We study the structure of eigenstates in two-body interaction random matrix ensembles and find significant deviations from random matrix theory expectations. The deviations are most prominent in the tails of the spectral density and indicate localization of the eigenstates in Fock space. Using ideas related to scar theory we derive an analytical formula that relates fluctuations in wave function intensities to fluctuations of the two-body interaction matrix elements. Numerical results for many-body fermion systems agree well with the theoretical predictions.
We study the structure of eigenstates in two-body interaction random matrix ensembles and find significant deviations from random matrix theory expectations. The deviations are most prominent in the tails of the spectral density and indicate localization of the eigenstates in Fock space. Using ideas related to scar theory we derive an analytical formula that relates fluctuations in wave function intensities to fluctuations of the two-body interaction matrix elements. Numerical results for many-body fermion systems agree well with the theoretical predictions.
Random matrix ensembles defined by a mean-field one body plus a chaos generating random two-body interaction [called embedded ensembles of (1+2)-body interactions] predict for wave functions, in the chaotic domain, an essentially one-parameter Gaussian forms for the energy dependence of the number of principal components (NPC) and the localization length l(H) (defined by information entropy), which are two important measures of chaos in finite interacting many-particle systems. Numerical embedded ensemble calculations and nuclear shell-model results, for NPC and l(H), are compared with the theory. These analyses clearly point out that for realistic finite interacting many-particle systems, in the chaotic domain, wave-function structure is given by (1+2)-body embedded random matrix ensembles.
Embedded random matrix ensembles generated by random interactions (of low body rank and usually two-body) in the presence of a one-body mean field, introduced in nuclear structure physics, are now established to be indispensable in describing statistical properties of a large number of isolated finite quantum many-particle systems. Lie algebra symmetries of the interactions, as identified from nuclear shell model and the interacting boson model, led to the introduction of a variety of embedded ensembles (EEs). These ensembles with a mean field and chaos generating two-body interaction generate in three different stages, delocalization of wave functions in the Fock space of the mean-field basis states. The last stage corresponds to what one may call thermalization and complex nuclei, as seen from many shell model calculations, lie in this region. Besides briefly describing them, their recent applications to nuclear structure are presented and they are (i) nuclear level densities with interactions; (ii) orbit occupancies; (iii) neutrinoless double beta decay nuclear transition matrix elements as transition strengths. In addition, their applications are also presented briefly that go beyond nuclear structure and they are (i) fidelity, decoherence, entanglement and thermalization in isolated finite quantum systems with interactions; (ii) quantum transport in disordered networks connected by many-body interactions with centrosymmetry; (iii) semicircle to Gaussian transition in eigenvalue densities with [Formula: see text]-body random interactions and its relation to the Sachdev–Ye–Kitaev (SYK) model for majorana fermions.
There is a newly emerging understanding that in the chaotic domain of isolated finite interacting many particle systems smoothed densities define the statistical description of these systems and these densities follow from embedded (two-body) random matrix ensembles and their various deformations. These ensembles predict that the smoothed form of matrix elements of a transition operator between the chaotic eigenstates weighted by the densities at the two ends (i.e. the bivariate strength density) will be a bivariate Gaussian with the bivariate correlation coefficient arising out of the non-commutability of the hamiltonian and the transition operator involved. The ensemble theory extends to systems with a mean-field and a chaos generating two-body interaction (as in nuclei, atoms and diffusive quantum dots). These developments in many-body quantum chaos are described with special reference to one-body transition operators.
We investigate the spin structure of many-fermion systems with a spin-conserving two-body random interaction. We find a strong dominance of spin-0 ground states and considerable correlations between energies and wave functions of low-lying states with different spin, but no indication of pairing. The spectral densities exhibit spin-dependent shapes and widths, and depend on the particle density as well as on the relative strengths of the spin-0 and spin-1 couplings in the two-body random matrix. The spin structure of low-lying states can largely be explained analytically.
We introduce a generic approach to study interaction effects in diffusive or chaotic quantum dots in the Coulomb blockade regime. The randomness of the single-particle wave functions induces randomness in the two-body interaction matrix elements. We classify the possible induced two-body ensembles, both in the presence and absence of spin degrees of freedom. The ensembles depend on the underlying space-time symmetries as well as on features of the two-body interaction. Confining ourselves to spinless electrons, we then use the Hartree-Fock (HF) approximation to calculate HF single-particle energies and HF wave functions for many realizations of the ensemble. We study the statistical properties of the resulting one-body HF ensemble for a fixed number of electrons. In particular, we determine the statistics of the interaction matrix elements in the HF basis, of the HF single-particle energies (including the HF gap between the last occupied and the first empty HF level), and of the HF single-particle wave functions. We also study the addition of electrons, and in particular the distribution of the distance between successive conductance peaks and of the conductance peak heights.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.