Exchange symmetry in a system of nonrelativistic spin-1/2 fermions in the Feynman quantum statistics representation

A compact representation is obtained for the quantum statistical sum of indistinguishable nonrelativistic spin-1/2 fermions in the form of Feynman path integrals which can be used as the basis to develop a fundamentally exact method of computer modeling for systems of strongly interacting electrons at nonzero temperature. A basis of symmetrized wave functions is constructed using Young symmetry operators. An exact permutation symmetrization procedure leads to an avalanche-like multiplication in the number of diagrams of linked Feynman integrals of the order of N!. The partition function can be simplified without introducing any approximations and this is performed numerically by computer by direct sorting of diagrams. The control tables obtained, containing combinatorial weights of diagrams, direct the Markov random walk process in virtual trajectory space which is achieved numerically by computer. The equilibrium characteristics of the quantum system are calculated by averaging. This approach is an expansion of the Monte Carlo-Metropolis method to systems of quantum indistinguishable particles with spin. Demonstration numerical calculations using this method were made for the simplest exchange systems, for a hydrogen molecule, a Be⁺ ion, and a Li atom. The ground state of the hydrogen molecule is reproduced with a statistical error of 0.2%. Exchange-correlation effects lead to nontrivial structural changes in the thermally excited electron shells of ions in a state of strong plasma compression.     

Calculation of the equation of state of a dense hydrogen plasma by the Feynman path integral method

A method is developed for calculating the equation of state of a system of quantum particles at a finite temperature, based on the Feynman formulation of quantum statistics. A general analytical expression is found for the virial estimator for the kinetic energy of a system with rigid boundaries at a finite pressure. An effective method is developed for eliminating the unphysical singularity in the electrostatic potential between a discretized Feynman path of an electron and a proton. It is shown that the “refinement” of an expansion of a quantum-mechanical propagator by addition of high powers of time exacerbates, rather than eliminates, the divergence of a Feynman path integral. A brief summary of the current status of the problem is presented. The proposed new approaches are presented in relation to progress made in this field. Path integral Monte Carlo simulations are performed for nonideal hydrogen plasmas in which both indistinguishability and spin of electrons are taken into account under conditions preceding the formation of the electron shells of atoms. The electron permutation symmetry is represented in terms of Young operators. It is shown that, owing to the singularity of the Coulomb potential, quantum effects on the behavior of the electron component cannot be reduced to small corrections even if the system must be treated as a classical system according to the formal de Broglie criterion. Quantum-mechanical delocalization of electrons substantially weakens the repulsion between electrons as compared to protons. In relatively cold plasmas, many-body correlations lead to complex behavior of the potential of the average force between particles and give rise to repulsive forces acting between protons and electrons at distances of about 5 angstroms. Plasma pressure drops with decreasing plasma temperature as the electron shells of atoms begin to form, and the electron kinetic energy reaches a minimum at a temperature of about 31000 K. The minimum point weakly depends on plasma density. Owing to quantum effects, the electron component is “heated” well before electrons are completely bound in the field of protons.     

Simulation of thermal ionization in a dense helium plasma by the Feynman path integral method

The region of equilibrium states is studied where the quantum nature of the electron component and a strong nonideality of a plasma play a key role. The problem of negative signs in the calculation of equilibrium averages a system of indistinguishable quantum particles with a spin is solved in the macroscopic limit. It is demonstrated that the calculation can be conducted up to a numerical result. The complete set of symmetrized basis wave functions is constructed based on the Young symmetry operators. The combinatorial weight coefficients of the states corresponding to different graphs of connected Feynman paths in multiparticle systems are calculated by the method of random walk over permutation classes. The kinetic energy is calculated using a viral estimator at a finite pressure in a statistical ensemble with flexible boundaries. Based on the methods developed in the paper, the computer simulation is performed for a dense helium plasma in the temperature range from 30000 to 40000 K. The equation of state, internal energy, ionization degree, and structural characteristic of the plasma are calculated in terms of spatial correlation functions. The parameters of a pseudopotential plasma model are estimated.     

Spin susceptibility of two-dimensional electrons in narrow AlAs quantum wells

We report measurements of the spin susceptibility in dilute two-dimensional electrons confined to a 45 A wide AlAs quantum well. The electrons in this well occupy an out-of-plane conduction-band valley, rendering a system similar to two-dimensional electrons in Si-MOSFETs but with only one valley occupied. We observe an enhancement of the spin susceptibility over the band value that increases as the density is decreased, following closely the prediction of quantum Monte Carlo calculations and continuing at finite values through the metal-insulator transition.     

Effects of thickness on the spin susceptibility of the two dimensional electron gas

Using available quantum Monte Carlo predictions for a strictly 2D electron gas, we estimate the spin susceptibility of electrons in actual devices taking into account the effect of the finite transverse thickness and finding very good agreement with experiments. A weak disorder, as found in very clean devices and/or at densities not too low, just brings about a minor enhancement of the susceptibility.     

Feynman integral and perturbation theory in quantum tomography

We present a definition for tomographic Feynman path integral as representation for quantum tomograms via Feynman path integral in the phase space. The proposed representation is the potential basis for investigation of Path Integral Monte Carlo numerical methods with quantum tomograms. Tomographic Feynman path integral is a representation of solution of initial problem for evolution equation for tomograms. The perturbation theory for quantum tomograms is constructed.     

Counting non-planar diagrams: an exact formula

We present an explicit solution of a simply stated, yet unsolved, combinatorial problem, of interest both in quantum field theory (Feynman diagrams enumeration, beyond the planar approximation) and in statistical mechanics (high temperature loop expansion of some frustrated lattice spin model).     

Order-disorder transition in nanoscopic semiconductor quantum rings

Using the path integral Monte Carlo technique we show that semiconductor quantum rings with up to six electrons exhibit a temperature, ring diameter, and particle number dependent transition between spin ordered and disordered Wigner crystals. Because of the small number of particles the transition extends over a broad temperature range and is clearly identifiable from the electron pair correlation functions.     

Control of electron correlations in a spherical quantum dot

Spherical quantum dots containing several electrons are considered for different values of the total spin. Numerical calculations are carried out using the quantum path-integral Monte Carlo method. The dependence of the electron correlations on the dimensionless control quantum parameter q associated with the steepness of the confinement potential is studied. The quantum transition from a Wigner crystal-like state (i.e., from the regime of strongly correlated electrons) to a Fermi-liquid state (“cold” melting) driven by the parameter q is studied in detail. The behavior of the radial and pair correlation functions, which characterize quantum delocalization of the electrons, is considered.     

Quantum statistical monte carlo methods and applications to spin systems

A short review is given concerning the quantum statistical Monte Carlo method based on the equivalence theorem⁽¹⁾ thatd-dimensional quantum systems are mapped onto (d+1)-dimensional classical systems. The convergence property of this approximate tansformation is discussed in detail. Some applications of this geneal appoach to quantum spin systems are reviewed. A new Monte Carlo method, “thermo field Monte Carlo method,” is presented, which is an extension of the projection Monte Carlo method at zero temperature to that at finite temperatures. Invited talk presented at “Frontiers of Quantum Monte Carlo,” Los Alamos National Laboratory, September 3–6, 1985.     

Induced quantum dots and wires: electron storage and delivery

We show that quantum dots and quantum wires are formed underneath metal electrodes deposited on a planar semiconductor heterostructure containing a quantum well. The confinement is due to the self-focusing mechanism of an electron wave packet interacting with the charge induced on the metal surface. Induced quantum wires guide the transfer of electrons along metal paths and induced quantum dots store the electrons in specific locations of the nanostructure. Induced dots and wires can be useful for devices operating on the electron spin. An application for a spin readout device is proposed.     

Interacting electrons in a two-dimensional disordered environment: effect of a zeeman magnetic field

The effect of a Zeeman magnetic field coupled to the spin of the electrons on the conducting properties of the disordered Hubbard model is studied. Using the determinant quantum Monte Carlo method, the temperature- and magnetic-field-dependent conductivity is calculated, as well as the degree of spin polarization. We find that the Zeeman magnetic field suppresses the metallic behavior present for certain values of interaction and disorder strength and is able to induce a metal-insulator transition at a critical field strength. It is argued that the qualitative features of magnetoconductance in this microscopic model containing both repulsive interactions and disorder are in agreement with experimental findings in two-dimensional electron and hole gases in semiconductor structures.     

Numerical evaluation of the Feynman integral-over-paths in real and imaginary-time

New techniques are described for Monte Carlo evaluation of the propagation of quantum mechanical systems in both real and imaginary-time using the Feynman integral-over-paths formulation of quantum mechanics. For imaginary-time calculations path translation is used to augment the technique of Lawande et. al. This simple-yet-powerful technique allows the equilibrium probability density to be accurately evaluated in the presence of multiple potential wells. It is shown that path translation permits the calculation of the unknown ground-state energy of one confining potential by comparison with the known ground-state energy of another. A double finite-square-well potential and a finite-square-well/parabolic-well pair are presented as examples. For real-time calculations, a weighted analytical averaging of the exponential in the classical action is performed over a region of paths. This “windowed action” has both real and imaginary components. The imaginary component yields an exponentially decaying probability for selecting paths, thereby providing a basis for the Monte Carlo evaluation of the real-time integral-over-paths. Examples of a wave-packet in a parabolic well and a wave-packet impinging upon a potential barrier are considered.     

Effective electron-electron interactions and magnetic phase transition in a two-dimensional electron liquid

We investigate the spin-dependent effective electron-electron interactions in a uniform system of two-dimensional electrons to understand the spontaneous magnetization expected to occur at very low density. For this purpose, we adopt the Kukkonen-Overhauser form for the effective interactions which are built by accurately determined local-field factors describing the charge and spin fluctuations. The critical behavior of the effective interaction for parallel spin electrons allows us to quantitatively locate the transition to the ferromagnetic state at r_s≈27. When the finite width effects are approximately taken into account the transition occurs at r_s≈30 in agreement with recent quantum Monte Carlo calculations.     

Crystallization and quantum melting of few electron system in a spherical quantum dot: quantum Monte Carlo simulation

Spherical quantum dots with a few charged Fermi particles (electrons or holes) are studied for different total spins. Simulation by quantum path integral Monte Carlo method is performed. The dependence of the electron correlations in the quantum dot is studied at different mean interelectron separation controlled by number of electrons in the quantum dot and by steepness of electron confinement (the latter parameter can be changed by the gate voltage). The ‘cold’ melting—quantum transition from Wigner crystal-like state (i.e. from regime of strongly correlated electrons) to a Fermi liquid-like state—driven by the steepness of electron confinement is studied. The pair correlation function and radial function characterizing electron quantum delocalization are analyzed.     

Spin of the ground quantum state of electrons from first principles in the representation of Feynman path integrals

A method for calculating the spin of the ground quantum state of nonrelativistic electrons and distance between energy levels of quantum states differing in the spin magnitude from first principles is proposed. The approach developed is free from the one-electron approximation and applicable in multielectron systems with allowance for all spatial correlations. The possibilities of the method are demonstrated by the example of calculating the energy gap between spin states in model ellipsoidal quantum dots with a harmonic confining field. The results of computations by the Monte Carlo method point to high sensitivity of the energy gap to the break of spherical symmetry of the quantum dot. For three electrons, the phenomenon of inversion has been revealed for levels corresponding to high and low values of the spin. The calculations demonstrate the practical possibility to obtain spin states with arbitrarily close energies by varying the shape of the quantum dot, which is a key condition for development prospects in technologies of storage systems based on spin qubits.     

Monte Carle simulation of quantum transport through nanostructures

We describe a numerical scheme of combining Monte Carlo procedure and quantum scattering theory to simulate electron transport processes through nanostructures. The transport of electrons through a nanostructure is a highly nontrivial nonequilibrium process in which we should consider the interplay of (i) complicated many-body quantum states in nanostructure, (ii) thermal relaxation processes keeping the leads (electron reservoirs) in local equilibrium, (iii) the coupling between the leads and the nanostructure, and (iv) the bias causing nonequilibrium, current, and evolution of quantum states in the nanostructure. Considering the quantum coherence within the nanostructure, we include the degrees of freedom of the nanostructure and a single tunneling electron and solve the Schrödinger equation for the many-body states to obtain the scattering matrix in the Fock space from which both the transmission of the electron and the variation of the states in nanostructure can be full quantum-mechanically calculated. The transport is investigated by the Monte Carlo simulation of successive scattering events of single electrons which are sampled with the Metropolis scheme governed by the scattering probabilities, the thermal statistics in the leads, and the applied bias. By this way from a given initial nanostructure state we can calculate the time evolutions of the current and the internal state. As examples we investigate the transmission of electrons through a two-level system. It is shown that the proposed method can properly deal with the inelastic effects in transport processes.     

Thermal ionization in hydrogen plasma simulated using Feynman path integrals

Thermal ionization of hydrogen at temperatures on the order of 10⁴–10⁵ K and densities within 10²⁴–10²⁸ m^?3 has been simulated using Feynman path integrals. This method has been realized for the first time under conditions of a statistical ensemble with fluctuating volume. Multidimensional integrals have been calculated using the Monte Carlo simulation method that was preliminarily tested numerically on a problem of the quantum ground state of a confined hydrogen atom, which admits analytical solution. The position of isolines of the degree of ionization has been determined on the p-T plane of plasma states. The spatial correlation functions for electrons and nuclei are calculated, and the quantum effects in behavior of the electron component are evaluated. It is shown that, owing to the presence of strong Coulomb interactions, plasma retains a substantially quantum character in a broad domain of thermodynamic states, where a formal use of the degeneracy criterion predicts a classical regime. A basically exact stochastic method is developed for calculating the equilibrium kinetic energy of a spatially bounded system of quantum particles free of the dispersion divergence.     

Nagaoka ferromagnetism in the two-dimensional infinite-U Hubbard model

We present different numerical calculations based on variational quantum Monte Carlo simulations supporting a ferromagnetic ground state for finite and small hole densities in the two-dimensional infinite-U Hubbard model. Moreover, by studying the energies of different total spin sectors, these calculations strongly suggest that the paramagnetic phase is unstable against a phase with a partial polarization for large hole densities delta approximately 0.40 with evidence for a second-order transition to the paramagnetic large doping phase.     

Simple classical mapping of the spin-polarized quantum electron gas: distribution functions and local-field corrections

We use the now well known spin unpolarized exchange-correlation energy E(xc) of the uniform electron gas as the basic "many-body" input to determine the temperature T(q) of a classical Coulomb fluid having the same correlation energy as the quantum system. It is shown that the spin-polarized pair distribution functions (SPDFs) of the classical fluid at T(q), obtained using the hypernetted chain equation, are in excellent agreement with those of the T = 0 quantum fluid obtained by quantum Monte Carlo (QMC) simulations. These methods are computationally simple and easily applied to problems which are currently beyond QMC simulations. Results are presented for the SPDFs and the local-field corrections to the response functions of the electron fluid at T = 0 and finite T.