Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

被引:57
作者
Baecker, Arnd [1 ,2 ,3 ]
Haque, Masudul [3 ,4 ]
Khaymovich, Ivan M. [3 ]
机构
[1] Tech Univ Dresden, Inst Theoret Phys, D-01062 Dresden, Germany
[2] Tech Univ Dresden, Ctr Dynam, D-01062 Dresden, Germany
[3] Max Planck Inst Phys Komplexer Syst, Nothnitzer Str 38, D-01187 Dresden, Germany
[4] Maynooth Univ, Dept Theoret Phys, Maynooth, Kildare, Ireland
基金
俄罗斯基础研究基金会;
关键词
INVERSE PARTICIPATION RATIO; EMBEDDED GAUSSIAN ENSEMBLES; STATISTICAL PROPERTIES; SPECTRAL PROPERTIES; EIGENFUNCTIONS; LOCALIZATION; THERMALIZATION; FLUCTUATIONS; LEVEL; DISTRIBUTIONS;
D O I
10.1103/PhysRevE.100.032117
中图分类号
O35 [流体力学]; O53 [等离子体物理学];
学科分类号
070204 ; 080103 ; 080704 ;
摘要
Multifractal dimensions allow for characterizing the localization properties of states in complex quantum systems. For ergodic states the finite-size versions of fractal dimensions converge to unity in the limit of large system size. However, the approach to the limiting behavior is remarkably slow. Thus, an understanding of the scaling and finite-size properties of fractal dimensions is essential. We present such a study for random matrix ensembles, and compare with two chaotic quantum systems-the kicked rotor and a spin chain. For random matrix ensembles we analytically obtain the finite-size dependence of the mean behavior of the multifractal dimensions, which provides a lower bound to the typical (logarithmic) averages. We show that finite statistics has remarkably strong effects, so that even random matrix computations deviate from analytic results (and show strong sample-to-sample variation), such that restoring agreement requires exponentially large sample sizes. For the quantized standard map (kicked rotor) the multifractal dimensions are found to follow the random matrix predictions closely, with the same finite statistics effects. For a XXZ spin-chain we find significant deviations from the random matrix prediction-the large-size scaling follows a system-specific path towards unity. This suggests that local many-body Hamiltonians are "weakly ergodic," in the sense that their eigenfunction statistics deviate from random matrix theory.
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页数:14
相关论文
共 140 条
[71]   Apparent slow dynamics in the ergodic phase of a driven many-body localized system without extensive conserved quantities [J].
Lezama, Talia L. M. ;
Bera, Soumya ;
Bardarson, Jens H. .
PHYSICAL REVIEW B, 2019, 99 (16)
[72]   STATISTICAL PROPERTIES OF HIGH-LYING CHAOTIC EIGENSTATES [J].
LI, BW ;
ROBNIK, M .
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 1994, 27 (16) :5509-5523
[73]   Multifractality in Fock Space of the Ground State of the Bose-Hubbard Hamiltonian [J].
Lindinger, J. ;
Rodriguez, A. .
ACTA PHYSICA POLONICA A, 2017, 132 (06) :1683-1687
[74]   Many-Body Multifractality throughout Bosonic Superfluid and Mott Insulator Phases [J].
Lindinger, Jakob ;
Buchleitner, Andreas ;
Rodriguez, Alberto .
PHYSICAL REVIEW LETTERS, 2019, 122 (10)
[75]   Multifractal finite-size scaling at the Anderson transition in the unitary symmetry class [J].
Lindinger, Jakob ;
Rodriguez, Alberto .
PHYSICAL REVIEW B, 2017, 96 (13)
[76]   The ergodic side of the many-body localization transition [J].
Luitz, David J. ;
Lev, Yevgeny Bar .
ANNALEN DER PHYSIK, 2017, 529 (07)
[77]   Many-body localization edge in the random-field Heisenberg chain [J].
Luitz, David J. ;
Laflorencie, Nicolas ;
Alet, Fabien .
PHYSICAL REVIEW B, 2015, 91 (08)
[78]   Universal Behavior beyond Multifractality in Quantum Many-Body Systems [J].
Luitz, David J. ;
Alet, Fabien ;
Laflorencie, Nicolas .
PHYSICAL REVIEW LETTERS, 2014, 112 (05)
[79]  
Mace N., ARXIV181210283CONDMA
[80]   Multifractality and intermediate statistics in quantum maps [J].
Martin, J. ;
Giraud, O. ;
Georgeot, B. .
PHYSICAL REVIEW E, 2008, 77 (03)