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Marcel Novaes

Publications and source records attributed to Marcel Novaes.

At least 37 records · Page 2Linked to original sources
arXiv2020

Semiclassical treatment of quantum chaotic transport with a tunnel barrier

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.

nlin.CD↗
arXiv2020

On the 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 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.

math-ph↗
arXiv2019

Random stochastic matrices from classical compact Lie groups and symmetric spaces

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.

math-ph↗
arXiv2018

Semiclassical calculation of spectral correlation functions of chaotic systems

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.

nlin.CD↗
arXiv2018

Full perturbative calculation of spectral correlation functions for chaotic systems in the unitary symmetry class

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.

nlin.CD↗
arXiv2017

Introduction to Random Matrices - Theory and Practice

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.

math-ph↗
arXiv2017

Energy-dependent correlations in the $S$-matrix of chaotic systems

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 $\left \langle S_{ij}(E+ε)S^†_{pq}(E-ε)\right\rangle$, 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 $ε$. From a mathematical point of view, this may be seen as a generalization of the Weingarten functions of circular ensembles.

math-ph↗
arXiv2016

Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations

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.

math-ph↗
arXiv2015

Statistics of time delay in quantum chaotic transport

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.

nlin.CD↗
arXiv2015

Statistics of time delay and scattering correlation functions in chaotic systems II. Semiclassical Approximation

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 ${\rm Tr}[S^†(E-ε)S(E+ε)]^n$, for general positive integer $n$. Our result is an infinite series in $ε$, whose coefficients are rational functions of $M$. From this we extract moments of the time delay matrix $Q=-i\hbar S^†dS/dE$, and check that the first 8 of them agree with the random matrix theory prediction from our previous paper.

nlin.CD↗
arXiv2015

Statistics of time delay and scattering correlation functions in chaotic systems I. Random Matrix Theory

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 $Q=-i\hbar S^†dS/dE$, 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.

nlin.CD↗
arXiv2015

Semiclassical matrix model for quantum chaotic transport with time-reversal symmetry

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.

nlin.CD↗
arXiv2013

A semiclassical matrix model for quantum chaotic transport

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.

nlin.CD↗
arXiv2013

Statistics of quantum transport in weakly non-ideal chaotic cavities

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 $Γ_i$. Using symmetric function expansions and a generalized Selberg integral, we develop a systematic perturbation theory in $1-Γ_i$ 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.

cond-mat.mes-hall↗
arXiv2013

Orbital entanglement production in Andreev billiards with time-reversal symmetry

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.

cond-mat.mes-hall↗
arXiv2013

Combinatorial problems in the semiclassical approach to quantum chaotic transport

A semiclassical approach to the calculation of transport moments $M_m={\rm Tr}[(t^†t)^m]$, 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.

nlin.CD↗
arXiv2012

Resonances in open quantum maps

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.

nlin.CD↗