Laboratory of Theoretical Physics and Statistical Models

Through a combination of rigorous analytical derivations and extensive numerical simulations, this work reports an exotic multifractal behavior, dubbed "logarithmic multifractality", in effectively infinite-dimensional systems undergoing the Anderson transition. In marked contrast to conventional multifractal critical properties observed at finite-dimensional Anderson transitions or scale-invariant second-order phase transitions, in the presence of logarithmic multifractality, eigenstate statistics, spatial correlations, and wave packet dynamics can all exhibit scaling laws which are algebraic in the logarithm of system size or time. Our findings offer crucial insights into strong finite-size effects and slow dynamics in complex systems undergoing the Anderson transition, such as the many-body localization transition.

Progress in understanding quantum systems has been driven by the exploration of the geometry, topology, and dimensionality of ultracold atomic systems. The NASA Cold Atom Laboratory (CAL) aboard the International Space Station has enabled the study of ultracold atomic bubbles, a terrestrially-inaccessible topology. Proof-of-principle bubble experiments have been performed on CAL with an radiofrequency-dressing technique; an alternate technique (dual-species interaction-driven bubbles) has also been proposed. Both techniques can drive discovery in the next decade of fundamental physics research in microgravity.

We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where \frac{N\beta}{2} \to c N β 2 → c , called the high-temperature regime and show that it can be used to construct a promising one-parameter interpolation family, with parameter c between the classical and the free convolution. This c-convolution has a simple interpretation in terms of another associated family of distribution indexed by c, called the Markov-Krein transform: the c-convolution of two distributions corresponds to the classical convolution of their Markov-Krein transforms. We derive first cumulant-moment relations, a central limit theorem, a Poisson limit theorem and show several numerical examples of c-convoluted distributions.

We determine exactly the short-distance effective potential between two ‘guest’ charges immersed in a two-dimensional two-component charge-asymmetric plasma composed of positively (q 1 = +1) and negatively (q 2 = −1/2) charged point particles. The result is valid over the whole regime of stability, where the Coulombic coupling (dimensionless inverse temperature) β < 4. At high Coulombic coupling β > 2, this model features like-charge attraction. Also, there cannot be repulsion between opposite-charges at short-distances, at variance with large-distance interactions.

We study the distribution of the largest eigenvalue in the "Pfaffian" classical ensembles of random matrix theory, namely in the Gaussian orthogonal (GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian" identities due to Knuth, we explicitly construct these semi-classical skew-orthogonal polynomials in terms of the semi-classical orthogonal polynomials studied by NM in the case of the GUE. With these polynomials we obtain expressions for the cumulative distribution functions of the largest eigenvalue in the GOE and the GSE. Further, by performing asymptotic analysis of these skew-orthogonal polynomials in the limit of large matrix size, we obtain an alternative derivation of the Tracy-Widom distributions for GOE and GSE. This asymptotic analysis relies on a certain Pfaffian identity, the proof of which employs the characterization of Pfaffians in terms of perfect matchings and link diagrams.

Liquids displaying strong virial-potential energy correlations conform to an approximate density scaling of their structural and dynamical observables. This scaling property does not extend to the entire phase diagram, in general. The validity of the scaling can be quantified by a correlation coefficient. In this work a simple scheme to predict the correlation coefficient and the density-scaling exponent is presented. Although this scheme is exact only in the dilute gas regime or in high dimension d, a comparison with results from molecular dynamics simulations in d=1 to 4 shows that it reproduces well the behavior of generalized Lennard-Jones systems in a large portion of the fluid phase.

We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.

We consider the time evolution of a state in an isolated quantum spin lattice system with energy cumulants proportional to the number of the sites L^dLd. We compute the distribution of the eigenvalues of the time averaged state over a time window [t_0,t_0+t][t0,t0+t] in the limit of large L. This allows us to infer the size of a subspace that captures time evolution in [t_0,t_0+t][t0,t0+t] with an accuracy 1-\epsilon1−ϵ. We estimate the size to be \frac{\sqrt{2{\mathfrak e}_2}}{\pi}erf^{-1}(1-\epsilon) L^{\frac{d}{2}}t2𝔢2πerf−1(1−ϵ)Ld2t, where {\mathfrak e}_2𝔢2 is the energy variance per site, and erf^{-1}erf−1 is the inverse error function.

We find stochastic equations governing eigenvalues and eigenvectors of a dynamical complex Ginibre ensemble reaffirming the intertwined role played between both sets of matrix degrees of freedom. We solve the accompanying Smoluchowski–Fokker–Planck equation valid for any initial matrix. We derive evolution equations for the averaged extended characteristic polynomial and for a class of k-point eigenvalue correlation functions. From the latter we obtain a novel formula for the eigenvector correlation function which we inspect for Ginibre and spiric initial conditions and obtain macro- and microscopic limiting laws.

In this paper, we study the large n asymptotics of the expected maximum of an n-step random walk/Lévy flight (characterized by a Lévy index ) on a line, in the presence of a constant drift c. For , the expected maximum is infinite, even for finite values of n. For , we obtain all the non-vanishing terms in the asymptotic expansion of the expected maximum for large n. For c < 0 and , the expected maximum approaches a non-trivial constant as n gets large, while for , it grows as a power law ∼ . For c > 0, the asymptotic expansion of the expected maximum is simply related to the one for c < 0 by adding to the latter the linear drift term cn, making the leading term grow linearly for large n, as expected. Finally, we derive a scaling form interpolating smoothly between the cases c = 0 and . These results are borne out by numerical simulations in excellent agreement with our analytical predictions.