Exactly solvable 1D frustrated quantum spin models

A class of the frustrated quantum s = ½ models with nearest and next nearest neighbor couplings is investigated. An exact wave function of the singlet ground state at the transition point from the ferromagnetic to the singlet state is presented. The recurrence technics of expectation value calculations is developed and the simple expressions for spin-correlation function at N → ∞ are obtained. A long range double-spiral ordering is demonstrated. We show that in one particular case the model reduces to the effective spin-1 model and the exact singlet ground state wave function is presented for this model. The behavior of the system in the vicinity of the transition point is investigated.     

Microcanonical relation between continuous and discrete spin models

A relation between a class of stationary points of the energy landscape of continuous spin models on a lattice and the configurations of an Ising model defined on the same lattice suggests an approximate expression for the microcanonical density of states. Based on this approximation we conjecture that if a O(n) model with ferromagnetic interactions on a lattice has a phase transition, its critical energy density is equal to that of the n=1 case, i.e., an Ising system with the same interactions. The conjecture holds true in the case of long-range interactions. For nearest-neighbor interactions, numerical results are consistent with the conjecture for n=2 and n=3 in three dimensions. For n=2 in two dimensions (XY model) the conjecture yields a prediction for the critical energy of the Bere?inskij-Kosterlitz-Thouless transition, which would be equal to that of the two-dimensional Ising model. We discuss available numerical data in this respect.     

Information storage capacity of discrete spin systems

Understanding the limits imposed on information storage capacity of physical systems is a problem of fundamental and practical importance which bridges physics and information science. There is a well-known upper bound on the amount of information that can be stored reliably in a given volume of discrete spin systems which are supported by gapped local Hamiltonians. However, all the previously known systems were far below this theoretical bound, and it remained open whether there exists a gapped spin system that saturates this bound. Here, we present a construction of spin systems which saturate this theoretical limit asymptotically by borrowing an idea from fractal properties arising in the Sierpinski triangle. Our construction provides not only the best classical error-correcting code which is physically realizable as the energy ground space of gapped frustration-free Hamiltonians, but also a new research avenue for correlated spin phases with fractal spin configurations.     

2D quantum spin models and Jordan-Wigner fermions

We apply the 2D Jordan-Wigner fermionization to examine the ground state and thermodynamic properties of the square-lattice s=1/2 anisotropic XY model. We compare our findings with the results of different analytical and numerical approaches.     

Kac type models of semi-finite spin systems

The Ising model for systems with free surfaces and Kac type interaction is exactly solved. The free energy and the transition point are similar to those obtained for Valenta's model of thin films. The spherical model for the same system is also investigated, and has the same critical temperature.     

Generalized percolation models for frustrated spin systems

Summary A brief review on how to study frustrated spin models by mapping them into generalized percolation models is given. The percolation models associated to a number of deterministic frustrated models are discussed with particular attention on the properties of critical clusters. Paper presented at the I Internatioanl Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Percolation for low energy clusters and discrete symmetry breaking in classical spin systems

We consider classical lattice systems in two or more dimensions with general state space and with short-range interactions. It is shown that percolation is a general feature of these systems: If the temperature is sufficiently low, then almost surely with respect to some equilibrium state there is an infinite cluster of spins trying to form a ground state. For systems having several stable sets of symmetry-related ground states we show that at low temperatures spontaneous symmetry breaking occurs because in a two-dimensional subsystem there is a unique infinite cluster of this type.     

The logarithmic sobolev inequality for discrete spin systems on a lattice

For finite range lattice gases with a finite spin space, it is shown that the Dobrushin-Shlosman mixing condition is equivalent to the existence of a logarithmic Sobolev inequality for the associated (unique) Gibbs state. In addition, implications of these considerations for the ergodic properties of the corresponding Glauber dynamics are examined.During the period of this research, both authors were partially supported by NSF grant DMS 8913328     

Magnetic resonance in systems with equivalent spin-1/2 nuclides. Part 2: energy values and spin states

In an earlier paper (Part 1), featuring group-theoretical analysis, it was shown that the isotropic EPR spectra of free radical (S=1/2) species XL(n), where the n equivalent nuclei also have spin 1/2, have a more complicated form than anticipated from the usual (first-order) oversimplified analysis. The nucleus of X is taken to be spin-less. The latter predicts n+1 lines with intensity ratios given by the coefficients of the binomial expansion; for systems with n=3, the EPR spectrum in fact consists of 6 lines. Analogous considerations hold for NMR spectroscopy of XL(n) non-radicals. For n > or = 3 systems, the degeneracy of the energy levels cannot be completely removed by the Zeeman electronic and nuclear interactions. Explicit solutions for n=3 (analytic, as well as computational) of the spin-hamiltonian for the energies and spin states have been obtained and are given in the present work. Discussion of the underlying theory, invoking exchange degeneracy, is included herein in some detail, focusing on line positions and relative spectral intensities.     

Controllability of spin 1 systems and realization of ternary SWAP gate in two spin 1 systems coupled with Ising interaction

In this paper,we investigate the controllability of spin 1 systems and the realization of ternary gates.Using dipole and quadrupole operators as the orthogonal basis of su(3) algebra,we discuss the controllability of one spin 1 systems and offer the concept of a complete set of control operators first.Then we present the controllability of two spin 1 systems coupled with Ising interaction and the transforming relations of the drift process of the system.Finally the specific realization of the ternary SWAP gate in these systems is discussed.It takes 9 drift processes and 25 basic control processes.     

A fixed-point equation for the high-temperature phase of discrete lattice spin systems

A fixed-point equation on an infinite-dimensional space is proposed as an alternative to the usual definition of the infinite-volume limit in discrete lattice spin systems in the high-temperature phase. It is argued heuristically that the free energy and correlation functions one obtains by solving this equation agree with the usual definitions of these quantities. A theorem is then proved that says that if a certain finite-volume condition is satisfied, then this fixed-point equation has a solution and the resulting free energy is analytic in the parameters in the Hamiltonian. For particular values of the temperature this finite-volume condition may be checked with the help of a computer. The two-dimensional Ising model is considered as a test case, and it is shown that the finite-volume condition is satisfied for0.77 _critical.     

Aging phenomena in spin glasses and frustrated ferromagnets

An experimental protocol developed for spin glasses has been applied to a reentrant system. We compare the main features of the aging phenomena in both spin glass and ferromagnetic phases.     

Calculation of spin correlations in two-dimensional Ising systems from one-dimensional kinetic models

Kinetic spin models of the type first introduced by Glauber are considered with the most general choice of transition rates. It is shown that their time evolution operator can be related to the transfer matrices of certain two-dimensional Ising lattices. This allows an exact calculation of spin correlation functions at certain temperatures. Specifically we show that for the triangular antiferromagnet this special temperature corresponds to the disorder point. For the Hamiltonian version of the ANNNI model the corresponding result restricts in an important way the possible structure of the phase diagram.     

New exact approach to models of diffusion in one-dimensional random systems and optimal network kinetics in spin glass models

It is shown that in one-dimensional stochastic models with gaussian random energy levels along a quantum reaction coordinate the dominant, rate-determining time-scale does not follow the conventional Arrhenius law, but rather has a much stronger temperature dependence, of the form τ ～ exp[(B/k_BT)²], where B is proportional to the width of the energy distribution. The new activation law can be ascribed to the large number of energy barriers of varying heights which exist in the random structure, as distinct from the conventional case of a single barrier, leading to the Arrhenius form τ ～ T^p × exp(A/k_BT). In systems with random structure and configuration space which are not strictly one-dimensional it is discussed if the thermal energy bias of detailed balance may lead to a kinetics that is essentially restricted to an energetically optimal network at low temperatures, thus leading to an essentially one-dimensional diffusion. Several recent studies of spin glass models appear to support the relevance of this principle, and include the observation of the new activation law in Monte Carlo experiments.     

Vertex correction and conductivity in 2D Dirac-like systems with non-conserving spin disorder

We determine the classical diffusion of two dimensional Dirac-like quasiparticles, in the presence of conserving spin disorder (scattering off electric impurities) and non-conserving spin disorder (scattering off magnetic impurities). We use the Kubo formula for the conductivity tensor and employ diagrammatic perturbation theory to calculate the vertex correction and the renormalisation of the current operator for both electric and magnetic scattering. Scattering off electric impurities is isotropic and the current operator renormalised to two times the bare current operator irrespective of the direction of the dynamics, as usual for Dirac-like fermions. For magnetic scattering the renormalisation of the current operator depends on the direction of the dynamics and on the polarisation of the magnetic impurities, making the system anisotropic. We calculate the anisotropic magnetoresistance (AMR) and analyse it as a function of the ratio of the strength of the electric to the magnetic scattering potentials, for short range Gaussian correlation.