Asymptotics of a τ-function arising in the two-dimensional Ising model

The short-distance assymptotics of the -function associated to the 2-point function of the two-dimensional Ising model is computed as a function of the integration constant defined from the long-distance behavior of the -function. The result is expressible in terms of the Barnes double gamma function (equivalently, the BarnesG-function).Supported in part by the National Science Foundation, Grant No. DMS-90-01794     

Pseudo-ε-expansion and the two-dimensional Ising model

The pseudo-ε-expansions for the coordinate of the fixed point g^*, the critical exponents, and the sextic effective coupling constant g₆ are determined for the two-dimensional Ising model on the basis of the five-loop renormalization group series. It is found that the pseudoe-expansions for the coordinate of the fixed point g^*, the inverse exponent γ^?1, and the constant g₆ possess a remarkable property, namely, the higher terms of these series are so small that reliable numerical results can be obtained without invoking Borel summation.     

Zero- and low-temperature behavior of the two-dimensional ±J Ising spin glass

Scaling arguments and precise simulations are used to study the square lattice ±J Ising spin glass, a prototypical model for glassy systems. Droplet theory explains, and our numerical results show, entropically stabilized long-range spin-glass order at zero temperature, which resembles the energetic stabilization of long-range order in higher-dimensional models at finite temperature. At low temperature, a temperature-dependent crossover length scale is used to predict the power-law dependence on temperature of the heat capacity and clarify the importance of disorder distributions.     

Exact ground states of two-dimensional ±J Ising spin glasses

In this paper we study the problem of finding an exact ground state of a two-dimensional ±J Ising spin glass on a square lattice with nearest neighbor interactions and periodic boundary conditions when there is a concentrationp of negative bonds, withp ranging between 0.1 and 0.9. With our exact algorithm we can determine ground states of grids of sizes up to 50×50 in a moderate amount of computation time (up to 1 hr each) for several values ofp. For the ground-state energy of an infinite spin-glass system withp=0.5 we estimateE _0.5 =–1.4015±0.0008. We report on extensive computational tests based on more than 22,000 experiments.     

On the ground-state threshold in random two-dimensional Ising ±J models

For square, triangular, and for hexagonal lattices there is numerical and theoretical support that the ground-state thresholdp _c between ferro- and paramagnetism in random 2D Ising ±J models, withp as the concentration of antiferromagnetic bonds, is identical top ^*which is characterized by minimal matching properties of frustrated plaquettes. From square lattices of size 100×100 we have got p_c,sq<0.117 by simulations which produced average groundstate magnetizations per spin by means of exact minimal matchings. Moreover, from the squareL×L-lattices treated (L = 10, 20, 50, 100) we obtained the estimatep _c,sq 0.1 which is in agreement with the Grinstein estimatep _c,sq 0.099 andp _c,sq 0.105 by Freund and Grassberger.     

Geometric localization of the threshold in two-dimensional Ising ±J spin glasses forT=0

Following an approach of Toulouse, ground states in random 2D Ising ±J spin glasses (without external magnetic field), on square lattices, and with concentrations 0p0.5 of antiferromagnetic bonds are studied by means of minimal matchings of frustrated plaquettes. Lete(p) be the ground-state energy per spin in the thermodynamic limit. Then the well-known equatione(p)=–2+(p)f(p) holds, wheref(p) is the concentration of frustrated plaquettes and(p) is the average connection length between paired frustrated plaquettes in minimal matchings. Introducing (p) as the probability that a frustrated plaquette is matched to another frustrated plaquette by a connection of length (in a minimal matching), the average length(p) can be rewritten asgl(p)=(p). The study of(p) and its components (p) leads to an intervalp ^*pp ₂ (p ^*0.121±0.008,p ₂0.161±0.008) where the threshold between ferromagnet and paramagnet forT=0 lies. Analyzing a similar so-called adjoined average lengthl(p) admits further insight.     

Tightness of the Ising–Kac Model on the Two-Dimensional Torus

We consider the sequence of Gibbs measures of Ising models with Kac interaction defined on a periodic two-dimensional discrete torus near criticality. Using the convergence of the Glauber dynamic proven by Mourrat and Weber (Commun Pure Appl Math 70:717–812, 2017) and a method by Tsatsoulis and Weber employed in (arXiv:1609.08447 2016), we show tightness for the sequence of Gibbs measures of the Ising–Kac model near criticality and characterise the law of the limit as the \(\Phi ^4_2\) measure on the torus. Our result is very similar to the one obtained by Cassandro et al. (J Stat Phys 78(3):1131–1138, 1995) on \(\mathbb {Z}^2\), but our strategy takes advantage of the dynamic, instead of correlation inequalities. In particular, our result covers the whole critical regime and does not require the large temperature/large mass/small coupling assumption present in earlier results.     

The Ising version of the t–J model

The t–J model is analysed in the limit of strong anisotropy, where the transverse components of electron spin are neglected. We propose a slave-particle-type approach that is valid, in contradiction to many of the standard approaches, in the low-doping regime and becomes exact for a half-filled system. We describe an effective method that allows to numerically study the system with the no-double-occupancy constraint rigorously taken into account at each lattice site. Then, we use this approach to demonstrate the destruction of the antiferromagnetic order by increasing the doping and formation of Nagaoka polarons in the strong interaction regime.     

Spin-correlation function of the fully frustrated Ising model and ± J Ising spin glass on the square lattice

In this work we study the correlation function of the ground state of two-dimensional fully frustrated Ising model as well as spin glass. The Pfaffian method is used to calculate free energy and entropy as well as correlation function. We estimate the exponent of spin correlation function for fully frustrated model and spin glass. In this paper an overview of the latest results on the spin correlation function is presented.     

Renormalization of the contribution of the electron—electron interaction to the conductivity of two-dimensional electron systems

The contribution of the electron—electron interaction to the conductivity of the two-dimensional electron gas in an In_x Ga_1-x As single quantum well with different disorder strengths was experimentally studied. It is shown that the data are described well within the framework of the one-loop approximation of the renormalization group theory so long as the conductivity of the system remains higher than around 15e ²/μh.     

Study of supersolidity in the two-dimensional Hubbard–Holstein model

We derive an effective Hamiltonian for the two-dimensional Hubbard–Holstein model in the regimes of strong electron–electron and strong electron–phonon interactions by using a nonperturbative approach. In the parameter region where the system manifests the existence of a correlated singlet phase, the effective Hamiltonian transforms to a t₁ ? V ₁ ? V ₂ ? V ₃ Hamiltonian for hard-core-bosons on a checkerboard lattice. We employ quantum Monte Carlo simulations, involving stochastic-series-expansion technique, to obtain the ground state phase diagram. At filling 1∕8, as the strength of off-site repulsion increases, the system undergoes a first-order transition from a superfluid to a diagonal striped solid with ordering wavevector \(\vec{Q}\) = (π∕4, 3π∕4) or (π∕4, 5π∕4). Unlike the one-dimensional situation, our results in the two-dimensional case reveal a supersolid phase (corresponding to the diagonal striped solid) around filling 1∕8 and at large off-site repulsions. Furthermore, for small off-site repulsions, we witness a valence bond solid at one-fourth filling and tiny phase-separated regions at slightly higher fillings.     

Cancellations of infrared divergences in the two-dimensional non-linear σ models

In the two-dimensionalO(N) nonlinear models, the expectation value of anyO(N) invariant observable is shown to have an infrared finite weak coupling perturbative expansion, although it is computed in the wrong spontaneously broken symmetry phase. This result is proved by extracting all infrared divergences of any bare Feynman amplitude atD=2– dimension. The divergences cancel at any order only for invariant observables. The renormalization atD=2 preserves the infrared finiteness of the theory.     

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.     

The Phase Transition of the Quantum Ising Model is Sharp

An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d+1 dimensions. A so-called ‘random-parity’ representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.     

Phase Transition of a Distance-Dependent Ising Model on the Barabasi-Albert Network

We report our investigation on the behaviour of distance-dependent Ising models, which are located on the BA model network. The interaction strength between two nodes （the spins） is considered to obey an exponential decay dependence on the geometrical distance. The Monte Carlo simulation shows a phase transition from ferromagnetism to paramagnetism, and the critical temperature approaches a constant temperature as the interaction decaying exponent increases.     

Spectral Analysis of the Disordered Stochastic¶1-D Ising Model

We consider the generator of the Glauber dynamics for a 1-D Ising model with random bounded potential at any temperature. We prove that for any realization of the potential the spectrum of the generator is the union of separate branches (so-called k-particles branches, k= 0,1,2,…), and with probability one it is a nonrandom set. We find the location of the spectrum and prove the localization for the one-particle branch of the spectrum. As a consequence we find a lower bound for the spectral gap for any realization of the random potential. Received: 22 September 1998 / Accepted: 12 February 1999     

Effect of the defect on the focusing in a two-dimensional photonic-crystal-based flat lens

We have investigated in detail the influence of defect on the focusing of electromagnetic waves in a two-dimensional photonic-crystal flat lens by using the finite-difference time-domain method. The result shows that many focusings can be observed at the symmetrical positions when a defect is introduced into the lens. Furthermore, the wave-guides in the lens can confine the transmission wave effectively and improve the quality of the focusing.     

Spin-correlation function of the fully frustrated Ising model and ±J Ising spin glass on a square lattice

In this work we study the correlation function of the ground state of a two-dimensional fully frustrated Ising model as well as spin glass. The Pfaffian method is used to calculate free energy and entropy as well as the correlation function. We estimate the exponent of spin correlation function for the fully frustrated model and spin glass. In this paper an overview of the latest results on the spin correlation function is presented.     

An improvement of the lattice theory of dislocation for a two-dimensional triangular crystal

The structure of dislocation in a two-dimensional triangular crystal has been studied theoretically on the basis of atomic interaction and lattice statics. The theory presented in this paper is an improvement to that published previously.Within a reasonable interaction approximation, a new dislocation equation is obtained, which remedies a fault existing in the lattice theory of dislocation. A better simplification of non-diagonal terms of the kernel is given. The solution of the new dislocation equation asymptotically becomes the same as that obtained in the elastic theory, and agrees with experimental data. It is found that the solution is formally identical with that proposed phenomenologically by Foreman et al, where the parameter can be chosen freely, but cannot uniquely determined from theory. Indeed, if the parameter in the expression of the solution is selected suitably, the expression can be well applied to describe the fine structure of the dislocation.