Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos

Sá, Lucas; Ribeiro, Pedro; Prosen, Tomaž

doi:10.1103/physrevx.10.021019

Cited by 158 publications

(157 citation statements)

References 111 publications

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“…We see that all curves r(W ) with growing W cross over from the "Wigner surmise value" 0.72 to the Poisson value 2/3 calculated for random points in a plane in Ref. 13. Remarkably, all curves r(W ) cross each other near W c = 6.…”

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confidence: 55%

“…Random lasers 6-9 with random dissipation and amplification regions are such prototypical non-Hermitian systems. The other parts of non-Hermitian disorder physics are related to Hatano-Nelson matrices 10 , their biological applications 11,12 or to spin chains [13][14][15] . All these works focus on onedimensional systems.A simple and elegant extension of the 2D Anderson localization problem was proposed in a recent paper by Tzortzakakis, Makris and Economou (TME) 16 .…”

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confidence: 99%

“…20 Our parameter r(W ) is the modulus of the more informative complex parameter introduced in Ref. 13. Similar to Ref.…”

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Anderson transition in three-dimensional systems with non-Hermitian disorder

Huang

Shklovskiǐ

2020

Phys. Rev. B

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We study the Anderson transition for three-dimensional (3D) N × N × N tightly bound cubic lattices where both real and imaginary parts of onsite energies are independent random variables distributed uniformly between −W/2 and W/2. Such a non-Hermitian analog of the Anderson model is used to describe random-laser medium with local loss and amplification. We employ eigenvalue statistics to search for the Anderson transition. For 25% smallest-modulus complex eigenvalues we find the average ratio r of distances to the first and the second nearest neighbor as a function of W . For a given N the function r(W ) crosses from 0.72 to 2/3 with a growing W demonstrating a transition from delocalized to localized states. When plotted at different N all r(W ) cross at Wc = 6.0±0.1 (in units of nearest neighbor overlap integral) clearly demonstrating the 3D Anderson transition. We find that in the non-Hermitian 2D Anderson model, the transition is replaced by a crossover.Anderson localization is the central concept of solid state physics for more than 60 years 1-3 . It determines electron conductivity of doped crystalline and amorphous semiconductors and many other disordered systems and is observed in experiments 4,5 .In recent years the problem of localization attracted renewed interest as research moved to formerly unexplored area of non-Hermitian systems. Random lasers 6-9 with random dissipation and amplification regions are such prototypical non-Hermitian systems. The other parts of non-Hermitian disorder physics are related to Hatano-Nelson matrices 10 , their biological applications 11,12 or to spin chains [13][14][15] . All these works focus on onedimensional systems.A simple and elegant extension of the 2D Anderson localization problem was proposed in a recent paper by Tzortzakakis, Makris and Economou (TME) 16 . They studied 50 × 50 tight-binding square lattices with real overlap energy I ij = I, and random complex onsite energies E i whose real and imaginary parts are independent random variables distributed uniformly between −W/2 and W/2. The Hamiltonian readswhere i, j in the second term are nearest neighbors, and the hard-wall boundary is employed (no bonds extended out the boundary). Below we call this non-Hermitian Hamiltonian the TME model. By calculating the participation ratio of eigenfunctions of such non-Hermitian Hamiltonian, TME noticed that they become progressively more localized when W (in units of I) grows from 1 to 5. Simultaneously the distribution function P (s) of nearest neighbor distances s between eigenvalues in the complex plane widens, which shows that the repulsion of eigenvalues weakens due to the progressive localization of eigenfunctions. This behavior is similar to what happens in the Anderson model 17 . TME, however, did not raise a question whether there is an Anderson transition or a crossover in the limit of large system.In this paper, we focus on the question of the existence of the Anderson transition in the TME model for 3D cubic and 2D square lattices. We show that in the TME m...

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Anderson transition in three-dimensional systems with non-Hermitian disorder

Huang

Shklovskiǐ

2020

Phys. Rev. B

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“…Complex spectra have no ordering, but it is straightforward to generalize level spacing ratios [30] (which get rid of the otherwise necessary unfolding of the local density of states) to the complex case by finding the nearest λ 1 and next nearest neighbor λ 2 of an eigenvalue λ 0 and defining r = |λ 0 − λ 1 |/|λ 0 − λ 2 |. This definition was recently generalized to include phase angles [31].…”

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confidence: 99%

Hierarchy of Relaxation Timescales in Local Random Liouvillians

2020

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To characterize the generic behavior of open quantum systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-1/2 system of size with up to n-body Lindblad operators, which are n-local in the complexity-theory sense. Without locality (n = ), the complex Liouvillian spectrum densely covers a "lemon"-shaped support, in agreement with recent findings [Phys. Rev. Lett. 123, 140403; arXiv:1905.02155]. However, for local Liouvillians (n < ), we find that the spectrum is composed of several dense clusters with random matrix spacing statistics, each featuring a lemon-shaped support wherein all eigenvectors correspond to n-body decay modes. This implies a hierarchy of relaxation timescales of n-body observables, which we verify to be robust in the thermodynamic limit.Introduction. For unitary quantum many-body dynamics, the characterization of generic features common to the vast majority of systems is well developed, in the form of an effective random matrix theory [1-6]. It is for example a crucial ingredient to our understanding of thermalization in unitary quantum systems, manifest in the eigenstate thermalization hypothesis (ETH) [7][8][9][10][11][12][13].For open quantum many-body systems analogous organizing principles are yet missing. Only very recently the first developments in this direction appeared with the investigation of spectral features of a purely random Liouvillian [14-18], describing generic properties of trace preserving positive quantum maps. The purely random Liouvillians considered in [14,15,17] constitute the least structured models aiming at describing generic features of open quantum many-body systems. A main result of these recent studies is that the spectrum of purely random Liouvillians is densely covering a lemon-shaped support whose form is universal, differentiating random Liouvillians from completely random Ginibre matrices with circular spectrum [1,[19][20][21].How close is a completely random Liouvillian to a physical system? In this Letter, we make a step towards answering this question by adding the minimal amount of structure to a purely random Liouvillian, namely a notion of locality. We consider a dissipative spin system of size with Lindblad operators given by Pauli strings classified by their length n ≤ , the latter being a measure of their n-locality in a complexity-theory sense [22].In this language, purely random Liouvillians are completely nonlocal and structureless, since they include the full operator basis including Pauli strings of all lengths 1 ≤ n ≤ .Here, we restrict the maximal number of non-identity operators in the Lindblad operators to n max and examine

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“…Although the speedup at the second level is not ideal, parallelization solves the main problem, allowing us to fit into the limitations of memory size. The proposed parallel algorithm opens up new perspectives to numerical studies of large open quantum models and allows us to advance further into the territory of “Dissipative Quantum Chaos” [ 17 , 27 , 28 , 29 ].…”

Section: Discussionmentioning

confidence: 99%

Transforming Lindblad Equations into Systems of Real-Valued Linear Equations: Performance Optimization and Parallelization of an Algorithm

Meyerov

Kozinov

Liniov

et al. 2020

Entropy

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With their constantly increasing peak performance and memory capacity, modern supercomputers offer new perspectives on numerical studies of open many-body quantum systems. These systems are often modeled by using Markovian quantum master equations describing the evolution of the system density operators. In this paper, we address master equations of the Lindblad form, which are a popular theoretical tools in quantum optics, cavity quantum electrodynamics, and optomechanics. By using the generalized Gell–Mann matrices as a basis, any Lindblad equation can be transformed into a system of ordinary differential equations with real coefficients. Recently, we presented an implementation of the transformation with the computational complexity, scaling as O(N5logN) for dense Lindbaldians and O(N3logN) for sparse ones. However, infeasible memory costs remains a serious obstacle on the way to large models. Here, we present a parallel cluster-based implementation of the algorithm and demonstrate that it allows us to integrate a sparse Lindbladian model of the dimension N=2000 and a dense random Lindbladian model of the dimension N=200 by using 25 nodes with 64 GB RAM per node.

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Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos

Cited by 158 publications

References 111 publications

Anderson transition in three-dimensional systems with non-Hermitian disorder

Anderson transition in three-dimensional systems with non-Hermitian disorder

Hierarchy of Relaxation Timescales in Local Random Liouvillians

Transforming Lindblad Equations into Systems of Real-Valued Linear Equations: Performance Optimization and Parallelization of an Algorithm

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