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arXiv:2309.16005 · Physical Review Research

Scale-invariant critical dynamics at eigenstate transitions

Miroslav Hopjan, Lev Vidmar

Published Dec 29, 2023• 23 citations
arXivHTMLPDF

Abstract

The notion of scale-invariant dynamics is well established at late times in quantum chaotic systems, as illustrated by the emergence of a ramp in the spectral form factor (SFF). Building on the results of the preceding Letter [Phys. Rev. Lett. 131, 060404 (2023)], we explore features of scale-invariant dynamics of survival probability and SFF at criticality, i.e., at eigenstate transitions from quantum chaos to localization. We show that, in contrast to the quantum chaotic regime, the quantum dynamics at criticality do not only exhibit scale invariance at late times, but also at much shorter times that we refer to as mid-time dynamics. Our results apply to both quadratic and interacting models. Specifically, we study Anderson models in dimensions three to five and power-law random banded matrices for the former, and the quantum sun model and the ultrametric model for the latter, as well as the Rosenzweig-Porter model.

DOI
10.1103/physrevresearch.5.043301

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The notion of scale-invariant dynamics is well established at late times in quantum chaotic systems, as illustrated by the emergence of a ramp in the spectral form factor (SFF). Building on the results of the preceding Letter [Phys. Rev. Lett. 131, 060404 (2023)], we explore features of scale-invariant dynamics of survival probability and SFF at criticality, i.e., at eigenstate transitions from quantum chaos to localization. We show that, in contrast to the quantum chaotic regime, the quantum dynamics at criticality do not only exhibit scale invariance at late times, but also at much shorter times that we refer to as mid-time dynamics. Our results apply to both quadratic and interacting models. Specifically, we study Anderson models in dimensions three to five and power-law random banded matrices for the former, and the quantum sun model and the ultrametric model for the latter, as well as the Rosenzweig-Porter model.
It is an outstanding goal to unveil the key features of quantum dynamics at eigenstate transitions. Focusing on quadratic fermionic Hamiltonians that exhibit localization transitions, we identify physical observables that exhibit scale-invariant critical dynamics at the transition when quenched from the initially localized charge-density-wave states. The identification is based on two ingredients: (a) a relationship between the time evolution of observables in a many-body state and the transition probabilities of single-particle states, and (b) scale invariance of transition probabilities, which generalizes the corresponding recent result for survival probabilities [M. Hopjan and L. Vidmar, Phys. Rev. Lett. 131, 060404 (2023); Phys. Rev. Res. 5, 043301 (2023)]. These properties suggest that the scale-invariant critical dynamics in the quantum-quench dynamics is also exhibited by the observables, which share the common eigenbasis with the Hamiltonian before the quench. Focusing on experimentally relevant observables such as site occupations and the particle imbalance, we numerically demonstrate their critical behavior at the eigenstate transitions in the three-dimensional Anderson model and the one-dimensional Aubry-Andr\'e model.
Random matrix spectral correlations is a defining feature of quantum chaos. Here, we study such correlations in a minimal model of chaotic many-body quantum dynamics where interactions are confined to the system’s boundary, dubbed , in terms of the spectral form factor and its fluctuations. We exactly calculate the latter in the limit of large local Hilbert space dimension q q for different classes of random boundary interactions and find it to coincide with random matrix theory, possibly after a non-zero Thouless time. The latter effect is due to a drastic enhancement of the spectral form factor, when integer time and system size fulfill a resonance condition. We compare our semiclassical (large q q ) results with numerics at small local Hilbert space dimension ( q=2,3 q = 2 , 3 ) and observe qualitatively similar features as in the semiclassical regime.
Understanding quantum phase transitions in highly excited Hamiltonian eigenstates is currently far from being complete. It is particularly important to establish tools for their characterization in time domain. Here, we argue that a scaled survival probability, where time is measured in units of a typical Heisenberg time, exhibits a scale-invariant behavior at eigenstate transitions. We first demonstrate this property in two paradigmatic quadratic models, the one-dimensional Aubry-Andre model and three-dimensional Anderson model. Surprisingly, we then show that similar phenomenology emerges in the interacting avalanche model of ergodicity breaking phase transitions. This establishes an intriguing similarity between localization transition in quadratic systems and ergodicity breaking phase transition in interacting systems.
The emergence of random matrix spectral correlations in interacting quantum systems is a defining feature of quantum chaos. We study such correlations in terms of the spectral form factor and its moments in interacting chaotic few- and many-body systems, modeled by suitable random-matrix ensembles. We obtain the spectral form factor exactly for large Hilbert space dimension. Extrapolating those results to finite Hilbert space dimension we find a universal transition from the noninteracting to the strongly interacting case, which can be described as a simple combination of these two limits. This transition is governed by a single scaling parameter. In the bipartite case we derive similar results also for all moments of the spectral form factor. We confirm our results by extensive numerical studies and demonstrate that they apply to more realistic systems given by a pair of quantized kicked rotors as well. Ultimately we complement our analysis by a perturbative approach covering the small-coupling regime.
The emergence of quantum chaos in a system of trapped interacting bosons with externally impressed rotation is studied through spectral form factor (SFF) and power spectrum using exact diagonalization. Two distinct interaction regimes are considered: the moderate, when the interaction energy is small compared to the trap energy and the strong, when the interaction energy is comparable to the trap energy. In the moderate interaction regime, the SFF for the non-rotating case exhibits a dip-plateau structure with absence of linear ramp, indicating integrable behavior, while for the single-vortex state the SFF exhibits a discernible linear ramp consistent with pseudo-integrable behavior. In the strong interaction regime, the non-rotating case exhibits emergence of a linear ramp with small time span in SFF, indicating that the system has moved further towards chaotic regime but continues to be pseudo-integrable. For the single-vortex and the multi-vortex states in strong interaction regime, the span of the linear ramp in SFF increases progressively with rotation, indicating the system has moved into strong chaotic regime consistent with Gaussian orthogonal ensemble. The power spectrum results with exponent lying in the interval 1≲α≲21 \lesssim α\lesssim 21≲α≲2 are consistent with the findings of SFF. An understanding of the observed crossover from integrable to quantum chaos is presented in terms of the macroscopic occupation of a single-particle quantum state---the Bose-Einstein condensation---and its depletion driven by interaction and rotation.
The emergence of random matrix spectral correlations in interacting quantum systems is a defining feature of quantum chaos. We study such correlations in terms of the spectral form factor and its moments in interacting chaotic few- and many-body systems, modeled by suitable random-matrix ensembles. We obtain the spectral form factor exactly for large Hilbert space dimension. Extrapolating those results to finite Hilbert space dimension we find a universal transition from the non-interacting to the strongly interacting case, which can be described as a simple combination of these two limits. This transition is governed by a single scaling parameter. In the bipartite case we derive similar results also for all moments of the spectral form factor. We confirm our results by extensive numerical studies and demonstrate that they apply to more realistic systems given by a pair of quantized kicked rotors as well. Ultimately we complement our analysis by a perturbative approach covering the small coupling regime.