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Physical Review

Ising-Model Reformulation. II. Relation to the Random-External-Field Method

Frank H. Stillinger

Published May 17, 1965• 2 citations
Publisher

Abstract

Starting with the previously derived vacuum-state expectation representation of the Ising partition function, an integral transform is applied to part of the Bose field operator arising from pair interactions, generating thereby the effect of random external fields. The remaining pair interaction is treated as a perturbation, whose effect is small in both the high- and the low-temperature extremes. The transformed part of the potential is selected first to generate a Markoff process, and the perturbation equations may then be regarded as conditions that the perturbation have no effect on either the partition function or the long-range order below the transition. In two dimensions, it is shown how the characteristic integral equation of the Markoff process reduces to the matrix eigenvalue problem solved by Onsager, in the limit of just nearest-neighbor interactions.

DOI
10.1103/physrev.138.a1174

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This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding $l$ particles out of their total number N not in the condensate is exponentially small in the number $l$. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.

•Jan 1, 2017•3 citations
Starting with the previously derived vacuum-state expectation representation of the Ising partition function, an integral transform is applied to part of the Bose field operator arising from pair interactions, generating thereby the effect of random external fields. The remaining pair interaction is treated as a perturbation, whose effect is small in both the high- and the low-temperature extremes. The transformed part of the potential is selected first to generate a Markoff process, and the perturbation equations may then be regarded as conditions that the perturbation have no effect on either the partition function or the long-range order below the transition. In two dimensions, it is shown how the characteristic integral equation of the Markoff process reduces to the matrix eigenvalue problem solved by Onsager, in the limit of just nearest-neighbor interactions.
The partition function (ZZZ) for the classical Ising model of cooperative phenomena in lattice of arbitrary dimensionality and with interactions of unspecified range is expressed as a vacuum-state expectation value of a product of two operators, each constructed from boson annihilation and creation operators. In the absence of external fields, ZZZ, and similarly the spin-pair correlation function $\ensuremath{\psi}(\mathrm{r})$, are expanded thereupon into a series of Feynman diagrams. In the case of $\ensuremath{\psi}(\mathrm{r})$, a formally exact diagram summation (1) shows how the spherical model may be recovered in low-order approximation, (2) suggests a way of introducing systematic corrections to this approximation, and (3) leads to a generalized criterion for suppression of antiferro-magnetic order in "nonfitting" lattices. A full topological-diagram reduction to restricted sets of "elementary" subdiagrams is carried out.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.
The partition functions of many lattice statistical problems, such as the Ising models, the dimer model, and the ferroelectric model, have been written in the following general form:[Formula: see text]For models that can be solved by the pfaffian technique, H(j) is a quadratic function of fermion operators, but otherwise H(j) also contains quartic functions of fermion operators, making the above expression intractable to exact evaluation. General field theoretical perturbation methods, such as the Green's function techniques and the diagonalization (by a Fourier transformation) of the Hamiltonian, Ho(j), are developed in relation to the above expression. It is shown that the partition function for unsolved problems can be expressed in terms of an "irreducible vertex part" for which low-order approximations can be obtained. This general formalism is applied to the next-nearest neighbor two-dimensional Ising lattice. A first-order approximation is obtained for the irreducible vertex part, and the critical properties are obtained. For the equivalent-neighbor model, the critical temperature obtained is in excellent agreement with the series expansion results, being only 0.4 % higher.
This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding lll particles out of their total number N not in the condensate is exponentially small in the number lll. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.