Quantum Many-particle Systems Research Articles

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless fermion (or boson) systems, with say m fermions (or bosons) in N single particle states and interacting via k-body interactions, we have EGUE(k) [embedded GUE of k-body interactions] with GUE embedding and the embedding algebra is U(N). A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have derived formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. Firstly, for a spinless fermion system, using EGUE(k) representation for a Hamiltonian that is k-body and an independent EGUE(t) representation for a transition operator that is t-body and employing the embedding U(N) algebra, finite-N formulas for moments up to order four are derived, for the first time, for the transition strength densities. Secondly, formulas for the moments up to order four are also derived for systems with two types of spinless fermions and a transition operator similar to beta decay and neutrinoless beta decay operators. In addition, moments formulas are also derived for a transition operator that removes k0 number of particles from a system of m spinless fermions. In the dilute limit, these formulas are shown to reduce to those for the EGOE version derived using the asymptotic limit theory of Mon and French (1975). Numerical results obtained using the exact formulas for two-body (k=2) Hamiltonians (in some examples for k=3 and 4) and the asymptotic formulas clearly establish that in general the smoothed (with respect to energy) form of the bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extensions of these results to bosonic systems and EGUE ensembles with further symmetries are discussed.

Group theory for embedded random matrix ensembles

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated quantum many-particle systems. For the simplest spinless fermion (or boson) systems with say m fermions (or bosons) in N single particle states and interacting with say k-body interactions, we have EGUE(k) [embedded GUE of k-body interactions) with GUE embedding and the embedding algebra is U(N). In this paper, using EGUE(k) representation for a Hamiltonian that is fc-body and an independent EGUE(t) representation for a transition operator that is t-body and employing the embedding U(N) algebra, finite-N formulas for moments up to order four are derived, for the first time, for the transition strength densities (transition strengths multiplied by the density of states at the initial and final energies). In the asymptotic limit, these formulas reduce to those derived for the EGOE version and establish that in general bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extension of these results for other types of transition operators and EGUE ensembles with further symmetries are discussed.

On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation

The rigorous derivation of the Uehling-Uhlenbeck equation from more fundamental quantum many-particle systems is a challenging open problem in mathematics. In this paper, we exam the weak coupling limit of quantum N-particle dynamics. We assume the integral of the microscopic interaction is zero and we assume W^{4,1} per-particle regularity on the coressponding BBGKY sequence so that we can rigorously commute limits and integrals. We prove that, if the BBGKY sequence does converge in some weak sense, then this weak-coupling limit must satisfy the infinite quantum Maxwell-Boltzmann hierarchy instead of the expected infinite Uehling-Uhlenbeck hierarchy, regardless of the statistics the particles obey. Our result indicates that, in order to derive the Uehling-Uhlenbeck equation, one must work with per-particle regularity bound below W^{4,1}.

“Light-Cone” Dynamics After Quantum Quenches in Spin Chains

Signal propagation in the nonequilibrium evolution after quantum quenches has recently attracted much experimental and theoretical interest. A key question arising in this context is what principles, and which of the properties of the quench, determine the characteristic propagation velocity. Here we investigate such issues for a class of quench protocols in one of the central paradigms of interacting many-particle quantum systems, the spin-1/2 Heisenberg XXZ chain. We consider quenches from a variety of initial thermal density matrices to the same final Hamiltonian using matrix product state methods. The spreading velocities are observed to vary substantially with the initial density matrix. However, we achieve a striking data collapse when the spreading velocity is considered to be a function of the excess energy. Using the fact that the XXZ chain is integrable, we present an explanation of the observed velocities in terms of "excitations" in an appropriately defined generalized Gibbs ensemble.

Dynamics of symmetry breaking during quantum real-time evolution in a minimal model system.

One necessary criterion for the thermalization of a nonequilibrium quantum many-particle system is ergodicity. It is, however, not sufficient in cases where the asymptotic long-time state lies in a symmetry-broken phase but the initial state of nonequilibrium time evolution is fully symmetric with respect to this symmetry. In equilibrium, one particular symmetry-broken state is chosen as a result of an infinitesimal symmetry-breaking perturbation. From a dynamical point of view the question is: Can such an infinitesimal perturbation be sufficient for the system to establish a nonvanishing order during quantum real-time evolution? We study this question analytically for a minimal model system that can be associated with symmetry breaking, the ferromagnetic Kondo model. We show that after a quantum quench from a completely symmetric state the system is able to break its symmetry dynamically and discuss how these features can be observed experimentally.

Focus on out-of-equilibrium dynamics in strongly interacting one-dimensional systems

In the past few years, there have been significant advances in understanding out-of-equilibrium dynamics in strongly interacting many-particle quantum systems. This is the case for 1D dynamics, where experimental advances—both with ultracold atomic gases and with solid state systems—have been accompanied by advances in theoretical methods, both analytical and numerical. This ‘focus on’ collection brings together 17 new papers, which together give a representative overview of the recent advances.

Absence of finite-temperature ballistic charge (and spin) transport in the 1D Hubbard model at half filling (and zero spin density)

Finite-temperature T > 0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is a prominent example of an integrable correlated system. For electronic densities n ≠ 1 (and spin densities m ≠ 0) it is an ideal charge (and spin) conductor, with ballistic charge (and spin) transport for T ⩾ 0. In spite of the fact that it is solvable by the Bethe ansatz, at n = 1 (and m = 0) its T > 0 charge (and spin) transport properties are an issue that remains poorly understood. Here we combine this solution with symmetry and the explicit calculation of current-operator matrix elements between energy eigenstates to show that for on-site repulsion U > 0 and at n = 1 the charge stiffness Dη(T) vanishes for T > 0 in the thermodynamic limit. A similar behavior is found by such methods for the spin stiffness Ds(T) for U > 0 and T > 0, which vanishes at m = 0. This absence of finite temperature n = 1 ballistic charge transport and m = 0 ballistic spin transport are exact results that clarify long-standing open problems.

Evolution of weakly correlated systems

A propagator with an effective action is constructed for a quantum many-particle system in the approximation taking into account only pair correlations between particles. The theory can easily be generalized to the case of two species of particles. In the case of a rarefied plasma and a gas, this propagator for a one-particle density matrix provides a solution to the kinetic problem over time intervals longer than the relaxation time.

Hydrodynamic long-time tails after a quantum quench

After a quantum quench, a sudden change of parameters, generic many-particle quantum systems are expected to equilibrate. A few collisions of quasiparticles are usually sufficient to establish approximately local equilibrium. Reaching global equilibrium is, however, much more difficult as conserved quantities have to be transported for long distances to build up a pattern of fluctuations characteristic for equilibrium. Here we investigate the quantum quench of the one-dimensional bosonic Hubbard model from infinite to finite interaction strength $U$ using semiclassical methods for weak and exact diagonalization for strong quenches. Equilibrium is approached only slowly, as ${t}^{\ensuremath{-}1/2}$ with subleading corrections proportional to ${t}^{\ensuremath{-}3/4}$, consistent with predictions from hydrodynamics. We show that these long-time tails determine the relaxation of a wide range of physical observables.

Expectation value based equation-of-motion approach for open quantum systems: A general formalism

We present a new method to formulate equations of motion for open quantum many-particle systems. Our approach allows for a numerically exact treatment as well as for approximations necessary in large systems and can be applied to systems involving both bosonic and fermionic particles. The method generalizes the cluster expansion technique by using expectation values instead of correlation functions. The use of expectation values not only makes the equations more transparent but also considerably reduces the amount of algebraic effort to derive the equations. The proposed formulation offers a unified view on various approximation techniques presented recently in the literature. The microscopic semiconductor model for quantum-dot-based microcavity lasers is extended to higher-order photon-autocorrelation functions and the validity of the cluster expansion is shown for this system. We study photon-autocorrelation functions up to fifth order and monitor the onset of lasing in quantum-dot-based microcavity lasers. We observe a successive vanishing of photon bunching in the higher-order photon-autocorrelation functions with increasing pump rates. Our results reveal that the laser threshold is not only softened in microcavity laser systems but is centered around different pump rates with respect to the photon-autocorrelation functions.

Nonequilibrium configuration interaction method for transport in correlated quantum systems

We present a new approach to treat correlations in nonequilibrium quantum many-particle system. The method is based on ideas of configuration interaction theory of exact nonperturbative ground state electronic structure calculations. We use superoperator techniques in Liouville–Fock space and represent the nonequilibrium density matrix as a linear combination of all possible nonequilibrium quasiparticle excitations built on the appropriate reference state. As an example we consider the electron transport through the system with electron–phonon interaction. The concept of embedding (buffer zones between the reservoirs and the correlated quantum system) is used to derive an exact master equation for the reduced density matrix. Using approximate (truncated) expansion of the trial density matrix we obtain the linear system of equations for two-quasiparticle amplitudes. Then we compute the steady-state current and compare the result with other approaches. The current conserving property of the method is proven.

Mechanism of collisionless sound damping in dilute Bose gas with condensate

We develop a microscopic theory of sound damping due to Landau mechanism in dilute gas with Bose condensate. It is based on the coupled evolution equations of the parameters describing the system. These equations have been derived in earlier works within a microscopic approach which employs the Peletminskii-Yatsenko reduced description method for quantum many-particle systems and Bogoliubov model for a weakly nonideal Bose gas with a separated condensate. The dispersion equations for sound oscillations were obtained by linearization of the mentioned evolution equations in the collisionless approximation. They were analyzed both analytically and numerically. The expressions for sound speed and decrement rate were obtained in high and low temperature limiting cases. We have shown that at low temperature the dependence of the obtained quantities on temperature significantly differs from those obtained by other authors in the semi-phenomenological approaches. Possible effects connected with non-analytic temperature dependence of dispersion characteristics of the system were also indicated.

Inequivalence of Canonical and Grand Canonical Ensembles for Bosonic Systems

For many-particle quantum systems, calculating thermodynamic quantities in the canonical ensemble is a very hard task, while this is tractable in the grand canonical ensemble. The second ensemble is then used. The results are supposed to be the same, at least in the thermodynamic limit. Is this actually the case? In this work, we consider a system of N noninteracting bosons distributed among few energy levels. We can calculate the canonical partition function in this case and deduce the canonical mean energy. We compare it to the mean energy deduced from the grand canonical ensemble for the same number of particles. We consider the case of a large number of particles.

Directed-polymer systems explored via their quantum analogs: Topological constraints and their consequences

The equilibrium statistical mechanics of classical directed polymers in two dimensions is well known to be equivalent to the imaginary-time quantum dynamics of a 1+1-dimensional many-particle system, with polymer configurations corresponding to particle world-lines. This equivalence motivates the application of techniques originally designed for one-dimensional many-particle quantum systems to the exploration of many-polymer systems, as first recognized and exploited by de Gennes [J. Chem. Phys. 48, 2257 (1968)]. In this low-dimensional setting interactions give rise to an emergent polymer fluid, and we examine how topological constraints on this polymer fluid (e.g., due to uncrossable pins or barriers) and their geometry give rise to strong, entropy-driven forces. In the limit of large polymer densities, in which a type of mean-field theory is accurate, we find that a pointlike pin causes a divergent pileup of polymer density on the high-density side of the pin and a zero-density region (or gap) of finite area on the low-density side. In addition, we find that the force acting on a pin that is only mildly displaced from its equilibrium position is sub-Hookean, growing less than linearly with the displacement, and that the gap created by the pin also grows sublinearly with the displacement. By contrast, the forces acting between multiple pins separated along the direction preferred by the polymers are super-Hookean. These nonlinear responses result from effective long-ranged interactions between polymer segments, which emerge via short-ranged interactions between distant segments of long polymer strands. In the present paper, we focus on the case of an infinitely strong, repulsive contact interaction, which ensures that the polymers completely avoid one another. In a companion paper, we consider the effects of a wider set of interpolymer interactions.

Some problems of the theory of quantum statistical systems with an energy spectrum of the fractional-power type

We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter α, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of α. We consider Bose—Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for 1 < α < 2 is greater in the case under consideration than in the case of an ideal system, where α = 2.

On quantum kinetic equations of many-particle systems in condensed states

We develop a rigorous formalism for the description of the evolution of states of quantum many-particle systems in terms of a one-particle density operator, which enables us to construct the kinetic equations in scaling limits in the presence of correlations of particle states at initial time, for instance, correlations characterizing the condensed states.

Erratum: Quantification of Correlations in Quantum Many-Particle Systems [Phys. Rev. Lett.108, 087004 (2012)

Received 12 March 2012DOI:https://doi.org/10.1103/PhysRevLett.108.189902© 2012 American Physical Society

Entanglement Entropy of Fermi Liquids via Multidimensional Bosonization

Highly deft technical work by a group of three theoretical physicists leads to much needed fundamental understanding of the ``area law'' governing the entanglement entropy of a many-particle quantum system and shows that the leading form of the law survives the addition of particle-particle interactions.

Quantification of Correlations in Quantum Many-Particle Systems

We introduce a well-defined and unbiased measure of the strength of correlations in quantum many-particle systems which is based on the relative von Neumann entropy computed from the density operator of correlated and uncorrelated states. The usefulness of this general concept is demonstrated by quantifying correlations of interacting electrons in the Hubbard model and in a series of transition-metal oxides using dynamical mean-field theory.

The FermiFab toolbox for fermionic many-particle quantum systems

This paper introduces the FermiFab toolbox for many-particle quantum systems. It is mainly concerned with the representation of (symbolic) fermionic wavefunctions and the calculation of corresponding reduced density matrices (RDMs). The toolbox transparently handles the inherent antisymmetrization of wavefunctions and incorporates the creation/annihilation formalism. Thus, it aims at providing a solid base for a broad audience to use fermionic wavefunctions with the same ease as matrices in Matlab, say. Leveraging symbolic computation, the toolbox can greatly simply tedious pen-and-paper calculations for concrete quantum mechanical systems, and serves as “sandbox” for theoretical hypothesis testing. FermiFab (including full source code) is freely available as a plugin for both Matlab and Mathematica. Program summary Program title: FermiFab Catalogue identifier: AEIN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEIN_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Special license provided by the author No. of lines in distributed program, including test data, etc.: 1 165 461 No. of bytes in distributed program, including test data, etc.: 15 557 308 Distribution format: tar.gz Programming language: MATLAB 7.9, Mathematica 7.0, C Computer: PCs, Sun Solaris workstation Operating system: Any platform supporting MATLAB or Mathematica; tested with Windows (32 and 64 bit) and Sun Solaris. RAM: Case dependent Classification: 4.15 Nature of problem: Representation of fermionic wavefunctions, computation of RDMs (reduced density matrices) and handing of the creation/annihilation operator formalism. Solution method: Mapping of Slater determinants to bitfields, implementation of the creation/annihilation and RDM formalism by bit operations. Running time: Depends on the problem size; several seconds for the provided demonstration files.