The extended Hubbard model in the ionic limit

In this paper, we study the Hubbard model with intersite Coulomb interaction in the ionic limit (i.e. no kinetic energy). It is shown that this model is isomorphic to the spin-1 Ising model in presence of a crystal field and an external magnetic field. We show that for such models it is possible to find, for any dimension, a finite complete set of eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of the equations of motion closes and analytical expressions for the relevant Green's functions and correlation functions can be obtained. These expressions are formal because these functions depend on a finite set of unknown parameters, and only a set of exact relations among the correlation functions can be derived. In the one-dimensional case we show that by means of algebraic constraints it is possible to obtain extra equations which close the set and allow us to obtain a complete exact solution of the model. The behavior of the relevant physical properties for the 1D system is reported.     

Polyakov loop and spin correlators on finite lattices A study beyond the mass gap

We derive an analytic expression for point-to-point correlation functions of the Polyakov loop based on the transfer matrix formalism. For the 21) Ising model we show that the results deduced from point-point spin correlators are coinciding with those from zero momentum correlators. We investigate the contributions from eigenvalues of the transfer matrix beyond the mass gap and discuss the limitations and possibilities of such an analysis. The finite size behaviour of the obtained 21) Ising model matrix elements is examined. The point-to-point correlator formula is then applied to Polyakov loop data in finite temperature SU(2) gauge theory. The leading matrix element shows all expected scaling properties. Just above the critical point we find a Debye screening mass μ_D/T ≈ 4 , independent of the volume.     

Variational method for lattice systems: General formalism and application to the two-dimensional Ising model in an external field

A simple variational approach to the eigenvalue problem of the transfer operator is proposed. After reducing the transfer operator according to the symmetries of the Hamiltonian, the leading eigenvalues of the irreducible blocks can be evaluated by elementary variational principles. Hence the thermodynamics and a large class of correlation functions of lattice systems can be calculated. Following a natural truncation scheme the results can be improved in a systematic way. The high accuracy and the convergence of the method is demonstrated by two-dimensional Ising model. As a first application, the thermodynamics of the two-dimensional Ising ferro-and antiferromagnet in an external field is studied. We show how the same method can be used to obtain zero-temperature properties of interacting quantum lattice systems.     

Dynamic properties of the Monte Carlo method in statistical mechanics

By means of the Monte Carlo sampling technique the equilibrium thermodynamics of fluids and magnets can be calculated numerically. We show that the questions of convergence and accuracy of this method can be understood in terms of the dynamics of the appropriate stochastic model. Also, we discuss to what extent various choices of transition probabilities lead to different dynamic properties of the system. As examples of applications, we consider Ising and Heisenberg spin systems. The numerical results about the dynamic correlation functions are compared to simple approximations taken from the theory of the kinetic Ising model.     

Auxiliary Vertices Method for Kagomé-Lattice Eight-Vertex Model

We investigate a class of eight-vertex models on a Kagomé lattice. With the help of auxiliary vertices, the Kagomé-lattice eight-vertex model (KEVM) is related to an inhomogeneous system which leads to a one-parameter family of commuting transfer matrices. Using an equation for commuting transfer matrices, we determine their eigenvalues. From calculated eigenvalues the correlation length of the KEVM is derived with its full anisotropy. There are two cases: In the first case the anisotropic correlation length (ACL) is the same as that of the triangular/honeycomb-lattice Ising model. By the use of an algebraic curve, it is shown that the Kagomé-lattice Ising model, the diced-lattice Ising model, and the hard-hexagon model also have (essentially) the same ACL as the KEVM. In the second case we find that the ACL displays 12fold rotational symmetry.     

Solving Generalized Polyomino Puzzles Using the Ising Model

In the polyomino puzzle, the aim is to fill a finite space using several polyomino pieces with no overlaps or blanks. Because it is an NP-complete combinatorial optimization problem, various probabilistic and approximated approaches have been applied to find solutions. Several previous studies embedded the polyomino puzzle in a QUBO problem, where the original objective function and constraints are transformed into the Hamiltonian function of the simulated Ising model. A solution to the puzzle is obtained by searching for a ground state of Hamiltonian by simulating the dynamics of the multiple-spin system. However, previous methods could solve only tiny polyomino puzzles considering a few combinations because their Hamiltonian designs were not efficient. We propose an improved Hamiltonian design that introduces new constraints and guiding terms to weakly encourage favorable spins and pairs in the early stages of computation. The proposed model solves the pentomino puzzle represented by approximately 2000 spins with >90% probability. Additionally, we extended the method to a generalized problem where each polyomino piece could be used zero or more times and solved it with approximately 100% probability. The proposed method also appeared to be effective for the 3D polycube puzzle, which is similar to applications in fragment-based drug discovery.     

Finite Size Corrections for the Ising Model on Higher Genus Triangular Lattices

We study the topology dependence of the finite size corrections to the Ising model partition function by considering the model on a triangular lattice embedded on a genus two surface. At criticality we observe a universal shape dependent correction, expressible in terms of Riemann theta functions, that reproduces the modular invariant partition function of the corresponding conformal field theory. The period matrix characterizing the moduli parameters of the limiting Riemann surface is obtained by a numerical study of the lattice continuum limit. The same results are reproduced using a discrete holomorphic structure.     

Finitized Conformal Spectra of the Ising Model on the Klein Bottle and Möbius Strip

We study the conformal spectra of the critical square lattice Ising model on the Klein bottle and Möbius strip using Yang–Baxter techniques and the solution of functional equations. In particular, we obtain expressions for the finitized conformal partition functions in terms of finitized Virasoro characters. This demonstrates that Yang–Baxter techniques and functional equations can be used to study the conformal spectra of more general exactly solvable lattice models in these topologies. The results rely on certain properties of the eigenvalues which are confirmed numerically.     

Boundary conditions and correlation functions in the ν-dimensional Ising model at low temperature

The boundary condition dependence of the correlation functions in a phase transition region of the thermodynamic parameters is of great importance to understand the character and properties of the phase transition itself. In this paper we study the boundary condition dependence of certain correlation functions in the Ising model at low temperature.     

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.     

An exactly soluble case of the triangular ising model in a magnetic field

An anisotropic triangular Ising model in which the first- and second-order parameters and the field parameters are functionally related is solved exactly by representing the distribution of the atom patterns in terms of a suitably constructed Markov process. The probabilities of patterns, defined as the probabilities generated by this process, are a mathematically tractable alternative to the classical representation of these probabilities in terms of the partition function. The interaction and field parameters of this Ising model, its magnetization, free energy, and its nearest neighbor correlation functions, are expressed in terms of the parameters of this Markov process. Special cases are worked out in detail and numerical examples are given.     

Exact solution of the one-dimensional spin-\frac{\mathsf 3}{\mathsf 2} Ising model in magnetic field

In this paper, we study the Ising model with general spin S in presence of an external magnetic field by means of the equations of motion method and of the Green's function formalism. First, the model is shown to be isomorphic to a fermionic one constituted of 2S species of localized particles interacting via an intersite Coulomb interaction. Then, an exact solution is found, for any dimension, in terms of a finite, complete set of eigenoperators of the latter Hamiltonian and of the corresponding eigenenergies. This explicit knowledge makes possible writing exact expressions for the corresponding Green's function and correlation functions, which turn out to depend on a finite set of parameters to be self-consistently determined. Finally, we present an original procedure, based on algebraic constraints, to exactly fix these latter parameters in the case of dimension 1 and spin . For this latter case and, just for comparison, for the cases of dimension 1 and spin [F. Mancini, Eur. Phys. J. B 45, 497 (2005)] and spin 1 [F. Mancini, Eur. Phys. J. B 47, 527 (2005)], relevant properties such as magnetization 〈S 〉 and square magnetic moment 〈S² 〉, susceptibility and specific heat are reported as functions of temperature and external magnetic field both for ferromagnetic and antiferromagnetic couplings. It is worth noticing the use we made of composite operators describing occupation transitions among the 3 species of localized particles and the related study of single, double and triple occupancy per site.     

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.     

Superconductivity in Hubbard models coupled to non-fermionic degrees of freedom

We explore the consequences of coupling between repulsive Hubbard models and Bosonic or spin degrees of freedom. In the regime where the characteristic energy of the non-fermionic part is large compared to the characteristic energy of the Fermions, the effective Hamiltonian corresponds to a generalized attractive Hubbard model. Superconducting properties are then calculated within the BCS scheme including the finite size dependence of correlation functions functions characterizings-wave pairing.     

Quantum Harmonic Oscillator Systems with Disorder

We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction correlators. In particular, we show that static correlations decay exponentially fast if the corresponding effective one-particle Hamiltonian exhibits localization at low energies, regardless of whether there is a gap in the spectrum above the ground state or not. Our results apply to finite as well as to infinite oscillator systems. The eigenfunction correlators that appear are more general than those previously studied in the literature. In particular, we must allow for functions of the Hamiltonian that have a singularity at the bottom of the spectrum. We prove exponential bounds for such correlators for some of the standard models.     

自我质疑机制下公共物品博弈模型的相变特性

公共物品博弈是研究群体相互作用的经典模型,广泛用于解释自私个体间合作的涌现和保持.本文从理论分析和蒙特卡罗模拟两个方面研究了二维正方格子上一个有偿惩罚机制下随自我质疑更新规则演化的公共物品博弈模型的相变特性.理论分析方面,将公共物品博弈模型转化为一个外场不为零的铁磁Ising模型.通过有效能量发现:不存在惩罚时,个体间的耦合强度为零,体系只有外场作用;存在惩罚时,个体间包含最近邻、次近邻和第三近邻相互作用且外场不为零.蒙特卡罗模拟方面,首先验证了理论分析的正确性,然后对公共物品博弈模型相关的一级相变和二级相变进行了有限尺度标度分析.研究发现:1)蒙特卡罗模拟所得结果与类Ising模型分析结果完全吻合;2)相比二维Ising模型,公共物品博弈的二级相变临界指数发生了变化;3)公共物品博弈的一级相变与二维Ising模型相同.     

Simulation of a critical state without critical slowing down by a renormalization group multi-grid method

We have implemented and tested a method to eliminate critical slowing down from Monte Carlo (and possibly other) simulations of very large systems, even for a “critical” state, where the correlation length of fluctuations is the size of the sample. Static correlation functions on all length scales (down to microscopic!) may thus be obtained with an amount of work asymptotically growing with size only as in conventional simulations of nearly uncorrelated states. In the simulation we use a finite but large set of approximated renormalized coupling constants, which describe very closely the coarse grained variable interactions of the simulated model. As a test bed, the 2D Ising model has been used. The method has the advantage that critical states of other types of systems can be simulated along the same line.     

How random are matrix elements of the nuclear shell model Hamiltonian?

In this paper we study the general behavior of matrix elements of the nuclear shell model Hamiltonian.We find that nonzero off-diagonal elements exhibit a regular pattern,if one sorts the diagonal matrix elements from smaller to larger values.The correlation between eigenvalues and diagonal matrix elements for the shell model Hamiltonian is more remarkable than that for random matrices with the same distribution unless the dimension is small.     

Exactly Solvable Model of Quantum Diffusion

We study the transport property of diffusion in a finite translationally invariant quantum subsystem described by a tight-binding Hamiltonian with a single energy band. The subsystem interacts with its environment by a coupling expressed in terms of correlation functions which are delta-correlated in space and time. For weak coupling, the time evolution of the subsystem density matrix is ruled by a quantum master equation of Lindblad type. Thanks to the invariance under spatial translations, we can apply the Bloch theorem to the subsystem density matrix and exactly diagonalize the time evolution superoperator to obtain the complete spectrum of its eigenvalues, which fully describe the relaxation to equilibrium. Above a critical coupling which is inversely proportional to the size of the subsystem, the spectrum at given wave number contains an isolated eigenvalue describing diffusion. The other eigenvalues rule the decay of the populations and quantum coherences with decay rates which are proportional to the intensity of the environmental noise. An analytical expression is obtained for the dispersion relation of diffusion. The diffusion coefficient is proportional to the square of the width of the energy band and inversely proportional to the intensity of the environmental noise because diffusion results from the perturbation of quantum tunneling by the environmental fluctuations in this model. Diffusion disappears below the critical coupling     

Ising Models with Four Spin Interaction at Criticality

We consider two bidimensional Ising models coupled by an interaction quartic in the spins, like in the spin representation of the Eight vertex or the Ashkin-Teller model. By Renormalization Group methods we write a convergent perturbative expansion for the specific heat and for the energy-energy correlation up to the critical temperature. A form of nonuniversality is proved, in the sense that the critical behaviour is described in terms of critical indices which are non trivial functions of the coupling. The logarithmic singularity of the specific heat of the Ising model is removed or changed in a power law (with a non universal critical index) depending on the sign of the interaction.