Circular Ensemble Research Articles

Joint Moments of Higher Order Derivatives of CUE Characteristic Polynomials I: Asymptotic Formulae

Abstract We derive explicit asymptotic formulae for the joint moments of the $n_{1}$-th and $n_{2}$-th derivatives of the characteristic polynomials of Circular Unitary Ensemble random matrices for any non-negative integers $n_{1}, n_{2}$. These formulae are expressed in terms of determinants whose entries involve modified Bessel functions of the first kind. We also express them in terms of two types of combinatorial sums. Similar results are obtained for the analogue of Hardy’s $Z$-function. We use these formulae to formulate general conjectures for the joint moments of the $n_{1}$-th and $n_{2}$-th derivatives of the Riemann zeta-function and of Hardy’s $Z$-function. Our conjectures are supported by comparison with results obtained previously in the number theory literature.

Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble

Abstract We find closed formulas for arbitrarily high mixed moments of characteristic polynomials of the Alternative Circular Unitary Ensemble, as well as closed formulas for the averages of ratios of characteristic polynomials in this ensemble. A comparison is made to analogous results for the Circular Unitary Ensemble. Both moments and ratios are studied via symmetric function theory and a general formula of Borodin-Olshanski-Strahov.

Expanding the Fourier Transform of the Scaled Circular Jacobi beta Ensemble Density

The family of circular Jacobi beta ensembles has a singularity of a type associated with Fisher and Hartwig in the theory of Toeplitz determinants. Our interest is in the Fourier transform of the corresponding N rightarrow infty bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobibeta ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for beta = 2 and beta = 4, and the loop equation hierarchy. The polynomials in the variable u=2/beta which occur in the expansion coefficents are found to have special properties analogous to those known for the structure function of the circular beta ensemble, specifically in relation to the zeros lying on the unit circle |u|=1 and interlacing. Comparison is also made with known results for the expanded Fourier transform of the density about a guest charge in the two-dimensional one-component plasma.

Toeplitz determinants with a one-cut regular potential and Fisher–Hartwig singularities I. Equilibrium measure supported on the unit circle

We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential $V$, (ii) Fisher–Hartwig singularities and (iii) a smooth function in the background. The potential $V$ is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials $V$, the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher–Hartwig singularities. For non-constant $V$, our results appear to be new even in the case of no Fisher–Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.

Signatures of a sampling quantum advantage in driven quantum many-body systems

A crucial milestone in the field of quantum simulation and computation is to demonstrate that a quantum device can perform a computation task that is classically intractable. A key question is to identify setups that can achieve such goal within current technologies. In this work, we provide formal evidence that sampling bit-strings from a periodic evolution of a unitary drawn from the circular orthogonal ensemble (COE) cannot be efficiently simulated with classical computers. As the statistical properties of COE coincide with a large class of driven analog quantum systems thanks to the Floquet eigenstate thermalization hypothesis, our results indicate the possibility that those driven systems could constitute practical candidates for a sampling quantum advantage. To further support this, we give numerical examples of driven disordered Ising chains and 1D driven Bose–Hubbard model.

Blocky Diagonalized Scattering Matrices in Chaotic Scattering with Direct Processes

Scattering matrices that can be diagonalized by a rotation through an angle θ in 2×2 blocks of independent scattering matrices of rank N, are considered. Assuming that the independent scattering matrices are chosen from one of the circular ensembles, or from the Poisson kernel, the 2N×2N scattering matrix may describe the scattering through chaotic cavities with reduced symmetry in the absence, or presence, of direct processes, respectively. To illustrate the effect of such symmetry, the statistical distribution of the dimensionless conductance through a ballistic chaotic cavity in the presence of direct processes is analyzed for N=1 using analytical calculations. We make a conjecture for N=2 in the absence of direct processes, which is verified by numerical random-matrix theory simulations, and the first two moments are calculated analytically for arbitrary N.

Power spectrum of the circular unitary ensemble

We study the power spectrum of eigen-angles of random matrices drawn from the circular unitary ensemble CUE(N) and show that it can be evaluated in terms of either a Fredholm determinant, or a Toeplitz determinant, or a sixth Painlevé function. In the limit of infinite-dimensional matrices, N→∞, we derive a concise parameter-free formula for the power spectrum which involves a fifth Painlevé transcendent and interpret it in terms of the Sine2 determinantal random point field. Further, we discuss a universality of the predicted power spectrum law and tabulate it for easy use by random-matrix-theory and quantum chaos practitioners.

The spectral form factor in the 't Hooft limit – Intermediacy versus universality

The Spectral Form Factor (SFF) is a convenient tool for the characterization of eigenvalue statistics of systems with discrete spectra, and thus serves as a proxy for quantum chaoticity. This work presents an analytical calculation of the SFF of the Chern-Simons Matrix Model (CSMM), which was first introduced to describe the intermediate level statistics of disordered electrons at the mobility edge . The CSMM is characterized by a parameter 0\leq q\leq 1 0≤q≤1, where the Circular Unitary Ensemble (CUE) is recovered for q\to 0 q→0. The CSMM was later found as a matrix model description of U(N) U(N) Chern-Simons theory on S^3 S3 , which is dual to a topological string theory characterized by string coupling g_s=-\log q gs=−logq. The spectral form factor is proportional to a colored HOMFLY invariant of a (2n,2) (2n,2)-torus link with its two components carrying the fundamental and antifundamental representations, respectively. We check explicitly that taking N \to \infty N→∞ whilst keeping q<1 q<1 reduces the connected SFF to an exact linear ramp of unit slope, thereby confirming the main result from for the specific case of the CSMM. We then consider the ’t Hooft limit, where N \to \infty N→∞ and q \to 1^- q→1− such that y=q^N y=qN remains finite. As we take q\to 1^- q→1−, this constitutes the opposite extreme of the CUE limit. In the 't Hooft limit, the connected SFF turns into a remarkable sequence of polynomials which, as far as the authors are aware, have not appeared in the literature thus far. A gap opens in the spectrum and, after unfolding by a constant rescaling, the connected SFF approximates a linear ramp of unit slope for all y y except y \approx 1 y≈1, where the connected SFF goes to zero. We thus find that, although the CSMM was introduced to describe intermediate statistics and the 't Hooft limit is the opposite limit of the CUE, we still recover Wigner-Dyson universality for all y y except y\approx 1 y≈1.

The many faces of the stochastic zeta function

We introduce a framework to study the random entire function $\zeta_\beta$ whose zeros are given by the Sine$_\beta$ process, the bulk limit of beta ensembles. We present several equivalent characterizations, including an explicit power series representation built from Brownian motion. We study related distributions using stochastic differential equations. Our function is a uniform limit of characteristic polynomials in the circular beta ensemble; we give upper bounds on the rate of convergence. Most of our results are new even for classical values of $\beta$. We provide explicit moment formulas for $\zeta$ and its variants, and we show that the Borodin-Strahov moment formulas hold for all $\beta$ both in the limit and for circular beta ensembles. We show a uniqueness theorem for $\zeta$ in the Cartwright class, and deduce some product identities between conjugate values of $\beta$. The proofs rely on the structure of the Sine$_\beta$ operator to express $\zeta$ in terms of a regularized determinant.

On the Critical–Subcritical Moments of Moments of Random Characteristic Polynomials: A GMC Perspective

We study the ‘critical moments’ of subcritical Gaussian multiplicative chaos (GMCs) in dimensions d le 2. In particular, we establish a fully explicit formula for the leading order asymptotics, which is closely related to large deviation results for GMCs and demonstrates a similar universality feature. We conjecture that our result correctly describes the behaviour of analogous moments of moments of random matrices, or more generally structures which are asymptotically Gaussian and log-correlated in the entire mesoscopic scale. This is verified for an integer case in the setting of circular unitary ensemble, extending and strengthening the results of Claeys et al. and Fahs to higher-order moments.

Three-fold way of entanglement dynamics in monitored quantum circuits

We investigate the measurement-induced entanglement transition in quantum circuits built upon Dyson’s three circular ensembles (circular unitary, orthogonal, and symplectic ensembles; CUE, COE and CSE). We utilise the established model of a one-dimensional circuit evolving under alternating local random unitary gates and projective measurements performed with tunable rate, which for gates drawn from the CUE is known to display a transition from extensive to intensive entanglement scaling as the measurement rate is increased. By contrasting this case to the COE and CSE, we obtain insights into the interplay between the local entanglement generation by the gates and the entanglement reduction by the measurements. For this, we combine exact analytical random-matrix results for the entanglement generated by the individual gates in the different ensembles, and numerical results for the complete quantum circuit. These considerations include an efficient rephrasing of the statistical entangling power in terms of a characteristic entanglement matrix capturing the essence of Cartan’s KAK decomposition, and a general result for the eigenvalue statistics of antisymmetric matrices associated with the CSE.

Hydrodynamic equations for the Ablowitz–Ladik discretization of the nonlinear Schrödinger equation

Ablowitz and Ladik discovered a discretization that preserves the integrability of the nonlinear Schrödinger equation in one dimension. We compute the generalized free energy of this model and determine the generalized Gibbs ensemble averaged fields and their currents. They are linked to the mean-field circular unitary matrix ensemble. The resulting hydrodynamic equations follow the pattern already known from other integrable many-body systems. The discretized modified Korteweg–de-Vries equation is also studied, which turns out to be related to the beta Jacobi log gas.

Central limit theorem for CβE pair dependent statistics in mesoscopic regime

We extend our results on the fluctuation of the pair counting statistic of the Circular Beta Ensemble ∑i≠jf(LN(θi−θj)) for arbitrary β>0 in the mesoscopic regime LN=O(N2∕3−ϵ). In addition, we obtain similar results for bipartite statistics.

Polarisation statistics of vector scattering matrices from the circular orthogonal ensemble

We study the polarisation properties of random N×N scattering matrices distributed according to the circular orthogonal ensemble. We interpret 2 × 2 sub-blocks of the scattering matrix as Jones matrices and study their statistical properties. Using the polar decomposition, we derive probability density functions for retardance and diattenuation from scattering matrices of arbitrary size and in the limit N→∞.

Direct product of random unitary matrices: Two-point correlations and fluctuations.

We study the ensembles of direct product of m random unitary matrices of size N drawn from a given circular ensemble. We calculate the statistical measures, viz. number variance and spacing distribution to investigate the level correlations and fluctuation properties of the eigenangle spectrum. Similar to the random unitary matrices, the level statistics is stationary for the ensemble constructed by their direct product. We find that the eigenangles are uncorrelated in the small spectral intervals. While, in large spectral intervals, the spectrum is rigid due to strong long-range correlations between the eigenangles. The analytical and numerical results are in good agreement. We also test our findings on the multipartite system of quantum kicked rotors.

Symmetric function theory and unitary invariant ensembles

Representation theory and the theory of symmetric functions have played a central role in random matrix theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the circular unitary ensemble and other circular ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulas, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble (GUE) of random matrices and other related unitary invariant matrix ensembles. This allows us to write down exact formulas in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces, we derive explicit asymptotic formulas for the rate of convergence of the moments of polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity.

Superposition and higher-order spacing ratios in random matrix theory with application to complex systems

The statistical properties of spectra of quantum systems within the framework of random matrix theory is widely used in many areas of physics. These properties are affected, if two or more sets of spectra are superposed, resulting from the discrete symmetries present in the system. Superposition of spectra of $m$ such circular orthogonal, unitary and symplectic ensembles are studied numerically using higher-order spacing ratios. For given $m$ and the Dyson index $\beta$, the modified index $\beta'$ is tabulated whose nearest neighbor spacing distribution is identical to that of $k$-th order spacing ratio. For the case of $m=2$ ($m=3$) in COE (CUE) a scaling relation between $\beta'$ and $k$ is given. For COE, it is conjectured that for $k=m+1$ ($m\geq2$) and $k=m-3$-th ($m\geq5$) order spacing ratio distribution the $\beta'$ is $m+2$ and $m-4$ respectively. Whereas in the case of CSE, for $k=m+1$ ($m\geq2$) and $k=m-1$-th ($m\geq3$) the $\beta'$ is $2m+3$ and $2(m-2)$ respectively. We also conjecture that for given $m$ ($k$) and $\beta$, the sequence of $\beta'$ as a function of $k$ ($m$) is unique. Strong numerical evidence in support of these results is presented. These results are tested on complex systems like the measured nuclear resonances, quantum chaotic kicked top and spin chains.

Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits

We investigate a class of brickwork-like quantum circuits on chains of $d-$level systems (qudits) that share the so-called `dual unitarity' property. Namely, these systems generate unitary dynamics not only when propagating in the time direction, but also when propagating in the space direction. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site disorder, i.e. by applying independent single site random unitaries drawn from arbitrary non-singular distribution over ${\rm SU}(d)$, e.g. one concentrated around the identity, after each layer of the circuit. We identify the spectral form factor at time $t$ in the limit of long chains as the dimension of the commutant of a finite set of operators on a qudit ring of $t$ sites. For general dual unitary circuits of qubits $(d=2)$ and a family of their extensions to higher $d>2$, we provide explicit construction of the commutant and prove that spectral form factor exactly matches the prediction of circular unitary ensemble for all $t$, if only the local 2-qubit gates are different from a SWAP (non-interacting gate). We discuss and partly prove possible extensions of our results to a weaker (more singular) forms of disorder averaging, as well as to quantum circuits with time-reversal symmetry, and to computing higher moments of the spectral form factor.

Immanants of blocks from random matrices in some unitary ensembles

The permanent of unitary matrices and their blocks has attracted increasing attention in quantum physics and quantum computation because of connections with the Hong–Ou–Mandel effect and the boson sampling problem. In that context, it would be useful to know the distribution of the permanent and other immanants for random matrices, but that seems a difficult problem. We advance this program by calculating the average of the squared modulus of a generic immanant for blocks from random matrices in the unitary group, in the orthogonal group and in the circular orthogonal ensemble. In the case of the permanent in the unitary group, we also compute the variance. Our approach is based on Weingarten functions and factorizations of permutations. In the course of our calculations we are led to two curious conjectures relating dimensions of irreducible representations of the orthogonal and symplectic groups to the value of zonal and symplectic zonal polynomials at the identity.

Convergence of moments of twisted COE matrices

We investigate eigenvalue moments of matrices from the circular orthogonal ensemble multiplicatively perturbed by a permutation matrix. More precisely, we investigate the variance of the sum of the eigenvalues raised to power k for arbitrary but fixed k and in the limit of a large matrix size. We find that when the permutation defining the perturbed ensemble has only long cycles, the answer is universal and approaches the corresponding moment of the circular unitary ensemble with a particularly fast rate: the error is of order 1/N3 and the terms of orders 1/N and 1/N2 disappear due to cancellations. We prove this rate of convergence using Weingarten calculus and classifying the contributing Weingarten functions first in terms of a graph model and then algebraically.