Completeness of the classical 2D Ising model and universal quantum computation

We prove that the 2D Ising model is complete in the sense that the partition function of any classical q-state spin model (on an arbitrary graph) can be expressed as a special instance of the partition function of a 2D Ising model with complex inhomogeneous couplings and external fields. In the case where the original model is an Ising or Potts-type model, we find that the corresponding 2D square lattice requires only polynomially more spins with respect to the original one, and we give a constructive method to map such models to the 2D Ising model. For more general models the overhead in system size may be exponential. The results are established by connecting classical spin models with measurement-based quantum computation and invoking the universality of the 2D cluster states.     

Cluster algorithms for anisotropic quantum spin models

We present cluster Monte Carlo algorithms for theXYZ quantum spin models. In the special case ofS=1/2, the new algorithm can be viewed as a cluster algorithm for the 8-vertex model. As an example, we study theS=1/2XY model in two dimensions with a representation in which the quantization axis lies in the easy plane. We find that the numerical autocorrelation time for the cluster algorithm remains of the order of unity and does not show any significant dependence on the temperature, the system size, or the Trotter number. On the other hand, the autocorrelation time for the conventional algorithm strongly depends on these parameters and can be very large. The use of improved estimators for thermodynamic averages further enhances the efficiency of the new algorithms.     

Variational principle for regular and random Ising models on the cactus tree or on the usual lattice in the “cactus approximation”

The distribution functions and the free energy are expressed in terms of the effective fields for the regular and random Ising models of an arbitrary spin S on the generalized cactus tree. The same expressions apply to systems on the usual lattice in the “cactus approximation” in the cluster variation method. For an ensemble of random Ising models of an arbitrary spin S on the generalized cactus tree, the equation for the probability distribution function of the effective fields is set up and the averaged free energy is expressed in terms of the probability distribution. The same expressions apply to the system on the usual lattice in the “cactus approximation”. We discuss the quantities on the usual lattice when the system or the ensemble of random systems has the translational symmetry. Variational properties of the free energy for a system and of the averaged free energy for an ensemble of random systems are noted. The “cactus approximations” are applicable to the Heisenberg model as well as to the Ising model of an arbitrary spin, and to ensembles of random systems of these models.     

The Percolation Signature of the Spin Glass Transition

Magnetic ordering at low temperature for Ising ferromagnets manifests itself within the associated Fortuin–Kasteleyn (FK) random cluster representation as the occurrence of a single positive density percolating network. In this paper we investigate the percolation signature for Ising spin glass ordering—both in short-range (EA) and infinite-range (SK) models—within a two-replica FK representation and also within the different Chayes–Machta–Redner two-replica graphical representation. Based on numerical studies of the ±J EA model in three dimensions and on rigorous results for the SK model, we conclude that the spin glass transition corresponds to the appearance of two percolating clusters of unequal densities.     

Efficient algorithm for random-bond ising models in 2D

We present an efficient algorithm for calculating the properties of Ising models in two dimensions, directly in the spin basis, without the need for mapping to fermion or dimer models. The algorithm computes the partition function and correlation functions at a single temperature on any planar network of N Ising spins in O(N;{3/2}) time or less. The method can handle continuous or discrete bond disorder and is especially efficient in the case of bond or site dilution, where it executes in O(NlnN) time near the percolation threshold. We demonstrate its feasibility on the ferromagnetic Ising model and the +/-J random-bond Ising model and discuss the regime of applicability in cases of full frustration such as the Ising antiferromagnet on a triangular lattice.     

Dynamical response of quantum spin-glass models at t = 0

We study the behavior of two archetypal quantum spin glasses at T = 0 by exact diagonalization techniques: the random Ising model in a transverse field and the random Heisenberg model. The behavior of the dynamical spin response is obtained in the spin-glass ordered phase. In both models it is gapless and has the general form chi(")(omega) = qdelta(omega)+chi(")(reg)(omega), with chi(")(reg)(omega) approximately omega for the Ising and chi(")(reg)(omega) approximately const for the Heisenberg, at low frequencies. The method provides new insight to the physical nature of the low-lying excitations.     

A new method for calculation of the magnetic properties in one- and two-dimensional spin systems with anisotropic Heisenberg exchange

A new approximation method is proposed for the calculation of the magnetic susceptibility of one-dimensional assembly of spins and the critical temperature of two-dimensional one both with the anisotropic Heisenberg exchange. In a linear chain system, every spins are grouped into pairs of adjacent spins (pair-approximation) or clusters of adjacent three spins ((q+1)-approximation), and the partition function of the total spin system is approximated as a sum of products of the partitions functions for the pairs or the clusters. Then the partition function of the anisotropic Heisenberg spin system is shown to reduce into a form of the Ising spin system with modified coupling constants. The exact result for the Ising chain system enables us to obtain an analytical expression for the magnetic susceptibility of anisotropic Heisenberg chain system. The same approximations are also applied to two-dimensional lattices, and the critical temperatures of the square, triangular, and honeycomb lattices with anisotropic Heisenberg exchange are calculated as a function of anisotropy parameter. The results are compared with those of the existing theories and shown to be quite excellent.     

Fermionic Representation of Two-Dimensional Dimer Models

Dimer models in two dimensions give rise to well-known statistical lattice problems, which can be exactly solved by the same combinatorial techniques associated with the Ising model and which have been used to account for the phase transitions in a number of physically interesting systems. More recently, dimer models have been regarded as classical limits of the quantum ground state of some antiferromagnetic systems. We then revisit an early transfer-matrix calculation for the dimer model on the simple square lattice. We write a spin representation for the transfer matrix associated with the canonical partition function of two paradigmatic dimers models, on the 4–8 lattice, with an Ising-type transition, and on the brick lattice, with a peculiar commensurate–incommensurate transition. Using standard techniques, the problem is reduced to the calculation of the eigenvalues of a system of free fermions.     

Low-concentration series in general dimension

We discuss recent work on the development and analysis of low-concentration series. For many models, the recent breakthrough in the extremely efficient no- free-end method of series generation facilitates the derivation of 15th-order series for multiple moments in general dimension. The 15th-order series have been obtained for lattice animals, percolation, and the Edwards-Anderson Ising spin glass. In the latter cases multiple moments have been found. From complete graph tables through to 13th order, general dimension 13th-order series have been derived for the resistive susceptibility, the moments of the logarithms of the distribution of currents in resistor networks, and the average transmission coefficient in the quantum percolation problem, 11th-order series have been found for several other systems, including the crossover from animals to percolation, the full resistance distribution, nonlinear resistive susceptibility and current distribution in dilute resistor networks, diffusion on percolation clusters, the dilute Ising model, dilute antiferromagnet in a field, and random field Ising model and self-avoiding walks on percolation clusters. Series for the dilute spin-1/2 quantum Heisenberg ferromagnet are in the process of development. Analysis of these series gives estimates for critical thresholds, amplitude ratios, and critical exponents for all dimensions. Where comparisons are possible, our series results are in good agreement with both-expansion results near the upper critical dimension and with exact results (when available) in low dimensions, and are competitive with other numerical approaches in intermediate realistic dimensions.     

Correlation inequalities for vector spin models

Correlation inequalities forn-vector spin models (n 2) are reviewed. A relatively simple and unified derivation of the inequalities is achieved, using duplicate variable methods, for spin dimensionalitiesn=2 (plane rotator model),n=3 (classical Heisenberg model), andn=4. Although correlation inequalities are lacking forn > 4, new proofs are presented for the comparison inequalities relating correlations for systems with arbitrary spin dimensionality to corresponding correlations for systems with low spin dimensionality (n = 1 or 2).Research supported by National Science Foundation under Grant DMR 76-23071.     

Spin quantum number in the ground state of the Mattis-Heisenberg model

Total spin quantum number is rigorously calculated for a quantum version of the Mattis model of random spin systems. Crossover between three universality classes of the Ising model, theXY model, and the Heisenberg model is explicitly worked out in the presence of randomness. The randomness of the type of the Mattis model is shown to have no thermodynamic effects even in quantum systems.     

Simulations on infinite-size lattices

We introduce a Monte Carlo method, as a modification of existing cluster algorithms, which allows simulations directly on systems of infinite size, and for quantum models also at beta = infinity. All two-point functions can be obtained, including dynamical information. When the number of iterations is increased, correlation functions at larger distances become available. Limits q-->0 and omega-->0 can be approached directly. As examples we calculate spectra for the d = 2 Ising model and for Heisenberg quantum spin ladders with two and four legs.     

The Heisenberg ferromagnet in spin coherent state representation

A new approach to Heisenberg ferromagnet using the spin coherent state representation is developed. The differential operator representation of spin angular momentum operators is used to derive thec-number analogs of the basic quantum mechanical equations, viz., the Schrödinger, Bloch and Liouville equations for the Heisenberg ferromagnet. As an important illustration of our formulation, which has noad hoc assumptions and does not use any boson representation, the excitation spectrum for one, two and three spin waves is obtained. In these cases it is also shown that eigenvalue spectrum can be obtained by completely ignoring the kinematical interactions.     

Exact eigenvalues of the Ising Hamiltonian in one-, two- and three-dimensions in the absence of a magnetic field

The Hamiltonian of the Ising model in one-, two- and three-dimensions has been analysed using unitary transformations and combinatorics. We have been able to obtain closed formulas for the eigenvalues of the Ising Hamiltonian for an arbitrary number of dimensions and sites. Although the solution provided assumes the absence of external magnetic fields an extension to include a magnetic field along the z-axis is readily extracted. Furthermore, generalisations to a higher number of spin components on each site are possible within this method. We made numerical comparisons with the partition function from the earlier analytical expressions known in the literature for one- and two-dimensional cases. We find complete agreement with these studies.     

Mathematical structure of the three-dimensional（3D） Ising model

An overview of the mathematical structure of the three-dimensional(3D) Ising model is given from the points of view of topology,algebra,and geometry.By analyzing the relationships among transfer matrices of the 3D Ising model,Reidemeister moves in the knot theory,Yang-Baxter and tetrahedron equations,the following facts are illustrated for the 3D Ising model.1) The complex quaternion basis constructed for the 3D Ising model naturally represents the rotation in a(3+1)-dimensional space-time as a relativistic quantum statistical mechanics model,which is consistent with the 4-fold integrand of the partition function obtained by taking the time average.2) A unitary transformation with a matrix that is a spin representation in 2 n·l·o-space corresponds to a rotation in 2n·l·o-space,which serves to smooth all the crossings in the transfer matrices and contributes the non-trivial topological part of the partition function of the 3D Ising model.3) A tetrahedron relationship would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model,and its existence is guaranteed by the Jordan algebra and the Jordan-von Neumann-Wigner procedures.4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φx,φy,and φz.The relationship with quantum field and gauge theories and the physical significance of the weight factors are discussed in detail.The conjectured exact solution is compared with numerical results,and the singularities at/near infinite temperature are inspected.The analyticity in β=1/(kBT) of both the hard-core and the Ising models has been proved only for β0,not for β=0.Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.     

Generalized transformation for decorated spin models

The paper discusses the transformation of decorated Ising models into an effective undecorated spin model, using the most general Hamiltonian for interacting Ising models including a long range and high order interactions. The inverse of a Vandermonde matrix with equidistant nodes [−s,s] is used to obtain an analytical expression of the transformation. This kind of transformation is very useful to obtain the partition function of decorated systems. The method presented by Fisher is also extended, in order to obtain the correlation functions of the decorated Ising models transforming into an effective undecorated Ising model. We apply this transformation to a particular mixed spin-(1/2, 1) and (1/2, 2) square lattice with only nearest site interaction. This model could be transformed into an effective uniform spin-S square lattice with nearest and next-nearest interaction, furthermore the effective Hamiltonian also includes combinations of three-body and four-body interactions; in particular we considered spin 1 and 2.     

Classical spin models and the quantum-stabilizer formalism

We relate a large class of classical spin models, including the inhomogeneous Ising, Potts, and clock models of q-state spins on arbitrary graphs, to problems in quantum physics. More precisely, we show how to express partition functions as inner products between certain quantum-stabilizer states and product states. This connection allows us to use powerful techniques developed in quantum-information theory, such as the stabilizer formalism and classical simulation techniques, to gain general insights into these models in a unified way. We recover and generalize several symmetries and high-low temperature dualities, and we provide an efficient classical evaluation of partition functions for all interaction graphs with a bounded tree-width.     

Exact ground state of a spin ladder with a quantum phase transition

We introduce a spin ladder with Ising interactions along the legs and intrinsically frustrated Heisenberg-like ferromagnetic interactions on the rungs. The model is solved exactly in the subspaces relevant for the ground state by mapping to the quantum Ising model, and we show that a first order quantum phase transition separates the classical from quantum regime, with the spin correlations on the rungs being either ferromagnetic or antiferromagnetic, and different spin excitations in both regimes. The present case resembles the quantum phase transition found in the compass model in one and two dimensions.     

New method for parameter estimation in probabilistic models: minimum probability flow

Fitting probabilistic models to data is often difficult, due to the general intractability of the partition function. We propose a new parameter fitting method, minimum probability flow (MPF), which is applicable to any parametric model. We demonstrate parameter estimation using MPF in two cases: a continuous state space model, and an Ising spin glass. In the latter case, MPF outperforms current techniques by at least an order of magnitude in convergence time with lower error in the recovered coupling parameters.     

Stabilization of a steady state in oscillators coupled by a digital delayed connection

We propose the mapping of polynomial of degree 2S constructed as a linear combination of powers of spin-S (for simplicity, we called as spin-S polynomial) onto spin-crossover state. The spin-S polynomial in general can be projected onto non-symmetric degenerated spin up (high-spin) and spin down (low-spin) momenta. The total number of mapping for each general spin-S is given by 2(2^2S ? 1). As an application of this mapping, we consider a general non-bilinear spin-S Ising model which can be transformed onto spin-crossover described by Wajnflasz model. Using a further transformation we obtain the partition function of the effective spin-1/2 Ising model, making a suitable mapping the non-symmetric contribution leads us to a spin-1/2 Ising model with a fixed external magnetic field, which in general cannot be solved exactly. However, for a particular case of non-bilinear spin-S Ising model could become equivalent to an exactly solvable Ising model. The transformed Ising model exhibits a residual entropy, then it should be understood also as a frustrated spin model, due to competing parameters coupling of the non-bilinear spin-S Ising model.