Dimers and the Ising model

We present a innovative relationship between ground states of the Ising model and dimer coverings which sheds new light on the Ising models with highly degenerate ground states and enables one to construct such models. Thanks to this relationship we also find the generating function for dimers as the appropriate limit of the free energy per spin for the Ising model.     

Ising model in the high density limit

It is proved that the free energy per spin in the thermodynamic limit of an Ising model on a lattice with coordination numberz approaches the classical Curie-Weiss free energy in the limitz→∞. The infinite spacial dimension limit of nearest neighbour lattice models is a special case of this result.     

Many-component limit of the quantum Potts model

The q-component quantum Potts model is solved exactly in the limit q→∞. The resulting free energy and its first partial derivatives (order parameters) are shown to be identical to the corresponding mean-field expressions taken in the same limit.     

Free energy of the Falicov-Kimball model in large dimensions

We show how to calculate the free energy of the spinless Falicov Kimball model exactly in the limit of infinite dimensions, and do it explicitly for the homogeneous and the chessboard phase. By comparing the free energies for those two cases we study the transitions between the pure phases, and by looking for concavities in the course of the free energy as a function of the meanf-particle density we examine the possibility of phase segregation.     

Investigation of magnetic field effect on surface and finite-site free energy in one-dimensional Ising model of nanosystems

We investigate a one-dimensional (1-D) Ising model for finite-site systems. The finite-site free energy and the surface free energy are calculated via the transfer matrix method. We show that, at high magnetic fields, the surface free energy has an asymptotic limit. The absolute surface energy increases when the value of f (the ratio of magnetic field to nearest-neighbor interactions) increases, and for f?≥?10 approaches a constant value. For the values of f?≥?0.2, the finite-site free energy also increases, but slowly. The thermodynamic limit in which physical properties approach the bulk value is also explored.     

Equilibrium state and free energy of an infinite mode dicke maser model

For an infinite mode Dicke maser model, we show that the infimum of the free energy over a class of translation-invariant product states is attained for a quasi-free state. Furthermore, the thermodynamic limit of the free energy is proved to exist and equals the quasi-free solution.     

The anisotropic Heisenberg model in the long-range interaction limit

We consider the general spin quantum mechanical anisotropic Heisenberg model on av-dimensional lattice with potential γ ^v ?(γ|r|) of Kac type. In the limit γ→0+ (after the thermodynamic limit) it is shown that the free energy is equivalent to the corresponding Curie-Weiss or mean-field expression.     

Free energy of the solvable chiral Potts model

Very recently, it has been shown that there are chiralN-state Potts models in statistical mechanics that satisfy the star-triangle relation. Here it is shown that the relation implies that the free energy (and its derivatives) satisfies certain functional relations. These can be used to obtain the free energy: in particular, we expand about the critical case and find that the exponent is 1–2/N.     

Exact solution of the Faddeev-Volkov model

The Faddeev-Volkov model is an Ising-type lattice model with positive Boltzmann weights where the spin variables take continuous values on the real line. It serves as a lattice analog of the sinh-Gordon and Liouville models and intimately connected with the modular double of the quantum group Uq(sl₂). The free energy of the model is exactly calculated in the thermodynamic limit. In the quasi-classical limit c→+∞ the model describes quantum fluctuations of discrete conformal transformations connected with the Thurston's discrete analogue of the Riemann mappings theorem. In the strongly-coupled limit c→1 the model turns into a discrete version of the D=2 Zamolodchikov's “fishing-net” model.     

Low temperature properties of the hierarchical classical vector model

We obtain low temperature properties of the classical vector model in a hierarchical formulation in three or more dimensions. We consider the lattice model in a zero or non-zero magnetic field, where the single site spin variable R ^v has a density proportional to for large . Using renormalization group methods we obtain a convergent expansion for the free energy with zero magnetic field. For non-zero fields a shift formula is used to obtain the effective action generated by the renormalization group transformation (RGT). To obtain the pure state zero field free energy and spontaneous magnetization we take the thermodynamic limit together with the zero field limit at a specified rate. The spontaneous magnetization,m, is calculated, is non-zero and the pure state free energy coincides, as expected, with the zero field free energy. Also the sequence of zero field actions does not have a limit but we show that the sequence of actions generated from the original action shifted bym does; the limiting action corresponds to a non-canonical Gaussian fixed point of the RGT.     

Curved polymer and polyelectrolyte brushes beyond the Daoud-Cotton model

We revise the classical Daoud-Cotton (DC) model to describe conformations of polymer and polyelectrolyte chains end-grafted to convex spherical and cylindrical surfaces. In the framework of the DC model, local stretching of chains in the brush does not depend on the degree of polymerization of grafted chains, and the polymer density profile follows a single-exponent power law. This model, however, does not correspond to a minimum in free energy of the curved brush. The nonlocal (NL) approximation exploited in the present paper implies the minimization of the overall free energy of the brush and predicts that the polymer density profile does not follow a single-exponent power law. In the limit of large surface curvature the NL approximation provides the same scaling laws for brush thickness and free energy as the local DC model. Numerical prefactors are however different. Extra extension of chains in the brush interior region leads to larger equilibrium brush thickness and lower free energy per chain. A significant difference between outcomes of the two models is found for brushes formed by ionic polymers, particularly for weakly dissociating (p H-sensitive) polyelectrolytes at low solution salinity.     

最近邻弱交换相互作用对纳米管中Blume-Capel模型热学性质的研究

利用有效场理论研究了纳米管上最近邻弱交换相互作用下spin-1纳米管中Blume-Capel模型的内能、比热和自由能,得到了系统的内能、比热和自由能与最近邻弱交换相互作用和晶场的关系。结果表明：最近邻弱交换相互作用和晶场强度等诸多因素相互竞争,使系统表现出比 = = =1时的BC模型更为复杂的热学性质;系统内能随温度的变化曲线表现出不连续性;比热随温度的变化出现奇异性;高温对自由能的影响更加明显。     

Large-n limit of the Heisenberg model: Random external field and random uniaxial anisotropy

The thermodynamic equivalence of the large-n limit of then-vector model in a random external field and the corresponding disordered spherical model is proved. An analytic expression for the free energy and a phase diagram of the large-n limit of then-vector model with random uniaxial anisotropy are obtained by rigorous argument. The ferromagnetic order in the large-n limit is proved to be stable against the switching on of an arbitrarily small random anisotropy.     

Dynamic model and stationary shapes of fluid vesicles

A phase-field model that takes into account the bending energy of fluid vesicles is presented. The Canham-Helfrich model is derived in the sharp-interface limit. A dynamic equation for the phase-field has been solved numerically to find stationary shapes of vesicles with different topologies and the dynamic evolution towards them. The results are in agreement with those found by minimization of the Canham-Helfrich free energy. This fact shows that our phase-field model could be applied to more complex problems of instabilities.     

Condensation and metastability in the 2D Potts model

For the first order transition of the Ising model below , Isakov has proven that the free energy possesses an essential singularity in the applied field. Such a singularity in the control parameter, anticipated by condensation theory, is believed to be a generic feature of first order transitions, but too weak to be observable. We study these issues for the temperature driven transition of the q states 2D Potts model at . Adapting the droplet model to this case, we relate its parameters to the critical properties at and confront the free energy to the many informations brought by previous works. The essential singularity predicted at the transition temperature leads to observable effects in numerical data. On a finite lattice, a metastability domain of temperatures is identified, which shrinks to zero in the thermodynamical limit. Received 30 March 1999     

On the application of the critical minimum energy subspace method to disordered systems

We discuss the recent application to strongly disordered systems of the Critical Minimum Energy Subspace (CrMES) method, used to limit the energy subspace of the Wang-Landau sampling. We compare with our results on the 3D random field Ising model obtained by a multi-range Wang-Landau simulation over the whole energy range. We point out some problems that may arise when applying the CrMES scheme to models having a complex free energy landscape.     

Exact results for a generalized classicalO(n) matrix spin model

A generalizedO(n) matrix version of the classical Heisenberg model, introduced by Fuller and Lenard as a classical limit of a quantum model, is solved exactly in one dimension. The free energy is analytic and the pair correlation functions decay exponentially for all finite temperatures. It is shown, however, that even for a finite number of spins the model has a phase transition in then limit. The transition features a specific heat jump, zero long-range order at all temperatures, and zero correlation length at the critical point. The Curie-Weiss version of the model is also solved exactly and shown to have standard mean-field type behavior for all finiten and to differ from the one-dimensional results in then limit.     

Quantum Ashkin-Teller model near the decoupling limit

The critical properties and the spectrum of the quantum Ashkin-Teller chain are investigated by perturbation expansion around the exactly soluble Ising decoupling limit. The critical exponents are found to satisfy the hyperscaling relation and they are consistent with the conjectured values. The critical Hamiltonian in the finite-size scaling limit is transformed into a two-band spin-1/2 Fermi system, where the interaction energy in first order is the product of the band magnetizations. The complete spectrum is shown to exhibit a conformal structure with infinite primary operators, the anomalous dimensions of which are consistent with a Gaussian form.     

An exact solution to the classical,anisotropic Heisenberg model with long-range Kac interactions

A rigorous derivation is given for the “constant-magnetization” free energy density of the classical, anisotropic Heisenberg model with long-range Kac interactions. The derivation involves bounding arguments similar to those used for a classical fluid by Lebowitz and Penrose. The present work is carried out in a constant-magnetization ensemble. The free energy density is determined exactly under a quadruple-limiting process. The limits involved are a Lebowitz-Penrose type of triple-limiting process, followed by a final limit,x→ 0, wherex is a parameter which represents the range over which each component of the net spin density can vary. Explicit equations of state are determined for the special case of zero short-range interactions plus pure Kac-type long-range interactions.     

Crust-core properties of neutron stars in the Nambu–Jona-Lasinio model

We adopt the Nambu–Jona-Lasinio(NJL) model to study the crust-core transition properties in neutron stars(NSs). For a given momentum cutoff and symmetry energy of saturation density in the NJL model, decreasing the slope of the symmetry energy gives rise to an increase in the crust-core transition density and transition pressure.Given the slope of the symmetry energy at saturation density, the transition density and corresponding transition pressure increase with increasing symmetry energy. The increasing trend between the fraction of the crustal moment of inertia and the slope of symmetry energy at saturation density indicates that a relatively large momentum cutoff of the NJL model is preferred. For a momentum cutoff of 500 Me V, the fraction of the crustal moment of inertia clearly increases with the slope of symmetry energy at saturation density. Thus, at the required fraction(7%) of the crustal moment of inertia, the NJL model with momentum cutoff of 500 Me V and a large slope of the symmetry energy of saturation density can give the upper limit of the mass of the Vela pulsar to be above 1.40 M_⊙.