[en] Open quantum systems have complex energy eigenvalues which are expected to follow non-Hermitian random matrix statistics, when chaotic, or two-dimensional (2d) Poisson statistics, when integrable. We investigate the spectral properties of a many-body quantum spin chain, i.e., the Hermitian XXZ Heisenberg model with imaginary disorder. Its rich complex eigenvalue statistics is found to separately break both Hermiticity and integrability at different scales of the disorder strength. With no disorder, the system is integrable and Hermitian, with spectral statistics corresponding to the 1d Poisson point process. At very small disorder, we find a transition from 1d Poisson statistics to an effective D-dimensional Poisson point process, showing Hermiticity breaking. At intermediate disorder, we find integrability breaking, as inferred from the statistics matching that of non-Hermitian complex symmetric random matrices in class AI†. For large disorder, as the spins align, we recover the expected integrability (now in the non-Hermitian setup), indicated by 2d Poisson statistics. These conclusions are based on fitting the spin-chain data of numerically generated nearest- and next-to-nearest-neighbor spacing distributions to an effective 2d Coulomb gas description at inverse temperature β. We confirm that such an effective description of random matrices also applies in classes AI† and AII† up to next-to-nearest-neighbor spacings.
Disciplines :
Physique
Auteur, co-auteur :
Akemann, Gernot ; Faculty of Physics, Bielefeld University, Germany ; School of Mathematics, University of Bristol, Bristol, United Kingdom
BALDUCCI, Federico ; University of Luxembourg > Faculty of Science, Technology and Medicine > Department of Physics and Materials Science > Team Aurélia CHENU ; Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
Päßler, Patricia ; Faculty of Physics, Bielefeld University, Germany
ROCCATI, Federico ; University of Luxembourg > Faculty of Science, Technology and Medicine > Department of Physics and Materials Science > Team Aurélia CHENU ; Department of Physics, Columbia University, New York, United States ; Max Planck Institute for the Science of Light, Erlangen, Germany
SHIR, Ruth ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Physics and Materials Science (DPHYMS)
CHENU, Aurélia ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Physics and Materials Science (DPHYMS)
Co-auteurs externes :
yes
Langue du document :
Anglais
Titre :
Two transitions in complex eigenvalue statistics: Hermiticity and integrability breaking
John Templeton Foundation Fonds National de la Recherche Luxembourg Deutsche Forschungsgemeinschaft Leverhulme Trust Los Alamos National Laboratory
Subventionnement (détails) :
We acknowledge funding from the John Templeton Foundation (JTF Grant No. 62171) and the Luxembourg National Research Fund (FNR, Attract Grant No. 15382998). F.R. acknowledges financial support from the Fulbright Research Scholar Program. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the JTF. The numerical simulations presented in this work were in part carried out using the HPC facilities of the University of Luxembourg. This work was partly supported by the Deutsche Forschungsgemeinschaft (DFG) Grant No. SFB 1283/2 2021\u2013317210226 (G.A., P.P.) and a Leverhulme Trust Visiting Professorship, Grant No. VP1-2023-007 (G.A.). G.A. is indebted to the School of Mathematics, University of Bristol, where this research was conducted. A.C. acknowledges the CNLS at Los Alamos National Laboratory, where part of this research was conducted.
Commentaire :
12+2 pages, 12+2 figures. Accepted for publication in Phys. Rev.
Research
M. V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. R. Soc. London, Ser. A: Math. Phys. Sci. 356, 375 (1977) 10.1098/rspa.1977.0140.
O. Bohigas, M.-J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52, 1 (1984) 0031-9007 10.1103/PhysRevLett.52.1.
G. Casati, F. Valz-Gris, and I. Guarnieri, On the connection between quantization of nonintegrable systems and statistical theory of spectra, Lett. Nuovo Cim. 28, 279 (1980) 1827-613X 10.1007/BF02798790.
L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016) 0001-8732 10.1080/00018732.2016.1198134.
S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, Semiclassical foundation of universality in quantum chaos, Phys. Rev. Lett. 93, 014103 (2004) 0031-9007 10.1103/PhysRevLett.93.014103.
F. Haake, Quantum Signatures of Chaos (Springer, New York, 2010).
W. Buijsman, V. Cheianov, and V. Gritsev, Random matrix ensemble for the level statistics of many-body localization, Phys. Rev. Lett. 122, 180601 (2019) 0031-9007 10.1103/PhysRevLett.122.180601.
F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps, Phys. Rev. A 100, 062131 (2019) 2469-9926 10.1103/PhysRevA.100.062131.
M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. W. Murch, Quantum state tomography across the exceptional point in a single dissipative qubit, Nat. Phys. 15, 1232 (2019) 1745-2473 10.1038/s41567-019-0652-z.
F. Roccati, G. M. Palma, F. Ciccarello, and F. Bagarello, Non-Hermitian physics and master equations, Open Syst. Inf. Dyn. 29, 2250004 (2022) 1230-1612 10.1142/S1230161222500044.
R. Grobe, F. Haake, and H.-J. Sommers, Quantum distinction of regular and chaotic dissipative motion, Phys. Rev. Lett. 61, 1899 (1988) 0031-9007 10.1103/PhysRevLett.61.1899.
H. Sompolinsky, A. Crisanti, and H.-J. Sommers, Chaos in random neural networks, Phys. Rev. Lett. 61, 259 (1988) 0031-9007 10.1103/PhysRevLett.61.259.
J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Math. Phy. 6, 440 (1965) 0022-2488 10.1063/1.1704292.
R. Hamazaki, K. Kawabata, N. Kura, and M. Ueda, Universality classes of non-Hermitian random matrices, Phys. Rev. Res. 2, 023286 (2020) 2643-1564 10.1103/PhysRevResearch.2.023286.
D. Bernard and A. LeClair, A Classification of non-Hermitian random matrices, Statistical Field Theories, edited by A. Cappelli and G. Mussardo (Springer, New York, 2002), Vol. 73, pp. 207-214.
U. Magnea, Random matrices beyond the Cartan classification, J. Phys. A: Math. Theor. 41, 045203 (2008) 1751-8113 10.1088/1751-8113/41/4/045203.
N. Hatano and D. R. Nelson, Localization transitions in non-Hermitian quantum mechanics, Phys. Rev. Lett. 77, 570 (1996) 0031-9007 10.1103/PhysRevLett.77.570.
H. Markum, R. Pullirsch, and T. Wettig, Non-Hermitian random matrix theory and lattice QCD with chemical potential, Phys. Rev. Lett. 83, 484 (1999) 0031-9007 10.1103/PhysRevLett.83.484.
T. Kanazawa and T. Wettig, New universality classes of the non-Hermitian Dirac operator in QCD-like theories, Phys. Rev. D 104, 014509 (2021) 2470-0010 10.1103/PhysRevD.104.014509.
B. Ye, L. Qiu, X. Wang, and T. Guhr, Spectral statistics in directed complex networks and universality of the Ginibre ensemble, Commun. Nonlinear Sci. Numer. Simul. 20, 1026 (2015) 1007-5704 10.1016/j.cnsns.2014.07.001.
G. Akemann, M. Kieburg, A. Mielke, and T. Prosen, Universal signature from integrability to chaos in dissipative open quantum systems, Phys. Rev. Lett. 123, 254101 (2019) 0031-9007 10.1103/PhysRevLett.123.254101.
A. Rubio-García, R. A. Molina, and J. Dukelsky, From integrability to chaos in quantum Liouvillians, SciPost Phys. Core 5, 026 (2022) 2666-9366 10.21468/SciPostPhysCore.5.2.026.
A. B. Jaiswal, A. Pandey, and R. Prakash, Universality classes of quantum chaotic dissipative systems, Europhys. Lett. 127, 30004 (2019) 1286-4854 10.1209/0295-5075/127/30004.
Y. Huang and B. I. Shklovskii, Anderson transition in three-dimensional systems with non-Hermitian disorder, Phys. Rev. B 101, 014204 (2020) 2469-9950 10.1103/PhysRevB.101.014204.
G. Akemann, N. Chakarov, O. Krüger, A. Mielke, M. Ottensmann, and P. Pässler, Interactions between different birds of prey as a random point process, J. Stat. Mech. (2024) 053501 1742-5468 10.1088/1742-5468/ad37be.
D. Basko, I. Aleiner, and B. Altshuler, Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states, Ann. Phys. 321, 1126 (2006) 0003-4916 10.1016/j.aop.2005.11.014.
V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007) 1098-0121 10.1103/PhysRevB.75.155111.
M. Žnidarič, T. Prosen, and P. Prelovšek, Many-body localization in the Heisenberg XXZ magnet in a random field, Phys. Rev. B 77, 064426 (2008) 1098-0121 10.1103/PhysRevB.77.064426.
A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010) 1098-0121 10.1103/PhysRevB.82.174411.
J. H. Bardarson, F. Pollmann, and J. E. Moore, Unbounded growth of entanglement in models of many-body localization, Phys. Rev. Lett. 109, 017202 (2012) 0031-9007 10.1103/PhysRevLett.109.017202.
A. D. Luca and A. Scardicchio, Ergodicity breaking in a model showing many-body localization, Europhys. Lett. 101, 37003 (2013) 0295-5075 10.1209/0295-5075/101/37003.
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019) 0034-6861 10.1103/RevModPhys.91.021001.
P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-Body localization in the age of classical computing, Rep. Prog. Phys. (2024) 10.1088/1361-6633/ad9756.
J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991) 1050-2947 10.1103/PhysRevA.43.2046.
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994) 1063-651X 10.1103/PhysRevE.50.888.
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 (2013) 0031-9007 10.1103/PhysRevLett.111.127201.
V. Ros, M. Müller, and A. Scardicchio, Integrals of motion in the many-body localized phase, Nucl. Phys. B 891, 420 (2015) 0550-3213 10.1016/j.nuclphysb.2014.12.014.
J. Z. Imbrie, Diagonalization and many-body localization for a disordered quantum spin chain, Phys. Rev. Lett. 117, 027201 (2016) 0031-9007 10.1103/PhysRevLett.117.027201.
R. Hamazaki, K. Kawabata, and M. Ueda, Non-Hermitian many-body localization, Phys. Rev. Lett. 123, 090603 (2019) 0031-9007 10.1103/PhysRevLett.123.090603.
L.-J. Zhai, S. Yin, and G.-Y. Huang, Many-body localization in a non-Hermitian quasiperiodic system, Phys. Rev. B 102, 064206 (2020) 2469-9950 10.1103/PhysRevB.102.064206.
K. Suthar, Y.-C. Wang, Y.-P. Huang, H. H. Jen, and J.-S. You, Non-Hermitian many-body localization with open boundaries, Phys. Rev. B 106, 064208 (2022) 2469-9950 10.1103/PhysRevB.106.064208.
S. Ghosh, S. Gupta, and M. Kulkarni, Spectral properties of disordered interacting non-Hermitian systems, Phys. Rev. B 106, 134202 (2022) 2469-9950 10.1103/PhysRevB.106.134202.
G. De Tomasi and I. M. Khaymovich, Stable many-body localization under random continuous measurements in the no-click limit, Phys. Rev. B 109, 174205 (2024) 2469-9950 10.1103/PhysRevB.109.174205.
F. Roccati, F. Balducci, R. Shir, and A. Chenu, Diagnosing non-Hermitian many-body localization and quantum chaos via singular value decomposition, Phys. Rev. B 109, L140201 (2024) 2469-9950 10.1103/PhysRevB.109.L140201.
S. Lapp, J. Ang'ong'a, F. A. An, and B. Gadway, Engineering tunable local loss in a synthetic lattice of momentum states, New J. Phys. 21, 045006 (2019) 1367-2630 10.1088/1367-2630/ab1147.
G. Akemann, A. Mielke, and P. Päßler, Spacing distribution in the two-dimensional Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at noninteger (Equation presented), Phys. Rev. E 106, 014146 (2022) 2470-0045 10.1103/PhysRevE.106.014146.
L. Sá, P. Ribeiro, and T. Prosen, Complex spacing ratios: A signature of dissipative quantum chaos, Phys. Rev. X 10, 021019 (2020) 10.1103/PhysRevX.10.021019.
C. N. Yang and C. P. Yang, One-dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe's hypothesis for ground state in a finite system, Phys. Rev. 150, 321 (1966) 0031-899X 10.1103/PhysRev.150.321.
D. E. Mahoney and J. Richter, Transport and integrability-breaking in non-Hermitian many-body quantum systems, Phys. Rev. B 110, 134302 (2024) 2469-9950 10.1103/PhysRevB.110.134302.
K. Joel, D. Kollmar, and L. F. Santos, An introduction to the spectrum, symmetries, and dynamics of spin-1/2 Heisenberg chains, Am. J. Phys. 81, 450 (2013) 0002-9505 10.1119/1.4798343.
W. Pauli, On Dirac's new method of field quantization, Rev. Mod. Phys. 15, 175 (1943) 0034-6861 10.1103/RevModPhys.15.175.
J. Feinberg and M. Znojil, Which metrics are consistent with a given pseudo-Hermitian matrix J. Math. Phys. 63, 013505 (2022) 0022-2488 10.1063/5.0079385.
A. Melkani, Degeneracies and symmetry breaking in pseudo-Hermitian matrices, Phys. Rev. Res. 5, 023035 (2023) 2643-1564 10.1103/PhysRevResearch.5.023035.
This is after removing the term proportional to the identity, which is responsible for the constant imaginary shift in the complex plane.
We found that for 1d, (Equation presented) gives good results.
D. Kundu, S. Kumar, and S. S. Gupta, Beyond nearest-neighbour universality of spectral fluctuations in quantum chaotic and complex many-body systems, arXiv:2408.04345.
J. Li, S. Yan, T. Prosen, and A. Chan, Spectral form factor in chaotic, localized, and integrable open quantum many-body systems, arXiv:2405.01641.
Such point may not be unique, but the event of finding two neighboring points at the very same distance will be of measure zero.
It is expected based on numerical simulations that the spacing distribution at the edge is different.
The effect of this rescaling on the logarithmic term amounts to adding a constant (Equation presented), which, in turn, becomes a multiplicative factor, (Equation presented), in the joint distribution. In the limit (Equation presented), this factor goes to 1.
For (Equation presented), the NN spacing distribution is known for classes (Equation presented) and (Equation presented) [14,23], which are, however, not a good approximation for the large-(Equation presented) limit, as was discussed in [46].
Equations (23) and (27) of Ref. [46] contain typos; we write down the corrected formula in (7).
M. L. Mehta, Random Matrices (Elsevier, Amsterdam, 2004).
P. J. Forrester, Log-gases and Random Matrices (LMS-34) (Princeton University Press, Princeton, NJ, 2010).
A. Borodin and C. D. Sinclair, The Ginibre ensemble of real random matrices and its scaling limits, Commun. Math. Phys. 291, 177 (2009) 0010-3616 10.1007/s00220-009-0874-5.
J. Sakhr and J. M. Nieminen, Wigner surmises and the two-dimensional homogeneous poisson point process, Phys. Rev. E 73, 047202 (2006) 1539-3755 10.1103/PhysRevE.73.047202.
H.-J. Sommers, Y. V. Fyodorov, and M. Titov, S-matrix poles for chaotic quantum systems as eigenvalues of complex symmetric random matrices: From isolated to overlapping resonances, J. Phys. A: Math. Gen. 32, L77 (1999) 0305-4470 10.1088/0305-4470/32/5/003.
T. Guhr, A. Müller-Groeling, and H. A. Weidenmüller, Random-matrix theories in quantum physics: Common concepts, Phys. Rep. 299, 189 (1998) 0370-1573 10.1016/S0370-1573(97)00088-4.
A. M. García-García, L. Sá, and J. J. M. Verbaarschot, Universality and its limits in non-Hermitian many-body quantum chaos using the Sachdev-Ye-Kitaev model, Phys. Rev. D 107, 066007 (2023) 2470-0010 10.1103/PhysRevD.107.066007.
In [21] it was suggested that (Equation presented) is a good choice and we use this value too.
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 (2013) 0031-9007 10.1103/PhysRevLett.110.084101.
I. G. Dusa and T. Wettig, Approximation formula for complex spacing ratios in the Ginibre ensemble, Phys. Rev. E 105, 044144 (2022) 2470-0045 10.1103/PhysRevE.105.044144.
It appears from numerics that classes (Equation presented) and (Equation presented) also have a flat spectrum in the bulk and thus do not require unfolding.
R. Grobe and F. Haake, Universality of cubic-level repulsion for dissipative quantum chaos, Phys. Rev. Lett. 62, 2893 (1989) 0031-9007 10.1103/PhysRevLett.62.2893.
G. Akemann, M. Phillips, and L. Shifrin, Gap probabilities in non-Hermitian random matrix theory, J. Math. Phys. 50, 063504 (2009) 0022-2488 10.1063/1.3133108.
