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arXiv:cond-mat/0004237 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Scaling near the quantum chaos border in interacting Fermi systems

Pil Hun Song

Published Dec 1, 2000• 9 citations
arXivHTMLPDF

Abstract

The emergence of quantum chaos for interacting Fermi systems is investigated by numerical calculation of the level spacing distribution $P(s)$ as a function of interaction strength U and the excitation energy $\ensuremath{\epsilon}$ above the Fermi level. As U increases, $P(s)$ undergoes a transition from Poissonian (nonchaotic) to Wigner-Dyson (chaotic) statistics and the transition is described by a single scaling parameter given by $Z=(U{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}\ensuremath{-}{u}_{0}){\ensuremath{\epsilon}}^{1/(2\ensuremath{\nu})},$ where ${u}_{0}$ is a constant. While the exponent $\ensuremath{\alpha},$ which determines the global change of the chaos border, is indecisive within a broad range of $0.9\ensuremath{\sim}2.0,$ the finiteness of $\ensuremath{\nu},$ which comes from the increase of the Fock space size with $\ensuremath{\epsilon},$ suggests that the transition becomes sharp as $\ensuremath{\epsilon}$ increases.

DOI
10.1103/physreve.62.r7575

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The emergence of quantum chaos for interacting Fermi systems is investigated by numerical calculation of the level spacing distribution P(s)P(s)P(s) as a function of interaction strength U and the excitation energy \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ above the Fermi level. As U increases, P(s)P(s)P(s) undergoes a transition from Poissonian (nonchaotic) to Wigner-Dyson (chaotic) statistics and the transition is described by a single scaling parameter given by Z=(U\ensuremathϵ\ensuremathα\ensuremath−u0)\ensuremathϵ1/(2\ensuremathν),Z=(U{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}\ensuremath{-}{u}_{0}){\ensuremath{\epsilon}}^{1/(2\ensuremath{\nu})},Z=(U\ensuremathϵ\ensuremathα\ensuremath−u0​)\ensuremathϵ1/(2\ensuremathν), where u0{u}_{0}u0​ is a constant. While the exponent \ensuremathα,\ensuremath{\alpha},\ensuremathα, which determines the global change of the chaos border, is indecisive within a broad range of 0.9\ensuremath∼2.0,0.9\ensuremath{\sim}2.0,0.9\ensuremath∼2.0, the finiteness of \ensuremathν,\ensuremath{\nu},\ensuremathν, which comes from the increase of the Fock space size with \ensuremathϵ,\ensuremath{\epsilon},\ensuremathϵ, suggests that the transition becomes sharp as \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ increases.
We study the level spacing statistics P(s)P(s)P(s) in many-body Fermi systems and determine a critical two-body interaction strength Uc{U}_{c}Uc​ at which a crossover from Poisson to Wigner-Dyson statistics takes place. Near the Fermi level, the results allow one to find a critical temperature Tch{T}_{\mathrm{ch}}Tch​ above which quantum chaos and thermalization set in.
The development of quantum chaos in finite interacting Fermi systems is considered. At sufficiently high excitation energy the direct two-particle interaction may mix into an eigenstate the exponentially large number of simple Slater determinant states. Nevertheless, the transition from Poisson to Wigner-Dyson statistics of energy levels is governed by the effective high-order interaction between states very distant in the Fock space. The concrete form of the transition depends on the way one chooses to work out the problem of factorial divergence of the number of Feynman diagrams. In the proposed scheme the change of statistics has a form of narrow phase transition and may happen even below the direct interaction threshold.
For many-body Fermi systems we determine the dependence of the Breit-Wigner width $\ensuremath{\gamma}$ and inverse participation ratio $\ensuremath{\xi}$ on interaction strength $U\ensuremath{\ge}{U}_{c}$ and energy excitation $\ensuremath{\delta}E\ensuremath{\ge}\ensuremath{\delta}{E}_{\mathrm{ch}}$ when a crossover from Poisson to Wigner-Dyson P(s)P(s)P(s) statistics takes place. At $U\ensuremath{\ge}{U}_{c}$ the eigenstates are composed of a large number of noninteracting states and even for U<UcU<{U}_{c}U<Uc​ there is a regime where P(s)P(s)P(s) is close to the Poisson distribution but $\ensuremath{\xi}\ensuremath{\gg}1$.
We study a generic one-dimensonal quantum model of two flavors (pseudospins) chiral complex fermions by exact diagonalization, which can have local interflavor interaction and superconducting pairings (with all irrelevant terms ignored). Analytically, the model has two solvable (integrable) points in the parameter space: it is a free fermion model when the fermion interaction is zero, and is a free boson Luttinger liquid when there is a global U(1)(↑)×^{(\uparrow)}\times(↑)×U(1)(↓)^{(\downarrow)}(↓) symmetry (with nonzero interaction). When the global symmetry of the interacting model is lowered by turning on symmetry breaking parameters, the model undergoes a transition from a quantum integrable model to a fully quantum chaotic model, as we demonstrate by examining the level spacing statistics (LSS) of the many-body energy spectrum. In particular, there is a possibly integrable regime with intermediate global symmetries, where the model is neither free bosons nor free fermions, but shows Poisson LSS in each global symmetry charge sector. This implies the existence of hidden (quasi)local conserved quantities. When the global symmetries are further lowered, the LSS in each charge sector becomes Wigner-Dyson, implying quantum chaos.
A theory of strongly interacting Fermi systems of a few particles is developed.At high excitation energies (a few times the single-particle level spacing) these systems are characterized by an extreme degree of complexity due to strong mixing of the shell-model-based many-particle basis states by the residual twobody interaction.This regime can be described as many-body quantum chaos.Practically, it occurs when the excitation energy of the system is greater than a few single-particle level spacings near the Fermi energy.Physical examples of such systems are compound nuclei, heavy open shell atoms (e.g.rare earths) and multicharged ions, molecules, clusters and quantum dots in solids.The main quantity of the theory is the strength function which describes spreading of the eigenstates over many-particle basis states (determinants) constructed using the shell-model orbital basis.A nonlinear equation for the strength function is derived, which enables one to describe the eigenstates without diagonalization of the Hamiltonian matrix.We show how to use this approach to calculate mean orbital occupation numbers and matrix elements between chaotic eigenstates and introduce typically statistical variables such as temperature in an isolated microscopic Fermi system of a few particles.
The standard generic quantum computer model is studied analytically and numerically and the border for emergence of quantum chaos, induced by imperfections and residual inter-qubit couplings, is determined. This phenomenon appears in an isolated quantum computer without any external decoherence. The onset of quantum chaos leads to quantum computer hardware melting, strong quantum entropy growth and destruction of computer operability. The time scales for development of quantum chaos and ergodicity are determined. In spite the fact that this phenomenon is rather dangerous for quantum computing it is shown that the quantum chaos border for inter-qubit coupling is exponentially larger than the energy level spacing between quantum computer eigenstates and drops only linearly with the number of qubits n. As a result the ideal multi-qubit structure of the computer remains rather robust against imperfections. This opens a broad parameter region for a possible realization of quantum computer. The obtained results are related to the recent studies of quantum chaos in such many-body systems as nuclei, complex atoms and molecules, finite Fermi systems and quantum spin glass shards which are also reviewed in the paper.