Two-level Correlation Function Research Articles

Analyticity constraints bound the decay of the spectral form factor

Quantum chaos cannot develop faster than λ≤2π/(ℏβ) for systems in thermal equilibrium [Maldacena, Shenker & Stanford, JHEP (2016)]. This `MSS bound' on the Lyapunov exponent λ is set by the width of the strip on which the regularized out-of-time-order correlator is analytic. We show that similar constraints also bound the decay of the spectral form factor (SFF), that measures spectral correlation and is defined from the Fourier transform of the two-level correlation function. Specifically, the inflection exponentη, that we introduce to characterize the early-time decay of the SFF, is bounded as η≤π/(2ℏβ). This bound is universal and exists outside of the chaotic regime. The results are illustrated in systems with regular, chaotic, and tunable dynamics, namely the single-particle harmonic oscillator, the many-particle Calogero-Sutherland model, an ensemble from random matrix theory, and the quantum kicked top. The relation of the derived bound with other known bounds, including quantum speed limits, is discussed.

Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model

We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, $N$ Majorana fermions with infinite range interactions in $0+1$ dimensions. We have found that, close to the ground state $E \approx 0$, discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite $N$, which we compute analytically and numerically, grows exponentially with $N$ for $E \approx 0$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $E \approx 0$, the spectral density grows exponential with the energy. Deep in the quantum regime, corresponding to the first $O(N)$ eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $E \approx 0$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalues separations smaller than the Thouless energy, which seems to scale linearly with $N$. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universality characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor $g(t)$, obtained from the connected two-level correlation function of the unfolded spectrum, decays as $1/t^2$ for times shorter but comparable to the Thouless time with $g(0)$ related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.

Virial expansion of the nonlinear sigma model in the strong coupling limit

We develop a perturbative approach to study the supersymmetric nonlinear sigma model characterized by a generic coupling matrix in the strong coupling limit. The method allows us to calculate explicitly the moments of the eigenfunctions and the two-level correlation function in the lowest order of the perturbative expansion. We find that the obtained expressions are equivalent to the results derived before for the corresponding random matrix ensembles. Such an equivalence is elucidated and generalized to all orders of the perturbative expansion by mapping the sigma model onto the field theory describing the almost diagonal random matrices.

Contour-integral method for transitions to the circular unitary ensemble

The representation of correlation functions as a contour integral has been useful in the study of transitions to the Gaussian unitary ensemble (GUE). We develop the formalism for transitions to the circular unitary ensemble (CUE) and consider the general ℓCUE to CUE transition where ℓCUE denotes a superposition of ℓ independent CUE spectra in an arbitrary ratio. For large matrices, we derive the two-level correlation function for all ℓ including ℓ = ∞ (the Poisson case). The results are useful in the study of weakly broken partitioning symmetries and weakly coupled mesoscopic cavities.

Dimensional dependence of the metal-insulator transition

We study the dependence on the spatial dimensionality of different quantities relevant in the description of the Anderson transition by combining numerical calculations in a $3 \leq d \leq 6$ disordered tight binding model with theoretical arguments. Our results indicate that, in agreement with the one parameter scaling theory, the upper critical dimension for localization is infinity. Typical properties of the spectral correlations at the Anderson transition such as level repulsion or a linear number variance are still present in higher dimensions though eigenvalues correlations get weaker as the dimensionality of the space increases. It is argued that such a critical behavior can be traced back to the exponential decay of the two-level correlation function in a certain range of eigenvalue separations. We also discuss to what extent different effective random matrix models proposed in the literature to describe the Anderson transition provide an accurate picture of this phenomenon. Finally, we study the effect of a random flux in our results.

Random matrix theory within superstatistics

We propose a generalization of the random matrix theory following the basic prescription of the recently suggested concept of superstatistics. Spectral characteristics of systems with mixed regular-chaotic dynamics are expressed as weighted averages of the corresponding quantities in the standard theory assuming that the mean level spacing itself is a stochastic variable. We illustrate the method by calculating the level density, the nearest-neighbor-spacing distributions, and the two-level correlation functions for systems in transition from order to chaos. The calculated spacing distribution fits the resonance statistics of random binary networks obtained in a recent numerical experiment.

Two-level correlation function of critical random-matrix ensembles

The two-level correlation function $R_{d,\beta}(s)$ of $d$-dimensional disordered models ($d=1$, 2, and 3) with long-range random-hopping amplitudes is investigated numerically at criticality. We focus on models with orthogonal ($\beta=1$) or unitary ($\beta=2$) symmetry in the strong ($b^d \ll 1$) coupling regime, where the parameter $b^{-d}$ plays the role of the coupling constant of the model. It is found that $R_{d,\beta}(s)$ is of the form $R_{d,\beta}(s)=1+\delta(s)-F_{\beta}(s^{\beta}/b^{d\beta})$, where $F_{1}(x)=\text{erfc}(a_{d,\beta} x)$ and $F_{2}(x)=\exp (-a_{d,\beta} x^2)$, with $a_{d,\beta}$ being a numerical coefficient depending on the dimensionality and the universality class. Finally, the level number variance and the spectral compressibility are also considerded.

Nonlinearσmodel approach for level correlations in chiral disordered systems

We study level correlations of disordered systems with chiral unitary symmetry (AIII symmetry). We use a random matrix model with a finite correlation length to derive a supersymmetric nonlinear sigma model. The result is compared with existing results based on other models. Using the methods of Sov. Phys. JETP 60, 656 (1994)] and Phys. Rev. Lett. 75, 902 (1995)], we calculate the density of states and two-level correlation function. The result is expressed using the spectral determinant as in traditional nonchiral systems. We discuss the renormalization of the mean level spacing which is not present in the traditional systems.

Long range disorder and Anderson transition in systems with chiral symmetry

We study the spectral properties of a chiral random banded matrix (chRBM) with elements decaying as a power-law H i j ∼ | i − j | − α . This model is equivalent to a chiral 1D Anderson Hamiltonian with long range power-law hopping. In the weak disorder limit we obtain explicit nonperturbative analytical results for the density of states (DoS) and the two-level correlation function (TLCF) by mapping the chRBM onto a nonlinear σ model. We also put forward, by exploiting the relation between the chRBM at α = 1 and a generalized chiral random matrix model, an exact expression for the above correlation functions. We give compelling analytical and numerical evidence that for this value the chRBM reproduces all the features of an Anderson transition. Finally we discuss possible applications of our results to quantum chromodynamics (QCD).

Brownian-Motion Model of Quantum-Chaotic Spectra: Exact Two-level Correlations for Transitions to CUE

Spectral fluctuations in quantum-chaotic systems undergo transition from one universality class to another as a good symmetry of the system is gradually broken. After suitable rescaling and limits, the transitions also have a universal classification which depends only on the initial and final systems and is describable in terms of Dyson's Brownian-motion ensembles of random matrices. In this article we consider Brownian ensembles which undergo transitions to the circular unitary ensemble (CUE). We give a general method of deriving two-level correlation functions exactly. As a first application, we derive results for the transition with initial spectrum being uniformly spaced. For the study of partitioning symmetries in time-reversal noninvariant quantum-chaotic systems, we obtain the two-level correlation functions for transitions starting with initial ensemble as direct sum of two independent CUEs with lower dimensionalities. For the study of time-reversal noninvariant quantum-chaotic systems whose classical analogs display an abrupt transition from integrability to chaos, we derive the two-level correlation function for the Poisson → CUE transition. We also rederive our earlier results for the COE → CUE and CSE* → CUE transitions, relevant for the study of weakly-broken time reversal symmetry, COE and CSE* denoting here respectively the circular orthogonal and symplectic ensembles with doubly-degenerate eigenvalues in the latter. These results, announced earlier without proofs, are also expected to hold for the corresponding Gaussian ensembles with large dimensionalities.

Time Asymmetry Structure of Intermediate Random Matrix Distributions

The method of quaternion matrices by Dyson is adopted here for the Yukawa distribution, and the statistical properties of energy levels for orthogonal, unitary and symplectic classes are studied. To obtain a smooth interpolation between the Wigner–Dyson and the Poisson statistics by a single parameter a (Yukawa's parameter of intermediacy), a parameter \tildeθ( a ) which represents the degree of time-asymmetry is introduced [\tildeθ(0)=1 for the Poisson region, and \tildeθ(∞)=0 for the Wigner–Dyson limit]. On this basis, the two-level correlation functions are expressed as a function of θ( a ). As a consequence, the level compressibility for three universality classes is obtained as a function of a .

Exact density of states for finite Gaussian random matrix ensembles via supersymmetry

We calculate the exact density of states (DOS) for the three classical and two non-classical Random Matrix Ensembles for finite matrix size N using supersymmetric integrals. The 1/N-Expansion yields already in lowest order good approximations to the exact result even for small values of N~5. We conjecture a connection between the N-dependence of the oscillating part of the DOS and the short-distance behavior of the two-level correlation function.

Grand canonical random matrices for intermediate level statistics

The grand canonical treatment of the energy level statistics for the unitary ensemble of Hermitians of the form H= H 0+ tV (Yukawa's level-dynamics) reported by Forrester is extended to the orthogonal and the symplectic ensembles by using Dyson's quaternion representation. As a result, the two-level correlation functions are analyzed as functions of a single parameter a (=( β y / γ) 1/2: Yukawa's parameter of intermediacy), leading to the compressibility of the level gas for the three ensembles near the Poisson and the Wigner–Dyson limits.

Weak localization in disordered systems at the ballistic limit

The weak localization contribution to the two-level correlation function $R(\ensuremath{\omega})$ is calculated for two-dimensional disordered conductors. Our analysis extends to the nondiffusive (ballistic) regime, where the elastic mean path is of order of the size of the system. In this regime, the structure factor $S(t)$ [the Fourier transform of $R(\ensuremath{\omega})]$ exhibits a singular behavior consisting of dips superimposed on a smooth positive background. The strongest dips appear at periods of the periodic orbits of the underlying clean system. Somewhat weaker singularities appear at times that are sums of periods of two such orbits. The results elucidate various aspects of the weak-localization physics of ballistic chaotic systems.

Level statistics in a two-dimensional system with strong spin-orbit coupling

We study level correlations in a two-dimensional system with a long-range random potential and strong spin-orbit (SO) splitting of the spectrum. The level correlations for sufficiently large splitting are shown to be described by orthogonal Wigner-Dyson statistics, in contrast to the common point of view that a system with the strong SO coupling belongs to the symplectic statistical ensemble. We demonstrate also that in a wide energy interval the level statistics is completely determined by transitions between two branches of the split spectrum. A sharp resonance is obtained in the two-level correlation function at energy equal to the value of SO splitting.

Random matrix model for disordered conductors

We present a random matrix ensemble where real, positive semi-definite matrix elements, x, are log-normal distributed, exp[−log2(x)]. We show that the level density varies with energy, E, as 2/(1+E) for large E, in the unitary family, consistent with the expectation for disordered conductors. The two-level correlation function is studied for the unitary family and found to be largely of the universal form despite the fact that the level density has a non-compact support. The results are based on the method of orthogonal polynomials (the Stieltjes-Wigert polynomials here). An interesting random walk problem associated with the joint probability distribution of the ensuing ensemble is discussed and its connection with level dynamics is brought out. It is further proved that Dyson’s Coulomb gas analogy breaks down whenever the confining potential is given by a transcendental function for which there exist orthogonal polynomials.

Critical spectral statistics as the Luttinger liquid of energy levels at a finite temperature

We review the recent results on the energy level statistics in disordered conductors near the Anderson localization transition and other critical systems with multifractal eigenstates. We focus on the critical random matrix ensemble (CRMT) suggested by Mirlin and Fyodorov as a possible generic description of the critical parametric spectral statistics (CPSS). We show that in the region of large energy and/or parameter separations and weak multifractality this CRMT is equivalent to the Luttinger liquid of energy levels at a finite temperature with the interaction between the fictitious 1D fermions (energy levels) being dependent on the Dyson symmetry parameter β = 1, 2, 4 and the temperature being proportional to the multifractality exponent η = 1 — D2. We show that the CRMT is the simplest extension of the classical Wigner-Dyson random matrix ensemble for the case of finite dimensionless conductance g ≫ 1 and suggest the form of the two-level correlation functions for all the symmetry classes and all energy separations. We argue that these correlation functions coincide with the density correlation functions for the Calogero-Sutherland model at finite temperatures.

Persistent current in a mesoscopic ring with diffuse surface scattering

The persistent current in a clean mesoscopic ring with ballistic electron motion is calculated. The particle dynamics inside a ring is assumed to be chaotic due to scattering at the surface irregularities of atomic size. This allows one to use the so-called ``ballistic'' supersymmetric \sigma model for calculation of the two-level correlation function in the presence of a nonzero magnetic flux.

Can the trace formula describe weak localization?

We attempt to systematically derive perturbative quantum corrections to the Berry diagonal approximation of the two-level correlation function (TLCF) for chaotic systems. To this end, we develop a ‘weak diagonal approximation’ based on a recent description of the first weak localization correction to conductance in terms of the Gutzwiller trace formula. This semiclassical method is tested by using it to derive the weak localization corrections to the TLCF for a semiclassically disordered system. Unfortunately, the method is unable to correctly reproduce the ‘Hikami boxes’ (the relatively small regions where classical paths are glued together by quantum processes). This results in the method failing to reproduce the well known weak localization expansion. It so happens that for the first-order correction it merely produces the wrong prefactor. However, for the second-order correction, it is unable to reproduce certain contributions, and leads to a result which is of a different form to the standard one.

Spectral statistics in disordered metals: A trajectories approach

We show that the perturbative expansion of the two-level correlation function, $R(\omega)$, in disordered conductors can be understood semiclassically in terms of self-intersecting particle trajectories. This requires the extension of the standard diagonal approximation to include pairs of paths which are non-identical but have almost identical action. The number of diagrams thus produced is much smaller than in a standard field-theoretical approach. We show that such a simplification occurs because $R(\omega)$ has a natural representation as the second derivative of free energy $F(\omega)$. We calculate $R(\omega)$ to 3-loop order, and verify a one-parameter scaling hypothesis for it in 2d. We discuss the possibility of applying our ``weak diagonal approximation'' to generic chaotic systems.