Unitary Universality Classes Research Articles

Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos

We introduce a complex-plane generalization of the consecutive level-spacing distribution, used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of complex-valued ratios between nearest- and next-to-nearest neighbor spacings. We show that this quantity can successfully detect the chaotic or regular nature of complex-valued spectra. This is done in two steps. First, we show that, if eigenvalues are uncorrelated, the distribution of complex spacing ratios is flat within the unit circle, whereas random matrices show a strong angular dependence in addition to the usual level repulsion. The universal fluctuations of Gaussian Unitary and Ginibre Unitary universality classes in the large-matrix-size limit are shown to be well described by Wigner-like surmises for small-size matrices with eigenvalues on the circle and on the two-torus, respectively. To study the latter case, we introduce the Toric Unitary Ensemble, characterized by a flat joint eigenvalue distribution on the two-torus. Second, we study different physical situations where nonhermitian matrices arise: dissipative quantum systems described by a Lindbladian, non-unitary quantum dynamics described by nonhermitian Hamiltonians, and classical stochastic processes. We show that known integrable models have a flat distribution of complex spacing ratios whereas generic cases, expected to be chaotic, conform to Random Matrix Theory predictions. Specifically, we were able to clearly distinguish chaotic from integrable dynamics in boundary-driven dissipative spin-chain Liouvillians and in the classical asymmetric simple exclusion process and to differentiate localized from delocalized phases in a nonhermitian disordered many-body system.

Fidelity susceptibility in Gaussian random ensembles

The fidelity susceptibility measures the sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here, we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. We find analytically the fidelity susceptibility distributions for Gaussian orthogonal and unitary universality classes for arbitrary system sizes. The results are verified by a comparison with numerical data.

Conductance Fluctuations in Disordered 2D Topological Insulator Wires: From Quantum Spin-Hall to Ordinary Phases

Impurities and defects are ubiquitous in topological insulators (TIs) and thus understanding the effects of disorder on electronic transport is important. We calculate the distribution of the random conductance fluctuations $P(G)$ of disordered 2D TI wires modeled by the Bernevig-Hughes-Zhang (BHZ) Hamiltonian with realistic parameters. As we show, the disorder drives the TIs into different regimes: metal (M), quantum spin-Hall insulator (QSHI), and ordinary insulator (OI). By varying the disorder strength and Fermi energy, we calculate analytically and numerically $P(G)$ across the entire phase diagram. The conductance fluctuations follow the statistics of the unitary universality class $\beta=2$. At strong disorder and high energy, however, the size of the fluctutations $\delta G$ reaches the universal value of the orthogonal symmetry class ($\beta=1$). At the QSHI-M and QSHI-OI crossovers, the interplay between edge and bulk states plays a key role in the statistical properties of the conductance.

Estimate of the Critical Exponent of the Anderson Transition in the Three and Four-Dimensional Unitary Universality Classes

Disordered non-interacting systems are classified into ten symmetry classes, with the unitary class being the most fundamental. The three and four dimensional unitary universality classes are attracting renewed interest because of their relation to three dimensional Weyl semi-metals and four dimensional topological insulators. Determining the critical exponent of the correlation/localistion length for the Anderson transition in these classes is important both theoretically and experimentally. Using the transfer matrix technique, we report numerical estimations of the critical exponent in a U(1) model in three and four dimensions.

Edge States in a Thin Film of Disordered Topological Insulator

A thin-film geometry of nonmagnetic or magnetic topological insulator (TI) alters its topological properties depending on the thickness of the film and the strength of the magnetic coupling, and can exhibit the quantum spin Hall (QSH) effect as well as the quantum anomalous Hall (QAH) effect. In this study, we pursue a quantitative investigation of effects of nonmagnetic disorder on thin-film TIs in all the possible topological phases. The results show that strong enough disorder induces a topological phase transition and generically creates metallic helical edge states featuring the QSH phase before all electronic states are eventually localized. In the thermodynamic limit, the localization–delocalization transition in the disorder-induced QSH phase is in the unitary universality class with nontrivial Chern numbers. In a finite-size system, topologically protected edge states persist unless the mobility gap collapses. It is also demonstrated that edge states extend over a macroscopic distance in the midd...

Statistical wave scattering through classically chaotic cavities in the presence of surface absorption

We propose a model to describe the statistical properties of wave scattering through a classically chaotic cavity in the presence of surface absorption. Experimentally, surface absorption could be realized by attaching an "absorbing patch" to the inner wall of the cavity. In our model, the cavity is connected to the outside by a waveguide with N open modes (or channels), while an experimental patch is simulated by an "absorbing mirror" attached to the inside wall of the cavity; the mirror, consisting of a waveguide that supports N(a) channels, with absorption inside and a perfectly reflecting wall at its end, is described by a subunitary scattering matrix S(a). The number of channels N(a) , as a measure of the geometric cross section of the mirror, and the lack of unitarity P(a) = [symbol: see text]N(a) - S(a)(+)S(a) , as a measure of absorption, are under our control: these parameters have an important physical significance for real experiments. The absorption strength in the cavity is quantified by gamma(a) = tr P(a). The statistical distribution of the resulting S matrix for N = 1 open channel and only one absorbing channel, N(a) = 1, is solved analytically for the orthogonal and unitary universality classes, beta = 1 and beta = 2, respectively, and the results are compared with those arising from numerical simulations. The relation with other models existing in the literature, in some of which absorption has a volumetric character, is also studied.

Density of states for theπ-flux state with bipartite real random hopping only: A weak disorder approach

Gade [R. Gade, Nucl. Phys. B \textbf{398}, 499 (1993)] has shown that the local density of states for a particle hopping on a two-dimensional bipartite lattice in the presence of weak disorder and in the absence of time-reversal symmetry(chiral unitary universality class) is anomalous in the vicinity of the band center $\epsilon=0$ whenever the disorder preserves the sublattice symmetry. More precisely, using a nonlinear-sigma-model that encodes the sublattice (chiral) symmetry and the absence of time-reversal symmetry she argues that the disorder average local density of states diverges as $|\epsilon|^{-1}\exp(-c|\ln\epsilon|^\kappa)$ with $c$ some non-universal positive constant and $\kappa=1/2$ a universal exponent. Her analysis has been extended to the case when time-reversal symmetry is present (chiral orthogonal universality class) for which the same exponent $\kappa=1/2$ was predicted. Motrunich \textit{et al.} [O. Motrunich, K. Damle, and D. A. Huse, Phys. Rev. B \textbf{65}, 064206 (2001)] have argued that the exponent $\kappa=1/2$ does not apply to the typical density of states in the chiral orthogonal universality class. They predict that $\kappa=2/3$ instead. We confirm the analysis of Motrunich \textit{et al.} within a field theory for two flavors of Dirac fermions subjected to two types of weak uncorrelated random potentials: a purely imaginary vector potential and a complex valued mass potential. This model is believed to belong to the chiral orthogonal universality class. Our calculation relies in an essential way on the existence of infinitely many local composite operators with negative anomalous scaling dimensions.

Crossover between universality classes in the statistics of rare events in disordered conductors.

The crossover from orthogonal to the unitary universality classes in the distribution of the anomalously localized states (ALS) in two-dimensional disordered conductors is traced as a function of magnetic field. We demonstrate that the microscopic origin of the crossover is the change in the symmetry of the underlying disorder configurations that are responsible for ALS. These disorder configurations are of weak magnitude (compared to the Fermi energy) and of small size (compared to the mean free path). We find their shape explicitly by means of the direct optimal fluctuation method.

Review of recent progress on numerical studies of the Anderson transition

AbstractA review of recent progress in numerical studies of the Anderson transition in three dimensional systems is presented. From high precision calculations the critical exponent ν for the divergence of the localization length is estimated to be ν = 1.57 ± 0.02 for the orthogonal universality class, which is clearly distinguished from ν = 1.43 ± 0.03 for the unitary universality class. The boundary condition dependences of some quantities at the Anderson transition are also discussed.

Review of recent progress on numerical studies of the Anderson transition

A review of recent progress in numerical studies of the Anderson transition in three dimensional systems is presented. From high precision calculations the critical exponent $\nu$ for the divergence of the localization length is estimated to be $\nu=1.57\pm 0.02$ for the orthogonal universality class, which is clearly distinguished from $\nu=1.43\pm 0.03$ for the unitary universality class. The boundary condition dependences of some quantities at the Anderson transition are also discussed.

Localization and Metal-Insulator Transition in Multilayer Quantum Hall Structures

We study the phase structure and Hall conductance quantization in weakly coupled multilayer electron systems in the integer quantum Hall regime. We derive an effective field theory and perform a two-loop renormalization group calculation. It is shown that (i) finite interlayer tunnelings (however small) give rise to successive metallic and insulating phases and metal-insulator transitions in the unitary universality class; (ii) the Hall conductivity is not renormalized in the metallic phases in the 3D regime; and (iii) the Hall conductances are quantized in the insulating phases. In the bulk quantum Hall phases, the effective field theory describes the transport on the surface.

Mesoscopic conductance and its fluctuations at a nonzero Hall angle

We consider the bilocal conductivity tensor, the two-probe conductance and its fluctuations for a disordered phase-coherent two-dimensional system of non-interacting electrons in the presence of a magnetic field, including correctly the edge effects. Analytical results are obtained by perturbation theory in the limit $\sigma_{xx} \gg 1$. For mesoscopic systems the conduction process is dominated by diffusion but we show that, due to the lack of time-reversal symmetry, the boundary condition for diffusion is altered at the reflecting edges. Instead of the usual condition, that the derivative along the direction normal to the wall of the diffusing variable vanishes, the derivative at the Hall angle to the normal vanishes. We demonstrate the origin of this boundary condition from different starting points, using (i) a simplified Chalker-Coddington network model, (ii) the standard diagrammatic perturbation expansion, and (iii) the nonlinear sigma-model with the topological term, thus establishing connections between the different approaches. Further boundary effects are found in quantum interference phenomena. We evaluate the mean bilocal conductivity tensor $\sigma_{\mu\nu}(r,r')$, and the mean and variance of the conductance, to leading order in $1/\sigma_{xx}$ and to order $(\sigma_{xy}/\sigma_{xx})^2$, and find that the variance of the conductance increases with the Hall ratio. Thus the conductance fluctuations are no longer simply described by the unitary universality class of the $\sigma_{xy}=0$ case, but instead there is a one-parameter family of probability distributions. In the quasi-one-dimensional limit, the usual universal result for the conductance fluctuations of the unitary ensemble is recovered, in contrast to results of previous authors. Also, a long discussion of current conservation.

The Anderson Transition: Time Reversal Symmetry and Universality

We report a finite size scaling study of the Anderson transition. Different scaling functions and different values for the critical exponent have been found, consistent with the existence of the orthogonal and unitary universality classes which occur in the field theory description of the transition. The critical conductance distribution at the Anderson transition has also been investigated and different distributions for the orthogonal and unitary classes obtained.

Crossover behavior for a noninteracting disordered electronic system in the presence of a weak magnetic field

Near the noninteracting localization phase transition, we investigate the cross-over between orthogonal and unitary universality classes as a weak magnetic field is applied. We first derive a field-theoretic description of the noninteracting disordered electronic system in the presence of a weak magnetic field. This description contains both symmetries corresponding to the system with and without a magnetic field. We obtain a unified generating functional, from which a variety of relevant physical quantities can be calculated. The theory is then renormalized. The renormalization group flow equations contain information on the fixed points of the theory and on the crossover between the two universality classes. Within ane (=d − 2) expansion the crossover exponent is obtained and previous phenomenological results from the literature are recovered. We also discuss the shift in the location of the mobility edge caused by the weak field.