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 order
ed 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.
The quantum critical behavior of the 2+1 dimensional Gross--Neveu model in the vicinity of its zero temperature critical point is considered. The model is known to be renormalisable in the large $N$ limit, which offers the possibility to obtain expre
ssions for various thermodynamic functions in closed form. We have used the concept of finite--size scaling to extract information about the leading temperature behavior of the free energy and the mass term, defined by the fermionic condensate and determined the crossover lines in the coupling ($g$) -- temperature ($T$) plane. These are given by $Tsim|g-g_c|$, where $g_c$ denotes the critical coupling at zero temperature. According to our analysis no spontaneous symmetry breaking survives at finite temperature. We have found that the leading temperature behavior of the fermionic condensate is proportional to the temperature with the critical amplitude $frac{sqrt{5}}3pi$. The scaling function of the singular part of the free energy is found to exhibit a maximum at $frac{ln2}{2pi}$ corresponding to one of the crossover lines. The critical amplitude of the singular part of the free energy is given by the universal number $frac13[frac1{2pi}zeta(3)-mathrm{Cl}_2(frac{pi}3)]=-0.274543...$, where $zeta(z)$ and $mathrm{Cl}_2(z)$ are the Riemann zeta and Clausens functions, respectively. Interpreted in terms the thermodynamic Casimir effect, this result implies an attractive Casimir force. This study is expected to be useful in shedding light on a broader class of four fermionic models.
The theory of phase transitions is based on the consideration of idealized models, such as the Ising model: a system of magnetic moments living on a cubic lattice and having only two accessible states. For simplicity the interaction is supposed to be
restricted to nearest--neighbour sites only. For these models, statistical physics gives a detailed description of the behaviour of various thermodynamic quantities in the vicinity of the transition temperature. These findings are confirmed by the most precise experiments. On the other hand, there exist other cases, where one must account for additional features, such as anisotropy, defects, dilution or any effect that may affect the nature and/or the range of the interaction. These features may have impact on the order of the phase transition in the ideal model or smear it out. Here we address two classes of models where the nature of the transition is altered by the presence of anisotropy or dilution.
At variance with the authors statement [L. P{a}lov{a}, P. Chandra and P. Coleman, Phys. Rev. B 79, 075101 (2009)], we show that the behavior of the universal scaling amplitude of the gap function in the phonon dispersion relation as a function of the
dimensionality $d$, obtained within a self--consistent one--loop approach, is consistent with some previous analytical results obtained in the framework of the $epsilon$--expansion in conjunction with the field theoretic renormalization group method [S. Sachdev, Phys. Rev. B 55, 142 (1997)] and the exact calculations corresponding to the spherical limit i.e. infinite number $N$ of the components of the order parameter [H. Chamati. and N. S. Tonchev, J. Phys. A: Math. Gen. 33, 873 (2000)]. Furthermore we determine numerically the behavior of the temporal Casimir amplitude as a function of the dimensionality $d$ between the lower and upper critical dimension and found a maximum at $d=2.9144$. This is confirmed via an expansion near the upper dimension $d=3$.
A detailed analysis of the finite-size effects on the bulk critical behaviour of the $d$-dimensional mean spherical model confined to a film geometry with finite thickness $L$ is reported. Along the finite direction different kinds of boundary condit
ions are applied: periodic $(p)$, antiperiodic $(a)$ and free surfaces with Dirichlet $(D)$, Neumann $(N)$ and a combination of Neumann and Dirichlet $(ND)$ on both surfaces. A systematic method for the evaluation of the finite-size corrections to the free energy for the different types of boundary conditions is proposed. The free energy density and the equation for the spherical field are computed for arbitrary $d$. It is found, for $2<d<4$, that the singular part of the free energy has the required finite-size scaling form at the bulk critical temperature only for $(p)$ and $(a)$. For the remaining boundary conditions the standard finite-size scaling hypothesis is not valid. At $d=3$, the critical amplitude of the singular part of the free energy (related to the so called Casimir amplitude) is estimated. We obtain $Delta^{(p)}=-2zeta(3)/(5pi)=-0.153051...$, $Delta^{(a)}=0.274543...$ and $Delta^{(ND)}=0.01922...$, implying a fluctuation--induced attraction between the surfaces for $(p)$ and repulsion in the other two cases. For $(D)$ and $(N)$ we find a logarithmic dependence on $L$.
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 im
plicitly 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.
