This paper discusses the examples of 0+1-dimensional Liouvillean dynamics instigated by the various deformations of the Sachdev–Ye–Kitaev (SYK) model. In reference to such deformations the main focus is on the regions of parameter space where the competing SYK couplings are of a comparable strength and cannot be treated as each other’s perturbations in the vicinity of the conformal fixed points corresponding to the pure SYK_q models with different values of q. Crossovers between such fixed points (‘SYK transits’) can be efficiently studied in the equivalent framework of single-particle quantum mechanics.

[1] S. Sachdev and Jinwu Ye, Gapless spin-fluid ground state in a random quantum Heisenberg magnet, Phys. Rev. Lett. 70, 3339 (1993), arXiv:cond-mat/9212030,
https://doi.org/10.1103/PhysRevLett.70.3339
[2] S. Sachdev, Holographic metals and the fractionalized Fermi liquid, Phys. Rev. Lett. 105, 151602 (2010),
https://doi.org/10.1103/PhysRevLett.105.151602
[3] S. Sachdev, Bekenstein-Hawking entropy and strange metals, Phys. Rev. X 5, 041025 (2015),
https://doi.org/10.1103/PhysRevX.5.041025
[4] A. Kitaev, KITP Seminars, 2015,
http://online.kitp.ucsb.edu/
[5] A. Kitaev, Notes on  $\widetilde{\mathrm{SL}}(2,ℝ)$ representations, arXiv:1711.08169,
https://doi.org/10.48550/arXiv.1711.08169
[6] A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 2018, 183 (2018), arXiv:1711.08467,
https://doi.org/10.1007/JHEP05(2018)183
[7] A. Kitaev and S.J. Suh, Statistical mechanics of a two-dimensional black hole, JHEP 2019, 198 (2019), arXiv:1808.07032,
https://doi.org/10.1007/JHEP05(2019)198
[8] S. Sachdev, Statistical mechanics of strange metals and black holes, arXiv:2205.02285,
https://doi.org/10.48550/arXiv.2205.02285
[9] G. Penington, Entanglement wedge reconstruction and the information paradox, arXiv:1905.08255,
https://doi.org/10.48550/arXiv.1905.08255
[10] A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole. JHEP 2019, 63 (2019), arXiv:1905.08762,
https://doi.org/10.1007/JHEP12(2019)063
[11] S. Banerjee and E. Altman, Solvable model for a dynamical quantum phase transition from fast to slow scrambling, Phys. Rev. B 95, 134302 (2017),
https://doi.org/10.1103/PhysRevB.95.134302
[12] Zhen Bi, Chao-Ming Jian, Yi-Zhuang You, K.A. Pawlak, and Cenke Xu, Instability of the non-Fermi-liquid state of the Sachdev-Ye-Kitaev model, Phys. Rev. B 95, 205105 , (2017),
https://doi.org/10.1103/PhysRevB.95.205105
[13] Shao-Kai Jian and Hong Yao, Solvable Sachdev-Ye-Kitaev models in higher dimensions: From diffusion to many-body localization, Phys. Rev. Lett. 119, 206602 (2017),
https://doi.org/10.1103/PhysRevLett.119.206602
[14] A. Haldar, S. Banerjee, and V.B. Shenoy, Higher-dimensional Sachdev-Ye-Kitaev non-Fermi liquids at Lifshitz transitions, Phys. Rev. B 97, 241106 (2018),
https://doi.org/10.1103/PhysRevB.97.241106
[15] Chao-Ming Jian, Zhen Bi, and Cenke Xu, Model for continuous thermal metal to insulator transition, Phys. Rev. B 96, 115122 (2017),
https://doi.org/10.1103/PhysRevB.96.115122
[16] Xue-Yang Song, Chao-Ming Jian, and L. Balents, Strongly correlated metal built from Sachdev-Ye-Kitaev models, Phys. Rev. Lett. 119, 216601 (2017),
https://doi.org/10.1103/PhysRevLett.119.216601
[17] Xin Chen, Ruihua Fan, Yiming Chen, Hui Zhai, and Pengfei Zhang, Competition between chaotic and nonchaotic phases in a quadratically coupled Sachdev-Ye-Kitaev model, Phys. Rev. Lett. 119, 207603 (2017),
https://doi.org/10.1103/PhysRevLett.119.207603
[18] Pengfei Zhang, Dispersive Sachdev-Ye-Kitaev model: Band structure and quantum chaos, Phys. Rev. B 96, 205138 (2017),
https://doi.org/10.1103/PhysRevB.96.205138
[19] Wenhe Cai, Xian-Hui Ge, and Guo-Hong Yang, Diffusion in higher dimensional SYK model with complex fermions, JHEP 2018(01), 76 (2018),
https://doi.org/10.1007/JHEP01(2018)076
[20] Zhong Yin, Periodic Anderson model meets Sachdev-Ye-Kitaev interaction: a solvable playground for heavy fermion physics, J. Phys. Commun. 2, 095014 (2018),
https://doi.org/10.1088/2399-6528/aae06b
[21] Xin Dai, Shao-Kai Jian, Hong Yao, Global phase diagram of the one-dimensional Sachdev-Ye-Kitaev model at finite N, Phys. Rev. B 100, 235144 (2019), arXiv:1802.10029,
https://doi.org/10.1103/PhysRevB.100.235144
[22] Pengfei Zhang and Hui Zhai, Topological Sachdev-Ye-Kitaev model, Phys. Rev. B 97, 201112(R) (2018),
https://doi.org/10.1103/PhysRevB.97.201112
[23] Xiaochuan Wu, Xiao Chen, Chao-Ming Jian, Yi-Zhuang You, and Cenke Xu, Candidate theory for the strange metal phase at a finite-energy window, Phys. Rev. B 98, 165117 (2018),
https://doi.org/10.1103/PhysRevB.98.165117
[24] D. Ben-Zion and J. McGreevy, Strange metal from local quantum chaos, Phys. Rev. B 97, 155117 (2018),
https://doi.org/10.1103/PhysRevB.97.155117
[25] A.A. Patel, J. McGreevy, D.P. Arovas, and S. Sachdev, Magnetotransport in a model of a disordered strange metal, Phys. Rev. X 8, 021049 (2018),
https://doi.org/10.1103/PhysRevX.8.021049
[26] D. Chowdhury, Y. Werman, E. Berg, and T. Senthil, Translationally invariant non-Fermi-liquid metals with critical Fermi surfaces: Solvable models, Phys. Rev. X 8, 031024 (2018),
https://doi.org/10.1103/PhysRevX.8.031024
[27] A.A. Patel and S. Sachdev, Theory of a Planckian metal, Phys. Rev. Lett. 123, 066601 (2019),
https://doi.org/10.1103/PhysRevLett.123.066601
[28] A.A. Patel and S. Sachdev, Critical strange metal from fluctuating gauge fields in a solvable random model, Phys. Rev. B 98, 125134 (2018),
https://doi.org/10.1103/PhysRevB.98.125134
[29] D. Miserev, J. Klinovaja, and D. Loss, Fermi surface resonance and quantum criticality in strongly interacting Fermi gases, Phys. Rev. B 103, 075104 (2021),
https://doi.org/10.1103/PhysRevB.103.075104
[30] D. Chowdhury, A. Georges, O. Parcollet, and S. Sachdev, Sachdev-Ye-Kitaev models and beyond: A window into non-Fermi liquids, arXiv:2109.05037,
https://doi.org/10.48550/arXiv.2109.05037
[31] I. Esterlis, Haoyu Guo, A.A. Patel, and S. Sachdev, Large-N theory of critical Fermi surfaces, Phys. Rev. B 103, 235129 (2021),
https://doi.org/10.1103/PhysRevB.103.235129
[32] D. Chowdhury and E. Berg, Intrinsic superconducting instabilities of a solvable model for an incoherent metal, Phys. Rev. Research 2, 013301 (2020),
https://doi.org/10.1103/PhysRevResearch.2.013301
[33] P. Cha, A.A. Patel, E. Gull, and Eun-Ah Kim, Slope invariant T-linear resistivity from local self-energy, Phys. Rev. Research 2, 033434 (2020),
https://doi.org/10.1103/PhysRevResearch.2.033434
[34] Haoyu Guo, Yingfei Gu, and S. Sachdev, Transport and chaos in lattice Sachdev-Ye-Kitaev models, Phys. Rev. B 100, 045140 (2019),
https://doi.org/10.1103/PhysRevB.100.045140
[35] I. Esterlis and J. Schmalian, Cooper pairing of incoherent electrons: An electron-phonon version of the Sachdev-Ye-Kitaev model, Phys. Rev. B 100, 115132 (2019),
https://doi.org/10.1103/PhysRevB.100.115132
[36] Yuxuan Wang and A. V. Chubukov, Quantum phase transition in the Yukawa-SYK model, Phys. Rev. Research 2, 033084 (2020),
https://doi.org/10.1103/PhysRevResearch.2.033084
[37] B. Douçot, A. Mukhopadhyay, G. Policastro, and S. Samanta, Linear-in-T resistivity from semiholographic non-Fermi liquid models, Phys. Rev. D 104, L081901 (2021),
https://doi.org/10.1103/PhysRevD.104.L081901
[38] L. Classen and A. Chubukov, Superconductivity of incoherent electrons in the Yukawa Sachdev-Ye-Kitaev model, Phys. Rev. B 104, 125120 (2021),
https://doi.org/10.1103/PhysRevB.104.125120
[39] Peter Cha, A.A. Patel, and Eun-Ah Kim, Strange metals from melting correlated insulators in twisted bilayer graphene, Phys. Rev. Lett. 127, 266601 (2021),
https://doi.org/10.1103/PhysRevLett.127.266601
[40] F. Salvati and A. Tagliacozzo, Superconducting critical temperature in the extended diffusive Sachdev-Ye-Kitaev model, Phys. Rev. Research 3, 033117 (2021),
https://doi.org/10.1103/PhysRevResearch.3.033117
[41] Gaopei Pan, Wei Wang, A. Davis, Yuxuan Wang, and Zi Yang Meng, Yukawa-SYK model and self-tuned quantum criticality, Phys. Rev. Research 3, 013250 (2021),
https://doi.org/10.1103/PhysRevResearch.3.013250
[42] G. Jose, Kangjun Seo, and B. Uchoa, Non-Fermi liquid behavior in the Sachdev-Ye-Kitaev model for a one-dimensional incoherent semimetal, Phys. Rev. Research 4, 013145 (2022),
https://doi.org/10.1103/PhysRevResearch.4.013145
[43] A.A. Patel, Haoyu Guo, I. Esterlis, and S. Sachdev, Universal, low temperature, T-linear resistivity in two-dimensional quantum-critical metals from spatially random interactions, arXiv:2203.04990,
https://doi.org/10.48550/arXiv.2203.04990
[44] S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26, 224002 (2009),
https://doi.org/10.1088/0264-9381/26/22/224002
[45] C.P. Herzog, Lectures on holographic superfluidity and superconductivity, J. Phys. A 42, 343001 (2009),
https://doi.org/10.1088/1751-8113/42/34/343001
[46] J. McGreevy, Holographic duality with a view toward many-body physics, Adv. High Energy Phys. 2010, 723105 (2010),
https://doi.org/10.1155/2010/723105
[47] S. Sachdev, What can gauge-gravity duality teach us about condensed matter physics?, Annu. Rev. Cond. Matt. Phys. 3, 9 (2012),
https://doi.org/10.1146/annurev-conmatphys-020911-125141
[48] J. Zaanen, Y. Liu, Y.-W. Sun, and K. Schalm, Holographic Duality in Condensed Matter Physics (Cambridge University Press, 2015),
https://doi.org/10.1017/CBO9781139942492
[49] M. Ammon and J. Erdmenger, Gauge/Gravity Duality (Cambridge University Press, 2015),
https://doi.org/10.1017/CBO9780511846373
[50] S.A. Hartnoll, A. Lucas, and S. Sachdev, Holographic Quantum Matter (MIT Press, 2018),
https://mitpress.mit.edu/books/holographic-quantum-matter
[51] D.V. Khveshchenko, Phase space holography with no strings attached, Lith. J. Phys. 61, 233 (2021), arXiv:2102.01617,
https://doi.org/10.3952/physics.v61i4.4642
[52] Chao-Ming Jian, Zhen Bi, and Cenke Xu, Model for continuous thermal metal to insulator transition, Phys. Rev. B 96, 115122 (2017), arXiv:1703.07793,
https://doi.org/10.1103/PhysRevB.96.115122
[53] D. Anninos and D.A. Galante, Constructing AdS₂ flow geometries, JHEP 2021, 45 (2021), arXiv:2011.01944,
https://doi.org/10.1007/JHEP02(2021)045
[54] Jiaqi Jiang and Zhenbin Yang, Thermodynamics and many body chaos for generalized large q SYK models, JHEP 2019 19 (2019), arXiv:1905.00811,
https://doi.org/10.1007/JHEP08(2019)019
[55] A.V. Lunkin, K.S. Tikhonov, and M.V. Feigel'man, Sachdev-Ye-Kitaev model with quadratic perturbations: The route to a non-Fermi liquid, Phys. Rev. Lett. 121, 236601 (2018), arXiv:1806.11211,
https://doi.org/10.1103/PhysRevLett.121.236601
[56] D.V. Khveshchenko, Thickening and sickening the SYK model, SciPost Phys. 5, 012 (2018), arXiv:1705.03956,
https://doi.org/10.21468/SciPostPhys.5.1.012
[57] D.V. Khveshchenko, Seeking to develop global SYK-ness, Condens. Matter 2018, 3(4), 40 (2018), arXiv:1805.00870,
https://doi.org/10.3390/condmat3040040
[58] J. Maldacena, S.H. Shenker, and D. Stanford, A bound on chaos, JHEP 2016(08), 106 (2016), arXiv:1503.01409,
https://doi.org/10.1007/JHEP08(2016)106
[59] J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016), arXiv:1604.07818,
https://doi.org/10.1103/PhysRevD.94.106002
[60] J. Maldacena, D. Stanford, and Zhebin Yang, Conformal symmetry and its breaking in two dimensional nearly Anti-de-Sitter space, PTEP 2016(12), 12C104 (2016), arXiv:1606.01857,
https://doi.org/10.1093/ptep/ptw124
[61] D. Stanford and E. Witten, Fermionic localization of the schwarzian theory, JHEP 2017(10), 8 (2017), arXiv:1703.04612,
https://doi.org/10.1007/JHEP10(2017)008
[62] J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 2016(04), 1 (2016), arXiv:1601.06768,
https://doi.org/10.1007/JHEP04(2016)001
[63] D.J. Gross and V. Rosenhaus, The bulk dual of SYK: cubic couplings, JHEP 2017(05), 92 (2017), arXiv:1702.08016,
https://doi.org/10.1007/JHEP05(2017)092
[64] D.J. Gross and V. Rosenhaus, All point correlation functions in SYK, JHEP 2017(12), 148 (2017), arXiv:1710.08113,
https://doi.org/10.1007/JHEP12(2017)148
[65] G. Sárosi, AdS₂ holography and the SYK model, in: XIII Modave Summer School in Mathematical Physics (Modave2017), Proc. Sci. vol. 323, arXiv:1711.08482,
https://doi.org/10.22323/1.323.0001
[66] Henry W. Lin, J. Maldacena, and Ying Zhao, Symmetries near the horizon, JHEP 2019, 49 (2019), arXiv:1904.12820,
https://doi.org/10.1007/JHEP08(2019)049
[67] E. Witten, An SYK-Like model without disorder, arXiv:1610.09758,
https://doi.org/10.48550/arXiv.1610.09758
[68] R. Gurau, The complete 1/N expansion of a SYK-like tensor model, Nucl. Phys. B 916, 386--401 (2017), arXiv:1611.04032,
https://doi.org/10.1016/j.nuclphysb.2017.01.015
[69] R. Gurau, The ıε prescription in the SYK model, arXiv:1705.08581,
https://doi.org/10.48550/arXiv.1705.08581
[70] I.R. Klebanov and G. Tarnopolsky, Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models, Phys. Rev. D 95, 046004 (2017),
https://doi.org/10.1103/PhysRevD.95.046004
[71] S. Giombi, I.R. Klebanov, and G. Tarnopolsky, Bosonic tensor models at large N and small ε, Phys. Rev. D 96, 106014 (2017),
https://doi.org/10.1103/PhysRevD.96.106014
[72] Yingfei Gu, Xiao-Liang Qi, and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP 2017, 125 (2017),
https://doi.org/10.1007/JHEP05(2017)125
[73] Yingfei Gu, A. Lucas, and Xiao-Liang Qi, Energy diffusion and the butterfly effect in inhomogeneous Sachdev-Ye-Kitaev chains, SciPost Phys. 2, 018 (2017),
https://doi.org/10.21468/SciPostPhys.2.3.018
[74] Yingfei Gu, A. Lucas, and Xiao-Liang Qi, Spread of entanglement in a Sachdev-Ye-Kitaev chain, JHEP 2017, 120 (2017),
https://doi.org/10.1007/JHEP09(2017)120
[75] Yingfei Gu and A. Kitaev, On the relation between the magnitude and exponent of OTOCs, JHEP 2019(02), 75 (2019),
https://doi.org/10.1007/JHEP02(2019)075
[76] Zhenbin Yang, The quantum gravity dynamics of near extremal black holes, JHEP 2019(05), 205 (2019), arXiv:1809.08647,
https://doi.org/10.1007/JHEP05(2019)205
[77] A. Blommaert, T.G. Mertens, and H. Verschelde, The Schwarzian theory — a Wilson line perspective, JHEP 2018(12), 22 (2018), arXiv:1806.07765,
https://doi.org/10.1007/JHEP12(2018)022
[78] A. Blommaert, T.G. Mertens, and H. Verschelde, Fine structure of Jackiw-Teitelboim quantum gravity, JHEP 2019(9), 66 (2019), arXiv:1812.00918,
https://doi.org/10.1007/JHEP09(2019)066
[79] G. Tarnopolsky, Large q expansion in the Sachdev-Ye-Kitaev model, Phys. Rev. D 99, 026010 (2019), arXiv:1801.06871,
https://doi.org/10.1103/PhysRevD.99.026010
[80] A. Milekhin, Coupled Sachdev-Ye-Kitaev models without Schwartzian dominance, arXiv:2102.06651,
https://doi.org/10.48550/arXiv.2102.06651
[81] D. Bagrets, A. Altland, and A. Kamenev, Sachdev–Ye–Kitaev model as Liouville quantum mechanics, Nucl. Phys. B 911, 191--205 (2016),
https://doi.org/10.1016/j.nuclphysb.2016.08.002
[82] D. Bagrets, A. Altland, and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nucl. Phys. B 921, 727 (2017), arXiv:1702.08902,
https://doi.org/10.1016/j.nuclphysb.2017.06.012
[83] N.V. Gnezdilov, J.A. Hutasoit, and C.W.J. Beenakker, Low-high voltage duality in tunneling spectroscopy of the Sachdev-Ye-Kitaev model, Phys. Rev. B 98, 081413,
https://doi.org/10.1103/PhysRevB.98.081413
[84] O. Can, E.M. Nica, and M. Franz, Charge transport in graphene-based mesoscopic realizations of Sachdev-Ye-Kitaev models, Phys. Rev. B 99, 045419 (2019),
https://doi.org/10.1103/PhysRevB.99.045419
[85] A. Altland, D. Bagrets, and A. Kamenev, Sachdev-Ye-Kitaev non-Fermi-liquid correlations in nanoscopic quantum transport, Phys. Rev. Lett. 123, 226801 (2019),
https://doi.org/10.1103/PhysRevLett.123.226801
[86] A. Kruchkov, A.A. Patel, P. Kim, and S. Sachdev, Thermoelectric power of Sachdev-Ye-Kitaev islands: Probing Bekenstein-Hawking entropy in quantum matter experiments, Phys. Rev. B 101, 205148 (2020), arXiv:1912.02835,
https://doi.org/10.1103/PhysRevB.101.205148
[87] D.I. Pikulin and M. Franz, Black hole on a chip: Proposal for a physical realization of the Sachdev-Ye-Kitaev model in a solid-state system, Phys. Rev. X 7, 031006 (2017),
https://doi.org/10.1103/PhysRevX.7.031006
[88] A. Chew, A. Essin, and J. Alicea, Approximating the Sachdev-Ye-Kitaev model with Majorana wires, Phys. Rev. B 96, 121119 (2017),
https://doi.org/10.1103/PhysRevB.96.121119
[89] Anffany Chen, R. Ilan, F. de Juan, D. I. Pikulin, and M. Franz, Quantum holography in a graphene flake with an irregular boundary, Phys. Rev. Lett. 121, 036403 (2018),
https://doi.org/10.1103/PhysRevLett.121.036403
[90] E. Lantagne-Hurtubise, Chengshu Li, and M. Franz, Family of Sachdev-Ye-Kitaev models motivated by experimental considerations, Phys. Rev. B 97, 235124 (2018),
https://doi.org/10.1103/PhysRevB.97.235124
[91] M. Franz and M. Rozali, Mimicking black hole event horizons in atomic and solid-state systems, arXiv:1808.00541,
https://doi.org/10.48550/arXiv.1808.00541
[92] D.V. Khveshchenko, Connecting the SYK dots, Condens. Matter 5(2), 37 (2020), arXiv:2004.06646,
https://doi.org/10.3390/condmat5020037
[93] D.V. Khveshchenko, One SYK single electron transistor, Lith. J. Phys. 60, 185 (2020), arXiv:1912.05691,
https://doi.org/10.3952/physics.v60i3.4305
[94] J. Maldacena and Xiao-Liang Qi, Eternal traversable wormhole, arXiv:1804.00491,
https://doi.org/10.48550/arXiv.1804.00491
[95] T.G. Mertens, G.J. Turiaci, and H.L. Verlinde, Solving the Schwarzian via the conformal bootstrap, JHEP 08, 136 (2017), arXiv:1705.08408,
https://doi.org/10.1007/JHEP08(2017)136
[96] T.G. Mertens, The Schwarzian theory — origins, JHEP 2018(05), 36 (2018), arXiv:1801.09605,
https://doi.org/10.1007/JHEP05(2018)036
[97] D.V. Khveshchenko, On a (pseudo)holographic nature of the SYK-like models, Lith. J. Phys. 59, 104 (2019), arXiv:1905.04381,
https://doi.org/10.3952/physics.v59i2.4013
[98] A.V. Lunkin, A.Yu. Kitaev, and M.V. Feigel'man, Perturbed Sachdev-Ye-Kitaev model: A polaron in the hyperbolic plane, Phys. Rev. Lett. 125, 196602 (2020), arXiv:2006.14535,
https://doi.org/10.1103/PhysRevLett.125.196602
[99] A.M. García-García, B. Loureiro, A. Romero-Bermúdez, and M. Tezuka, Chaotic-integrable transition in the Sachdev-Ye-Kitaev model, Phys. Rev. Lett. 120 241603, (2018), arXiv:1707.02197,
https://doi.org/10.1103/PhysRevLett.120.241603
[100] A.M. García-García, B. Loureiro, A. Romero-Bermúdez, and M. Tezuka, García-García et al. Reply:, Phys. Rev. Lett. 126, 109102 (2021),
https://doi.org/10.1103/PhysRevLett.126.109102
[101] Jaewon Kim and Xiangyu Cao, Comment on “Chaotic-integrable transition in the Sachdev-Ye-Kitaev model”, Phys. Rev. Lett. 126, 109101 (2021), arXiv:2004.05313,
https://doi.org/10.1103/PhysRevLett.126.109101