[Show abstract][Hide abstract] ABSTRACT: We present a comprehensive study of the destruction of quantum multifractality in the presence of perturbations. We study diverse representative models displaying multifractality, including a pseudointegrable system, the Anderson model and a random matrix model. We apply several types of natural perturbations which can be relevant for experimental implementations. We construct an analytical theory for certain cases, and perform extensive large-scale numerical simulations in other cases. The data are analyzed through refined methods including double scaling analysis. Our results confirm the recent conjecture that multifractality breaks down following two scenarios. In the first one, multifractality is preserved unchanged below a certain characteristic length which decreases with perturbation strength. In the second one, multifractality is affected at all scales and disappears uniformly for a strong enough perturbation. Our refined analysis shows that subtle variants of these scenarios can be present in certain cases. This study could guide experimental implementations in order to observe quantum multifractality in real systems.
[Show abstract][Hide abstract] ABSTRACT: We propose a generalisation of the Bloch sphere representation for mixed spin
states based on covariant matrices introduced by Weinberg in 1964 in the
context of quantum field theory. We show that these matrices form a tight
frame, enabling a compact and transparent representation of density matrices of
arbitrary spin in terms of tensors that share the most important properties of
Bloch vectors. We investigate the properties of this representation, and give
various applications, notably a novel characterization of anti-coherent spin
[Show abstract][Hide abstract] ABSTRACT: In the introductory section of the article we give a brief account of recent
insights into statistics of high and extreme values of disorder-generated
multifractals following a recent work by the first author with P. Le Doussal
and A. Rosso (FLR) employing a close relation between multifractality and
logarithmically correlated random fields. We then substantiate some aspects of
the FLR approach analytically for multifractal eigenvectors in the
Ruijsenaars-Schneider ensemble (RSE) of random matrices introduced by E.
Bogomolny and the second author by providing an ab initio calculation that
reveals hidden logarithmic correlations at the background of the
disorder-generated multifractality. In the rest we investigate numerically a
few representative models of that class, including the study of the highest
component of multifractal eigenvectors in the Ruijsenaars-Schneider ensemble.
[Show abstract][Hide abstract] ABSTRACT: We expose two scenarios for the breakdown of quantum multifractality under
the effect of perturbations. In the first scenario, multifractality survives
below a certain scale of the quantum fluctuations. In the other one, the
fluctuations of the wave functions are changed at every scale and each
multifractal dimension smoothly goes to the ergodic value. We use as generic
examples a one-dimensional dynamical system and the three-dimensional Anderson
model at the metal-insulator transition. Our results suggest that quantum
multifractality breakdown is universal and follows one of these two scenarios
depending on the perturbation. We also discuss the experimental implications.
[Show abstract][Hide abstract] ABSTRACT: We analyze the game of go from the point of view of complex networks. We
construct three different directed networks of increasing complexity, defining
nodes as local patterns on plaquettes of increasing sizes, and links as actual
successions of these patterns in databases of real games. We discuss the
peculiarities of these networks compared to other types of networks. We explore
the ranking vectors and community structure of the networks and show that this
approach enables to extract groups of moves with common strategic properties.
We also investigate different networks built from games with players of
different levels or from different phases of the game. We discuss how the study
of the community structure of these networks may help to improve the computer
simulations of the game. More generally, we believe such studies may help to
improve the understanding of human decision process.
[Show abstract][Hide abstract] ABSTRACT: For random matrices with tree-like structure there exists a recursive
relation for the local Green functions whose solution permits to find directly
many important quantities in the limit of infinite matrix dimensions. The
purpose of this note is to investigate and compare expressions for the spectral
density of random regular graphs, based on easy approximations for real
solutions of the recursive relation valid for trees with large coordination
number. The obtained formulas are in a good agreement with the results of
numerical calculations even for small coordination number.
Physical Review E 12/2013; 88(6-1):062811. DOI:10.1103/PhysRevE.88.062811 · 2.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We investigate the set of quantum channels acting on a single qubit. We
provide a compact generalization of the Fujiwara-Algoet conditions for complete
positivity to non-unital qubit channels, which we then use to characterize the
possible geometric forms of the pure output of the channel. We provide
universal sets of quantum channels for all unital qubit channels as well as for
all extremal (not necessarily unital) qubit channels, in the sense that all
qubit channels in these sets can be obtained by concatenation of channels in
the corresponding universal set. We also show that our universal sets are
Journal of Physics A Mathematical and Theoretical 06/2013; 47(13). DOI:10.1088/1751-8113/47/13/135302 · 1.69 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We calculate analytically, for finite-size matrices, joint probability
densities of ratios of level spacings in ensembles of random matrices
characterized by their associated confining potential. We focus on the ratios
of two spacings between three consecutive real eigenvalues, as well as certain
generalizations such as the overlapping ratios. The resulting formulas are
further analyzed in detail in two specific cases: the beta-Hermite and the
beta-Laguerre cases, for which we offer explicit calculations for small N. The
analytical results are in excellent agreement with numerical simulations of
usual random matrix ensembles, and with the level statistics of a quantum
many-body lattice model and zeros of the Riemann zeta function.
Journal of Physics A Mathematical and Theoretical 05/2013; 46(35). DOI:10.1088/1751-8113/46/35/355204 · 1.69 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices. This ratio distribution was recently introduced to study spectral properties of many-body problems, as, contrary to the standard level spacing distributions, it does not depend on the local density of states. Our Wigner-like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit. Quantitative improvements are found through a polynomial expansion. Examples from a quantum many-body lattice model and from zeros of the Riemann zeta function are presented.
[Show abstract][Hide abstract] ABSTRACT: We study a version of the mathematical Ruijsenaars-Schneider model and reinterpret it physically in order to describe the spreading with time of quantum wave packets in a system where multifractality can be tuned by varying a parameter. We compare different methods to measure the multifractality of wave packets and identify the best one. We find the multifractality to decrease with time until it reaches an asymptotic limit, which is different from the multifractality of eigenvectors but related to it, as is the rate of the decrease. Our results could guide the study of experimental situations where multifractality is present in quantum systems.
Physical Review E 11/2012; 86(5-2):056215. DOI:10.1103/PhysRevE.86.056215 · 2.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We construct perturbation series for the qth moment of eigenfunctions of various critical random-matrix ensembles in the strong multifractality regime close to localization. Contrary to previous investigations, our results are valid in the region q<1/2. Our findings allow one to verify, at first leading orders in the strong multifractality limit, the symmetry relation for anomalous fractal dimensions Δ(q)=Δ(1-q), recently conjectured for critical models where an analog of the metal-insulator transition takes place. It is known that this relation is verified at leading order in the weak multifractality regime. Our results thus indicate that this symmetry holds in both limits of small and large coupling constant. For general values of the coupling constant we present careful numerical verifications of this symmetry relation for different critical random-matrix ensembles. We also present an example of a system closely related to one of these critical ensembles, but where the symmetry relation, at least numerically, is not fulfilled.
[Show abstract][Hide abstract] ABSTRACT: We give an explicit parametrization of the set of mixed quantum states and of
the set of mixed classical states for a spin--1. Classical states are defined
as states with a positive Glauber-Sudarshan P-function. They are at the same
time the separable symmetric states of two qubits. We explore the geometry of
this set, and show that its boundary consists of a two-parameter family of
ellipsoids. The boundary does not contain any facets, but includes
straight-lines corresponding to mixtures of pure classical states.
Physical Review A 10/2011; 85(3). DOI:10.1103/PhysRevA.85.032101 · 2.99 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Fractal dimensions of eigenfunctions for various critical random matrix ensembles are investigated in perturbation series in the regimes of strong and weak multifractality. In both regimes, we obtain expressions similar to those of the critical banded random matrix ensemble extensively discussed in the literature. For certain ensembles, the leading-order term for weak multifractality can be calculated within standard perturbation theory. For other models, such a direct approach requires modifications, which are briefly discussed. Our analytical formulas are in good agreement with numerical calculations.
[Show abstract][Hide abstract] ABSTRACT: We study the game of go from a complex network perspective. We construct a
directed network using a suitable definition of tactical moves including local
patterns, and study this network for different datasets of professional
tournaments and amateur games. The move distribution follows Zipf's law and the
network is scale free, with statistical peculiarities different from other real
directed networks, such as e. g. the World Wide Web. These specificities
reflect in the outcome of ranking algorithms applied to it. The fine study of
the eigenvalues and eigenvectors of matrices used by the ranking algorithms
singles out certain strategic situations. Our results should pave the way to a
better modelization of board games and other types of human strategic scheming.
[Show abstract][Hide abstract] ABSTRACT: We propose new classes of random matrix ensembles whose statistical
properties are intermediate between statistics of Wigner-Dyson random matrices
and Poisson statistics. The construction is based on integrable N-body
classical systems with a random distribution of momenta and coordinates of the
particles. The Lax matrices of these systems yield random matrix ensembles
whose joint distribution of eigenvalues can be calculated analytically thanks
to integrability of the underlying system. Formulas for spacing distributions
and level compressibility are obtained for various instances of such ensembles.
[Show abstract][Hide abstract] ABSTRACT: Based on numerical and perturbation series arguments we conjecture that for certain critical random matrix models the information dimension of eigenfunctions D(1) and the spectral compressibility χ are related by the simple equation χ+D(1)/d=1, where d is system dimensionality.
[Show abstract][Hide abstract] ABSTRACT: In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer.
Math. Monthly 73, 1 (1966)] asked his famous question about the existence of
nonisometric billiards having the same spectrum of the Laplacian. This question
was eventually answered positively in 1992 by the construction of noncongruent
planar isospectral pairs. This review highlights mathematical and physical
aspects of isospectrality.
Review of Modern Physics 01/2011; 82(3). DOI:10.1103/RevModPhys.82.2213 · 42.86 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We study numerically multifractal properties of two models of one-dimensional quantum maps: a map with pseudointegrable dynamics and intermediate spectral statistics and a map with an Anderson-like transition recently implemented with cold atoms. Using extensive numerical simulations, we compute the multifractal exponents of quantum wave functions and study their properties, with the help of two different numerical methods used for classical multifractal systems (box-counting and wavelet methods). We compare the results of the two methods over a wide range of values. We show that the wave functions of the Anderson map display a multifractal behavior similar to eigenfunctions of the three-dimensional Anderson transition but of a weaker type. Wave functions of the intermediate map share some common properties with eigenfunctions at the Anderson transition (two sets of multifractal exponents, with similar asymptotic behavior), but other properties are markedly different (large linear regime for multifractal exponents even for strong multifractality, different distributions of moments of wave functions, and absence of symmetry of the exponents). Our results thus indicate that the intermediate map presents original properties, different from certain characteristics of the Anderson transition derived from the nonlinear sigma model. We also discuss the importance of finite-size effects.
[Show abstract][Hide abstract] ABSTRACT: We build a quantum algorithm which uses the Grover quantum search procedure in order to sample the exact equilibrium distribution of a wide range of classical statistical mechanics systems. The algorithm is based on recently developed exact Monte Carlo sampling methods, and yields a polynomial gain compared to classical procedures.