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.
We analyze several ground state related properties of mesoscopic systems using the random interaction matrix model EGOE(1+2)-$\cs$ (or RIMM) for many fermion systems with spin degree of freedom and the Hamiltonian containing pairing and exchange interactions in addition to the mean-field one-body and random two-body parts. RIMM reproduces the essential features of various properties: odd-even staggering in ground state energies as a function of particle number, delay in ground state magnetization and conductance peak spacing distributions. The analytical formula, we have derived, for the ensemble averaged spectral variances provides a simple understanding of some of these properties.
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.
It is shown that statistical mechanics is applicable to isolated quantum systems with finite numbers of particles, such as complex atoms, atomic clusters, or quantum dots in solids, where the residual two-body interaction is sufficiently strong. This interaction mixes the unperturbed shell-model (Hartree-Fock) basis states and produces chaotic many-body eigenstates. As a result, an interaction-induced statistical equilibrium emerges in the system. This equilibrium is due to the off-diagonal matrix elements of the Hamiltonian. We show that it can be described by means of temperature introduced through the canonical-type distribution. However, the interaction between the particles can lead to prominent deviations of the equilibrium distribution of the occupation numbers from the Fermi-Dirac shape. Besides that, the off-diagonal part of the Hamiltonian gives rise to an increase of the effective temperature of the system (statistical effect of the interaction). For example, this takes place in the cerium atom, which has four valence electrons and which is used in our work to compare the theory with realistic numerical calculations.
For a system of spinless fermions in a disordered mesoscopic ring, interactions can give rise to an enhancement of the persistent current by orders of magnitude. The increase in the current is associated with a charge reorganization of the ground state. The interaction strength for which this reorganization takes place is sample dependent and the logarithm averages over the ensemble are not representative. In this paper we demonstrate that the Hartree-Fock method closely reproduces results obtained by exact diagonalization. For spinless fermions subject to a short-range Coulomb repulsion U, we show that plots of phase sensitivity versus U exhibit sharp peaks associated with significant charge reorganization. Their positions are highly sample dependent and they disappear upon ensemble averaging. Having established that the Hartree-Fock method works well in one dimension, we present corresponding results for the phase sensitivity in two coupled chains.
We provide a framework for analyzing the problem of interacting electrons in a ballistic quantum dot with chaotic boundary conditions within an energy
ET (the Thouless energy) of the Fermi energy. Within this window we show that the interactions can be characterized by Landau Fermi-liquid parameters. When g, the Thouless number (also the dimensionless conductance when it has strong coupling to leads) of the dot, is large, we find that the disordered interacting problem can be solved in a saddle-point approximation which becomes exact as $\stackrel{\ensuremath{\rightarrow}}{g}\ensuremath{\infty}$ (as in a $\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\ensuremath{-}N$ theory). The infinite g theory has two phases as a function of the Landau parameter
um in a channel with angular momentum m: A weak-coupling phase where constant charging and exchange interactions dominate the low-energy physics, as in previous ``universal Hamiltonian'' treatments, and a strong-coupling phase characterized by the same order parameter as in the Pomeranchuk transition in clean systems (a spontaneous interaction-induced Fermi-surface distortion), but smeared and pinned by disorder. Thus, both interactions and disorder are crucial to the existence of these phases. At finite g, the two phases and critical point evolve into three regimes in the ${u}_{m}\ensuremath{-}1/g$ plane---weak- and strong-coupling regimes separated by crossover lines from a quantum-critical regime controlled by the quantum-critical point. In this, the first of a two part series, we focus on the consequences of this picture for Coulomb Blockade experiments. We employ analytical and numerical methods to predict the statistics of single-particle levels, Coulomb Blockade peak spacings, conductance peak heights, and quasiparticle widths. We show that in the strong-coupling and quantum-critical regions, the quasiparticle acquires a width of the same order as the level spacing $\ensuremath{\Delta}$ within a few $\ensuremath{\Delta}\mathrm{'}\mathrm{s}$ of the Fermi energy due to coupling to collective excitations. In the strong-coupling regime if m is odd, the dot will (if isolated) crossover from the orthogonal to unitary ensemble for an exponentially small external flux or will (if strongly coupled to leads) break time-reversal symmetry spontaneously. For any m, the peak-spacing distribution becomes broader than expected in previous works and even has support at negative values, which in turn is correlated with small peak heights. Ballistic/chaotic quantum dots afford us unrivalled theoretical and experimental control over the problem of simultaneous disorder and interactions due to the
1/g expansion and our ability to vary disorder and interaction much more readily than in the bulk.
For
m number of bosons, carrying spin ($\cs=\spin$) degree of freedom, in
Ω number of single particle orbitals, each doubly degenerate, we introduce and analyze embedded Gaussian orthogonal ensemble of random matrices generated by random two-body interactions that are spin (
S) scalar [BEGOE(2)-$\cs$]. Embedding algebra for the BEGOE(2)-$\cs$ ensemble and also for BEGOE(1+2)-$\cs$ that includes the mean-field one-body part is
U(2Ω)⊃U(Ω)⊗SU(2) with SU(2) generating spin. A method for constructing the ensembles in fixed-(
m,S) spaces has been developed. Numerical calculations show that for BEGOE(2)-$\cs$, the fixed-
(m,S) density of states is close to Gaussian and level fluctuations follow GOE in the dense limit. For BEGOE(1+2)-$\cs$, generically there is Poisson to GOE transition in level fluctuations as the interaction strength (measured in the units of the average spacing of the single particle levels defining the mean-field) is increased. The interaction strength needed for the onset of the transition is found to decrease with increasing
S. Covariances in energy centroids and spectral variances are analyzed. Propagation formula is derived for the variance propagator for the fixed-
(m,S) ensemble averaged spectral variances. Variance propagator clearly shows, by applying the Jacquod and Stone prescription, that the BEGOE(2)-$\cs$ ensemble generates ground states with spin
S=Smax. This is further corroborated by analyzing the structure of the ground states in the presence of the exchange interaction
S^2 in BEGOE(1+2)-$\cs$. Natural spin ordering is also observed with random interactions. Going beyond these, we also introduce pairing symmetry in the space defined by BEGOE(2)-$\cs$. Expectation values of the pairing Hamiltonian show that random interactions exhibit pairing correlations in the ground state region.