Inhomogeneous Ising models with diagonal structure on a square lattice

We study inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate the partition function and free energy for a random coupling distribution of finite period. The phase transition is universally of Ising type. The transition temperature is independent of specific details of the coupling distribution. In particular, unexpected results for the absence of a phase transition are derived. Special examples are considered in detail, phase diagrams and critical temperature are determined. We calculate ground state energy and ground state degeneracy or, equivalently, rest entropy for pure frustration models, i.e. models with couplings of fixed strength but arbitrary sign, which never show a phase transition at a finite temperature.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln, FRG     

Layered inhomogeneous Ising models with frustration on a square lattice

Generalizing a previous treatment we study inhomogeneous Ising models on a square lattice. In particular, we consider models with a layered structure, i.e. translational invariance in the horizontal direction. Otherwise, couplings are allowed to be of arbitrary strength and sign. The partition function and free energy are calculated excatly for a random coupling distribution of finite period. The phase transition is universally of Ising type. The critical temperature can be calculated from a remarkably simple formula and depends only on the mean couplings, averaged over the period. Generally ground state properties do not reflect the singular structure of the free energy at finite temperature. This is shown by a special example.Work performed within the research program of the Sonder-forschungsbereich 125 Aachen-Jülich-Köln     

Layered Ising models on triangular and honeycomb lattices

We study inhomogeneous Ising models on triangular and honeycomb lattices. The nearest neighbour couplings can have arbitrary strength and sign such that the coupling distribution is translationally invariant in the direction of one lattice axis, i.e. the models have a layered structure. By using a transfer matrix method we derive closed form expressions for the partition functions and free energies. The critical temperatures are calculated. Phase transitions at a finite critical temperature are universally of Ising type. Models with no phase transition may show different behaviour atT=0, which is explicitly shown for fully frustrated models on square, triangular and honeycomb lattices. Finally, generalizations to layered Ising models on more general lattices are discussed.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Random layered frustration models

We study inhomogeneous two-dimensional Ising models with a random distribution of ferro- and antiferromagnetic couplings,K _ij =±K, or equivalently a random distribution of frustrations. In particular, we considerRandom Layered Frustration models (RLF) where randomness is confined to the vertical direction. These RLF-models are solved exactly, i.e., partition function and free energy are obtained in closed form for an arbitrary random distribution of finite period. The phase transition is of Ising type. A simple formula for the transition temperature is derived which depends only on the mean coupling , but not on other details of the distribution. Both cases,T _c =0 andT _c 0, are possible. Groundstate energy and groundstate degeneracy, or equivalently the rest entropy, are determined. It is found that both the occurence or absence of a phase transition may be accompanied with vanishing or nonvanishing rest entropy. We also show that for the RLF-models a phase transition is excluded when all groundstates are connected with one another by local transformations which presumably holds generally. A remarkable result is that the transition of the ferromagnetic Ising model can be destroyed completely if one replaces an arbitrarily small fraction of ferromagnetic couplings by antiferromagnetic ones in a suitable way.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Role of the spin anisotropy of the interchain interaction in weakly coupled antiferromagnetic Heisenberg chains

In quasi-one-dimensional(q1D) quantum antiferromagnets, the complicated interplay of intrachain and interchain exchange couplings may give rise to rich phenomena. Motivated by recent progress on field-induced phase transitions in the q1D antiferromagnetic(AFM) compound YbAlO_3, we study the phase diagram of spin-1/2 Heisenberg chains with Ising anisotropic interchain couplings under a longitudinal magnetic field via large-scale quantum Monte Carlo simulations,and investigate the role of the spin anisotropy of the interchain coupling on the ground state of the system. We find that the Ising anisotropy of the interchain coupling can significantly enhance the longitudinal spin correlations and drive the system to an incommensurate AFM phase at intermediate magnetic fields, which is understood as a longitudinal spin density wave(LSDW). With increasing field, the ground state changes to a canted AFM order with transverse spin correlations. We further provide a global phase diagram showing how the competition between the LSDW and the canted AFM states is tuned by the Ising anisotropy of the interchain coupling.     

Rigorous analysis of singularities and absence of analytic continuation at first-order phase-transition points in lattice-spin models

We report about two new rigorous results on the nonanalytic properties of thermodynamic potentials at first-order phase transition. For lattice models (d>or=2) with arbitrary finite state space, finite-range interactions which have two ground states, we prove that the pressure has no analytic continuation at the first-order phase-transition point, under the only further assumptions that the Peierls condition is satisfied for the ground states and that the temperature is sufficiently low. For Ising models with Kac potentials J(gamma)(x)=gamma(d)phi(gammax), where 00) and analyticity in the mean field limit (gamma SE pointing arrow 0).     

The Ising model and percolation on trees and tree-like graphs

We calculate the exact temperature of phase transition for the Ising model on an arbitrary infinite tree with arbitrary interaction strengths and no external field. In the same setting, we calculate the critical temperature for spin percolation. The same problems are solved for the diluted models and for more general random interaction strengths. In the case of no interaction, we generalize to percolation on certain tree-like graphs. This last calculation supports a general conjecture on the coincidence of two critical probabilities in percolation theory.Research partially supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship     

Correlations in inhomogeneous Ising models

We study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings are allowed to be of arbitrary strength and sign such that the coupling distribution is translationally invariant either in horizontal or in diagonal direction, i.e. the models have a layered structure. By using transfer matrix techniques the spin-spin correlations are calculated parallel to the layering and are expressed as Toeplitz determinants. After working out the general methods we discuss two special examples in detail: the fully frustrated square lattice (FFS) and the chessboard model, both having no phase transition. At zero temperature correlations in the chessboard model decay exponentially, while in the FFS model one has algebraic decay with a critical index =1/2, i.e.T=0 is a critical point. At finite temperature we find exponential decay in both models with a correlation length determined by the excitation gap in the fermion spectrum. Due to frustration correlations may develop on oscillatory structure and spins separated by an odd diagonal distance are totally uncorrelated at all temperatures.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köla     

A systematic expansion for random Ising models

An approximation scheme for random Ising models on a square lattice is derived for the case of a symmetric probability distribution of the couplings. The calculation is explicitly performed for the binary distribution. It is suggested that the quenched free energy is an analytic function of the temperature down toT=0, and that a spin glass phase as a phase in the usual sense does not exist. Rigorous nontrivial bounds for the ground state energy and the ground state entropy are also obtained.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Ground state(s) of the spin-boson hamiltonian

We establish that the finite temperature KMS states of the spin-boson hamiltonian have a limit as the temperature drops to zero. Using recent advances on the one-dimensional Ising model with long range, 1/r ², interactions, we prove qualitative properties of the ground state(s) in the ohmic case. We show (i) the asymptotics of the critical coupling as the bare energy gap goes to zero and to infinity, (ii) a jump in the order parameter, and (iii) that the number of bosons is finite below and infinite at and above the critical coupling strength.     

Exact solutions of an Ising spin chain with a spin-1 impurity

An exact solution of a single impurity model is hard to derive since it breaks translation invariance symmetry. We present the exact solution of the spin-1/2 transverse Ising chain imbedded by a spin-1 impurity. Using the hole decomposition scheme, we exactly solve the spin-1 impurity in two subspaces which are generated by a conserved hole operator.The impurity enlarges the energy deformation of the ground state above a pure transverse Ising system without impurity.The specific heat coefficient shows a small anomaly at low temperature for finite size. This indicates that the impurity can tune the ground state from a magnetic impurity space to a non-magnetic impurity space, which only exists for spin-1impurity comparing with spin-1/2 impurity and a pure transverse Ising chain without impurity. These behaviors essentially come from adding impurity freedom, which induces a competition between hole and fermion excitation depending on the coupling strength with its neighbor and the single-ion anisotropy.     

On the critical temperature of non-periodic Ising models on hexagonal lattices

The critical temperature of layered Ising models on triangular and honeycomb lattices are calculated in simple, explicit form for arbitrary distribution of the couplings.     

Monte Carlo study of low/high spin transitions

We study the finite temperature property of a model on two dimensional square lattices with two Ising spins at each lattice site by Monte Carlo simulations. When those Ising spins at a lattice site are parallel the site is said to be in the high-spin state (HS), while when they are antiparallel the site is said to be in the low-spin state (LS). Throughout the study, the energy of HS is presumed to be higher than that of LS. Two Ising spins at each site are added to form a total spin, which interacts with its nearest neighbour total spins via spin-spin couplings. The spin-phonon coupling also is introduced via harmonic springs between nearest neighbour sites with spring constants and equilibrium distances depending on the spin states of the sites involved. In this system, we investigate the feature of transitions between LS and HS (to be called low/high spin transition (LHST)) by varying the temperature. As for the ferromagnetic interaction between total spins, the second order phase transition: pure HSmixed state of HS and LS is possible to occur in a pure spin system, as is expected from mean field calculations. The role of lattice distortions by the change of lattice spacings is shown to be essential for LHST: pure LS(pure)HS. In the model investigated, there appears an indication of the strong first order phase transition which reveals a conspicuous hysteresis.     

On the critical temperature of non-periodic Ising models on hexagonal lattices

The critical temperature of layered Ising models on triangular and honeycomb lattices are calculated in simple, explicit form for arbitrary distribution of the couplings.     

Response factorization of simply connected Ising lattices with application to bethe lattice spin glasses

The thermodynamics of a classical lattice gas in Ising form, with arbitrary interaction, is set up in entropy format, with multipoint magnetizations as control parameters. It is specialized to the case of one- and two-point interactions on a simply connected lattice; both entropy and profile equations are written down explicitly. Linear response functions are expressed in Wertheim-Baxter factorization and used to derive the Jacobian of the transformation from couplings to magnetizations. An arbitrary spin-glass coupling distribution is transformed to the corresponding magnetization distribution, whose effect on thermodynamic properties is assessed. A Gaussian coupling-fluctuation expansion diverges at sufficiently large fluctuation amplitude, suggesting the possibility of a phase transition.     

Elastic Ising-like model for the nucleation and domain formation in spin crossover molecular solids

We investigate the nucleation, domain formation and propagation mechanisms observed in Spin Crossover materials, in the framework of an Ising-like model taking into account the elastic nature of the interactions. In Spin Crossover materials, the intermolecular coupling originates from a volume difference between the High Spin and the Low Spin molecular states and is simulated by anharmonic interaction potentials whose strengths are molecular-state-dependent. Using Monte Carlo methods, the phase diagram has been established. We show that the model contains both Ising short-range couplings and long-range elastic interactions. In particular, the results of long-range elastic models are reproduced. The introduction of lattice dynamics leads to the existence of spatial distributions of interaction energy and crystal field, corresponding to a local definition of physical properties. The nucleation process becomes highly dependent on the structural inhomogeneities induced by the spin transition. In this approach, connections strength between neighboring molecules are no more equivalent and have different ability to propagate domains. The presence of short-range Ising couplings gives rise to the occurrence of strong bonds forming a volume in which domains of the daughter phase can grow; in this case a macroscopic phase separation appears during the first order transition, even in a system with periodic boundary conditions. By contrast, in the case of a model with only long-range elastic interactions; strong bonds are uniformly spread in the lattice and a homogeneous phase transformation is observed, in good agreement with previous theoretical investigations.     

Thermodynamics of some layered Ising models

Thermodynamic properties of some inhomogeneous Ising models with layered distribution of couplings are studied. In particular, the specific heat is investigated in detail, both analytically and numerically. It is shown that different ordering mechanisms, namely ordering of finite regions and global ordering of infinite range, can occur in different temperature ranges. This results in remarkable structures of the specific heat curves. In addition we investigate the case where the random distribution of couplings extends over an infinite distance in one space direction. The ordinary Ising singularity then changes to one of infinite order provided the transition temperature remains finite.     

Hubbard模型中的相位弦效应与交互Chern-Simons理论

费米子符号在费米液体理论中至关重要. 然而, 在Mott绝缘体中, 很强的电子Coulomb相互作用抑制了体系的电荷涨落并消除了电子交换带来的费米子符号问题. 本文首先回顾二分晶格上Hubbard模型的相位弦理论, 从弱关联的费米液体到强关联的反铁磁Mott绝缘体的转变可以由此得到统一理解. 在任意Coulomb作用强度U下, 我们首先导出Hubbard模型的严格的符号结构. 在小U极限下, 它回到通常的费米子符号; 在大U极限下, 它给出了t-J模型的相位弦符号. 在半满情形下, 我们构造了一种电子分数化的表象, 其中, 电荷子与自旋子通过演生的交互Chern-Simons规范场相互耦合. 由此导出的基态波函数拟设与低能有效理论可以定性刻画Hubbard模型的基态相图. 在弱关联区域, 费米液体的准粒子由电荷子与自旋子的束缚态构成, 其长程相位相干性取决于背景自旋的关联性质. 体系的Mott转变可以通过电荷子打开能隙或是通过自旋子玻色凝聚来实现.     

The phase diagrams of a spin-1 transverse Ising model

The phase diagrams of the spin-1 transverse Ising model with the presence of a crystal field is investigated by using an effective-field theory (EFT). We give a method to calculate the Gibbs free energy numerically at finite temperature within the EFT. The first-order transition lines are obtained by comparing the Gibbs free energy. The phase diagrams and the Gibbs free energy are also compared with those given using the mean-field theory (MFT).     

Renormalization group studies of phase diagrams of Ising and XY-model films of variable thickness

We compare the critical behavior of $\text{spin}\text{-}\text{1}\text{2}$ Ising model (ordinary phase transition in two dimensions) and classical XY-model (topological phase transition in two dimensions) films, with two flat surfaces and nearest neighbor couplings K_S between surface spins and K_B between all others, as a function of film thickness. We carry out a real space Migdal-style renormalization in two stages. In the bulk stage the film is first renormalized towards a double layer, with the renormalized parameters as inter- and intra-layer couplings. Then the double layer is renormalized with those couplings as initial parameters. From the RG-equations for the bulk stage we find a tricritical point, not only for the Ising model (in which case it is well known) but also for the XY-model. It signals the existence of distinct surface and bulk transitions for sufficiently large values of $\text{K}{}_{\mathrm{<mtext>S</mtext>}}\text{K}{}_{\mathrm{<mtext>B</mtext>}}$. For the Ising model the complete program can be carried out and the phase diagram for films of arbitrary thickness is constructed. For the double layer XY-model a sufficiently complex Migdal-style renormalization appears to be unfeasible, presumably due to the possible presence of strings between the layers. Therefore, in an alternative approach, two representations for the partition function of the double layer XY-model are given. The system can be described in terms of its topological excitations, i.e., vortices on each layer and strings, either closed or terminating in vortices, between them. The system is also written as three coupled Coulomb gases. Based on this representation a renormalization group is found, and used, together with a Griffiths inequality for the correlation functions, to obtain information on the phase diagram. If there is a single phase transition, there is one phase where the correlations exhibit power law behavior and another where they fall off exponentially. The transition temperature increases monotonously with interlayer coupling to twice its value in two dimensions, but the nature of the phase transition for any finite inter-layer coupling appears to be different from that at zero coupling. We suggest that this is associated with the behavior of the internal strings. These results should be relevant for the renormalization of the film with isotropic bulk couplings as well as for layered systems of finite thickness, with different inter- and intra-layer couplings.