Dynamical temperature for spin systems

The paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on Euclidean spaces. Existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted Hilbert and Banach spaces. This paper is an enlarged and rewritten version of the paper [7].

We study S=1/2 quantum spin chains with shift-invariant and inversion-symmetric next-nearest-neighbor interaction, also known as zigzag spin chains. We completely classify the integrability and non-integrability of the above class of spin systems. We prove that in this class there are only two integrable models, a classical model and a model solvable by the Bethe ansatz, and all the remaining systems are non-integrable. Our classification theorem confirms that within this class of spin chains, there is no missing integrable model. This theorem also implies the absence of intermediate models with a finite number of local conserved quantities.

We utilize the graphon—a continuous mathematical object which represents the limit of convergent sequences of dense graphs—to formulate a general, continuous description of quantum spin systems in thermal equilibrium when the average coordination number grows extensively in the system size. Specifically, we derive a closed set of coupled nonlinear Fredholm integral equations which governs the properties of the system. The graphon forms the kernel of these equations, and their solution yields exact expressions for the macroscopic observables in the system in the thermodynamic limit. We analyze these equations for both quantum and classical spin systems, recovering known results and providing analytical solutions for a range of more complex cases. We supplement this with controlled, finite-size numerical calculations using Monte Carlo and tensor network methods, showing their convergence towards our analytical results with increasing system size. Published by the American Physical Society 2024

We study by a Monte Carlo method, a three-dimensional classical antiferromagnetic random Heisenberg spin system with an exchange interaction which decreases exponentially with distance. We find no indication of a spin glass transition when only isotropic exchange exists. However, a gradual spin freezing is observed as T→0. In the presence of a strong enough Ising-type uniaxial magnetic anisotropy, we observe a peak in the specific heat and a stable order parameter. However, no true thermoremanent magnetization is observed.

The purpose of this article is to obtain a better understanding of the extended variational principle (EVP). The EVP is a formula for the thermodynamic pressure of a statistical mechanical system as a limit of a sequence of minimization problems. It was developed for disordered mean-field spin systems, spin systems where the underlying Hamiltonian is itself random, and whose distribution is permutation invariant. We present the EVP in the simpler setting of classical mean-field spin systems, where the Hamiltonian is non-random and symmetric. The EVP essentially solves these models. We compare the EVP with another method for mean-field spin systems: the self-consistent mean-field equations. The two approaches lead to dual convex optimization problems. This is a new connection, and it permits a generalization of the EVP.

Alternating metalloporphyrin-acceptor systems provide a ‘laboratory’ for study of quasi-one-dimensional magnetic phenomena. For example, the [octaethylporphyrinatomanganate(III)][hexacyanobutadiene], [MnOEP][HCBD], compound is a model alternating classical (spin 2) and quantum (spin 1/2) spin system. While the high temperature susceptibility is well fit to Seiden's model for alternating S = 2 and S = 1/2 chains with antiferromagnetic exchange J/kB = 172 K, and g for [MnOEP] = 1.92, the low temperature data reveal a transition to a two-dimensional antiferromagnet, followed by a transition to a three-dimensional canted antiferromagnet. The similarly structured [tetraphenyl-porphyrinatomanganate(III)][tetracyanoethenide], [MnTPP][TCNE], also has ferrimagnetic behavior within the chains, though it has ferromagnetic interactions between the chains. We associate the change in sign of interchain exchange (from antiferromagnetic to ferromagnetic) with an increase in interchain separation and a change in the interchain exchange pathways. Subtle disorder, the magnitude of which depends upon the solvent incorporated within the structure, causes cluster glass-like behavior at low temperatures. The roles of the various constituents of these compounds in determining the ferromagnetic or antiferromagnetic nature of the ground state are discussed.

Dimensionality is a critical factor in determining the properties of solids and is an apparent built-in character of the crystal structure. However, it can be an emergent and tunable property in geometrically frustrated spin systems. Here, we study the spin dynamics of the tetrahedral cluster antiferromagnet, pharmacosiderite, via muon spin resonance and neutron scattering. We find that the spin correlation exhibits a two-dimensional characteristic despite the isotropic connectivity of tetrahedral clusters made of spin 5/2 Fe3+ ions in the three-dimensional cubic crystal, which we ascribe to two-dimensionalisation by geometrical frustration based on spin wave calculations. Moreover, we suggest that even one-dimensionalisation occurs in the decoupled layers, generating low-energy and one-dimensional excitation modes, causing large spin fluctuation in the classical spin system. Pharmacosiderite facilitates studying the emergence of low-dimensionality and manipulating anisotropic responses arising from the dimensionality using an external magnetic field.

The ground-state properties of a generalized quantum $s=\frac{1}{2}$ $\mathrm{XY}$ model (including the ferromagnetic and antiferromagnetic cases) in a $Z$ field on a triangular lattice are examined. The exact values of critical fields above which the gap opens are obtained. A real-space renormalization-group method is used to estimate the ground-state energy and to analyze the critical behavior. There is some evidence that certain features of frustration in classical spin systems still persist in quantum systems.

A one-dimensional system of electrons and classical isotropic $n$-component spins both interacting with a single phonon mode of the harmonic lattice is investigated. The spin system is treated exactly. Self-consistent equations for spin- and electron-phonon coupling induced lattice distortion (Peierls transition) and its critical temperatures are obtained and discussed for various values of the coupling constants. In the absence of the electron-phonon interaction it is concluded that at $T=0$ a distorted and spin dimerized ground state is obtained for $|{J}_{0}|l\frac{{J}_{1}^{2}}{4{\ensuremath{\omega}}_{0}}$, where ${J}_{0}$, ${J}_{1}$, and ${\ensuremath{\omega}}_{0}$ are the spin-spin, spin-lattice, and characteristic lattice frequency constants, respectively. For $|{J}_{0}|g\frac{{J}_{1}^{2}}{4{\ensuremath{\omega}}_{0}}$, the ground state is undistorted. In the presence of electron-phonon interaction large ${J}_{0}$ suppresses the dimerized state but does not affect the lattice distortion due to electron-phonon interaction. The relation to the Peierls instability on the $S=\frac{1}{2}$ Heisenberg chain is discussed.

For an exactly soluble classical spin model with long-range inhomogeneous coupling it is proved that in the absence of external magnetic field the free energy is aC ∞ function of the temperature at the critical point.

In this paper we pay attention to the classical one-dimensional planar spin system and, in particular, to the dynamics of such a model. We use the Monte Carlo method to calculate the static correlation functions, needed to determine the relaxation functions completely. We are then able to give the results for the in- and out-of-plane component of the relaxation function. At low temperatures the excitations of the out-of-plane component behave as free particles while the excitations of the in-plane component are very similar to the excitations in the Heisenberg chain.

We studied the quantum dynamics of ferromagnetic domain walls (topological kink-type solitons) in one-dimensional ferromagnetic spin chains. We show that the tunneling probability does not depend on the number of spins in a domain wall; thus, this probability can be large even for a domain wall containing a large number of spins. We also predict that there is a strong interplay between the tunneling of a wall from one lattice site to another (tunneling of the kink coordinate) and the tunneling of the kink topological charge (so-called chirality). Both of these elementary processes are suppressed for kinks in one-dimensional ferromagnets with half-integer spin. The dispersion law (i.e., the domain wall energy versus momentum) is essentially different for chains with either integer or half-integer spins. The predicted quantum effects could be observed for mesoscopic magnetic structures, e.g., chains of magnetic clusters, large-spin molecules, or nanosize magnetic dots.

The dynamic phase behavior of a classical Heisenberg spin system with a bilinear exchange anisotropy in a planar thin film geometry has been investigated by Monte Carlo simulations using different forms for the stochastic dynamics. In simulations of the dynamic phase transition (DPT) in films subject to a pulsed oscillatory external field with competing surface fields, both Glauber and Metropolis dynamics show a continuous DPT. But while the field amplitude dependence of the DPT is similar in both cases, the transition region for the DPT as a function of temperature is more extended with Metropolis dynamics. The difference arises from a decoupling of the surface and bulk responses of the film near the dynamic phase transition with Metropolis dynamics that is not evident for Glauber dynamics.

At low enough temperatures solitons (domain walls) are at lattice sites in the one­ dimensional, antiferromagnetic, classical XY model in a field. A crossover behavior due to the depinning of solitons by thermal fluctuations is obtained for a correlation function of the spin component perpendicular to the field. In one-dimensional systems with a discrete set of degenerate ground states, there appear solitons as elementary nonlinear excitations/) solitons in these systems are domain walls which separate two domains associated with distinct ground states. Examples of such systems may be found in some magnetic materials. 2 ),3) The influences of solitons on various physical quantities of these systems have extensively been studied in the literature. 4 ) In particular, it has been shown that the long range behavior of certain correlation functions is strongly affected by solitons. Most of the theories which are intended to calculate the effects of .solitons on statistical properties of those systems are based on continuum models or continuum approximations to lattice models. One of the qualitative differences between a lattice model and a continuum model is that a soliton in the former model is more or less pinned at a lattice site whereas a soliton in the latter is free to slide without extra cost of energy. There have been a few attempts 5 ),6) to investigate the lattice­ discreteness corrections to the continuum approximation, but no evidence showing the effects of soliton pinning has been obtained. The purpose of this pape,r is to study the effects of soliton pinning by examining a soliton-sensitive correlation function in a classical spin model, based on the transfer-matrix integral equation, (2) below. We consider a one-dimensional system of classical planar spins in a field, whose Hamiltonian is given by H =~cos(Bj+l- Bj)-B~cosBj, j

An order parameter, termed the maximal row correlation, is proposed for classical spin systems. Monte Carlo simulations on various Potts models suggest that this order parameter is applicable to a broad range of spin systems, including those defined on irregular lattices, systems with frustration, and systems exhibiting partial orders, provided some degree of spin ordering is present. This approach offers a unified framework for investigating phase transitions in such complex systems. The associated critical exponents are estimated via finite-size scaling analysis and show good agreement with established values.