Capacitance and charge relaxation resistance of chaotic cavities - Joint distribution of two linear statistics in the Laguerre ensemble of random matrices

We consider the AC transport in a quantum RC circuit made of a coherent chaotic cavity with a top gate. Within a random matrix approach, we study the joint distribution for the mesoscopic capacitance $C_\mu=(1/C+1/C_q)^{-1}$ and the charge relaxation resistance $R_q$, where $C$ is the geometric capacitance and $C_q$ the quantum capacitance. We study the limit of a large number of conducting channels $N$ with a Coulomb gas method. We obtain $\langle R_q\rangle\simeq h/(Ne^2)=R_\mathrm{dc}$ and show that the relative fluctuations are of order $1/N$ both for $C_q$ and $R_q$, with strong correlations $\langle \delta C_q\delta R_q\rangle/\sqrt{\langle \delta C_q^2\rangle\,\langle \delta R_q^2\rangle}\simeq+0.707$. The detailed analysis of large deviations involves a second order phase transition in the Coulomb gas. The two dimensional phase diagram is obtained.Comment: LaTex, 6 pages, 3 pdf figures ; v2 : minor corrections & Refs. adde

Topological phase transitions in the 1D multichannel Dirac equation with random mass and a random matrix model

We establish the connection between a multichannel disordered model --the 1D Dirac equation with $N\times N$ matricial random mass-- and a random matrix model corresponding to a deformation of the Laguerre ensemble. This allows us to derive exact determinantal representations for the density of states and identify its low energy ($\varepsilon\to0$) behaviour $\rho(\varepsilon)\sim|\varepsilon|^{\alpha-1}$. The vanishing of the exponent $\alpha$ for $N$ specific values of the averaged mass over disorder ratio corresponds to $N$ phase transitions of topological nature characterised by the change of a quantum number (Witten index) which is deduced straightforwardly in the matrix model.Comment: 7+4 pages, 9+1 pdf figures ; v2: paper reorganised, discussion of non-isotropic case adde

One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers

We study the one-dimensional Schr\"odinger equation with a disordered potential of the form $V (x) = \phi(x)^2+\phi'(x) + \kappa(x)$ where $\phi(x)$ is a Gaussian white noise with mean $\mu g$ and variance $g$, and $\kappa(x)$ is a random superposition of delta functions distributed uniformly on the real line with mean density $\rho$ and mean strength $v$. Our study is motivated by the close connection between this problem and classical diffusion in a random environment (the Sinai problem) in the presence of random absorbers~: $\phi(x)$ models the force field acting on the diffusing particle and $\kappa(x)$ models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponent $\Omega(E) = \gamma(E) - \mathrm{i} \pi N(E)$, where $N$ is the integrated density of states per unit length and $\gamma$ the reciprocal of the localisation length. By using the continuous version of the Dyson-Schmidt method, we find an exact formula, in terms of a Hankel function, in the particular case where the strength of the delta functions is exponentially-distributed with mean $v=2g$. Building on this result, we then solve the general case -- in the low-energy limit -- in terms of an infinite sum of Hankel functions. Our main result, valid without restrictions on the parameters of the model, is that the integrated density of states exhibits the power law behaviour N(E) \underset{E\to0+}{\sim} E^\nu \hspace{0.5cm} \mbox{where } \nu=\sqrt{\mu^2+2\rho/g}\:. This confirms and extends several results obtained previously by approximate methods.Comment: LaTeX, 44 pages, 17 pdf figure

Correlations of occupation numbers in the canonical ensemble and application to BEC in a 1D harmonic trap

We study statistical properties of $N$ non-interacting identical bosons or fermions in the canonical ensemble. We derive several general representations for the $p$-point correlation function of occupation numbers $\overline{n_1\cdots n_p}$. We demonstrate that it can be expressed as a ratio of two $p\times p$ determinants involving the (canonical) mean occupations $\overline{n_1}$, ..., $\overline{n_p}$, which can themselves be conveniently expressed in terms of the $k$-body partition functions (with $k\leq N$). We draw some connection with the theory of symmetric functions, and obtain an expression of the correlation function in terms of Schur functions. Our findings are illustrated by revisiting the problem of Bose-Einstein condensation in a 1D harmonic trap, for which we get analytical results. We get the moments of the occupation numbers and the correlation between ground state and excited state occupancies. In the temperature regime dominated by quantum correlations, the distribution of the ground state occupancy is shown to be a truncated Gumbel law. The Gumbel law, describing extreme value statistics, is obtained when the temperature is much smaller than the Bose-Einstein temperature.Comment: RevTex, 13 pages, 6 pdf figures ; v2: minor corrections (eqs. 40,41 added

Truncated linear statistics associated with the top eigenvalues of random matrices

Given a certain invariant random matrix ensemble characterised by the joint probability distribution of eigenvalues $P(\lambda_1,\ldots,\lambda_N)$, many important questions have been related to the study of linear statistics of eigenvalues $L=\sum_{i=1}^Nf(\lambda_i)$, where $f(\lambda)$ is a known function. We study here truncated linear statistics where the sum is restricted to the $N_1<N$ largest eigenvalues: $\tilde{L}=\sum_{i=1}^{N_1}f(\lambda_i)$. Motivated by the analysis of the statistical physics of fluctuating one-dimensional interfaces, we consider the case of the Laguerre ensemble of random matrices with $f(\lambda)=\sqrt{\lambda}$. Using the Coulomb gas technique, we study the $N\to\infty$ limit with $N_1/N$ fixed. We show that the constraint that $\tilde{L}=\sum_{i=1}^{N_1}f(\lambda_i)$ is fixed drives an infinite order phase transition in the underlying Coulomb gas. This transition corresponds to a change in the density of the gas, from a density defined on two disjoint intervals to a single interval. In this latter case the density presents a logarithmic divergence inside the bulk. Assuming that $f(\lambda)$ is monotonous, we show that these features arise for any random matrix ensemble and truncated linear statitics, which makes the scenario described here robust and universal.Comment: LaTeX, 30 pages, 20 pdf figures. Updated version: a typo has been corrected in Eq. (3.30) and more details are provided in the Appendi

Fluctuations of observables for free fermions in a harmonic trap at finite temperature

We study a system of 1D noninteracting spinless fermions in a confining trap at finite temperature. We first derive a useful and general relation for the fluctuations of the occupation numbers valid for arbitrary confining trap, as well as for both canonical and grand canonical ensembles. Using this relation, we obtain compact expressions, in the case of the harmonic trap, for the variance of certain observables of the form of sums of a function of the fermions' positions, $\mathcal{L}=\sum_n h(x_n)$. Such observables are also called linear statistics of the positions. As anticipated, we demonstrate explicitly that these fluctuations do depend on the ensemble in the thermodynamic limit, as opposed to averaged quantities, which are ensemble independent. We have applied our general formalism to compute the fluctuations of the number of fermions $\mathcal{N}_+$ on the positive axis at finite temperature. Our analytical results are compared to numerical simulations. We discuss the universality of the results with respect to the nature of the confinement.Comment: 36 pages, 6 pdf figure

Distribution of the Wigner-Smith time-delay matrix for chaotic cavities with absorption and coupled Coulomb gases

Within the random matrix theory approach to quantum scattering, we derive the distribution of the Wigner-Smith time delay matrix $\mathcal{Q}$ for a chaotic cavity with uniform absorption, coupled via $N$ perfect channels. In the unitary class $\beta=2$ we obtain a compact expression for the distribution of the full matrix in terms of a matrix integral. In the other symmetry classes we derive the joint distribution of the eigenvalues. We show how the large $N$ properties of this distribution can be analysed in terms of two interacting Coulomb gases living on two different supports. As an application of our results, we study the statistical properties of the Wigner time delay $\tau_{\mathrm{W}} = \mathrm{tr}[\mathcal{Q}]/N$ in the presence of absorption.Comment: 27 pages, 4 pdf figure

Extremes of 2d2d Coulomb gas: universal intermediate deviation regime

In this paper, we study the extreme statistics in the complex Ginibre ensemble of $N \times N$ random matrices with complex Gaussian entries, but with no other symmetries. All the $N$ eigenvalues are complex random variables and their joint distribution can be interpreted as a $2d$ Coulomb gas with a logarithmic repulsion between any pair of particles and in presence of a confining harmonic potential $v(r) \propto r^2$. We study the statistics of the eigenvalue with the largest modulus $r_{\max}$ in the complex plane. The typical and large fluctuations of $r_{\max}$ around its mean had been studied before, and they match smoothly to the right of the mean. However, it remained a puzzle to understand why the large and typical fluctuations to the left of the mean did not match. In this paper, we show that there is indeed an intermediate fluctuation regime that interpolates smoothly between the large and the typical fluctuations to the left of the mean. Moreover, we compute explicitly this "intermediate deviation function" (IDF) and show that it is universal, i.e. independent of the confining potential $v(r)$ as long as it is spherically symmetric and increases faster than $\ln r^2$ for large $r$ with an unbounded support. If the confining potential $v(r)$ has a finite support, i.e. becomes infinite beyond a finite radius, we show via explicit computation that the corresponding IDF is different. Interestingly, in the borderline case where the confining potential grows very slowly as $v(r) \sim \ln r^2$ for $r \gg 1$ with an unbounded support, the intermediate regime disappears and there is a smooth matching between the central part and the left large deviation regime.Comment: 36 pages, 7 figure

Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity

We consider a multichannel wire with a disordered region of length $L$ and a reflecting boundary. The reflection of a wave of frequency $\omega$ is described by the scattering matrix $\mathcal{S}(\omega)$, encoding the probability amplitudes to be scattered from one channel to another. The Wigner-Smith time delay matrix $\mathcal{Q}=-\mathrm{i}\, \mathcal{S}^\dagger\partial_\omega\mathcal{S}$ is another important matrix encoding temporal aspects of the scattering process. In order to study its statistical properties, we split the scattering matrix in terms of two unitary matrices, $\mathcal{S}=\mathrm{e}^{2\mathrm{i}kL}\mathcal{U}_L\mathcal{U}_R$ (with $\mathcal{U}_L=\mathcal{U}_R^\mathrm{T}$ in the presence of TRS), and introduce a novel symmetrisation procedure for the Wigner-Smith matrix: $\widetilde{\mathcal{Q}} =\mathcal{U}_R\,\mathcal{Q}\,\mathcal{U}_R^\dagger = (2L/v)\,\mathbf{1}_N -\mathrm{i}\,\mathcal{U}_L^\dagger\partial_\omega\big(\mathcal{U}_L\mathcal{U}_R\big)\,\mathcal{U}_R^\dagger$, where $k$ is the wave vector and $v$ the group velocity. We demonstrate that $\widetilde{\mathcal{Q}}$ can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires, $L\to\infty$, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for $\mathcal{Q}$'s eigenvalues of Brouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length $L$, the exponential functional representation is used to calculate the first moments $\langle\mathrm{tr}(\mathcal{Q})\rangle$, $\langle\mathrm{tr}(\mathcal{Q}^2)\rangle$ and $\langle\big[\mathrm{tr}(\mathcal{Q})\big]^2\rangle$. Finally we derive a partial differential equation for the resolvent $g(z;L)=\lim_{N\to\infty}(1/N)\,\mathrm{tr}\big\{\big( z\,\mathbf{1}_N - N\,\mathcal{Q}\big)^{-1}\big\}$ in the large $N$ limit.Comment: 30 pages, LaTe