Universal Entanglement Transitions of Free Fermions with Long-range Non-unitary Dynamics

Pengfei Zhang1, Chunxiao Liu2, Shao-Kai Jian3, and Xiao Chen4

1Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
2Department of Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA
3Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, MD 20742, USA
4Department of Physics, Boston College, Chestnut Hill, MA 02467, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Non-unitary evolution can give rise to novel steady states classified by their entanglement properties. In this work, we aim to understand the effect of long-range hopping that decays with $r^{-\alpha}$ in non-Hermitian free-fermion systems. We first study two solvable Brownian models with long-range non-unitary dynamics: a large-$N$ SYK$_2$ chain and a single-flavor fermion chain and we show that they share the same phase diagram. When $\alpha\gt0.5$, we observe two critical phases with subvolume entanglement scaling: (i) $\alpha\gt1.5$, a logarithmic phase with dynamical exponent $z=1$ and logarithmic subsystem entanglement, and (ii) $0.5 \lt \alpha \lt 1.5$, a fractal phase with $z=\frac{2\alpha-1}{2}$ and subsystem entanglement $S_A\propto L_A^{1-z}$, where $L_A$ is the length of the subsystem $A$. These two phases cannot be distinguished by the purification dynamics, in which the entropy always decays as $L/T$. We then confirm that the results are also valid for the static SYK$_2$ chain, indicating the phase diagram is universal for general free-fermion systems. We also discuss phase diagrams in higher dimensions and the implication in measurement-induced phase transitions.

We study the effect of long-range hopping in non-unitary non-Hermitian free-fermion systems. We first show that two solvable Brownian models: a large-$N$ SYK$_2$ chain and a single-flavor fermion chain share the same phase diagram, which contains novel critical phases including a logarithmic phase and a fractal phase. We then confirm that the results are also valid for the static SYK$_2$ chain, indicating the phase diagram is universal for general free-fermion systems.

► BibTeX data

► References

[1] Xiangyu Cao, Antoine Tilloy, and Andrea De Luca. Entanglement in a fermion chain under continuous monitoring. SciPost Phys., 7: 24, 2019. 10.21468/​SciPostPhys.7.2.024.

[2] Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. Quantum zeno effect and the many-body entanglement transition. Phys. Rev. B, 98: 205136, Nov 2018. 10.1103/​PhysRevB.98.205136.

[3] Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. Measurement-driven entanglement transition in hybrid quantum circuits. Phys. Rev. B, 100: 134306, Oct 2019. 10.1103/​PhysRevB.100.134306.

[4] Brian Skinner, Jonathan Ruhman, and Adam Nahum. Measurement-induced phase transitions in the dynamics of entanglement. Phys. Rev. X, 9: 031009, Jul 2019. 10.1103/​PhysRevX.9.031009.

[5] Amos Chan, Rahul M. Nandkishore, Michael Pretko, and Graeme Smith. Unitary-projective entanglement dynamics. Phys. Rev. B, 99: 224307, Jun 2019. 10.1103/​PhysRevB.99.224307.

[6] Yimu Bao, Soonwon Choi, and Ehud Altman. Theory of the phase transition in random unitary circuits with measurements. Physical Review B, 101 (10), Mar 2020. ISSN 2469-9969. 10.1103/​physrevb.101.104301.

[7] Soonwon Choi, Yimu Bao, Xiao-Liang Qi, and Ehud Altman. Quantum error correction in scrambling dynamics and measurement-induced phase transition. Physical Review Letters, 125 (3), Jul 2020. ISSN 1079-7114. 10.1103/​physrevlett.125.030505.

[8] Michael J. Gullans and David A. Huse. Dynamical purification phase transition induced by quantum measurements. Phys. Rev. X, 10: 041020, Oct 2020a. 10.1103/​PhysRevX.10.041020.

[9] Michael J. Gullans and David A. Huse. Scalable probes of measurement-induced criticality. Phys. Rev. Lett., 125: 070606, Aug 2020b. 10.1103/​PhysRevLett.125.070606.

[10] Chao-Ming Jian, Yi-Zhuang You, Romain Vasseur, and Andreas W. W. Ludwig. Measurement-induced criticality in random quantum circuits. Phys. Rev. B, 101: 104302, Mar 2020a. 10.1103/​PhysRevB.101.104302.

[11] Aidan Zabalo, Michael J. Gullans, Justin H. Wilson, Sarang Gopalakrishnan, David A. Huse, and J. H. Pixley. Critical properties of the measurement-induced transition in random quantum circuits. Phys. Rev. B, 101: 060301, Feb 2020. 10.1103/​PhysRevB.101.060301.

[12] Qicheng Tang and W. Zhu. Measurement-induced phase transition: A case study in the nonintegrable model by density-matrix renormalization group calculations. Phys. Rev. Research, 2: 013022, Jan 2020. 10.1103/​PhysRevResearch.2.013022.

[13] M. Szyniszewski, A. Romito, and H. Schomerus. Entanglement transition from variable-strength weak measurements. Phys. Rev. B, 100: 064204, Aug 2019. 10.1103/​PhysRevB.100.064204.

[14] Lei Zhang, Justin A. Reyes, Stefanos Kourtis, Claudio Chamon, Eduardo R. Mucciolo, and Andrei E. Ruckenstein. Nonuniversal entanglement level statistics in projection-driven quantum circuits. Physical Review B, 101 (23), Jun 2020a. ISSN 2469-9969. 10.1103/​physrevb.101.235104.

[15] Shimpei Goto and Ippei Danshita. Measurement-induced transitions of the entanglement scaling law in ultracold gases with controllable dissipation. Phys. Rev. A, 102 (3): 033316, 2020. 10.1103/​PhysRevA.102.033316.

[16] Shao-Kai Jian, Zhi-Cheng Yang, Zhen Bi, and Xiao Chen. Yang-lee edge singularity triggered entanglement transition. Phys. Rev. B, 104: L161107, Oct 2021a. 10.1103/​PhysRevB.104.L161107.

[17] M. Buchhold, Y. Minoguchi, A. Altland, and S. Diehl. Effective theory for the measurement-induced phase transition of dirac fermions. Phys. Rev. X, 11: 041004, Oct 2021. 10.1103/​PhysRevX.11.041004.

[18] Yimu Bao, Soonwon Choi, and Ehud Altman. Symmetry enriched phases of quantum circuits. Annals of Physics, 435: 168618, 2021. ISSN 0003-4916. https:/​/​doi.org/​10.1016/​j.aop.2021.168618. Special issue on Philip W. Anderson.

[19] Shengqi Sang and Timothy H. Hsieh. Measurement-protected quantum phases. Phys. Rev. Research, 3: 023200, Jun 2021. 10.1103/​PhysRevResearch.3.023200.

[20] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli. Measurement-induced topological entanglement transitions in symmetric random quantum circuits. Nature Physics, 17 (3): 342–347, Jan 2021. ISSN 1745-2481. 10.1038/​s41567-020-01112-z.

[21] Matteo Ippoliti, Tibor Rakovszky, and Vedika Khemani. Fractal, logarithmic, and volume-law entangled nonthermal steady states via spacetime duality. Phys. Rev. X, 12: 011045, Mar 2022. 10.1103/​PhysRevX.12.011045.

[22] Tsung-Cheng Lu and Tarun Grover. Spacetime duality between localization transitions and measurement-induced transitions. PRX Quantum, 2: 040319, Oct 2021. 10.1103/​PRXQuantum.2.040319. URL https:/​/​link.aps.org/​doi/​10.1103/​PRXQuantum.2.040319.

[23] Chao-Ming Jian, Bela Bauer, Anna Keselman, and Andreas W. W. Ludwig. Criticality and entanglement in non-unitary quantum circuits and tensor networks of non-interacting fermions. 12 2020b.

[24] Matteo Ippoliti, Michael J. Gullans, Sarang Gopalakrishnan, David A. Huse, and Vedika Khemani. Entanglement phase transitions in measurement-only dynamics. Physical Review X, 11 (1), Feb 2021. ISSN 2160-3308. 10.1103/​physrevx.11.011030.

[25] O. Alberton, M. Buchhold, and S. Diehl. Entanglement transition in a monitored free-fermion chain: From extended criticality to area law. Phys. Rev. Lett., 126: 170602, Apr 2021. 10.1103/​PhysRevLett.126.170602.

[26] Xiao Chen, Yaodong Li, Matthew P. A. Fisher, and Andrew Lucas. Emergent conformal symmetry in nonunitary random dynamics of free fermions. Physical Review Research, 2 (3), Jul 2020a. ISSN 2643-1564. 10.1103/​PhysRevResearch.2.033017.

[27] Chunxiao Liu, Pengfei Zhang, and Xiao Chen. Non-unitary dynamics of sachdev-ye-kitaev chain. SciPost Phys., 10: 48, 2021. 10.21468/​SciPostPhys.10.2.048.

[28] Pengfei Zhang, Shao-Kai Jian, Chunxiao Liu, and Xiao Chen. Emergent Replica Conformal Symmetry in Non-Hermitian SYK$_2$ Chains. Quantum, 5: 579, 2021. 10.22331/​q-2021-11-16-579.

[29] Shao-Kai Jian, Chunxiao Liu, Xiao Chen, Brian Swingle, and Pengfei Zhang. Measurement-Induced Phase Transition in the Monitored Sachdev-Ye-Kitaev Model. Phys. Rev. Lett., 127 (14): 140601, 2021b. 10.1103/​PhysRevLett.127.140601.

[30] Adam Nahum and Brian Skinner. Entanglement and dynamics of diffusion-annihilation processes with majorana defects. Physical Review Research, 2 (2), Jun 2020. ISSN 2643-1564. 10.1103/​physrevresearch.2.023288.

[31] Qicheng Tang, Xiao Chen, and W. Zhu. Quantum criticality in the nonunitary dynamics of (2+1) -dimensional free fermions. Physical Review B, 103 (17), May 2021. ISSN 2469-9969. 10.1103/​physrevb.103.174303.

[32] Alberto Biella and Marco Schiró. Many-body quantum zeno effect and measurement-induced subradiance transition. Quantum, 5: 528, 2021. 10.22331/​q-2021-08-19-528.

[33] Xhek Turkeshi, Alberto Biella, Rosario Fazio, Marcello Dalmonte, and Marco Schiró. Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks. Phys. Rev. B, 103: 224210, Jun 2021a. 10.1103/​PhysRevB.103.224210.

[34] Xhek Turkeshi, Marcello Dalmonte, Rosario Fazio, and Marco Schirò. Entanglement transitions from stochastic resetting of non-hermitian quasiparticles. arXiv preprint arXiv:2111.03500, 2021b.

[35] Xhek Turkeshi and Marco Schiró. Entanglement and correlation spreading in non-hermitian spin chains. arXiv preprint arXiv:2201.09895, 2022.

[36] O. Viyuela, D. Vodola, G. Pupillo, and M. A. Martin-Delgado. Topological massive dirac edge modes and long-range superconducting hamiltonians. Phys. Rev. B, 94: 125121, Sep 2016. 10.1103/​PhysRevB.94.125121.

[37] Oscar Viyuela, Liang Fu, and Miguel Angel Martin-Delgado. Chiral Topological Superconductors Enhanced by Long-Range Interactions. Phys. Rev. Lett., 120 (1): 017001, 2018. 10.1103/​PhysRevLett.120.017001.

[38] Matthew B. Hastings and Tohru Koma. Spectral gap and exponential decay of correlations. Communications in Mathematical Physics, 265 (3): 781–804, Apr 2006. ISSN 1432-0916. 10.1007/​s00220-006-0030-4.

[39] Takuro Matsuta, Tohru Koma, and Shu Nakamura. Improving the lieb–robinson bound for long-range interactions. In Annales Henri Poincaré, volume 18, pages 519–528. Springer, 2017. 10.1007/​s00023-016-0526-1.

[40] Xiao Chen and Tianci Zhou. Quantum chaos dynamics in long-range power law interaction systems. Physical Review B, 100 (6), Aug 2019. ISSN 2469-9969. 10.1103/​physrevb.100.064305.

[41] Chi-Fang Chen and Andrew Lucas. Finite speed of quantum scrambling with long range interactions. Phys. Rev. Lett., 123: 250605, Dec 2019. 10.1103/​PhysRevLett.123.250605.

[42] Tianci Zhou, Shenglong Xu, Xiao Chen, Andrew Guo, and Brian Swingle. Operator lévy flight: Light cones in chaotic long-range interacting systems. Phys. Rev. Lett., 124: 180601, May 2020. 10.1103/​PhysRevLett.124.180601.

[43] Minh C. Tran, Andrew Y. Guo, Christopher L. Baldwin, Adam Ehrenberg, Alexey V. Gorshkov, and Andrew Lucas. Lieb-robinson light cone for power-law interactions. Phys. Rev. Lett., 127: 160401, Oct 2021. 10.1103/​PhysRevLett.127.160401.

[44] Tomotaka Kuwahara and Keiji Saito. Strictly linear light cones in long-range interacting systems of arbitrary dimensions. Phys. Rev. X, 10: 031010, Jul 2020. 10.1103/​PhysRevX.10.031010.

[45] Maxwell Block, Yimu Bao, Soonwon Choi, Ehud Altman, and Norman Y. Yao. Measurement-induced transition in long-range interacting quantum circuits. Phys. Rev. Lett., 128: 010604, Jan 2022. 10.1103/​PhysRevLett.128.010604.

[46] Takaaki Minato, Koudai Sugimoto, Tomotaka Kuwahara, and Keiji Saito. Fate of measurement-induced phase transition in long-range interactions. Phys. Rev. Lett., 128: 010603, Jan 2022. 10.1103/​PhysRevLett.128.010603.

[47] Alexei Kitaev. A simple model of quantum holography, talk given at the kitp program: entanglement in strongly-correlated quantum matter. talk given at the KITP Program: entanglement in strongly-correlated quantum matter, 2015.

[48] Juan Maldacena and Douglas Stanford. Remarks on the sachdev-ye-kitaev model. Phys. Rev. D, 94: 106002, Nov 2016. 10.1103/​PhysRevD.94.106002.

[49] Subir Sachdev and Jinwu Ye. Gapless spin-fluid ground state in a random quantum heisenberg magnet. Phys. Rev. Lett., 70: 3339–3342, May 1993. 10.1103/​PhysRevLett.70.3339.

[50] Yingfei Gu, Xiao-Liang Qi, and Douglas Stanford. Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models. JHEP, 05: 125, 2017a. 10.1007/​JHEP05(2017)125.

[51] Richard A. Davison, Wenbo Fu, Antoine Georges, Yingfei Gu, Kristan Jensen, and Subir Sachdev. Thermoelectric transport in disordered metals without quasiparticles: The sachdev-ye-kitaev models and holography. Phys. Rev. B, 95: 155131, Apr 2017. 10.1103/​PhysRevB.95.155131.

[52] 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, Nov 2017a. 10.1103/​PhysRevLett.119.207603.

[53] Xue-Yang Song, Chao-Ming Jian, and Leon Balents. Strongly correlated metal built from sachdev-ye-kitaev models. Phys. Rev. Lett., 119: 216601, Nov 2017. 10.1103/​PhysRevLett.119.216601.

[54] Pengfei Zhang. Dispersive sachdev-ye-kitaev model: Band structure and quantum chaos. Phys. Rev. B, 96: 205138, Nov 2017. 10.1103/​PhysRevB.96.205138.

[55] Chao-Ming Jian, Zhen Bi, and Cenke Xu. Model for continuous thermal metal to insulator transition. Phys. Rev. B, 96: 115122, Sep 2017. 10.1103/​PhysRevB.96.115122.

[56] Yiming Chen, Hui Zhai, and Pengfei Zhang. Tunable Quantum Chaos in the Sachdev-Ye-Kitaev Model Coupled to a Thermal Bath. JHEP, 07: 150, 2017b. 10.1007/​JHEP07(2017)150.

[57] Phil Saad, Stephen H Shenker, and Douglas Stanford. A semiclassical ramp in syk and in gravity. arXiv preprint arXiv:1806.06840, 2018.

[58] Christoph Sünderhauf, Lorenzo Piroli, Xiao-Liang Qi, Norbert Schuch, and J. Ignacio Cirac. Quantum chaos in the Brownian SYK model with large finite $N$: OTOCs and tripartite information. JHEP, 11: 038, 2019. 10.1007/​JHEP11(2019)038.

[59] Zhihuang Luo, Yi-Zhuang You, Jun Li, Chao-Ming Jian, Dawei Lu, Cenke Xu, Bei Zeng, and Raymond Laflamme. Quantum simulation of the non-fermi-liquid state of sachdev-ye-kitaev model. npj Quantum Information, 5 (1): 1–6, 2019. 10.1038/​s41534-019-0166-7.

[60] 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, Jul 2018. 10.1103/​PhysRevLett.121.036403.

[61] Xiao Chen, Yaodong Li, Matthew PA Fisher, and Andrew Lucas. Emergent conformal symmetry in nonunitary random dynamics of free fermions. Physical Review Research, 2 (3): 033017, 2020b.

[62] Yuto Ashida, Shunsuke Furukawa, and Masahito Ueda. Quantum critical behavior influenced by measurement backaction in ultracold gases. Phys. Rev. A, 94: 053615, Nov 2016. 10.1103/​PhysRevA.94.053615.

[63] Yuto Ashida, Shunsuke Furukawa, and Masahito Ueda. Parity-time-symmetric quantum critical phenomena. Nature communications, 8 (1): 1–6, 2017. 10.1038/​ncomms15791.

[64] Gabriel Mazzucchi, Wojciech Kozlowski, Santiago F. Caballero-Benitez, Thomas J. Elliott, and Igor B. Mekhov. Quantum measurement-induced dynamics of many-body ultracold bosonic and fermionic systems in optical lattices. Phys. Rev. A, 93: 023632, Feb 2016a. 10.1103/​PhysRevA.93.023632.

[65] Gabriel Mazzucchi, Santiago F. Caballero-Benitez, Denis A. Ivanov, and Igor B. Mekhov. Quantum optical feedback control for creating strong correlations in many-body systems. Optica, 3 (11): 1213–1219, Nov 2016b. 10.1364/​OPTICA.3.001213.

[66] Shrabanti Dhar and Subinay Dasgupta. Measurement-induced phase transition in a quantum spin system. Phys. Rev. A, 93: 050103, May 2016. 10.1103/​PhysRevA.93.050103.

[67] D. A. Ivanov, T. Yu. Ivanova, S. F. Caballero-Benitez, and I. B. Mekhov. Feedback-induced quantum phase transitions using weak measurements. Phys. Rev. Lett., 124: 010603, Jan 2020. 10.1103/​PhysRevLett.124.010603.

[68] Giuseppe Buonaiuto, Federico Carollo, Beatriz Olmos, and Igor Lesanovsky. Dynamical phases and quantum correlations in an emitter-waveguide system with feedback. arXiv preprint arXiv:2102.02719, 2021. 10.1103/​PhysRevLett.127.133601.

[69] Alexei Kitaev and S. Josephine Suh. The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual. JHEP, 05: 183, 2018. 10.1007/​JHEP05(2018)183.

[70] Yingfei Gu, Alexei Kitaev, Subir Sachdev, and Grigory Tarnopolsky. Notes on the complex Sachdev-Ye-Kitaev model. JHEP, 02: 157, 2020. 10.1007/​JHEP02(2020)157.

[71] Chunxiao Liu, Xiao Chen, and Leon Balents. Quantum entanglement of the sachdev-ye-kitaev models. Phys. Rev. B, 97: 245126, Jun 2018. 10.1103/​PhysRevB.97.245126.

[72] Yingfei Gu, Andrew Lucas, and Xiao-Liang Qi. Spread of entanglement in a Sachdev-Ye-Kitaev chain. JHEP, 09: 120, 2017b. 10.1007/​JHEP09(2017)120.

[73] Yichen Huang and Yingfei Gu. Eigenstate entanglement in the sachdev-ye-kitaev model. Phys. Rev. D, 100: 041901, Aug 2019. 10.1103/​PhysRevD.100.041901.

[74] Pengfei Zhang, Chunxiao Liu, and Xiao Chen. Subsystem Rényi Entropy of Thermal Ensembles for SYK-like models. SciPost Phys., 8: 94, 2020b. 10.21468/​SciPostPhys.8.6.094.

[75] Arijit Haldar, Surajit Bera, and Sumilan Banerjee. Rényi entanglement entropy of Fermi and non-Fermi liquids: Sachdev-Ye-Kitaev model and dynamical mean field theories. Phys. Rev. Res., 2 (3): 033505, 2020. 10.1103/​PhysRevResearch.2.033505.

[76] Pengfei Zhang. Entanglement Entropy and its Quench Dynamics for Pure States of the Sachdev-Ye-Kitaev model. JHEP, 06: 143, 2020. 10.1007/​JHEP06(2020)143.

[77] Yiming Chen, Xiao-Liang Qi, and Pengfei Zhang. Replica wormhole and information retrieval in the SYK model coupled to Majorana chains. JHEP, 06: 121, 2020c. 10.1007/​JHEP06(2020)121.

[78] Shao-Kai Jian and Brian Swingle. Note on entropy dynamics in the Brownian SYK model. JHEP, 03: 042, 2021. 10.1007/​JHEP03(2021)042.

[79] Luca Lepori, Davide Vodola, Guido Pupillo, Giacomo Gori, and Andrea Trombettoni. Effective theory and breakdown of conformal symmetry in a long-range quantum chain. Annals of Physics, 374: 35–66, 2016. https:/​/​doi.org/​10.1016/​j.aop.2016.07.026.

[80] Lukasz Fidkowski, Jeongwan Haah, and Matthew B. Hastings. How dynamical quantum memories forget. Quantum, 5: 382, Jan 2021. ISSN 2521-327X. 10.22331/​q-2021-01-17-382.

[81] One may worry about using Majorana fermions in the SYK case but complex fermions in the single-flavor case. In fact, the complex SYK model at half-filling take exactly same entanglement properties as the Majorana SYK model up to a factor of $2$. Please see references liu2018quantum,10.21468/​SciPostPhys.8.6.094 for more details.

[82] Sergey Bravyi. Lagrangian representation for fermionic linear optics. Quantum Info. Comput., 5 (3): 216–238, may 2005. ISSN 1533-7146.

[83] Thomas Müller, Sebastian Diehl, and Michael Buchhold. Measurement-induced dark state phase transitions in long-ranged fermion systems. arXiv preprint arXiv:2105.08076, 2021. 10.1103/​PhysRevLett.128.010605.

[84] Hilary M Hurst and IB Spielman. Measurement-induced dynamics and stabilization of spinor-condensate domain walls. Physical Review A, 99 (5): 053612, 2019. 10.1103/​PhysRevA.99.053612.

[85] Rajibul Islam, Ruichao Ma, Philipp M Preiss, M Eric Tai, Alexander Lukin, Matthew Rispoli, and Markus Greiner. Measuring entanglement entropy in a quantum many-body system. Nature, 528 (7580): 77–83, 2015. 10.1038/​nature15750.

Cited by

[1] Xhek Turkeshi, "Measurement-induced criticality as a data-structure transition", Physical Review B 106 14, 144313 (2022).

[2] Davide Rossini and Ettore Vicari, "Coherent and dissipative dynamics at quantum phase transitions", Physics Reports 936, 1 (2021).

[3] Y. Minoguchi, P. Rabl, and M. Buchhold, "Continuous Gaussian Measurements of the Free Boson CFT: A model for Exactly Solvable and Detectable Measurement-Induced Dynamics", arXiv:2108.04256.

[4] Piotr Sierant, Giuliano Chiriacò, Federica M. Surace, Shraddha Sharma, Xhek Turkeshi, Marcello Dalmonte, Rosario Fazio, and Guido Pagano, "Dissipative Floquet Dynamics: from Steady State to Measurement Induced Criticality in Trapped-ion Chains", arXiv:2107.05669.

[5] Piotr Sierant and Xhek Turkeshi, "Universal Behavior beyond Multifractality of Wave Functions at Measurement-Induced Phase Transitions", Physical Review Letters 128 13, 130605 (2022).

[6] Tony Jin, João S. Ferreira, Michele Filippone, and Thierry Giamarchi, "Exact description of quantum stochastic models as quantum resistors", Physical Review Research 4 1, 013109 (2022).

[7] Pengfei Zhang, "Quantum entanglement in the Sachdev—Ye—Kitaev model and its generalizations", Frontiers of Physics 17 4, 43201 (2022).

[8] T. Botzung, S. Diehl, and M. Müller, "Engineered dissipation induced entanglement transition in quantum spin chains: From logarithmic growth to area law", Physical Review B 104 18, 184422 (2021).

[9] T. Boorman, M. Szyniszewski, H. Schomerus, and A. Romito, "Diagnostics of entanglement dynamics in noisy and disordered spin chains via the measurement-induced steady-state entanglement transition", Physical Review B 105 14, 144202 (2022).

[10] Shao-Kai Jian, Chunxiao Liu, Xiao Chen, Brian Swingle, and Pengfei Zhang, "Quantum error as an emergent magnetic field", arXiv:2106.09635.

[11] Tomohiro Hashizume, Gregory Bentsen, and Andrew J. Daley, "Measurement-induced phase transitions in sparse nonlocal scramblers", Physical Review Research 4 1, 013174 (2022).

[12] T. Kalsi, A. Romito, and H. Schomerus, "Three-fold way of entanglement dynamics in monitored quantum circuits", Journal of Physics A Mathematical General 55 26, 264009 (2022).

[13] D. A. Ivanov, T. Yu. Ivanova, S. F. Caballero-Benitez, and I. B. Mekhov, "Tuning the universality class of phase transitions by feedback: Open quantum systems beyond dissipation", Physical Review A 104 3, 033719 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2022-11-30 04:00:53) and SAO/NASA ADS (last updated successfully 2022-11-30 04:00:54). The list may be incomplete as not all publishers provide suitable and complete citation data.