The specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of
n parallel layers with free boundary conditions and a surface magnetic field
H1=kBTh1 imposed on the last layer, is analyzed for large
n in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form $\ensuremath{\Delta}f(n,t,{h}_{1})\ensuremath{\approx}{n}^{\ensuremath{-}2}(\mathrm{ln}n){W}_{1}(nt,{n}^{\frac{1}{2}}{h}_{1})+{n}^{\ensuremath{-}2}{W}_{2}(nt,{n}^{\frac{1}{2}}{h}_{1})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The scaling functions
Wi(x,y) are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed $t\ensuremath{\ne}0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order ${e}^{\frac{\mathrm{pna}}{\ensuremath{\xi}(T)}}$, where
a is the lattice spacing and $\ensuremath{\xi}(T)$ is the bulk correlation length; the value of the small rational constant
p is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").
The partition functions of the two- and three-dimensional finite Ising models in the presence of a magnetic field have been calculated with use of a high-speed computer. Periodic boundary conditions were imposed. All the numbers of configurations for the 3×3×3, 5×5, and 4×6 lattices are tabulated. The distribution of zeros of the partition functions in the complex magnetic-field plane has been obtained both for the ferro- and antiferromagnetic cases. The zeros in the ferromagnetic cases are distributed on the unit circle according to the Yang-Lee theorem, and the dependence on temperature of the distribution functions has been studied. In the antiferromagnetic case most of zeros are on the negative real axis, but a certain number of complex roots appear in the left half of the plane. The magnetization and the magnetic susceptibility have been calculated as functions of temperature.
Interest in finite-size systems has risen in the last decades, due to the focus on nanotechnological applications and because they are convenient for numerical treatment that can subsequently be extrapolated to infinite lattices.Independently of the envisioned application, special attention must be given to boundary condition, which may or may not preserve the symmetry of the infinite lattice.Here we present a detailed study of the compatibility between boundary conditions and conservation laws.The conflict between open boundary conditions and momentum conservation is well understood, but we examine other symmetries, as well: we discuss gauge invariance, inversion, spin, and particle-hole symmetry and their compatibility with open, periodic, and twisted boundary conditions.In the interest of clarity, we develop the reasoning in the framework of the one-dimensional half-filled Hubbard model, whose Hamiltonian displays a variety of symmetries.Our discussion includes analytical and numerical results.Our analytical survey shows that, as a rule, boundary conditions break one or more symmetries of the infinite-lattice Hamiltonian.The exception is twisted boundary condition with the special torsion Θ = πL/2, where L is the lattice size.Our numerical results for the ground-state energy at half-filling and the energy gap for L = 2-7 show how the breaking of symmetry affects the convergence to the L → ∞ limit.We compare the computed energies and gaps with the exact results for the infinite lattice drawn from the Bethe-Ansatz solution.The deviations are boundary-condition dependent.The special torsion yields more rapid convergence than open or periodic boundary conditions.For sizes as small as L = 7, the numerical results for twisted condition are very close to the L → ∞ limit.We also discuss the ground-state electronic density and magnetization at half filling under the three boundary conditions.
In contrast to an infinite chain, the low-temperature expansion of a one-dimensional free-field Ising model has a strong dependence on boundary conditions. I derive an explicit formula for the leading term of the expansion both under open and periodic boundary conditions and show that they are related to different distributions of the partition function zeros on the complex temperature plane. In particular, when a periodic boundary condition is imposed, the leading coefficient of the expansion grows with increasing size of the chain, due to the zeros approaching the origin.
Abstract The infinite interaction range model (IIRM) for ferromagnetic systems is presented in its spin‐free formulation. In this formulation the states are labelled by partitions which provide a natural variable for thermodynamic computation. We have extended the calculations of Kittel and Shore by computing to a practical thermodynamic limit (N ∼ 100,000). The heat capacity, magnetic susceptibility and the magnetization of the first two functions exhibit a critical temperature while the magnetization is zero at zero field for all temperatures. Spontaneous magnetization is obtained by linear extrapolation from high field or equivalently by a polarized partition function. Relationships are explored among IIRM, the Heisenberg model and the mean field model. Application to IIRM of the Yang‐Lee condition for a phase transition yields a critical temperature identical to that obtained by the direct calculation.