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
Thola leli phepha elithakazelisayo noma ufuna ukuxoxa ngalo? Scite noma shiya amazwana ku-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
Isifinyezo esidumile
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.
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► Izinkomba
[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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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].
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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].
i-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
i-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
i-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
i-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
i-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].
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-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
i-arXiv: 1601.07185
[154] D. Aasen, P. Fendley and R. S. K. Mong, Topological Defects on the Lattice: Dualities and Degeneracies, [2008.08598].
i-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
i-arXiv: 2111.04534
[156] A. Roy and H. Saleur, Entanglement entropy in critical quantum spin chains with boundaries and defects, [2111.07927].
i-arXiv: 2111.07927
[157] M. T. Tan, Y. Wang and A. Mitra, Topological Defects in Floquet Circuits, [ 2206.06272].
i-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
i-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
i-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
i-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
i-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
i-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
i-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
Icashunwe ngu
[1] David Pérez-García, Leonardo Santilli, and Miguel Tierz, “Hawking-Page transition on a spin chain”, i-arXiv: 2401.13963, (2024).
[2] Gilles Parez, “Symmetry-resolved Rényi fidelities and quantum phase transitions”, Ukubuyekezwa Komzimba B 106 23, 235101 (2022).
[3] Gilles Parez, “Symmetry-resolved Rényi fidelities and quantum phase transitions”, i-arXiv: 2208.09457, (2022).
[4] Ward L. Vleeshouwers and Vladimir Gritsev, “Unitary matrix integrals, symmetric polynomials, and long-range random walks”, Ijenali yeFiziksi A Okujwayelekile Kwezibalo 56 18, 185002 (2023).
[5] Elliott Gesteau and Leonardo Santilli, “Explicit large $N$ von Neumann algebras from matrix models”, i-arXiv: 2402.10262, (2024).
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