Many-body Hilbert Space Research Articles

Emergent exclusion statistics of quasiparticles in two-dimensional topological phases

We demonstrate how the generalized Pauli exclusion principle emerges for quasiparticle excitations in 2d topological phases. As an example, we examine the Levin-Wen model with the Fibonacci data (specified in the text), and construct the number operator for fluxons living on plaquettes. By numerically counting the many-body states with fluxon number fixed, the matrix of exclusion statistics parameters is identified and is shown to depend on the spatial topology (sphere or torus) of the system. Our work reveals the structure of the (many-body) Hilbert space and some general features of thermodynamics for quasiparticle excitations in topological matter.

Generalizing the self-healing diffusion Monte Carlo approach to finite temperature: A path for the optimization of low-energy many-body bases

A statistical method is derived for the calculation of thermodynamic properties of many-body systems at low temperatures. This method is based on the self-healing diffusion Monte Carlo method for complex functions [F. A. Reboredo, J. Chem. Phys. 136, 204101 (2012)] and some ideas of the correlation function Monte Carlo approach [D. M. Ceperley and B. Bernu, J. Chem. Phys. 89, 6316 (1988)]. In order to allow the evolution in imaginary time to describe the density matrix, we remove the fixed-node restriction using complex antisymmetric guiding wave functions. In the process we obtain a parallel algorithm that optimizes a small subspace of the many-body Hilbert space to provide maximum overlap with the subspace spanned by the lowest-energy eigenstates of a many-body Hamiltonian. We show in a model system that the partition function is progressively maximized within this subspace. We show that the subspace spanned by the small basis systematically converges towards the subspace spanned by the lowest energy eigenstates. Possible applications of this method for calculating the thermodynamic properties of many-body systems near the ground state are discussed. The resulting basis can also be used to accelerate the calculation of the ground or excited states with quantum Monte Carlo.

Time-dependent multiconfiguration methods for the numerical simulation of photoionization processes of many-electron atoms

Numerical simulations present an indispensable way to the understanding of complex physical processes. In quantum mechanics where the theoretical description is given in terms of the time-dependent Schrodinger equation the road is, however, difficult for any but the simplest systems. This is particularly true if one considers photoionization processes of atoms and molecules which, at the same time, require an accurate description of bound and continuum states and, therefore, an extensive region of space to be sampled during the calculation. As a consequence, direct simulations of photoionization processes are currently only feasible for systems containing up to three electrons. Despite this fundamental restriction, many physical effects can be essentially described by single- and two-electron models, among them are high-order harmonic generation and non-sequential double-ionization of atoms and molecules. A plethora of numerical investigations have been performed on atomic and molecular hydrogen and helium in the last two decades, and these have had a strong impact on the current understanding of photoionization. On the other hand, there are processes which are characterized by the interplay of a larger number of electrons, such as tunnel ionization, the Auger effect and, to give a more recent example, the temporal delay between the photo-emission of electrons from different shells of neon and krypton. The many-electron character of these effects complicates the accurate, time-resolved simulation and, so far, no universally applicable method exists. This review presents two theoretical methods which are promising candidates for closing this gap–the multiconfigurational time-dependent Hartree-Fock (MCTDHF) method and the time-dependent restricted active space configuration interaction (TD-RASCI) method. Both represent the wavefunction in a linear subspace of the many-body Hilbert space and follow particular strategies to avoid the exponential problem. This makes it possible to treat a much larger number of electrons than with the direct techniques mentioned above.

Clusters, Halos, And S-Factors In Fermionic Molecular Dynamics*

In Fermionic Molecular Dynamics antisymmetrized products of Gaussian wave packets are projected on angular momentum, linear momentum, and parity. An appropriately chosen set of these states span the many-body Hilbert space in which the Hamiltonian is diagonalized. The wave packet parameters - position, momentum, width and spin - are obtained by variation under constraints. The great flexibility of this basis allows to describe not only shell-model like states but also exotic states like halos, e.g. the two-proton halo in 17Ne, or cluster states as they appear for example in 12C close to the \alpha-breakup threshold where the Hoyle state is located. Even a fully microscopic calculation of the 3He(\alpha,\gamma)7Be capture reaction is possible and yields an astrophysical S-factor that compares very well with newer data. As representatives of numerous results these cases will be discussed in this contribution, some of them not published so far. The Hamiltonian is based on the realistic Argonne V18 nucleon-nucleon interaction.

Deterministic preparation of Dicke states of donor nuclear spins in silicon by cooperative pumping

For donor nuclear spins in silicon, we show how to deterministically prepare various symmetric and asymmetric Dicke states which span a complete basis of the many-body Hilbert space. The state preparation is realized by cooperative pumping of nuclear spins by coupled donor electrons, and the required controls are in situ to the prototype Kane proposal for quantum computation. This scheme only requires a subgigahertz donor exchange coupling, which can be readily achieved without atomically precise donor placement, and hence it offers a practical way to prepare multipartite entanglement of spins in silicon with current technology. All desired Dicke states appear as the steady state under various pumping scenarios, and therefore the preparation is robust and does not require accurate temporal controls. Numerical simulations with realistic parameters show that Dicke states of 10--20 qubits can be prepared with high fidelity in the presence of decoherence and unwanted dynamics.

Microscopic Calculation of theHe3(α,γ)Be7andH3(α,γ)Li7Capture Cross Sections Using Realistic Interactions

The radiative capture cross sections for the 3He(alpha,gamma)7Be and 3H(alpha,gamma)7Li reactions are calculated in the fully microscopic fermionic molecular dynamics approach using a realistic effective interaction that reproduces the nucleon-nucleon scattering data. At large distances bound and scattering states are described by antisymmetrized products of 4He and 3He/3H ground states. At short distances the many-body Hilbert space is extended with additional many-body wave functions needed to represent polarized clusters and shell-model-like configurations. Properties of the bound states are described well, as are the scattering phase shifts. The calculated S factor for the 3He(alpha,gamma)7Be reaction agrees very well with recent experimental data both in absolute normalization and energy dependence. In case of the 3H(alpha,gamma)7Li reaction the calculated S factor is larger than available experimental data by about 15%.

Mapping the braiding properties of the Moore-Read state

In this paper we explore the braiding properties of the Moore-Read fractional Hall sequence, which amounts to computing the adiabatic evolution of the Hall liquid when the anyons are moved along various trajectories. In this work, the anyons are pinned to precise spatial configurations by using specific external potentials. Such external potentials break the translational symmetry and it appears that one will be forced to simulate the braidings on the entire many-body Hilbert space, an absolutely prohibitive scenario. We demonstrate how to overcome this difficulty and obtain the exact braidings for fairly large Hall systems. For this, we show that the incompressible state of a general $(k,m)$ fractional Hall sequence can be viewed as the unique zero mode of a specific Hamiltonian $H^{(k,m)}$, whose form is explicitly derived by using k-particles creation operators. The compressible Hall states corresponding to $n$$\times$$k$ anyons fixed at $w_1$,...,$w_{nk}$ are shown to be the zero modes of a pinning Hamiltonian $H^{(k,m)}_{w_1,...,w_{nk}}$, which is also explicitly derived. The zero modes of $H^{(k,m)}_{w_1,...,w_{nk}}$ are shown to be contained in the space of the zero modes of $H^{(k,m)}$. Therefore, the computation of the braidings can be done entirely within this space, which we map out for a number of Hall systems. Using this efficient computational method, we study various properties of the Moore-Read states. In particular, we give direct confirmation of their topological and non-abelian properties that were previously implied from the underlying Conformal Field Theory (CFT) structure of the Moore-Read state.

Optical spectra of charged anisotropic quantum boxes

We have developed a numerical scheme for exact diagonalization of the few-particle electron–hole interaction Hamiltonians with a simultaneous resolution of several discrete conserved quantum numbers such as total electron or hole spins, total angular momentum, or parity. The method was applied to a simple model of a two-dimensional rectangular quantum box of arbitrary size and anisotropy, and containing either several electrons or several electrons and a single valence hole. In this model, effects of box size and anisotropy on the photoluminescence spectra are analyzed. Compared to earlier calculations, we were able to include enough single-particle shells in the many-body Hilbert space to observe the transitions between strong and weak spatial quantization regimes.

Joule Expansion of a Pure Many-Body State

We derive the Joule expansion of an isolated perfect gas from the principles of quantum mechanics. Contrary to most studies of irreversible processes which consider composite systems, the interesting and ignored degrees of freedom are here described by operators acting in the same many-body Hilbert space. Moreover, the expansion of the gas into the entire accessible volume is obtained for pure states. Still, the number particle density is characterized by a chemical potential and a temperature. We discuss the special case of a boson gas below the Bose condensation temperature.

Dynamical1/Napproach to time-dependent currents through quantum dots

A systematic truncation of the many-body Hilbert space is implemented to study how electrons in a quantum dot attached to conducting leads respond to time-dependent biases. The method, which we call the dynamical 1/N approach, is first tested in the most unfavorable case, the case of spinless fermions (N=1). We recover the expected behavior, including transient ringing of the current in response to an abrupt change of bias. We then apply the approach to the physical case of spinning electrons, N=2, in the Kondo regime for the case of infinite intradot Coulomb repulsion. In agreement with previous calculations based on the non-crossing approximation (NCA), we find current oscillations associated with transitions between Kondo resonances situated at the Fermi levels of each lead. We show that this behavior persists for a more realistic model of semiconducting quantum dots in which the Coulomb repulsion is finite.

A Workstation Farm Optimized for Monte Carlo Shell Model Calculations : Alphleet

The Quantum Monte Carlo Diagonalization (QMCD) method which is heart of the MCSM, has a peculiar characteristic as converting the original large matrix into a small one giving nearly equal eigenvalues by means of transforming basis vectors, as mentioned by another paper on this proceedings. 1) In the QMCD, most important basis vectors of the many-body Hilbert space are searched in a stochastic way, and the Hamiltonian is diagonalized with respect to those basis vectors. Certainly, it takes almost all cpu time to calculate each Hamiltonian matrix element with many trial basis vectors to find out the (nearly) best transformation. This characteristic is suitable to implement on parallel calculation with using many CPUs, especially on loosely coupled system because of the calculation of each matrix element can be divided into individual segments not talking among themselves. We decided to make such a computer system as known as “cluster” or “farm” by ourselves because in absence of the computer system which realize such simple parallelism in the market. Every parallel computer in the market is so sophisticate, that is, too expensive for our purpose. Performance of floating point calculation is also so crucial. Vector CPUs, state of the art architecture for fastest floating point calculation, are however not suitable for our calculation which does not treat vector long enough for the CPUs. We chose the Alpha 21264 which has highest performance of the floating point calculation (58.7 SPECfp95 at 500 MHz) as scaler CPU about 4 times better than most popular CPU in the commodity market. The workstation farm that we have built is named “Alphleet”, which consists of many “Alpha” CPUs like as a “Fleet”. Figure 1 shows a schematic diagram and an actual photo (half nodes) of the Alphleet. It consists of a server and 70 clients. It is our goal of the system to maximize performance for the MCSM calculation and to minimize management task. The latter means that easy management against such a large scale system is another big issue for non-professionals of computer itself

A GENERALIZATION OF QUANTUM STATISTICS

We examine a new fractional statistics for arbitrary dimensions, based on an extension of Pauli’s exclusion principle, to allow for finite multi-occupancies of a single quantum state. By explicitly constructing the many-body Hilbert space, we obtain a new algebra of operators and a new thermodynamics. The new statistics is different from fractional exclusion statistics. For a single species of particles it is equivalent to parafermi statistics; but their algebras are very different for the case of multiple species.

Haldane's fractional statistics and the lowest Landau level on a torus.

The lowest Landau level on a torus is studied. The dimension of the many-body Hilbert space is obtained and is found to be different from the formula given by Haldane. Our result can be tested in numerical investigations of the low-energy spectrum of fractional quantum Hall states on a torus.

A microscopic approach to nuclear reactions: General formalism

In this first of two successive papers we present a microscopic model for nuclear reactions. It allows the description of elastic and inelastic nucleon scattering as well as photonuclear reactions with real and virtual photons. The model is based on a particular classification scheme of the many-body Hilbert space according to the number of continuum nucleons. By its very definition it is restricted to processes with at most one continuum particle in the entrance and/or exit channel. In a large model space, containing all single-particle resonances as bound states, nuclear wave functions are calculated by means of the structure model MONSTER(VAMPIR). Arbitrary general effective two-body forces may be used for the structure calculations as well as for the coupling of the resulting highly correlated wave functions to the continuum. It is the coupling to a modified resonance free continuum in the framework of a general coupled-channel calculation which allows the description of the intermediate structure of the nuclear continuum. In the second paper results obtained from an application to the reaction 29Si( e, e′x i ) 27Al will be presented, while in a forthcoming paper the particular role of a fifth response function arising from polarized electron scattering will be discussed.

Dynamic structure and embedded representation in physics: The group theory of the adiabatic approximation

The group theoretical concepts of embedded representations and dynamical structure groups, distinct from dynamical symmetry groups, are introduced in order to describe the common physical situation in which collective bands of states of a many-body system are well described by an algebraic collective model even though the states may not span an invariant subspace of the many-body Hilbert space.

The Path Integral Approach for the Nuclear Collective Motion

A procedure is developed for treating a nuclear collective motion via the path integral formalism by defining a collective propagator in the "phase space" of the collective subspace. This collective subspace is projected out of the full many-body Hilbert space by an approach of the Generator Coordinate Method. Under the approximation of quadratic "momentum" dependence we write the propagator in the coordinate representation from which we derive the collective Schrödinger equation in terms of a canonical pair P̂ and Q̂.

Relationship between the semiclassical and the generator coordinate methods

Using a very simple type of wave packet, which is obtained by letting unitary displacement operators having as generators canonical operators Q and P in the many-body Hilbert space act on a reference state, we investigate the relationship between the semiclassical and the generator coordinate methods. The semiclassical method is based on the time-dependent variational principle, whereas in the generator coordinate method the wave packets are taken as generator states. To establish the equivalence of the two methods, we examine in detail, using tools developed in previous works, the concept of redundancy of the wave packet and the importance of the zero-point energy effects. We make a numerical application to the case of the Goldhaber-Teller mode in /sup 4/He.

Properties of Griffin-Hill-Wheeler spaces. II. One-parameter and two-conjugate-parameter families of generator states

The properties of the subspaces of the many-body Hilbert space, which are associated with the use of the generator coordinate method in connection with one-parameter and with two-conjugate-parameter families of generator states, are examined in detail. These families are obtained by letting unitary displacement operators, having as generators canonical operators $\stackrel{^}{P}$ and $\stackrel{^}{Q}$, defined in the many-body Hilbert space, act on a reference state. We show that natural orthonormal base vectors in each case are immediately related to Peierls-Yoccoz and Peierls-Thouless projections, respectively. Through the formal consideration of a canonical transformation to collective, $\stackrel{^}{P}$ and $\stackrel{^}{Q}$, and intrinsic degrees of freedom, we discuss in detail the properties of the generator-coordinate-method subspaces with respect to the kinematical separation of these degrees of freedom. An application is made, using the ideas developed in this paper, (a) to translations, (b) to illustration of the qualitative understanding of the content of existing generator-coordinate-method calculations of giant resonances in light nuclei, and (c) to the definition of appropriate asymptotic states in current generator-coordinate-method descriptions of scattering.

The kinematics of generator co-ordinates

The generator co-ordinate approach is used to construct a collective subspace of the full many-body Hilbert space. The construction has a purely kinematical character, given the apriori (e.g. phenomenological) specification of a set of generator many-body states. It is based on the analysis of the properties of the overlaps of the generator states and on the use of the standard spaces of square-integrable functions of quantum mechanics. Some well-known misbehaviours of the generator co-ordinate weight functions are clearly identified as of kinematical origin. A standard representation in the collective phase space is introduced which eliminates them. It is also indicated how appropriate collective dynamical variables can be defined aposteriori. Hilbert-Schmidt overlap kernels and the Gaussian overlap approximation are treated as special examples.