arXiv subjects

H. Chamati

Publications and source records attributed to H. Chamati.

30 records · Page 2Linked to original sources

arXiv2005

Generalized Mittag-Leffler functions in the theory of finite-size scaling for systems with strong anisotropy and/or long-range interaction

The difficulties arising in the investigation of finite-size scaling in $d$--dimensional O(n) systems with strong anisotropy and/or long-range interaction, decaying with the interparticle distance $r$ as $r^{-d-σ}$ ($0<σ\leq2$), are discussed. Some integral representations aiming at the simplification of the investigations are presented for the classical and quantum lattice sums that take place in the theory. Special attention is paid to a more general form allowing to treat both cases on an equal footing and in addition cases with strong anisotropic interactions and different geometries. The analysis is simplified further by expressing this general form in terms of a generalization of the Mittag-Leffler special functions. This turned out to be very useful for the extraction of asymptotic finite-size behaviours of the thermodynamic functions.

cond-mat.stat-mech↗

arXiv2004

Critical Casimir forces for ${\cal O}(n)$ systems with long-range interaction in the spherical limit

We present exact results on the behavior of the thermodynamic Casimir force and the excess free energy in the framework of the $d$-dimensional spherical model with a power law long-range interaction decaying at large distances $r$ as $r^{-d-σ}$, where $σ T_c$ and $T T_c$ it decays as $L^{-d-σ}$, where $L$ is the thickness of the film. We consider both the case of a finite system that has no phase transition of its own, when $d-1<σ$, as well as the case with $d-1>σ$, when one observes a dimensional crossover from $d$ to a $d-1$ dimensional critical behavior. The behavior of the force along the phase coexistence line for a magnetic field H=0 and $T 1$ and a decreasing one for $σ<1$. For any $d$ and $σ$ the minimum of the force at $T=T_c$ is always achieved at some $H\ne 0$.

cond-mat.stat-mech↗

arXiv2004

The black-body radiation in Tsallis statistics

Some results for the black-body radiation obtained in the context of the $q$-thermostatistics are analyzed on both thermodynamical and statistical-mechanical levels. Since the thermodynamic potentials can be expressed in terms of the Wright's special function an useful asymptotic expansion can be obtained. This allows the consideration of the problem away from the Boltzmann-Gibbs limit $q=1$. The role of non-extensivity, $q<1$, on the possible deviation from the Stefan-Boltzmann $T^{4}$ behavior is considered. The application of some approximation schemes widely used in the literature to analyze the cosmic radiation is discussed.

cond-mat.stat-mech↗

arXiv2003

Critical behavior of systems with long-range interaction in restricted geometry

The present review is devoted to the problems of finite-size scaling due to the presence of long-range interaction decaying at large distance as $1/r^{d+σ}$, $σ>0$. The attention is focused mainly on the renormalization group results in the framework of ${\cal O}(n)$ $ϕ^{4}$ - theory for systems with fully finite (block) geometry under periodic boundary conditions. Some bulk critical properties and Monte Carlo results also are reviewed. The role of the cutoff effects as well their relation with those originating from the long-range interaction is also discussed. Special attention is paid to the description of the adequate mathematical technique that allows to treat the long-range and short-range interactions on equal ground. The review closes with short discussion of some open problems.

cond-mat.stat-mech↗

arXiv2001

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 $ε$-expansion scheme under periodic boundary conditions. We suppose a van der Waals type long-range interaction falling apart with the distance r as $r^{-(d+σ)}$, where $2<σ<4$, 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.

cond-mat.stat-mech↗

arXiv2001

On the Finite Size Scaling in Disordered Systems

The critical behavior of a quenched random hypercubic sample of linear size $L$ is considered, within the ``random-$T_{c}$'' field-theoretical mode, by using the renormalization group method. A finite-size scaling behavior is established and analyzed near the upper critical dimension $d=4-ε$ and some universal results are obtained. The problem of self-averaging is clarified for different critical regimes.

cond-mat.stat-mech↗

arXiv2000

Scaling behavior for finite O(n) systems with long-range interaction

A detailed investigation of the scaling properties of the fully finite ${\cal O}(n)$ systems with long-range interaction, decaying algebraically with the interparticle distance $r$ like $r^{-d-σ}$, below their upper critical dimension is presented. The computation of the scaling functions is done to one loop order in the non-zero modes. The results are obtained in an expansion of powers of $\sqrtε$, where $ε=2σ-d$ up to ${\cal O}(ε^{3/2})$. The thermodynamic functions are found to be functions the scaling variable $z=RL^{2-η-ε/2}U^{-1/2}$, where $R$ and $U$ are the coupling constants of the constructed effective theory, and $L$ is the linear size of the system. Some simple universal results are obtained.

cond-mat.stat-mech↗

arXiv2000

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

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

cond-mat.stat-mech↗

arXiv2000

Casimir amplitudes in a quantum spherical model with long-range interaction

A $d$-dimensional quantum model system confined to a general hypercubical geometry with linear spatial size $L$ and ``temporal size'' $1/T $ ($T$ - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. For a film geometry in different space dimensions $\frac 12σ<d<\frac 32σ$, where $0<σ\leq 2$ is a parameter controlling the decay of the long-range interaction, the free energy and the Casimir amplitudes are given. We have proven that, if $d=σ$, the Casimir amplitude of the model, characterizing the leading temperature corrections to its ground state, is $Δ=-16ζ(3)/[5σ(4π)^{σ/2}Γ(σ/2)]$. The last implies that the universal constant $\tilde{c}=4/5$ of the model remains the same for both short, as well as long-range interactions, if one takes the normalization factor for the Gaussian model to be such that $\tilde{c}=1$. This is a generalization to the case of long-range interaction of the well known result due to Sachdev. That constant differs from the corresponding one characterizing the leading finite-size corrections at zero temperature which for $d=σ=1$ is $\tilde c=0.606$.

cond-mat.stat-mech↗

arXiv2000

Some new exact critical-point amplitudes

The scaling properties of the free energy and some of universal amplitudes of a group of models belonging to the universality class of the quantum nonlinear sigma model and the O(n) quantum $ϕ^4$ model in the limit $n\to \infty$ as well as the quantum spherical model, with nearest-neighbor and long-range interactions (decreasing at long distances $r$ as $1/r^{d+σ}$) is presented.

cond-mat.stat-mech↗

arXiv1999

Finite-size scaling properties and Casimir forces in an exactly solvable quantum statistical-mechanical model

A d-dimensional finite quantum model system confined to a general hypercubical geometry with linear spatial size L and ``temporal size'' 1/T (T - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. Because of its close relation with the system of quantum rotors it represents an effective model for studying the low-temperature behaviour of quantum Heisenberg antiferromagnets. Close to the zero-temperature quantum critical point the ideas of finite-size scaling are used for studying the critical behaviour of the model. For a film geometry in different space dimensions $\halfσ<d<\threehalfσ$, where $0<σ\leq2$ controls the long-ranginess of the interactions, an analysis of the free energy and the Casimir forces is given.

cond-mat↗

arXiv1997

Low-temperature regimes and finite-size scaling in a quantum spherical model

A $d$--dimensional quantum model in the spherical approximation confined to a general geometry of the form $L^{d-d^{\prime}} \times\infty^{d^{\prime}}\times L_τ^{z}$ ($L$--linear space size and $L_τ$--temporal size) and subjected to periodic boundary conditions is considered. Because of its close relation with the quantum rotors model it can be regarded as an effective model for studying the low-temperature behavior of the quantum Heisenberg antiferromagnets. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zero-temperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions $1<d<3$ and $0\leq d^{\prime}\leq d$ a detailed analysis, in terms of the special functions of classical mathematics, for the free energy, the susceptibility and the equation of state is given. Particular attention is paid to the two-dimensional case.

cond-mat↗