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Finite-size scaling for the correlation function of the spherical model with long-range interactions

J.G. Brankov, Daniel M. Danchev

Finite-size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry Ld−d′×∞d′, periodic boundary conditions, and long-range interactions decaying like R−d−σ at large distances R, with arbitrary real parameters 0<σ≤2, σ≤d≤2σ, d′≥0. The analytical technique used is based on integral transformations with kernels of Mittag–Leffler type. It makes it possible to easily generalize a number of results available only for short-range interactions or in special cases of the Ld−d′×∞d′ geometry. The effective correlation length is identified at arbitrary temperatures from the finite-size large-distance asymptotic behavior of the pair correlation function. It is explicitly shown that the finite-size scaling functions are not singular in the ε-expansion when ε→0+ neither at d=σ+ε, nor at d=2σ−ε. Moreover, the applicability of the ε-expansion is tested in the specific finite-size case of d′=σ±ε. The definition of the scaled field variable and the finite-size scaling in the regime of the first-order phase transition are also considered.Finite-size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry Ld−d′×∞d′, periodic boundary conditions, and long-range interactions decaying like R−d−σ at large distances R, with arbitrary real parameters 0<σ≤2, σ≤d≤2σ, d′≥0. The analytical technique used is based on integral transformations with kernels of Mittag–Leffler type. It makes it possible to easily generalize a number of results available only for short-range interactions or in special cases of the Ld−d′×∞d′ geometry. The effective correlation length is identified at arbitrary temperatures from the finite-size large-distance asymptotic behavior of the pair correlation function. It is explicitly shown that the finite-size scaling functions are not singular in the ε-expansion when ε→0+ neither at d=σ+ε, nor at d=2σ−ε. Moreover, the applicability of the ε-expansion is tested in the specific finite-size case of d′=σ±ε. The definition of the scaled field variable and the finite-size scaling in the regime of the first-order phase transition are also considered.

Journal of Mathematical PhysicsSep 1, 199121 citations

Fractals in quantum theory: Analytical and numerical approaches

P. K. Silaev, Eugene Tyurin, Олег Антонинович Хрусталев

Theoretical and Mathematical PhysicsOct 1, 19911 citation

Ordering, Symbols and Finite-Dimensional Approximations of Path Integrals

T. Kashiwa, S. Sakoda, S. V. Zenkin

We derive general form of finite-dimensional approximations of path integrals for both bosonic and fermionic canonical systems in terms of symbols of operators determined by operator ordering. We argue that for a system with a given quantum Hamiltonian such approximations are independent of the type of symbols up to terms of O(ε), where ε is infinitesimal time interval determining the accuracy of the approximations. A new class of such approximations is found for both c-number and Grassmannian dynamical variables. The actions determined by the approximations are non-local and have no classical continuum limit except the cases of pq- and qp-ordering. As an explicit example the fermionic oscillator is considered in detail.We derive general form of finite-dimensional approximations of path integrals for both bosonic and fermionic canonical systems in terms of symbols of operators determined by operator ordering. We argue that for a system with a given quantum Hamiltonian such approximations are independent of the type of symbols up to terms of O(ε), where ε is infinitesimal time interval determining the accuracy of the approximations. A new class of such approximations is found for both c-number and Grassmannian dynamical variables. The actions determined by the approximations are non-local and have no classical continuum limit except the cases of pq- and qp-ordering. As an explicit example the fermionic oscillator is considered in detail.

arXiv:hep-th/9401063Progress of Theoretical PhysicsSep 1, 19942 citations

An Algebra of Effects in the Formalism of Quantum Mechanics on Phase Space

Franklin E. Schroeck

International Journal of Theoretical PhysicsNov 1, 20057 citations

Superconductor-insulator quantum critical point in 1+ε dimensions

Igor F. Herbut

A system of disordered spinless fermions infinitesimally above one dimension at zero temperature is studied using the perturbative renormalization-group theory to first order in disorder, and to second order in interaction. We find a superconductor-to-Anderson insulator quantum fixed point at an infinitesimal value of disorder, and calculate the correlation length and the dynamical exponents to the lowest order in \ensuremath{\epsilon} and in interaction. The scaling of conductivity with temperature and the behavior of characteristic temperature scales on both sides of the transition is determined.A system of disordered spinless fermions infinitesimally above one dimension at zero temperature is studied using the perturbative renormalization-group theory to first order in disorder, and to second order in interaction. We find a superconductor-to-Anderson insulator quantum fixed point at an infinitesimal value of disorder, and calculate the correlation length and the dynamical exponents to the lowest order in \ensuremath{\epsilon} and in interaction. The scaling of conductivity with temperature and the behavior of characteristic temperature scales on both sides of the transition is determined.

arXiv:cond-mat/9702116Physical review. B, Condensed matterJan 15, 19983 citations

Fractal Paths and Schrödingers Equation in an Electromagnetic Field

Garnet N. Ord, J. Anthony Gualtieri

Chaos Solitons & FractalsNov 1, 19981 citation

Quantum Effects Through a Fractal Theory of Motion

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