Finite-size scaling in systems with long-range interaction

The finite-size critical properties of the ${\cal O}(n)$ vector $\phi^4$ model, with long-range interaction decaying algebraically with the interparticle distance $r$ like $r^{-d-\sigma}$, are investigated. The system is confined to a finite geometry subject to periodic boundary condition. Special attention is paid to the finite-size correction to the bulk susceptibility above the critical temperature $T_c$. We show that this correction has a power-law nature in the case of pure long-range interaction i.e. $0<\sigma<2$ and it turns out to be exponential in case of short-range interaction i.e. $\sigma=2$. The results are valid for arbitrary dimension $d$, between the lower ($d_=2\sigma$) critical dimensions

Nematic order by thermal disorder in a three-dimensional lattice-spin model with dipolar-like interactions

At low temperatures, some lattice spin models with simple ferromagnetic or antiferromagnetic interactions (for example nearest-neighbour interaction being isotropic in spin space on a bipartite three-dimensional lattice) produce orientationally ordered phases exhibiting nematic (second--rank) order, in addition to the primary first-rank one; on the other hand, in the Literature, they have been rather seldom investigated in this respect. Here we study the thermodynamic properties of a three-dimensional model with dipolar-like interaction. Its ground state is found to exhibit full orientational order with respect to a suitably defined staggered magnetization (polarization), but no nematic second-rank order. Extensive Monte Carlo simulations, in conjunction with Finite-Size Scaling analysis have been used for characterizing its critical behaviour; on the other hand, it has been found that nematic order does indeed set in at low temperatures, via a mechanism of order by disorder.Comment: 24 pages, 9 figure

First order phase transitions in classical lattice gas spin models

The present paper considers some classical ferromagnetic lattice--gas models, consisting of particles that carry $n$--component spins ($n=2,3$) and associated with a $D$--dimensional lattice ($D=2,3$); each site can host one particle at most, thus implicitly allowing for hard--core repulsion; the pair interaction, restricted to nearest neighbors, is ferromagnetic, and site occupation is also controlled by the chemical potential $\mu$. The models had previously been investigated by Mean Field and Two--Site Cluster treatments (when D=3), as well as Grand--Canonical Monte Carlo simulation in the case $\mu=0$, for both D=2 and D=3; the obtained results showed the same kind of critical behaviour as the one known for their saturated lattice counterparts, corresponding to one particle per site. Here we addressed by Grand--Canonical Monte Carlo simulation the case where the chemical potential is negative and sufficiently large in magnitude; the value $\mu=-D/2$ was chosen for each of the four previously investigated counterparts, together with $\mu=-3D/4$ in an additional instance. We mostly found evidence of first order transitions, both for D=2 and D=3, and quantitatively characterized their behaviour. Comparisons are also made with recent experimental results.Comment: 9 pages, 12 figure

Renormalization group treatment of the scaling properties of finite systems with subleading long-range interaction

The finite size behavior of the susceptibility, Binder cumulant and some even moments of the magnetization of a fully finite O(n) cubic system of size L are analyzed and the corresponding scaling functions are derived within a field-theoretic $\epsilon$-expansion scheme under periodic boundary conditions. We suppose a van der Waals type long-range interaction falling apart with the distance r as $r^{-(d+\sigma)}$, where $2<\sigma<4$, which does not change the short-range critical exponents of the system. Despite that the system belongs to the short-range universality class it is shown that above the bulk critical temperature $T_c$ the finite-size corrections decay in a power-in-L, and not in an exponential-in-L law, which is normally believed to be a characteristic feature for such systems.Comment: 14 pages, revte

Exact results for some Madelung type constants in the finite-size scaling theory

A general formula is obtained from which the madelung type constant: $$C(d|\nu)=\int_0^\infty dx x^{d/2-\nu-1}[(\sum_{l=-\infty}^\infty e^{-xl^2})^d-1-(\frac\pi x)^{d/2}]$$ extensively used in the finite-size scaling theory is computed analytically for some particular cases of the parameters $d$ and $\nu$. By adjusting these parameters one can obtain different physical situations corresponding to different geometries and magnitudes of the interparticle interaction.Comment: IOP- macros, 5 pages, replaced with amended version (1 ref. added