Non-ergodic delocalized phase with Poisson level statistics

Weichen Tang1 and Ivan M. Khaymovich2,3,4

1Department of Physics, University of California, Berkeley, California 94720, USA
2Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187-Dresden, Germany
3Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia
4Nordita, Stockholm University and KTH Royal Institute of Technology Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

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

Updated version: The authors have uploaded version v5 of this work to the arXiv which may contain updates or corrections not contained in the published version v4. The authors left the following comment on the arXiv:
24 pages, 6 figures, 73 references + 3 pages in Appendix, accepted to Quantum journal


Motivated by the many-body localization (MBL) phase in generic interacting disordered quantum systems, we develop a model simulating the same eigenstate structure like in MBL, but in the random-matrix setting. Demonstrating the absence of energy level repulsion (Poisson statistics), this model carries non-ergodic eigenstates, delocalized over the extensive number of configurations in the Hilbert space. On the above example, we formulate general conditions to a single-particle and random-matrix models in order to carry such states, based on the transparent generalization of the Anderson localization of single-particle states and multiple resonances.

🇺🇦 Quantum strongly condemns the 2022 invasion of Ukraine, the loss of life and war crimes inflicted by Russian forces. For more information on our policy on publishing articles by authors based in Russian institutions, see this post

Many-body localization is a phenomenon, providing a unique possibility for an isolated interacting quantum system to avoid thermalization and to keep the information about its initial state.
Being the localization in the real space, many-body localization not only suppresses the transport, but also avoids repulsion of energy levels and makes eigenstates of an interacting system to be non-ergodic, but extended in the space of configurations (Hilbert space).
Facing the difficulties of describing many-body quantum systems, many researchers focus on the universal statistical description of the above phenomenon in a random-matrix setting.

Motivated by the many-body localization phase, we develop a random-matrix model simulating the same eigenstate structure.
Demonstrating the absence of energy level repulsion (Poisson statistics), this model carries non-ergodic eigenstates, delocalized over the extensive number of configurations in the Hilbert space.
Using the above example and a transparent generalization of the Anderson localization to multiple resonances, we formulate general conditions to realize non-ergodic states with Poisson level statistics.

► BibTeX data

► References

[1] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn. Colloquium: Many-body localization, thermalization, and entanglement. Rev. Mod. Phys., 91: 021001, May 2019. 10.1103/​RevModPhys.91.021001. URL https:/​/​​10.1103/​RevModPhys.91.021001.

[2] F. Alet and N. Laflorencie. Many-body localization: An introduction and selected topics. Comptes Rendus Physique, 19 (6): 498 – 525, 2018. ISSN 1631-0705. 10.1016/​j.crhy.2018.03.003. URL https:/​/​​10.1016/​j.crhy.2018.03.003. Quantum simulation /​ Simulation quantique.

[3] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux. Distribution of the ratio of consecutive level spacings in random matrix ensembles. Phys. Rev. Lett., 110: 084101, 2013a. 10.1103/​PhysRevLett.110.084101. URL https:/​/​​10.1103/​PhysRevLett.110.084101.

[4] Y. Y. Atas, E. Bogomolny, O. Giraud, P. Vivo, and E. Vivo. Joint probability densities of level spacing ratios in random matrices. Journal of Physics A: Mathematical and Theoretical, 46 (35): 355204, aug 2013b. 10.1088/​1751-8113/​46/​35/​355204. URL https:/​/​​10.1088.

[5] J. H. Bardarson, F. Pollmann, and J. E. Moore. Unbounded growth of entanglement in models of many-body localization. Phys. Rev. Lett., 109: 017202, Jul 2012. 10.1103/​PhysRevLett.109.017202. URL https:/​/​​10.1103/​PhysRevLett.109.017202.

[6] D. M. Basko, I. L. Aleiner, and B. L. Altshuler. Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states. Annals of Physics, 321 (5): 1126 – 1205, 2006. ISSN 0003-4916. 10.1016/​j.aop.2005.11.014. URL https:/​/​​10.1016/​j.aop.2005.11.014.

[7] S. Bera, G. De Tomasi, I. M. Khaymovich, and A. Scardicchio. Return probability for the Anderson model on the random regular graph. Phys. Rev. B, 98: 134205, 2018. 10.1103/​PhysRevB.98.134205. URL https:/​/​​10.1103/​PhysRevB.98.134205.

[8] R. Berkovits. Super-Poissonian behavior of the Rosenzweig-Porter model in the nonergodic extended regime. Phys. Rev. B, 102: 165140, Oct 2020. 10.1103/​PhysRevB.102.165140. URL https:/​/​​10.1103/​PhysRevB.102.165140.

[9] R. Berkovits. Probing the metallic energy spectrum beyond the thouless energy scale using singular value decomposition. Phys. Rev. B, 104: 054207, Aug 2021. 10.1103/​PhysRevB.104.054207. URL https:/​/​​10.1103/​PhysRevB.104.054207.

[10] G. Biroli and M. Tarzia. Delocalized glassy dynamics and many-body localization. Phys. Rev. B, 96: 201114(R), Nov 2017. 10.1103/​PhysRevB.96.201114. URL https:/​/​​10.1103/​PhysRevB.96.201114.

[11] G. Biroli and M. Tarzia. Anomalous dynamics on the ergodic side of the many-body localization transition and the glassy phase of directed polymers in random media. Phys. Rev. B, 102: 064211, Aug 2020. 10.1103/​PhysRevB.102.064211. URL https:/​/​​10.1103/​PhysRevB.102.064211.

[12] G. Biroli and M. Tarzia. Lévy-Rosenzweig-Porter random matrix ensemble. Phys. Rev. B, 103: 104205, Mar 2021. 10.1103/​PhysRevB.103.104205. URL https:/​/​​10.1103/​PhysRevB.103.104205.

[13] A. Burin. Localization and chaos in a quantum spin glass model in random longitudinal fields: Mapping to the localization problem in a Bethe lattice with a correlated disorder. Annalen der Physik, 529 (7): 1600292, 2017. 10.1002/​andp.201600292. URL https:/​/​​10.1002/​andp.201600292.

[14] A. L. Burin. Many-body delocalization in a strongly disordered system with long-range interactions: Finite-size scaling. Phys. Rev. B, 91: 094202, 2015. 10.1103/​PhysRevB.91.094202. URL https:/​/​​10.1103/​PhysRevB.91.094202.

[15] A. L. Burin and L. A. Maksimov. Localization and delocalization of particles in disordered lattice with tunneling amplitude with $r^{-3}$ decay. JETP Lett., 50: 338, 1989. URL http:/​/​​ps/​1129/​article_17116.shtml.

[16] G. L. Celardo, R. Kaiser, and F. Borgonovi. Shielding and localization in the presence of long-range hopping. Phys. Rev. B, 94: 144206, 2016. 10.1103/​PhysRevB.94.144206. URL https:/​/​​10.1103/​PhysRevB.94.144206.

[17] X. Chen, X. Yu, G. Y. Cho, B. K. Clark, and E. Fradkin. Many-body localization transition in Rokhsar-Kivelson-type wave functions. Phys. Rev. B, 92: 214204, Dec 2015. 10.1103/​PhysRevB.92.214204. URL https:/​/​​10.1103/​PhysRevB.92.214204.

[18] L. Colmenarez, D. J. Luitz, I. M. Khaymovich, and G. De Tomasi. Subdiffusive thouless time scaling in the anderson model on random regular graphs. Phys. Rev. B, 105: 174207, May 2022. 10.1103/​PhysRevB.105.174207. URL https:/​/​​10.1103/​PhysRevB.105.174207.

[19] L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Advances in Physics, 65 (3): 239–362, 2016. 10.1080/​00018732.2016.1198134. URL http:/​/​​10.1080/​00018732.2016.1198134.

[20] F. A. B. F. de Moura, A. V. Malyshev, M. L. Lyra, V. A. Malyshev, and F. Dominguez-Adame. Localization properties of a one-dimensional tight-binding model with nonrandom long-range intersite interactions. Phys. Rev. B, 71: 174203, 2005. 10.1103/​PhysRevB.71.174203. URL https:/​/​​10.1103/​PhysRevB.71.174203.

[21] G. De Tomasi and I. M. Khaymovich. Multifractality meets entanglement: Relation for nonergodic extended states. Phys. Rev. Lett., 124: 200602, May 2020. 10.1103/​PhysRevLett.124.200602. URL https:/​/​​10.1103/​PhysRevLett.124.200602.

[22] G. De Tomasi and I. M. Khaymovich. Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase, 2022. URL https:/​/​​abs/​2204.00669.

[23] G. De Tomasi, S. Bera, A. Scardicchio, and I. M. Khaymovich. Subdiffusion in the Anderson model on the random regular graph. Phys. Rev. B, 101: 100201(R), Mar 2020. 10.1103/​PhysRevB.101.100201. URL https:/​/​​10.1103/​PhysRevB.101.100201.

[24] G. De Tomasi, I. M. Khaymovich, F. Pollmann, and S. Warzel. Rare thermal bubbles at the many-body localization transition from the Fock space point of view. Phys. Rev. B, 104: 024202, Jul 2021. 10.1103/​PhysRevB.104.024202. URL https:/​/​​10.1103/​PhysRevB.104.024202.

[25] X. Deng, V. E. Kravtsov, G. V. Shlyapnikov, and L. Santos. Duality in power-law localization in disordered one-dimensional systems. Phys. Rev. Lett., 120 (11): 110602, 2018. 10.1103/​PhysRevLett.120.110602. URL https:/​/​​10.1103/​PhysRevLett.120.110602.

[26] Xiaolong Deng, Alexander L. Burin, and Ivan M. Khaymovich. Anisotropy-mediated reentrant localization, 2020. URL https:/​/​​abs/​2002.00013.

[27] J. M. Deutsch. Quantum statistical mechanics in a closed system. Phys. Rev. A, 43: 2046–2049, Feb 1991. 10.1103/​PhysRevA.43.2046. URL https:/​/​​10.1103/​PhysRevA.43.2046.

[28] F. Evers and A. D. Mirlin. Anderson transitions. Rev. Mod. Phys, 80: 1355, 2008. 10.1103/​RevModPhys.80.1355. URL https:/​/​​10.1103/​RevModPhys.80.1355.

[29] R. Fan, P. Zhang, H. Shen, and H. Zhai. Out-of-time-order correlation for many-body localization. Science Bulletin, 62 (10): 707–711, 2017. ISSN 2095-9273. 10.1016/​j.scib.2017.04.011. URL https:/​/​​10.1016/​j.scib.2017.04.011.

[30] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov. Interacting electrons in disordered wires: Anderson localization and low-$t$ transport. Phys. Rev. Lett., 95: 206603, Nov 2005. 10.1103/​PhysRevLett.95.206603. URL https:/​/​​10.1103/​PhysRevLett.95.206603.

[31] M. Haque, P. A. McClarty, and I. M. Khaymovich. Entanglement of midspectrum eigenstates of chaotic many-body systems: Reasons for deviation from random ensembles. Phys. Rev. E, 105: 014109, Jan 2022. 10.1103/​PhysRevE.105.014109. URL https:/​/​​10.1103/​PhysRevE.105.014109.

[32] Y. Huang, Y.-L. Zhang, and X. Chen. Out-of-time-ordered correlators in many-body localized systems. Annalen der Physik, 529 (7): 1600318, 2016. 10.1002/​andp.201600318. URL https:/​/​​10.1002/​andp.201600318.

[33] D. A. Huse, R. Nandkishore, and V. Oganesyan. Phenomenology of fully many-body-localized systems. Phys. Rev. B, 90: 174202, Nov 2014. 10.1103/​PhysRevB.90.174202. URL https:/​/​​10.1103/​PhysRevB.90.174202.

[34] I. M. Khaymovich and V. E. Kravtsov. Dynamical phases in a ``multifractal'' Rosenzweig-Porter model. SciPost Phys., 11: 45, 2021. 10.21468/​SciPostPhys.11.2.045. URL https:/​/​​10.21468/​SciPostPhys.11.2.045.

[35] I. M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, and L. B. Ioffe. Fragile ergodic phases in logarithmically-normal Rosenzweig-Porter model. Phys. Rev. Research, 2: 043346, 2020. 10.1103/​PhysRevResearch.2.043346. URL https:/​/​​10.1103/​PhysRevResearch.2.043346.

[36] V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, and M. Amini. A random matrix model with localization and ergodic transitions. New J. Phys., 17: 122002, 2015. 10.1088/​1367-2630/​17/​12/​122002. URL https:/​/​​10.1088.

[37] V. E. Kravtsov, I. M. Khaymovich, B. L. Altshuler, and L. B. Ioffe. Localization transition on the random regular graph as an unstable tricritical point in a log-normal rosenzweig-porter random matrix ensemble, 2020. URL https:/​/​​abs/​2002.02979.

[38] A. G. Kutlin and I. M. Khaymovich. Renormalization to localization without a small parameter. SciPost Phys., 8: 49, 2020. 10.21468/​SciPostPhys.8.4.049. URL https:/​/​​10.21468/​SciPostPhys.8.4.049.

[39] A. G. Kutlin and I. M. Khaymovich. Emergent fractal phase in energy stratified random models. SciPost Phys., 11: 101, 2021. 10.21468/​SciPostPhys.11.6.101. URL https:/​/​​10.21468/​SciPostPhys.11.6.101.

[40] A. G. Kutlin and I. M. Khaymovich. Multifractal phase in real and energy spaces, 2022. in preparation.

[41] L. S. Levitov. Absence of localization of vibrational modes due to dipole-dipole interaction. Europhys. Lett., 9: 83, 1989. 10.1209/​0295-5075/​9/​1/​015. URL https:/​/​​10.1209.

[42] L. S. Levitov. Delocalization of vibrational modes caused by electric dipole interaction. Phys. Rev. Lett., 64: 547, 1990. 10.1103/​PhysRevLett.64.547. URL https:/​/​​10.1103/​PhysRevLett.64.547.

[43] D. J. Luitz, N. Laflorencie, and F. Alet. Many-body localization edge in the random-field Heisenberg chain. Phys. Rev. B, 91: 081103, Feb 2015. 10.1103/​PhysRevB.91.081103. URL https:/​/​​10.1103/​PhysRevB.91.081103.

[44] D. J. Luitz, I. M. Khaymovich, and Y. Bar Lev. Multifractality and its role in anomalous transport in the disordered XXZ spin-chain. SciPost Phys. Core, 2: 6, 2020. 10.21468/​SciPostPhysCore.2.2.006. URL https:/​/​​10.21468/​SciPostPhysCore.2.2.006.

[45] N. Macé, F. Alet, and N. Laflorencie. Multifractal scalings across the many-body localization transition. Phys. Rev. Lett., 123: 180601, Oct 2019. 10.1103/​PhysRevLett.123.180601. URL https:/​/​​10.1103/​PhysRevLett.123.180601.

[46] M. L. Mehta. Random matrices. Elsevier, 2004. 10.1016/​C2009-0-22297-5. URL https:/​/​​10.1016/​C2009-0-22297-5.

[47] A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman. Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices. Phys. Rev. E, 54: 3221, 1996. 10.1103/​PhysRevE.54.3221. URL https:/​/​​10.1103/​PhysRevE.54.3221.

[48] A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, and D. A. Huse. Avalanches and many-body resonances in many-body localized systems. Phys. Rev. B, 105: 174205, May 2022. 10.1103/​PhysRevB.105.174205. URL https:/​/​​10.1103/​PhysRevB.105.174205.

[49] Vedant Motamarri, Alexander S. Gorsky, and Ivan M. Khaymovich. Localization and fractality in disordered russian doll model, 2021. URL https:/​/​​abs/​2112.05066.

[50] P. A. Nosov and I. M. Khaymovich. Robustness of delocalization to the inclusion of soft constraints in long-range random models. Phys. Rev. B, 99: 224208, Jun 2019. 10.1103/​PhysRevB.99.224208. URL https:/​/​​10.1103/​PhysRevB.99.224208.

[51] P. A. Nosov, I. M. Khaymovich, and V. E. Kravtsov. Correlation-induced localization. Physical Review B, 99 (10): 104203, 2019. 10.1103/​PhysRevB.99.104203. URL https:/​/​​10.1103/​PhysRevB.99.104203.

[52] P. A. Nosov, I. M. Khaymovich, A. Kudlis, and V. E. Kravtsov. Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation. SciPost Phys., 12: 48, 2022. 10.21468/​SciPostPhys.12.2.048. URL https:/​/​​10.21468/​SciPostPhys.12.2.048.

[53] V. Oganesyan and D. A. Huse. Localization of interacting fermions at high temperature. Phys. Rev. B, 75: 155111, Apr 2007. 10.1103/​PhysRevB.75.155111. URL https:/​/​​10.1103/​PhysRevB.75.155111.

[54] A. Pal and D. A. Huse. Many-body localization phase transition. Phys. Rev. B, 82: 174411, Nov 2010. 10.1103/​PhysRevB.82.174411. URL https:/​/​​10.1103/​PhysRevB.82.174411.

[55] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore. Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys., 83: 863–883, Aug 2011. 10.1103/​RevModPhys.83.863. URL https:/​/​​10.1103/​RevModPhys.83.863.

[56] S. Ray, A. Ghosh, and S. Sinha. Drive-induced delocalization in the Aubry-André model. Phys. Rev. E, 97: 010101, Jan 2018. 10.1103/​PhysRevE.97.010101. URL https:/​/​​10.1103/​PhysRevE.97.010101.

[57] M. Rigol, V. Dunjko, and M. Olshanii. Thermalization and its mechanism for generic isolated quantum systems. Nature, 452 (7189): 854, apr 2008. 10.1038/​nature06838. URL https:/​/​​10.1038/​nature06838.

[58] R. Riser and E. Kanzieper. Power spectrum and form factor in random diagonal matrices and integrable billiards. Annals of Physics, 425: 168393, 2021. ISSN 0003-4916. 10.1016/​j.aop.2020.168393. URL https:/​/​​10.1016/​j.aop.2020.168393.

[59] R. Riser, V. Al. Osipov, and E. Kanzieper. Power spectrum of long eigenlevel sequences in quantum chaotic systems. Phys. Rev. Lett., 118: 204101, May 2017. 10.1103/​PhysRevLett.118.204101. URL https:/​/​​10.1103/​PhysRevLett.118.204101.

[60] R. Riser, V. Al. Osipov, and E. Kanzieper. Nonperturbative theory of power spectrum in complex systems. Annals of Physics, 413: 168065, 2020. ISSN 0003-4916. 10.1016/​j.aop.2019.168065. URL https:/​/​​10.1016/​j.aop.2019.168065.

[61] N. Rosenzweig and C. E. Porter. "repulsion of energy levels" in complex atomic spectra. Phys. Rev. B, 120: 1698, 1960. 10.1103/​PhysRev.120.1698. URL https:/​/​​10.1103/​PhysRev.120.1698.

[62] S. Roy, I. M. Khaymovich, A. Das, and R. Moessner. Multifractality without fine-tuning in a Floquet quasiperiodic chain. SciPost Phys., 4: 25, 2018. 10.21468/​SciPostPhys.4.5.025. URL https:/​/​​10.21468/​SciPostPhys.4.5.025.

[63] M. Sarkar, R. Ghosh, A. Sen, and K. Sengupta. Mobility edge and multifractality in a periodically driven Aubry-André model. Phys. Rev. B, 103: 184309, May 2021. 10.1103/​PhysRevB.103.184309. URL https:/​/​​10.1103/​PhysRevB.103.184309.

[64] M. Sarkar, R. Ghosh, A. Sen, and K. Sengupta. Signatures of multifractality in a periodically driven interacting aubry-andré model. Phys. Rev. B, 105: 024301, Jan 2022. 10.1103/​PhysRevB.105.024301. URL https:/​/​​10.1103/​PhysRevB.105.024301.

[65] M. Serbyn, Z. Papić, and D. A. Abanin. Local conservation laws and the structure of the many-body localized states. Phys. Rev. Lett., 111: 127201, Sep 2013. 10.1103/​PhysRevLett.111.127201. URL https:/​/​​10.1103/​PhysRevLett.111.127201.

[66] M. Srednicki. Chaos and quantum thermalization. Phys. Rev. E, 50: 888–901, Aug 1994. 10.1103/​PhysRevE.50.888. URL https:/​/​​10.1103/​PhysRevE.50.888.

[67] M. Srednicki. Thermal fluctuations in quantized chaotic systems. J. Phys. A: Mathematical and General, 29 (4): L75, 1996. 10.1088/​0305-4470/​29/​4/​003. URL https:/​/​​10.1088/​0305-4470/​29/​4/​003.

[68] M. Tarzia. Many-body localization transition in Hilbert space. Phys. Rev. B, 102: 014208, Jul 2020. 10.1103/​PhysRevB.102.014208. URL https:/​/​​10.1103/​PhysRevB.102.014208.

[69] S. H. Tekur, U. T. Bhosale, and M. S. Santhanam. Higher-order spacing ratios in random matrix theory and complex quantum systems. Phys. Rev. B, 98: 104305, Sep 2018. 10.1103/​PhysRevB.98.104305. URL https:/​/​​10.1103/​PhysRevB.98.104305.

[70] K. S. Tikhonov and A. D. Mirlin. Many-body localization transition with power-law interactions: Statistics of eigenstates. Phys. Rev. B, 97: 214205, Jun 2018. 10.1103/​PhysRevB.97.214205. URL https:/​/​​10.1103/​PhysRevB.97.214205.

[71] G. Torres-Vargas, R. Fossion, C. Tapia-Ignacio, and J. C. López-Vieyra. Determination of scale invariance in random-matrix spectral fluctuations without unfolding. Phys. Rev. E, 96: 012110, Jul 2017. 10.1103/​PhysRevE.96.012110. URL https:/​/​​10.1103/​PhysRevE.96.012110.

[72] G. Torres-Vargas, J. A. Méndez-Bermúdez, J. C. López-Vieyra, and R. Fossion. Crossover in nonstandard random-matrix spectral fluctuations without unfolding. Phys. Rev. E, 98: 022110, Aug 2018. 10.1103/​PhysRevE.98.022110. URL https:/​/​​10.1103/​PhysRevE.98.022110.

[73] F. Yonezawa and K. Morigaki. Coherent potential approximation. Basic concepts and applications. Progress of Theoretical Physics Supplement, 53: 1–76, 01 1973. ISSN 0375-9687. 10.1143/​PTPS.53.1. URL https:/​/​​10.1143/​PTPS.53.1.

Cited by

[1] Davide Venturelli, Leticia F. Cugliandolo, Grégory Schehr, and Marco Tarzia, "Replica approach to the generalized Rosenzweig-Porter model", SciPost Physics 14 5, 110 (2023).

[2] Pietro Brighi, Marko Ljubotina, Dmitry A. Abanin, and Maksym Serbyn, "Many-body localization proximity effect in a two-species bosonic Hubbard model", Physical Review B 108 5, 054201 (2023).

[3] Aamna Ahmed, Ajith Ramachandran, Ivan M. Khaymovich, and Auditya Sharma, "Flat band based multifractality in the all-band-flat diamond chain", Physical Review B 106 20, 205119 (2022).

[4] Dillip Kumar Nandy, Tilen Čadež, Barbara Dietz, Alexei Andreanov, and Dario Rosa, "Delayed thermalization in the mass-deformed Sachdev-Ye-Kitaev model", Physical Review B 106 24, 245147 (2022).

[5] Vedant Motamarri, Alexander S. Gorsky, and Ivan Khaymovich, "Localization and fractality in disordered Russian Doll model", SciPost Physics 13 5, 117 (2022).

[6] Stefano Piemontese, Tommaso Roscilde, and Alioscia Hamma, "Entanglement complexity of the Rokhsar-Kivelson-sign wavefunctions", Physical Review B 107 13, 134202 (2023).

[7] Nilanjan Roy, Jagannath Sutradhar, and Sumilan Banerjee, "Diagnostics of nonergodic extended states and many body localization proximity effect through real-space and Fock-space excitations", Physical Review B 107 11, 115155 (2023).

[8] Xiaolong Deng, Alexander Burin, and Ivan Khaymovich, "Anisotropy-mediated reentrant localization", SciPost Physics 13 5, 116 (2022).

[9] Miguel Gonçalves, Pedro Ribeiro, and Ivan M. Khaymovich, "Quasiperiodicity hinders ergodic Floquet eigenstates", Physical Review B 108 10, 104201 (2023).

[10] Francesco Mattiotti, Jérôme Dubail, David Hagenmüller, Johannes Schachenmayer, Jean-Philippe Brantut, and Guido Pupillo, "Multifractality in the interacting disordered Tavis-Cummings model", arXiv:2302.14718, (2023).

[11] Xiaolong Deng, Ivan Khaymovich, and Alexander L. Burin, "Superdiffusion in random two dimensional system with ubiquitous long-range hopping", arXiv:2205.14715, (2022).

[12] Anton Kutlin and Ivan M. Khaymovich, "Anatomy of the eigenstates distribution: a quest for a genuine multifractality", arXiv:2309.06468, (2023).

[13] Giuseppe De Tomasi and Ivan M. Khaymovich, "Non-Hermiticity induces localization: good and bad resonances in power-law random banded matrices", arXiv:2302.00015, (2023).

[14] Shilpi Roy, Saurabh Basu, and Ivan M. Khaymovich, "Ergodicity-breaking phase diagram and fractal dimensions in long-range models with generically correlated disorder", arXiv:2307.03085, (2023).

[15] Adway Kumar Das, Anandamohan Ghosh, and Ivan M. Khaymovich, "Absence of Mobility Edge in Short-range Uncorrelated Disordered Model: Coexistence of Localized and Extended States", arXiv:2305.02351, (2023).

[16] Adway Kumar Das and Anandamohan Ghosh, "Nonergodic extended states in the β ensemble", Physical Review E 105 5, 054121 (2022).

[17] M. Tarzia, "Fully localized and partially delocalized states in the tails of Erdös-Rényi graphs in the critical regime", Physical Review B 105 17, 174201 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2023-10-01 02:27:35) and SAO/NASA ADS (last updated successfully 2023-10-01 02:27:35). The list may be incomplete as not all publishers provide suitable and complete citation data.