Fermionic Many-body Systems Research Articles

Gaussian phase-space representations for fermions

We introduce a positive phase-space representation for fermions, using the most general possible multi-mode Gaussian operator basis. The representation generalizes previous bosonic quantum phase-space methods to Fermi systems. We derive equivalences between quantum and stochastic moments, as well as operator correspondences that map quantum operator evolution onto stochastic processes in phase space. The representation thus enables first-principles quantum dynamical or equilibrium calculations in many-body Fermi systems. Potential applications are to strongly interacting and correlated Fermi gases, including coherent behaviour in open systems and nanostructures described by master equations. Examples of an ideal gas and the Hubbard model are given, as well as a generic open system, in order to illustrate these ideas.

Exact Transformation for Spin-Charge Separation of Spin-1/2Fermions without Constraints

We demonstrate an exact local transformation which maps a purely Fermionic many-body system to a system of spinful bosons and spinless fermions, demonstrating a possible path to a non-Fermi-liquid state. We apply this to the half-filled Hubbard model and show how the transformation maps the ordinary spin half Fermionic degrees of freedom exactly and without introducing Hilbert space constraints to a chargelike quasicharge fermion and a spinlike quasispin Boson while preserving all the symmetries of the model. We present approximate solutions with localized charge which emerge naturally from the Hubbard model in this form. Our results strongly suggest that charge tends to remain localized for large values of the Hubbard U.

Exact BCS stochastic schemes for a time-dependent many-body fermionic system

The exact quantum state evolution of a fermionic gas with binary interactions is obtained as the stochastic average of BCS-state trajectories. We find the most general Ito stochastic equations which reproduce exactly the dynamics of the system and we obtain some conditions to minimize the stochastic spreading of the trajectories in the Hilbert space. The relation between the optimized equations and mean-field equations is analyzed. The method is applied to a simple two-site model. The simulations display effects that cannot be obtained in the mean-field approximation.

Gaussian operator bases for correlated fermions

We formulate a general multi-mode Gaussian operator basis for fermions, to enable a positive phase-space representation of correlated Fermi states. The Gaussian basis extends existing bosonic phase-space methods to Fermi systems and thus enables first-principles dynamical or equilibrium calculations in quantum many-body Fermi systems. We prove the completeness and positivity of the basis, and derive differential forms for products with one- and two-body operators. Because the basis satisfies fermionic superselection rules, the resulting phase space involves only c-numbers, without requiring anti-commuting Grassmann variables.

On integrability and pseudo-Hermitian systems with spin-coupling point interactions

We study the pseudo-Hermitian systems with general spin-coupling point interactions and give a systematic description of the corresponding boundary conditions for PT-symmetric systems. The corresponding integrability for both bosonic and fermionic many-body systems with PT-symmetric contact interactions is investigated.

Gaussian Quantum Monte Carlo Methods for Fermions and Bosons

We introduce a new class of quantum Monte Carlo methods, based on a Gaussian quantum operator representation of fermionic states. The methods enable first-principles dynamical or equilibrium calculations in many-body Fermi systems, and, combined with the existing Gaussian representation for bosons, provide a unified method of simulating Bose-Fermi systems. As an application relevant to the Fermi sign problem, we calculate finite-temperature properties of the two dimensional Hubbard model and the dynamics in a simple model of coherent molecular dissociation.

Group-theoretical deduction of a dyadic Tamm–Dancoff equation by using a matrix-valued generator coordinate

The traditional Tamm–Dancoff (TD) method is one of the standard procedures for solving the Schrodinger equation of fermion many-body systems. However, it meets a serious difficulty when an instability occurs in the symmetry-adapted ground state of the independent particle approximation (IPA) and when the stable IPA ground state becomes of broken symmetry. If one uses the stable but broken symmetry IPA ground state as the starting approximation, TD wave functions also become of broken symmetry. On the contrary, if we start from a symmetry-adapted but unstable wave function, the convergence of the TD expansion becomes bad. Thus, the requirements of symmetry and rapid convergence are not in general compatible in the conventional TD expansion of the systems with strong collective correlations. Along the same line as Fukutome's, we give a group-theoretical deduction of a U(n) dyadic TD equation by using a matrix-valued generator coordinate.

Orbital approximation for the reduced Bloch equations: Fermi–Dirac distribution for interacting fermions and Hartree–Fock equation at finite temperature

In this paper, we solve a set of hierarchy equations for the reduced statistical density operator in a grand canonical ensemble for an identical many-body fermion system without or with a two-body interaction. We take the single-particle approximation, and obtain an eigenequation for the single-particle states. For the case of no interaction, it is an eigenequation for the free particles, and the solutions are therefore the plane waves. For the case with a two-body interaction, however, it is an equation which is the extension of the usual Hartree–Fock equation at zero temperature to the case of any finite temperature. The average occupation number for the single-particle states with mean field interaction is also obtained, which has the same Fermi–Dirac distribution form as that for the free fermion gas. The derivation demonstrates that even for an interacting fermion system, only the lowest N-orbitals, where N is the number of particles, are occupied at zero temperature. In addition, we discuss their practical applications in such fields as studying the temperature effects on the average structure and electronic spectra for macromolecules.

Derivation of the Euler Equations from Quantum Dynamics

We derive the Euler equations from quantum dynamics for a class of fermionic many-body systems. We make two types of assumptions. The first type are physical assumptions on the solution of the Euler equations for the given initial data. The second type are a number of reasonable conjectures on the statistical mechanics and dynamics of the Fermion Hamiltonian.

Integrability and PT-Symmetry of N-Body Systems with Spin-Coupling -Interactions

We study the PT-symmetric boundary conditions for "spin"-related $\delta$-interactions and the corresponding integrability for both bosonic and fermionic many-body systems. The spectra and bound states are discussed in detail for spin-1/2 particle systems.

An exact integral equation for the renormalizedFermi surface

The true Fermi surface of a fermionic many-body system can be viewed as a fixedpoint manifold of the renormalization group (RG). Within the framework of theexact functional RG we show that the fixed point condition implies an exactintegral equation for the counter-term which is needed for a self-consistentcalculation of the Fermi surface. In the simplest approximation, our integralequation reduces to the self-consistent Hartree–Fock equation for thecounter-term.

Multi-particle–hole excitations in many-body Fermi systems

We formulate a microscopic theory for calculating the dynamic structure function of 3 He and other strongly interacting Fermi liquids. The theory is a generalization of the correlated basis function theory for the dynamics of 4 He and includes time-dependent two-particle—two-hole excitations. It may also be interpreted as a systematic extension of the random-phase approximation (RPA). Comparison of numerical results with experiments show a significant improvement of the excitation spectrum over RPA predictions while maintaining the accuracy of prediction of ground-state quantities like the static structure function.

Stochastic properties of many-body systems

The k-body embedded ensembles of random matrices originally defined by Mon and French are investigated as paradigmatic models of stochasticity in Fermionic many-body systems. In these ensembles, m Fermions in l degenerate single-particle states, interact via a random k-body interaction which obeys unitary or orthogonal symmetry. We focus attention on the spectral properties of these ensembles. We always take the limit l→∞. For 2 k> m, the spectral properties of the k-body embedded unitary and orthogonal ensembles coincide with those of the canonical Gaussian unitary and orthogonal random-matrix ensemble, respectively. For k⪡ m⪡ l, the spectral fluctuations become Poissonian. The reason for this behavior is displayed by constructing limiting ensembles. The case of embedded Bosonic ensembles ( m Bosons in l degenerate single-particle states interact via a random k-body interaction which obeys unitary or orthogonal symmetry) is also considered and compared with the case of Fermions.

Toward a unified algebraic understanding of concepts of particle and collective motions in fermion many-body systems

Toward a unified algebraic understanding of physical concepts of the so-called particle and collective motions, a new theory for unified description of both the motions is proposed with the use of time-dependent Hartree-Fock theory on a circle S1. The theory simply and clearly elucidates a collective motion induced by a TD mean-field potential. It also describes symmetry breaking of fermion systems and successive occurrence of the collective motion due to recovery of the symmetry. The theoretical frame asserts that the Fock space of finite-dimensional fermions has an algebraic structure to be embedded into that of infinite-dimensional fermions.

A link of information entropy and kinetic energy for quantum many-body systems

A direct connection of information entropy S and kinetic energy T is obtained for nuclei and atomic clusters, which establishes T as a measure of the information in a distribution. It is conjectured that this is a universal property for fermionic many-body systems. We also check rigorous inequalities previously found to hold between S and T for atoms and verify that they hold for nuclei and atomic clusters as well. These inequalities give a relationship of Shannon's information entropy in position-space with an experimental quantity, i.e., the rms radius of nuclei and clusters.

Partial dynamical symmetry in an interacting fermion system

The concept of partial symmetry is shown to be relevant to fermionic many-body systems. The associated Hamiltonians are closely related to a realistic nuclear quadrupolequadrupole interaction as applications to various nuclei confirm. As an illustration, the case of the deformed light nucleus 24Mg is presented.

Self-consistent field method from a τ-functional viewpoint

A unified aspect of the self-consistent field (SCF) method and the τ-functional method is presented. SCF theory in the τ-functional space F∞ manifestly results in a gauge theory of fermions and then a collective motion appears as a motion of fermion gauges with a common factor. This provides a new algebraic tool for the microscopic understanding of fermion many-body systems.

Wave function structure in two-body random matrix ensembles

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.

Properties of Low-Lying States in a Diffusive Quantum Dot and Fock-Space Localization

Motivated by an experiment of Sivan et al. and by subsequent theoretical work on localization in Fock space, a hierarchical model is studied numerically for a finite many-body system of fermions moving in a disordered potential and coupled by a two-body interaction. Attention is focussed on the low-lying states close to the Fermi energy. Both the spreading width and the participation number depend smoothly on excitation energy. This behavior can be fully understood in terms of the density of available 1p–0h and 2p–1h states. The system does not display localization of Anderson type. The numerical model reproduces essential features of the experiment by Sivan et al.

Properties of Low-Lying States in a Diffusive Quantum Dot and Fock-Space Localization

Motivated by an experiment by Sivan et al. (Europhys. Lett. 25, 605 (1994)) and by subsequent theoretical work on localization in Fock space, we study numerically a hierarchical model for a finite many-body system of Fermions moving in a disordered potential and coupled by a two-body interaction. We focus attention on the low-lying states close to the Fermi energy. Both the spreading width and the participation number depend smoothly on excitation energy. This behavior is in keeping with naive expectations and does not display Anderson localization. We show that the model reproduces essential features of the experiment by Sivan et al.