arXiv subjects
Publications and source records attributed to N. S. Tonchev.
Within Takahashi's spin-wave theory we study finite size and temperature effects near the quantum critical point in the $J_{1}-J_{2}$ Heisenberg antiferromagnet defined on a strip ($L\times\infty$). In the continuum limit, the theory predicts universal finite size and temperature corrections and describes the dimensional crossover in magnetic properties from 2+1 to 1+1 space-time dimensions.
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.
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.
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.
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 $\tilde c=0.606$.
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.
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.
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.