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We generalize the Kuramoto model by interpreting the $N$ variables on the unit circle as eigenvalues of a $N$-dimensional unitary matrix $U$, in three versions: general unitary, symmetric unitary and special orthogonal. The time evolution is generated by $N^2$ coupled differential equations for the matrix elements of $U$, and synchronization happens when $U$ evolves into a multiple of the identity. The Ott-Antonsen ansatz is related to the Poisson kernels that are so useful in quantum transport, and we prove it in the case of identical natural frequencies. When the coupling constant is a matrix, we find some surprising new dynamical behaviors.

We investigate the entanglement in the ground state of systems comprising two and three qubits with random interactions. Since the Hamiltonians also contain deterministic one-body terms, by varying the interaction strength, one can continuously interpolate between deterministic separable eigenstates and fully random entangled eigenstates, with non-trivial intermediate behavior. Entanglement strongly depends on the underlying topology of the interaction among the qubits. For a certain class of interactions GHZ entanglement is favoured by a non-separable collective interaction, while for fully separable pairwise interactions the ground states concentrate in the vicinity of W states.

We present a semiclassical approach for time delay statistics in quantum chaotic systems, in the presence of absorption, for broken time-reversal symmetry. We derive three kinds of expressions for Schur-moments of the time delay operator: as a power series in inverse channel number, $1/M$, whose coefficients are rational functions of absorption time, $\tau_a$; as a power series in $\tau_a$, tailored to strong absorption, whose coefficients are rational functions of $M$; as a power series in $1/\tau_a$, tailored to weak absorption, whose coefficients are rational functions of $M$.

We develop a semiclassical approach for the statistics of the time delay in quantum chaotic systems in the presence of a tunnel barrier, for broken time-reversal symmetry. Results are obtained as asymptotic series in powers of the reflectivity of the barrier, with coefficients that are rational functions of the channel number. Exact expressions, valid for arbitrary reflectivity and channel number, are conjectured and numerically verified for specific families of statistical moments., Comment: 9 pages, 2 figures

We uncover a hidden Gaussian ensemble inside each of the three circular ensembles of random matrices, which provide novel diagrammatic rules for the calculation of moments. The matrices involved are generic complex for $\beta=2$, complex symmetric for $\beta=1$ and complex self-dual for $\beta=4$, and their dimension must be set to $1-2/\beta$. As an application, we compute moments of traces of submatrices., Comment: 12 pages, 4 figures

We present a semiclassical calculation, based on classical action correlations implemented by means of a matrix integral, of all moments of the Wigner--Smith time delay matrix, $Q$, in the context of quantum scattering through systems with chaotic dynamics. Our results are valid for broken time reversal symmetry and depend only on the classical dwell time and the number of open channels, $M$, which is arbitrary. Agreement with corresponding random matrix theory reduces to an identity involving some combinatorial concepts, which can be proved in special cases., Comment: 9 pages, no figures

When time-reversal symmetry is broken, the average conductance through a chaotic cavity, from an entrance lead with $N_1$ open channels to an exit lead with $N_2$ open channels, is given by $N_1N_2/M$, where $M=N_1+N_2$. We show that, when tunnel barriers of reflectivity $\gamma$ are placed on the leads, two correction terms appear in the average conductance, and that one of them is proportional to $\gamma^{M}$. Since $M\sim \hbar^{-1}$, this correction is exponentially small in the semiclassical limit. Surprisingly, we derive this term from a semiclassical approximation, generally expected to give only leading orders in powers of $\hbar$. Even though the theory is built perturbatively both in $\gamma$ and in $1/M$, the final result is exact., Comment: 9 pages, 2 figures

We consider the problem of electronic quantum transport through ballistic mesoscopic systems with chaotic dynamics, connected to a three-terminal architecture in which one of the terminals has a tunnel barrier. Using a semiclassical approximation based on matrix integrals, we calculate several transport statistics, such as average and variance of conductance, average shot-noise power, among others, that give access to the extreme quantum regime (small channel numbers in the terminal) for broken and intact time-reversal symmetry, which the traditional random matrix approach does not access. As an application, we treat the dephasing regime., Comment: 24 pages, 5 figures. Accepted by Journal of Physics A

Within a random matrix theory approach, we obtain spectral statistics of the Wigner time delay matrix $Q$, for arbitrary channels number $M$ and for all symmetry classes, in fact for general Dyson parameter $\beta$. We also put forth two conjectures: one is related to the large-$M$ expansion of joint cumulants of traces of powers of $Q$, which generalizes and implies a previous conjecture of Cunden, Mezzadri, Vivo and Simm; the other concerns the tail of the distribution of traces of powers of $Q$., Comment: 6 pages, no figures. Accepted by Europhysics Letters

We bring together the semiclassical approximation, matrix integrals and the theory of symmetric polynomials in order to solve a long standing problem in the field of quantum chaos: to compute transport moments when tunnel barriers are present and the number of open channels, $M$, is small. In contrast to previous approaches, ours is non-perturbative in $M$; instead, we arrive at an explicit expression in the form of a power series in the barrier's reflectivity, whose coefficients are rational functions of $M$. For general moments we must require that the barriers are equal and time reversal symmetry is broken, but for conductance we treat the general situation. Our method accounts for exponentially small non-perturbative terms that were not accessible to previous semiclassical approaches. We also show how to include more than two leads in the system., Comment: 13 pages, 3 figures

The $M$-dimensional scattering matrix $S(E)$ which connects incoming to outgoing waves in a chaotic systyem is always unitary, but shows complicated dependence on the energy. This is partly encoded in correlators constructed from traces of powers of $S(E+\epsilon)S^\dagger(E-\epsilon)$, averaged over $E$, and by the statistical properties of the time delay operator, $Q(E)=-i\hbar S^\dagger dS/dE$. Using a semiclassical approach for systems with broken time reversal symmetry, we derive two kind of expressions for the energy correlators: one as a power series in $1/M$ whose coefficients are rational functions of $\epsilon$, and another as a power series in $\epsilon$ whose coefficients are rational functions of $M$. From the latter we extract an explicit formula for $\rm{Tr}(Q^n)$ which is valid for all $n$ and is in agreement with random matrix theory predictions., Comment: 6 pages

We consider the problem of recovering equations of motion from multivariate time series of oscillators interacting on sparse networks. We reconstruct the network from an initial guess which can include expert knowledge about the system such as main motifs and hubs. When sparsity is taken into account the number of data points needed is drastically reduced when compared to the least-squares recovery. We show that the sparse solution is stable under basis extensions, that is, once the correct network topology is obtained, the result does not change if further motifs are considered., Comment: 22 pages, 5 figure

We consider the problem of a semiclassical description of quantum chaotic transport, when a tunnel barrier is present in one of the leads. Using a semiclassical approach formulated in terms of a matrix model, we obtain transport moments as power series in the reflection probability of the barrier, whose coefficients are rational functions of the number of open channels M. Our results are therefore valid in the quantum regime and not only when $M\gg 1$. The expressions we arrive at are not identical with the corresponding predictions from random matrix theory, but are in fact much simpler. Both theories agree as far as we can test., Comment: 17 pages, 5 figures

We investigate the statistical properties of $C=uvu^{-1}v^{-1}$, when $u$ and $v$ are independent random matrices, uniformly distributed with respect to the Haar measure of the groups $U(N)$ and $O(N)$. An exact formula is derived for the average value of power sum symmetric functions of $C$, and also for products of the matrix elements of $C$, similar to Weingarten functions. The density of eigenvalues of $C$ is shown to become constant in the large-$N$ limit, and the first $N^{-1}$ correction is found., Comment: 26 pages, 4 figures

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 or 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 a conjecture relating dimensions of irreducible representations of the orthogonal group to the value of zonal polynomials at the identity.

We present a semiclassical approach to n-point spectral correlation functions of quantum systems whose classical dynamics is chaotic, for arbitrary n. The basic ingredients are sets of periodic orbits that have nearly the same action and therefore provide constructive interference. We calculate explicitly the first correlation functions, to leading orders in their energy arguments, for both unitary and orthogonal symmetry classes. The results agree with corresponding predictions from random matrix theory, thereby giving solid support to the conjecture of universality., Comment: 13 pages, 5 figures

Starting from a semiclassical approach recently developed for spectral correlation functions of quantum systems whose classical dynamics is chaotic, we focus on the case of broken time-reversal symmetry, the so-called unitary class. We obtain to all orders in perturbation theory the non-oscillatory parts of all correlation functions, showing that the off-diagonal contributions to these correlation functions cancel and the conjectured universailty holds. The innovation that allows this calculation to be performed is the introduction of an auxiliary matrix model which is governed by the same diagrammatic rules as the semiclassical approach and which can be exactly solved., Comment: 6 pages, 1 figure

We consider random stochastic matrices $M$ with elements given by $M_{ij}=|U_{ij}|^2$, with $U$ being uniformly distributed on one of the classical compact Lie groups or associated symmetric spaces. We observe numerically that, for large dimensions, the spectral statistics of $M$, discarding the Perron-Frobenius eigenvalue $1$, are similar to those of the Gaussian Orthogonal ensemble for symmetric matrices and to those of the real Ginibre ensemble for non-symmetric matrices. Using Weingarten functions, we compute some spectral statistics that corroborate this universality. We also establish connections with some difficult enumerative problems involving permutations., Comment: 27 pages, 4 figures

This is a book for absolute beginners. If you have heard about random matrix theory, commonly denoted RMT, but you do not know what that is, then welcome!, this is the place for you. Our aim is to provide a truly accessible introductory account of RMT for physicists and mathematicians at the beginning of their research career. We tried to write the sort of text we would have loved to read when we were beginning Ph.D. students ourselves. Our book is structured with light and short chapters, and the style is informal. The calculations we found most instructive are spelt out in full. Particular attention is paid to the numerical verification of most analytical results. Our book covers standard material - classical ensembles, orthogonal polynomial techniques, spectral densities and spacings - but also more advanced and modern topics - replica approach and free probability - that are not normally included in elementary accounts on RMT. This book is dedicated to the fond memory of Oriol Bohigas.

The $M$-dimensional unitary matrix $S(E)$, which describes scattering of waves, is a strongly fluctuating function of the energy for complex systems such as ballistic cavities, whose geometry induces chaotic ray dynamics. Its statistical behaviour can be expressed by means of correlation functions of the kind $\left \langle S_{ij}(E+\epsilon)S^\dag_{pq}(E-\epsilon)\right\rangle$, which have been much studied within the random matrix approach. In this work, we consider correlations involving an arbitrary number of matrix elements and express them as infinite series in $1/M$, whose coefficients are rational functions of $\epsilon$. From a mathematical point of view, this may be seen as a generalization of the Weingarten functions of circular ensembles.

Using the diagrammatic approach to integrals over Gaussian random matrices, we find a representation for polynomial Lie group integrals as infinite sums over certain maps on surfaces. The maps involved satisfy a specific condition: they have some marked vertices, and no closed walks that avoid these vertices. We also formulate our results in terms of permutations, arriving at new kinds of factorization problems., Comment: 21 pages, 7 figures

We consider $S$-matrix correlation functions for a chaotic cavity having $M$ open channels, in the absence of time-reversal invariance. Relying on a semiclassical approximation, we compute the average over $E$ of the quantities ${\rm Tr}[S^\dag(E-\epsilon)S(E+\epsilon)]^n$, for general positive integer $n$. Our result is an infinite series in $\epsilon$, whose coefficients are rational functions of $M$. From this we extract moments of the time delay matrix $Q=-i\hbar S^\dag dS/dE$, and check that the first 8 of them agree with the random matrix theory prediction from our previous paper., Comment: 20 pages, 3 figures. A previous paper, arXiv:1408.1669, has been split in two parts upon publication, each part containing different results, in order to make the presentation more consistent. This is the second part

We consider the statistics of time delay in a chaotic cavity having $M$ open channels, in the absence of time-reversal invariance. In the random matrix theory approach, we compute the average value of polynomial functions of the time delay matrix $Q=-i\hbar S^\dag dS/dE$, where $S$ is the scattering matrix. Our results do not assume $M$ to be large. In a companion paper, we develop a semiclassical approximation to $S$-matrix correlation functions, from which the statistics of $Q$ can also be derived. Together, these papers contribute to establishing the conjectured equivalence between the random matrix and the semiclassical approaches., Comment: 9 pages, no figures. A previous paper, arXiv:1408.1669, has been split in two parts upon publication, each part containing different results, in order to make the presentation more consistent. This is the first part

We show that the semiclassical approach to chaotic quantum transport in the presence of time-reversal symmetry can be described by a matrix model, i.e. a matrix integral whose perturbative expansion satisfies the semiclassical diagrammatic rules for the calculation of transport statistics. This approach leads very naturally to the semiclassical derivation of universal predictions from random matrix theory., Comment: 12 pages, 4 figures

We study the statistical properties of the time delay matrix $Q$ in the context of quantum transport through a chaotic cavity, in the absence of time-reversal invariance. First, we approach the problem from the point of view of random matrix theory, and obtain exact results that provide the average value of any polynomial function of $Q$. We then consider the problem from the point of view of the semiclassical approximation, obtaining the entire perturbation series for some energy-dependent correlation functions. Using these correlation functions, we show agreement between the random matrix and the semiclassical approaches for several statistical properties., Comment: 20 pages, 3 figures. This paper has been split in two parts upon publication, each part containing different results, in order to make the presentation more consistent. First part is arXiv:1507.05524, second part is arXiv:1507.05526

Integration of polynomials over the classical groups of unitary, orthogonal and symplectic matrices can be reduced to basic building blocks known as Weingarten functions. We present an elementary derivation of these functions., Comment: v2 fixes some imprecisions

We propose a matrix model which embodies the semiclassical approach to the problem of quantum transport in chaotic systems. Specifically, a matrix integral is presented whose perturbative expansion satisfies precisely the semiclassical diagrammatic rules for the calculation of general counting statistics. Evaluating it exactly, we show that it agrees with corresponding predictions from random matrix theory. This uncovers the algebraic structure behind the equivalence between these two approaches, and opens the way for further semiclassical calculations., Comment: 8 pages, 2 figures

We consider statistics of electronic transport in chaotic cavities where time-reversal symmetry is broken and one of the leads is weakly non-ideal, i.e. it contains tunnel barriers characterized by tunneling probabilities $\Gamma_i$. Using symmetric function expansions and a generalized Selberg integral, we develop a systematic perturbation theory in $1-\Gamma_i$ valid for arbitrary number of channels, and obtain explicit formulas up to second order for the average and variance of the conductance, and for the average shot-noise. Higher moments of the conductance are considered to leading order., Comment: 6 pag., 2 fig. - submitted to PRB

We study orbital entanglement production in a chaotic cavity connected to four single-channel normal-metal leads and one superconducting lead, assuming the presence of time-reversal symmetry (TRS). The scattered state of two incident electrons is written as the superposition of several two-outgoing quasi-particle states, four of which are orbitally entangled in a left-right bipartition. We calculate numerically the mean value of the squared norm of each scattered state's component, as functions of the number of channels in the superconducting lead. Its behavior is explained as resulting from the proximity effect. We also study statistically the amount of entanglement carried by each pair of outgoing quasi-particles. When the influence of the superconductor is more intense, the average entanglement is found to be considerably larger than that obtained using normal cavities.

We review recent studies about the resonance spectrum of quantum scattering systems, in the semiclassical limit and assuming chaotic classical dynamics. Stationary quantum properties are related to fractal structures in the classical phase space. We focus attention on a particular class of problems that are chaotic maps in the torus with holes. Among the topics considered are the fractal Weyl law, the formation of a spectral gap and the morphology of eigenstates. We also discuss the situation when the holes are only partially transparent and the use of random matrices for a statistical description., Comment: 30 pages, 14 figures

A semiclassical approach to the calculation of transport moments $M_m={\rm Tr}[(t^\dag t)^m]$, where $t$ is the transmission matrix, was developed in [M. Novaes, Europhys. Lett. 98, 20006 (2012)] for chaotic cavities with two leads and broken time-reversal symmetry. The result is an expression for $M_m$ as a perturbation series in 1/N, where N is the total number of open channels, which is in agreement with random matrix theory predictions. The coefficients in this series were related to two open combinatorial problems. Here we expand on this work, including the solution to one of the combinatorial problems. As a by-product, we also present a conjecture relating two kinds of factorizations of permutations., Comment: 13 pages, 2 figures. v2-minor changes, including title