Peierls Method Research Articles

The combination of the random interaction fields’ method and the Bethe–Peierls method for studying two-sublattice magnets

In this paper, the concentration magnetic phase transitions are described using the random interaction fields’ method in combination with the Bethe–Peierls method. This approach allows us to take into account the correlation between magnetic moments and obtain a more accurate solution for the calculation of the Curie point in comparison with the use of the Kikuchi method, which is considered as the most accurate method in the molecular field theory. As an example, we study the dependence of the Curie point on the concentration of titanium in titanomagnetite. This material is one of the typical natural ferrimagnetics.

Spin systems on hypercubic Bethe lattices: a Bethe–Peierls approach

We study spin systems on Bethe lattices constructed from d-dimensional hypercubes. Although these lattices are not tree-like, and therefore closer to real cubic lattices than Bethe lattices or regular random graphs, one can still use the Bethe–Peierls method to derive exact equations for the magnetization and other thermodynamic quantities. We compute phase diagrams for ferromagnetic Ising models on hypercubic Bethe lattices with dimension d = 2, 3, and 4. Our results are in good agreement with the results of the same models on d-dimensional cubic lattices, for low and high temperatures, and offer an improvement over the conventional Bethe lattice with connectivity .

Six–vertex model with domain wall boundary conditions in the Bethe–Peierls approximation

We use the Bethe–Peierls method combined with the belief propagation algorithm to study the arctic curves in the six vertex model on a square lattice with domain-wall boundary conditions, and the six vertex model on a rectangular lattice with partial domain-wall boundary conditions. We show that this rather simple approximation yields results that are remarkably close to the exact ones when these are known, and allows one to estimate the location of the phase boundaries with relative little effort in cases in which exact results are not available.

Complex replica zeros of ±J Ising spin glass at zero temperature

Zeros of the nth moment of the partition function [Zn] are investigated in a vanishing temperature limit β → ∞, n → 0 keeping y = βn ∼ O(1). In this limit, the moment parameterized by y characterizes the distribution of the ground-state energy. We numerically investigate the zeros for ±J Ising spin-glass models with tree and several other systems, which can be carried out with a feasible computational cost by a symbolic operation based on the Bethe–Peierls method. For several tree systems we find that the zeros tend to approach the real axis of y in the thermodynamic limit implying that the moment cannot be described by a single analytic function of y as the system size tends to infinity, which may be associated with breaking of the replica symmetry. However, examination of the analytical properties of the moment function and assessment of the spin-glass susceptibility indicate that the breaking of analyticity is relevant to neither one-step nor full replica symmetry breaking.

Variational solution of the one-dimensional t–J model

A new variational scheme based on a modified Bethe–Peierls method is used to study the ground state properties of the one-dimensional t–J model. Expectation values are evaluated by cutting out a four-site cluster from a correlated Fermi sea, the ground state of which is described by a variational trial wave function. We study a generalized Gutzwiller state where nearest-neighbour hole–hole correlations are controlled variationally. From the electron concentration dependence of the ground state energy, we determine the true thermodynamic boundary where segregation into an electron-rich, and purely hole phase sets in. We also determine the spinodal line and pair susceptibilities. The variational method is applied also to an extended t–J–V model, where V is the coupling constant of the charge interaction term.

Classical transverse Ising spin glass with short-range interactions beyond the mean-field approximation

The classical transverse field Ising spin- glass model with short-range interactions is investigated beyond the mean- field approximation for a real d- dimensional lattice. We use an appropriate nontrivial modification of the Bethe- Peierls method recently formulated for the Ising spin- glass. The zero- temperature critical value of the transverse field and the linear susceptibility in the paramagnetic phase are obtained analytically as functions of dimensionality d. The phase diagram is also calculated numerically for different values of d. In the limit d -> infinity, known mean- field results are consistently reproduced.

K+production in baryon-baryon and heavy-ion collisions

Kaon production cross sections in nucleon-nucleon, nucleon-$\ensuremath{\Delta}$, and $\ensuremath{\Delta}$-$\ensuremath{\Delta}$ interactions are studied in a boson exchange model. For the latter two interactions, the exchanged pion can be on-mass shell; only contributions due to a virtual pion are included via the Peierls method by taking into account the finite $\ensuremath{\Delta}$ width. With these cross sections and also those for pion-baryon interactions, subthreshold kaon production from heavy-ion collisions is studied in the relativistic transport model.

Study of Blume–Emery–Griffiths model by a modified Bethe-Peierls method

We present here a study of the Blume–Emery–Griffiths (BEG) model for general spins using a modified Bethe–Peierls (BP) method calling it the Bethe–Peierls–Lines (BPL) method, incorporating the well known Lines approximation. We compare our results of BPL as well as BP methods for the ferro-to para-magnetic transition temperatures Tc with those available from other theories such as the Correlated-Effective-Field Theory (CEFT), Cluster Variation Method (CVM) and the Monte-Carlo (MC) method. We find that our BPL method gives results for variation of Tc with the single-ion uniaxial anisotropy (D) and biquadratic interaction (J') coefficients close to those obtained by CVM method for spin S = 1 on a honeycomb lattice. We also find that the BPL results are in good agreement for the cubic lattice for S = 3/2 and J' = 0 with those of the MC results, and for S = 1 and J' = −0.5J with those of the CVM results. For these three dimensional lattices our BPL method gives the correct reentrant behaviour for negative values of J'. We also compare our BP and BPL results for magnetic susceptibility with those of the CEFT for S = 3/2 and 3 nearest neighbours and find fairly good agreement for D > 0.

Phi meson production cross sections in pion-baryon and baryon-baryon interactions

Phi meson production in pion-baryon and baryon-baryon interactions near the production threshold is studied in a boson-exchange model. Parameters in this model are either taken from the Bonn model for the nucleon-nucleon potential or determined from the measured phi meson decay width and the experimental data for the reactions π − p→ φn and pp→ ppφ. The isospin-averaged cross sections for the reactions πN→ φN, πΔ→ φN, NN→ NNφ, which are needed in transport models to study phi meson production in heavy-ion collisions at subthreshold energies, have been evaluated. For the reactions NΔ→ NNφ and ΔΔ→ NNφ, which become singular when the exchanged pion is on-shell, the Peierls method is used to calculate their cross sections. We find that the cross section for the reaction πN→ φN is much larger than the other cross sections.

Covariant description of the canonical formalism.

In a gauge theory, one can define the Poisson brackets of gauge-invariant functions ("observables") by three different methods. The first method is based on the constrained Hamiltonian reformulation of the theory. The other two methods, namely, the Peierls method and the covariant symplectic approach, deal directly with the Lagrangian. It is explicitly shown that these three methods are equivalent for an arbitrary gauge theory. The equivalence proof relies on the invariance of the Poisson structure among the observables under the introduction of auxiliary fields.

Removal of Non-Physical Components of Kemmer-β Formalism for Spin-1/2 Particles

The non-physical components of Kemmer-β formalism for spin-1/2 particles have been removed. It is shown that in the presence of an external electromagnetic field the formalism is identical with that of the second-order spinor equation of Feynman and Gell-Mann. Application of the Peierls method of quantization to this formalism is discussed.

Quantization of the Yang-Mills field and Mandelstam's theory in gauge-independent path-dependent formalism

We derive commutation relations (CR's) between gauge-invariant quantities in the Yang-Mills field theory by applying the Peierls method. The CR's obtained are different from those given by Mandelstam in his gauge-independent, path-dependent formalism. However, our CR's are shown to give a consistently quantized field theory, while his CR's do not. In fact, there exist systematic errors in Mandelstam's treatment of the covariant Green's functions. On the other hand, if we correctly treat covariant Green's functions guided by his procedure, our CR's are shown to lead to the same Feynman rules for the Yang-Mills field as prescribed by Feynman, DeWitt, Faddeev and Popov, and Mandelstam.

Approach to the theory of unstable states

In this work, a new method introduced by one of us (A.A.) has been followed, in order to analyze the effects produced by an internal (virtual) state |φ> on the transition amplitudes and, in particular, to single out the necessary conditions that |φ> is connected with a resonance in the cross section. In the case of potential scattering, for instance, the resonance determination by means of the Kapur and Peierls method is shown to be equivalent to the determination of the spectral function of a quasi self-adjoint operator (whose anti-Hermitian part is defined by means of a single state). In general, by associating unstable states with the eigenvalues of quasi self-adjoint Hamiltonians, a new definition of Resonance may be given. The application of this method to the study of resonances related to a spherical-well potential seems to confirm the validity of our method.

Antiferromagnetism

The ground state of a lattice with one electron per atom and antiferromagnetic interactions between nearest neighbours only is examined by a variational method similar in principle to the treatments by Hulthén (1938) and Kasteleijn (1952) of the linear-chain problem. The calculation involves a statistical problem which is shown to be exactly equivalent to the Ising ferromagnetic problem. This cannot be solved exactly, except in the one-dimensional case, and so the Bethe–Peierls method is used to solve it approximately. In complete contradiction to the Kubo (1953) variational calculation it is concluded that all lattices have disordered ground states. The energy differences between ordered and disordered states are small and so the nature of the ground state is likely to be sensitive to small additional ordering forces.