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arXiv:cond-mat/0509687 · Physical Review E

Incipience of quantum chaos in the Jahn-Teller model

Eva Majerníková, Serge Shpyrko

Published Jun 19, 2006• 13 citations
arXivHTMLPDF

Abstract

We studied complex spectra of a two-level electron system coupled to two phonon (vibron) modes represented by the product of E and e Jahn-Teller model. For particular rotation quantum numbers we found a coexistence of up to three regions of the spectra: (i) the dimerized region of long-range-ordered (extended) pairs of oscillating levels, (ii) the short-range-ordered (localized) "kink lattice" of avoiding levels, and (iii) the intermediate region of kink nucleation with variable range of ordering. This structure appears above a certain critical line as a function of interaction strength. The level clustering and level avoiding generic patterns reflect themselves in several intermittent regions between up to three branches of spectral entropies. Linear scaling behavior of the widths of level curvature probability distributions provides the conventionally adopted indication for the presence of quantum chaos. Level spacing probability distributions show peculiarities of the partial (for fixed quantum angular momentum) as well as of the cumulative (all angular momenta) case. The clustering of levels with two and three dominant spacings at fixed angular momenta causes notable deviations of the cumulative distributions from the Poissonian one.

DOI
10.1103/physreve.73.066215

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We studied complex spectra of a two-level electron system coupled to two phonon (vibron) modes represented by the product of E and e Jahn-Teller model. For particular rotation quantum numbers we found a coexistence of up to three regions of the spectra: (i) the dimerized region of long-range-ordered (extended) pairs of oscillating levels, (ii) the short-range-ordered (localized) "kink lattice" of avoiding levels, and (iii) the intermediate region of kink nucleation with variable range of ordering. This structure appears above a certain critical line as a function of interaction strength. The level clustering and level avoiding generic patterns reflect themselves in several intermittent regions between up to three branches of spectral entropies. Linear scaling behavior of the widths of level curvature probability distributions provides the conventionally adopted indication for the presence of quantum chaos. Level spacing probability distributions show peculiarities of the partial (for fixed quantum angular momentum) as well as of the cumulative (all angular momenta) case. The clustering of levels with two and three dominant spacings at fixed angular momenta causes notable deviations of the cumulative distributions from the Poissonian one.
The generalized multispin Jahn-Teller model on a finite lattice or formally equivalent Dicke model extended to two long-wavelength coherent bosons of different frequencies is shown to exhibit a crossover between the polaron-modified "quasi-normal" and the squeezed "radiation" domains. We investigate effects of two kinds of interfering fluctuations on the phase crossover and on statistical characteristics of boson complex spectra: (i) Fluctuations in the electron subsystem- finite-size quantum fluctuations- are responsible for the dephasing of the coherence in the radiation domain and for the moderate occupation of the excited states in the normal domain. In the quasiclassical limit, radiation phase implies existence of a coherent acoustic super-radiant phase. (ii) Level-spacing fluctuations in excited boson level subsystem with strong level repulsions. Related probability distributions are shown to be non-universally spread between the limiting universal Wigner-Dyson and Poisson distributions. We proved that the difference in boson frequencies is responsible for reaching the most stochastic limit of the Wigner-Dyson distribution. Instanton lattice as a sequence of tunneling events in the most chaotic radiation domain exhibits maximal number of level-avoidings (repulsions). The non-universality of the distributions is caused by boson correlations which compete the level repulsions.
The full spectrum of a single hole in a quantum antiferromagnetic background (t-Jz{\mathit{J}}_{\mathit{z}}Jz​-${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ model) is obtained by complete exact diagonalization of small two-dimensional clusters. Various statistical properties of the spectrum are investigated. On a very wide range of the parameters the level-spacing distribution follows Gaussian-orthogonal-ensemble Wigner law characteristic of chaotic spectra. At small separation, the spectral rigidity follows the universal behavior described by random-matrix theory and presents deviations at higher energies. We argue that quantum chaos is a generic feature of complex (i.e., nonintegrable) strongly correlated fermion systems. Our results suggest that random-matrix theory might be useful to investigate the incoherent part of dynamical correlation functions (such as the hole spectral density or the spin structure factor).
The Dicke model extended to two bosons of different frequencies or equivalent generalized Jahn-Teller lattice model are shown to exhibit a spontaneous quantum phase transition between the polaron-modified "quasi-normal" and squeezed "radiation" phase with the transition point dependent on the frequencies. In a finite lattice a mixed domain of coexistence of the quasi-normal and modified radiation phase is created within the quasi-normal phase domain. There occurs a field-directed oscillation-assisted tunneling (hopping). The field is driven by simultaneous squeezing and polaron-dressing of the collective boson level mode due to the additional boson mode. In a finite lattice in the radiation domain there occurs a sequence of local tunnelings (oscillations) between two minima of a local potential weakly coupled to two assisting oscillations. The "radiation" phase reveals itself as an almost ideal instanton--anti-instanton gas phase. The correlations among the energy levels mediated by the additional mode in the mixed domain considerably reduce the level repulsions. As a consequence, the Wigner level spacing probability distribution of the two-boson Dicke model is non-universally reduced from the Wigner to the semi-Poisson and asymptotically to the Poisson distribution of level spacings. The correlations cause a suppression of the coherence of the radiation phase as finite-size effect. Possible applications of the present theory are suggested.
We consider a coupled top model describing two interacting large spins, which is studied semiclassically as well as quantum mechanically. This model exhibits a variety of interesting phenomena such as a quantum phase transition (QPT), a dynamical transition, and excited-state quantum phase transitions above a critical coupling strength. Both classical dynamics and entanglement entropy reveal ergodic behavior at the center of the energy density band for an intermediate range of coupling strength above QPT, where the level spacing distribution changes from Poissonian to Wigner-Dyson statistics. Interestingly, in this model we identify quantum scars as reminiscent of unstable collective dynamics even in the presence of an interaction. The statistical properties of such scarred states deviate from the ergodic limit corresponding to the random matrix theory and violate Berry's conjecture. In contrast to ergodic evolution, the oscillatory behavior in the dynamics of the unequal time commutator and survival probability is observed as the dynamical signature of a quantum scar, which can be relevant for its detection.
We present an exact diagonalization study of the spectral properties of bosons harmonically confined in a quasi-2D plane and interacting via repulsive Gaussian potential. We consider the lowest 100 energy levels for systems of N=12,16, and 20 bosons in two distinct regimes: (a) when the interaction energy is small compared to the trap energy (moderate interaction) and (b) when the interaction energy is comparable to the trap energy (strong interaction), for the nonrotating (L_{z}=0) as well as the rotating single-vortex state (L_{z}=N). For higher angular momenta, L_{z}=2N and L_{z}=3N, only the strong interaction regime is considered. While the nearest-neighbor spacing distribution (NNSD) P(s) and the ratios of consecutive level spacings distribution P(r) are used to study the short-range correlations, the Dyson-Mehta Δ_{3} statistic and the level number variance Σ^{2}(L) are used to examine the long-range correlations. In the moderate interaction regime, the nonrotating system exhibits Poisson distribution, a characteristic of the regular energy spectra. In the strong interaction regime, the nonrotating system exhibits chaotic behavior signified by GOE distribution. Furthermore, in the rotating case for the single-vortex state (L_{z}=N) in the moderate interaction regime, the system exhibits signatures of weak chaos with some degree of regularity in the energy-level spectra. However, in the strong interaction regime for the rotating case with L_{z}=N, 2N and 3N, the system exhibits strong chaotic behavior. The rotation is found to contribute to enhancement of chaotic behavior in the system for both the moderate and the strong interaction regimes. Our results of NNSD analysis are supported by the analysis of the ratios of consecutive level spacings distribution P(r), which does not involve unfolding.
The problem of random interactions leading to regular spectra in shell model type simulations is described. The key results are reviewed alnog with a selection of the explanations. A model system of N particles on 2 spin levels having random 2-body collisions that conserve angular momentum is examined. Preliminary results are described, including the ground state spin distributions peaked at extreme values of angular momentum, signatures of rotational bands, and smooth parabolic yrast lines. A simple random matrix theory analysis shows signatures of quantum chaos in the level spacing distribution and the Δ3 statistic.