.1998.v5.n1.a6
[98] 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 (1999) 1119 [math/9810105].
https://doi.org/10.1090/S0894-0347-99-00307-0
arXiv:math/9810105
[99] S. Lu, M. C. Banuls and J. I. Cirac, Algorithms for quantum simulation at finite energies, PRX Quantum 2 (2021) 020321.
https://doi.org/10.1103/PRXQuantum.2.020321
[100] Y. Yang, A. Christianen, S. Coll-Vinent, V. Smelyanskiy, M. C. Bañuls, T. E. O’Brien, D. S. Wild and J. I. Cirac, Simulating Prethermalization Using Near-Term Quantum Computers, PRX Quantum 4 (2023) 030320 [2303.08461].
https://doi.org/10.1103/PRXQuantum.4.030320
arXiv:2303.08461
[101] C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357 (2017) 995.
https://doi.org/10.1126/science.aal383
[102] J. Vijayan, P. Sompet, G. Salomon, J. Koepsell, S. Hirthe, A. Bohrdt, F. Grusdt, I. Bloch and C. Gross, Time-resolved observation of spin-charge deconfinement in fermionic Hubbard chains, Science 367 (2020) 186 [ 1905.13638].
https://doi.org/10.1126/science.aay2354
arXiv:1905.13638
[103] E. Lieb, T. Schultz and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals Phys. 16 (1961) 407.
https://doi.org/10.1016/0003-4916(61)90115-4
[104] J. A. Muniz, D. Barberena, R. J. Lewis-Swan, D. J. Young, J. R. K. Cline, A. M. Rey and J. K. Thompson, Exploring dynamical phase transitions with cold atoms in an optical cavity, Nature 580 (2020) 602.
https://doi.org/10.1038/s41586-020-2224-x
[105] 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 (2011) 359 [0912.1138].
https://doi.org/10.1090/S1061-0022-2011-01146-X
arXiv:0912.1138
[106] C. Andréief, Note sur une relation entre les intégrales définies des produits des fonctions, Mém . Soc. Sci. Phys. Nat. Bordeaux 2 (1886) 1.
[107] C. Copetti, A. Grassi, Z. Komargodski and L. Tizzano, Delayed deconfinement and the Hawking-Page transition, JHEP 04 (2022) 132 [ 2008.04950].
https://doi.org/10.1007/JHEP04(2022)132
arXiv:2008.04950
[108] A. Deaño, Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval, J. Approx. Theory 186 (2014) 33 [ 1402.2085].
https://doi.org/10.1016/j.jat.2014.07.004
arXiv:1402.2085
[109] J. Baik and Z. Liu, Discrete Toeplitz/Hankel determinants and the width of non-intersecting processes, Int. Math. Research Not. 20 (2014) 5737 [1212.4467].
https://doi.org/10.1093/imrn/rnt143
arXiv:1212.4467
[110] L. Mandelstam and I. Tamm, The uncertainty relation between energy and time in non-relativistic quantum mechanics, in Selected papers (B. Bolotovskii, V. Frenkel and R. Peierls, eds.), pp. 115–123. Springer, Berlin, Heidelberg, 1991. DOI.
https://doi.org/10.1007/978-3-642-74626-0_8
[111] N. Margolus and L. B. Levitin, The maximum speed of dynamical evolution, Physica D 120 (1998) 188 [ quant-ph/9710043].
https://doi.org/10.1016/S0167-2789(98)00054-2
arXiv:quant-ph/9710043
[112] G. Ness, M. R. Lam, W. Alt, D. Meschede, Y. Sagi and A. Alberti, Observing crossover between quantum speed limits, Sci. Adv. 7 (2021) eabj9119.
https://doi.org/10.1126/sciadv.abj9119
[113] S. Deffner and S. Campbell, Quantum speed limits: from heisenberg’s uncertainty principle to optimal quantum control, J. Phys. A 50 (2017) 453001 [ 1705.08023].
https://doi.org/10.1088/1751-8121/aa86c6
arXiv:1705.08023
[114] L. Vaidman, Minimum time for the evolution to an orthogonal quantum state, Am. J. Phys. 60 (1992) 182.
https://doi.org/10.1119/1.16940
[115] 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 (2021) 094311 [2107.02709].
https://doi.org/10.1103/PhysRevB.104.094311
arXiv:2107.02709
[116] G. Szegő, On certain Hermitian forms associated with the Fourier series of a positive function, Comm. Sém. Math. Univ. Lund Tome Supplémentaire (1952) 228–238.
[117] M. Adler and P. van Moerbeke, Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Commun. Pure Appl. Math. 54 (2001) 153 [math/9912143].
<a href="https://doi.org/10.1002/1097-0312(200102)54:23.0.CO;2-5″>https://doi.org/10.1002/1097-0312(200102)54:2<153::AID-CPA2>3.0.CO;2-5
arXiv:math/9912143
[118] N. M. Bogoliubov, XX0 Heisenberg chain and random walks, J. Math. Sci. 138 (2006) 5636–5643.
https://doi.org/10.1007/s10958-006-0332-2
[119] N. M. Bogoliubov, Integrable models for vicious and friendly walkers, J. Math. Sci. 143 (2007) 2729.
https://doi.org/10.1007/s10958-007-0160-z
[120] C. Andréief, Note sur une relation entre les intégrales définies des produits des fonctions, Mém . Soc. Sci. Phys. Nat. Bordeaux 2 (1886) 1.
[121] P. J. Forrester, Meet Andréief, Bordeaux 1886, and Andreev, Kharkov 1882–-1883, Random Matrices: Theory and Applications 08 (2019) 1930001 [1806.10411].
https://doi.org/10.1142/S2010326319300018
arXiv:1806.10411
[122] D. Bump and P. Diaconis, Toeplitz Minors, J. Combin. Theory Ser. A 97 (2002) 252.
https://doi.org/10.1006/jcta.2001.3214
[123] P. J. Forrester, Log-gases and random matrices, vol. 34 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2010, 10.1515/9781400835416.
https://doi.org/10.1515/9781400835416
[124] T. Kimura and S. Purkayastha, Classical group matrix models and universal criticality, JHEP 09 (2022) 163 [ 2205.01236].
https://doi.org/10.1007/JHEP09(2022)163
arXiv:2205.01236
[125] P. Di Francesco, P. H. Ginsparg and J. Zinn-Justin, 2-D Gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153].
https://doi.org/10.1016/0370-1573(94)00084-G
arXiv:hep-th/9306153
[126] M. Mariño, Les Houches lectures on matrix models and topological strings, [ hep-th/0410165].
arXiv:hep-th/0410165
[127] B. Eynard, T. Kimura and S. Ribault, Random matrices, [1510.04430].
arXiv:1510.04430
[128] G. Mandal, Phase Structure of Unitary Matrix Models, Mod. Phys. Lett. A 5 (1990) 1147.
https://doi.org/10.1142/S0217732390001281
[129] S. Jain, S. Minwalla, T. Sharma, T. Takimi, S. R. Wadia and S. Yokoyama, Phases of large $N$ vector Chern-Simons theories on $S^2 times S^1$, JHEP 09 (2013) 009 [ 1301.6169].
https://doi.org/10.1007/JHEP09(2013)009
arXiv:1301.6169
[130] L. Santilli and M. Tierz, Exact equivalences and phase discrepancies between random matrix ensembles, J. Stat. Mech. 2008 (2020) 083107 [2003.10475].
https://doi.org/10.1088/1742-5468/aba594
arXiv:2003.10475
[131] G. ‘t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461.
https://doi.org/10.1016/0550-3213(74)90154-0
[132] P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, vol. 3 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.
[133] F. G. Tricomi, Integral equations, vol. 5 of Pure and Applied Mathematics. Courier Corporation, 1985.
[134] K. Johansson, On random matrices from the compact classical groups, Annals Math. 145 (1997) 519.
https://doi.org/10.2307/2951843
[135] D. García-García and M. Tierz, Matrix models for classical groups and Toeplitz$pm $Hankel minors with applications to Chern-Simons theory and fermionic models, J. Phys. A 53 (2020) 345201 [1901.08922].
https://doi.org/10.1088/1751-8121/ab9b4d
arXiv:1901.08922
[136] S. Garcia, Z. Guralnik and G. S. Guralnik, Theta vacua and boundary conditions of the Schwinger-Dyson equations, [hep-th/9612079].
arXiv:hep-th/9612079
[137] G. Guralnik and Z. Guralnik, Complexified path integrals and the phases of quantum field theory, Annals Phys. 325 (2010) 2486 [0710.1256].
https://doi.org/10.1016/j.aop.2010.06.001
arXiv:0710.1256
[138] D. D. Ferrante, G. S. Guralnik, Z. Guralnik and C. Pehlevan, Complex Path Integrals and the Space of Theories, in Miami 2010: Topical Conference on Elementary Particles, Astrophysics, and Cosmology, 1, 2013, [1301.4233].
arXiv:1301.4233
[139] M. Marino, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [ 0805.3033].
https://doi.org/10.1088/1126-6708/2008/12/114
arXiv:0805.3033
[140] M. Mariño, Lectures on non-perturbative effects in large $N$ gauge theories, matrix models and strings, Fortsch. Phys. 62 (2014) 455 [ 1206.6272].
https://doi.org/10.1002/prop.201400005
arXiv:1206.6272
[141] G. Penington, S. H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205 [ 1911.11977].
https://doi.org/10.1007/JHEP03(2022)205
arXiv:1911.11977
[142] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [ 1911.12333].
https://doi.org/10.1007/JHEP05(2020)013
arXiv:1911.12333
[143] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [2006.06872].
https://doi.org/10.1103/RevModPhys.93.035002
arXiv:2006.06872
[144] F. David, Phases of the large N matrix model and nonperturbative effects in 2-d gravity, Nucl. Phys. B 348 (1991) 507.
https://doi.org/10.1016/0550-3213(91)90202-9
[145] 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 (2019) 1262 [1810.12593].
https://doi.org/10.1007/s10955-019-02281-9
arXiv:1810.12593
[146] A. F. Celsus, A. Deaño, D. Huybrechs and A. Iserles, The kissing polynomials and their Hankel determinants, Trans. Math. Appl. 6 (2022) [ 1504.07297].
https://doi.org/10.1093/imatrm/tnab005
arXiv:1504.07297
[147] A. F. Celsus and G. L. Silva, Supercritical regime for the kissing polynomials, J. Approx. Theory 255 (2020) 105408 [1903.00960].
https://doi.org/10.1016/j.jat.2020.105408
arXiv:1903.00960
[148] L. Santilli and M. Tierz, Multiple phases and meromorphic deformations of unitary matrix models, Nucl. Phys. B 976 (2022) 115694 [2102.11305].
https://doi.org/10.1016/j.nuclphysb.2022.115694
arXiv:2102.11305
[149] J. Baik, Random vicious walks and random matrices, Comm. Pure Appl. Math. 53 (2000) 1385 [math/0001022].
<a href="https://doi.org/10.1002/1097-0312(200011)53:113.3.CO;2-K”>https://doi.org/10.1002/1097-0312(200011)53:11<1385::AID-CPA3>3.3.CO;2-K
arXiv:math/0001022
[150] E. Brezin and V. A. Kazakov, Exactly Solvable Field Theories of Closed Strings, Phys. Lett. B 236 (1990) 144.
https://doi.org/10.1016/0370-2693(90)90818-Q
[151] D. J. Gross and A. A. Migdal, Nonperturbative Two-Dimensional Quantum Gravity, Phys. Rev. Lett. 64 (1990) 127.
https://doi.org/10.1103/PhysRevLett.64.127
[152] M. R. Douglas and S. H. Shenker, Strings in Less Than One-Dimension, Nucl. Phys. B 335 (1990) 635.
https://doi.org/10.1016/0550-3213(90)90522-F
[153] D. Aasen, R. S. K. Mong and P. Fendley, Topological Defects on the Lattice I: The Ising model, J. Phys. A 49 (2016) 354001 [1601.07185].
https://doi.org/10.1088/1751-8113/49/35/354001
arXiv:1601.07185
[154] D. Aasen, P. Fendley and R. S. K. Mong, Topological Defects on the Lattice: Dualities and Degeneracies, [2008.08598].
arXiv:2008.08598
[155] A. Roy and H. Saleur, Entanglement Entropy in the Ising Model with Topological Defects, Phys. Rev. Lett. 128 (2022) 090603 [2111.04534].
https://doi.org/10.1103/PhysRevLett.128.090603
arXiv:2111.04534
[156] A. Roy and H. Saleur, Entanglement entropy in critical quantum spin chains with boundaries and defects, [2111.07927].
arXiv:2111.07927
[157] M. T. Tan, Y. Wang and A. Mitra, Topological Defects in Floquet Circuits, [ 2206.06272].
arXiv:2206.06272
[158] S. A. Hartnoll and S. Kumar, Higher rank Wilson loops from a matrix model, JHEP 08 (2006) 026 [hep-th/0605027].
https://doi.org/10.1088/1126-6708/2006/08/026
arXiv:hep-th/0605027
[159] J. G. Russo and K. Zarembo, Wilson loops in antisymmetric representations from localization in supersymmetric gauge theories, Rev. Math. Phys. 30 (2018) 1840014 [1712.07186].
https://doi.org/10.1142/S0129055X18400147
arXiv:1712.07186
[160] L. Santilli and M. Tierz, Phase transitions and Wilson loops in antisymmetric representations in Chern-Simons-matter theory, J. Phys. A 52 (2019) 385401 [ 1808.02855].
https://doi.org/10.1088/1751-8121/ab335c
arXiv:1808.02855
[161] L. Santilli, Phases of five-dimensional supersymmetric gauge theories, JHEP 07 (2021) 088 [ 2103.14049].
https://doi.org/10.1007/JHEP07(2021)088
arXiv:2103.14049
[162] M. R. Douglas and V. A. Kazakov, Large N phase transition in continuum QCD in two-dimensions, Phys. Lett. B 319 (1993) 219 [hep-th/9305047].
https://doi.org/10.1016/0370-2693(93)90806-S
arXiv:hep-th/9305047
[163] C. Lupo and M. Schiró, Transient Loschmidt echo in quenched Ising chains, Phys. Rev. B 94 (2016) [1604.01312].
https://doi.org/10.1103/physrevb.94.014310
arXiv:1604.01312
[164] T. Fogarty, S. Deffner, T. Busch and S. Campbell, Orthogonality Catastrophe as a Consequence of the Quantum Speed Limit, Phys. Rev. Lett. 124 (2020) [ 1910.10728].
https://doi.org/10.1103/physrevlett.124.110601
arXiv:1910.10728
[165] 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 (2018) 091404 [ 1805.08811].
https://doi.org/10.1063/1.5038658
arXiv:1805.08811
[166] M. Adler and P. van Moerbeke, Virasoro action on Schur function expansions, skew Young tableaux and random walks, Commun. Pure Appl. Math. 58 (2005) 362 [math/0309202].
https://doi.org/10.1002/cpa.20062
arXiv:math/0309202
[167] V. Periwal and D. Shevitz, Unitary matrix models as exactly solvable string theories, Phys. Rev. Lett. 64 (1990) 1326.
https://doi.org/10.1103/PhysRevLett.64.1326
Cited by
[1] David Pérez-García, Leonardo Santilli, and Miguel Tierz, “Hawking-Page transition on a spin chain”, arXiv:2401.13963, (2024).
[2] Ward L. Vleeshouwers and Vladimir Gritsev, “Unitary matrix integrals, symmetric polynomials, and long-range random walks”, Journal of Physics A Mathematical General 56 18, 185002 (2023).
[3] Gilles Parez, “Symmetry-resolved Rényi fidelities and quantum phase transitions”, Physical Review B 106 23, 235101 (2022).
[4] Gilles Parez, “Symmetry-resolved Rényi fidelities and quantum phase transitions”, arXiv:2208.09457, (2022).
[5] Elliott Gesteau and Leonardo Santilli, “Explicit large $N$ von Neumann algebras from matrix models”, arXiv:2402.10262, (2024).
The above citations are from SAO/NASA ADS (last updated successfully 2024-03-01 15:09:57). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref’s cited-by service no data on citing works was found (last attempt 2024-03-01 15:09:56).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
- SEO Powered Content & PR Distribution. Get Amplified Today.
- PlatoData.Network Vertical Generative Ai. Empower Yourself. Access Here.
- PlatoAiStream. Web3 Intelligence. Knowledge Amplified. Access Here.
- PlatoESG. Carbon, CleanTech, Energy, Environment, Solar, Waste Management. Access Here.
- PlatoHealth. Biotech and Clinical Trials Intelligence. Access Here.
- Source: https://quantum-journal.org/papers/q-2024-02-29-1271/
