The extreme-value statistics of the entanglement spectrum in disordered spin chains possessing a many-body localization transition is examined. It is expected that eigenstates in the metallic or ergodic phase, behave as random states and hence the eigenvalues of the reduced density matrix, commonly referred to as the entanglement spectrum, are expected to follow the eigenvalue statistics of a trace normalized Wishart ensemble. In particular, the density of eigenvalues is supposed to follow the universal Marchenko-Pastur distribution. We find deviations in the tails both for the disordered XXZ with total
Sz conserved in the half-filled sector as well as in a model that breaks this conservation. A sensitive measure of deviations is provided by the largest eigenvalue, which in the case of the Wishart ensemble after appropriate shift and scaling follows the universal Tracy-Widom distribution. We show that for the models considered, in the metallic phase, the largest eigenvalue of the reduced density matrix of eigenvector, instead follows the generalized extreme-value statistics bordering on the Fisher-Tipett-Gumbel distribution indicating that the correlations between eigenvalues are much weaker compared to the Wishart ensemble. We show by means of distributions conditional on the high entropy and normalized participation ratio of eigenstates that the conditional entanglement spectrum still follows generalized extreme value distribution. In the deeply localized phase we find heavy tailed distributions and Lévy stable laws in an appropriately scaled function of the largest and second largest eigenvalues. The scaling is motivated by a recently developed perturbation theory of weakly coupled chaotic systems.
We theoretically study quantum spin transport in a one-dimensional folded XXZ model with an alternating domain-wall initial state via the Bethe ansatz technique, exactly demonstrating that a probability distribution of finding a leftmost up spin with an appropriate scaling variable converges to the Tracy-Widom distribution for the Gaussian unitary ensemble (GUE), which is a universal distribution for the largest eigenvalue of GUE under a soft-edge scaling limit. Our finding presented here offers a first exact derivation of the GUE Tracy-Widom distribution in the dynamics of the interacting quantum model not being mapped to noninteracting fermions via the Jordan-Wigner transformation. On the basis of the exact solution of the folded XXZ model and our numerical analysis of the XXZ model, we discuss a universal behavior for the probability of finding the leftmost up spin in the XXZ model.
We study a system of
N noninteracting spinless fermions trapped in a confining potential, in arbitrary dimensions
d and arbitrary temperature
T. The presence of the confining trap breaks the translational invariance and introduces an edge where the average density of fermions vanishes. Far from the edge, near the center of the trap (the so-called ``bulk regime''), where the fermions do not feel the curvature of the trap, physical properties of the fermions have traditionally been understood using the local density (or Thomas-Fermi) approximation. However, these approximations drastically fail near the edge where the density vanishes and thermal and quantum fluctuations are thus enhanced. The main goal of this paper is to show that, even near the edge, novel universal properties emerge, independently of the details of the shape of the confining potential. We present a unified framework to investigate both the bulk and the edge properties of the fermions. We show that for large
N, these fermions in a confining trap, in arbitrary dimensions and at finite temperature, form a determinantal point process. As a result, any
n-point correlation function, including the average density profile, can be expressed as an $n\ifmmode\times\else\texttimes\fi{}n$ determinant whose entry is called the kernel, a central object for such processes. Near the edge, we derive the large-
N scaling form of the kernels, parametrized by
d and
T. In
d=1 and
T=0, this reduces to the so-called Airy kernel, that appears in the Gaussian unitary ensemble (GUE) of random matrix theory. In
d=1 and
T>0 we show a remarkable connection between our kernel and the one appearing in the (
1+1)-dimensional Kardar-Parisi-Zhang equation at finite time. Consequently, our result provides a finite-
T generalization of the Tracy-Widom distribution, that describes the fluctuations of the position of the rightmost fermion at
T=0, or those of the largest single-fermion momentum. In
d>1 and $T\ensuremath{\ge}0$, while the connection to GUE no longer holds, the process is still determinantal whose analysis provides a new class of kernels, generalizing the
1d Airy kernel at
T=0 obtained in random matrix theory. Some of our finite-temperature results should be testable in present-day cold-atom experiments, most notably our detailed predictions for the temperature dependence of the fluctuations near the edge.
Random matrix ensembles are introduced that respect the local tensor structure of Hamiltonians describing a chain of
n distinguishable spin-half particles with nearest-neighbour interactions. We prove a central limit theorem for the density of states when
n→∞, giving explicit bounds on the rate of approach to the limit. Universality within a class of probability measures and the extension to more general interaction geometries are established. The level spacing distributions of the Gaussian Orthogonal, Unitary and Symplectic Ensembles are observed numerically for the energy levels in these ensembles.
We study the edge behavior of inhomogeneous one-dimensional quantum systems, such as Lieb-Liniger models in traps or spin chains in spatially varying fields. For free systems these fall into several universality classes, the most generic one being governed by the Tracy-Widom distribution. We investigate in this paper the effect of interactions. Using semiclassical arguments, we show that since the density vanishes to leading order, the strong interactions in the bulk are renormalized to zero at the edge, which simply explains the survival of Tracy-Widom scaling in general. For integrable systems, it is possible to push this argument further, and determine exactly the remaining length scale which controls the variance of the edge distribution. This analytical prediction is checked numerically, with excellent agreement. We also study numerically the edge scaling at fronts generated by quantum quenches, which provide new universality classes awaiting theoretical explanation.
It is often expected (and assumed) for a quantum chaotic system that the presence of correlated eigenvalues implies that all the other properties as dictated by random matrix theory are satisfied. We demonstrate using the spin-
1/2 kicked field Ising model that this is not necessarily true. We study the properties of eigenvalues of the reduced density matrix for this model, which constitutes the entanglement spectrum. It is shown that the largest eigenvalue does not follow the expected Tracy--Widom distribution even for the large system sizes considered. The distribution instead follows the extreme value distribution of Weibull type. Furthermore, we also show that such deviations do not lead to drastic change in the thermalization property of this system by showing that the models satisfy the diagonal and off-diagonal eigenstate thermalization hypothesis. Finally, we study the spin-spin autocorrelation function and numerically show that it has the characteristic behavior for chaotic systems: it decreases exponentially and saturates to a value at late time that decreases with system size.