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Journal of Physical Studies

A simple interpolation procedure for dimensional crossover phenomena

L. De Cesare, L. Craco, Ileana Rabuffo, Dimo I. Uzunov

Published Jan 1, 1998• 7 citations
Publisher

Abstract

A method of investigation of dimensional crossover phenomena in the theory of phase transitions is illustrated for the nite{size crossover.The dimensional crossover phenomena in systems with quenched extended impurities and inhomogeneities as well as the classical{to{quantum crossover at extremely low temperatures are also discussed.The fundamental interrelationship between the effective dimensionality and the ratio of the relevant characteristic lengths of the system is established and applied to the description of dimensional crossover phenomena.

DOI
10.30970/jps.02.175

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A method of investigation of dimensional crossover phenomena in the theory of phase transitions is illustrated for the nite{size crossover.The dimensional crossover phenomena in systems with quenched extended impurities and inhomogeneities as well as the classical{to{quantum crossover at extremely low temperatures are also discussed.The fundamental interrelationship between the effective dimensionality and the ratio of the relevant characteristic lengths of the system is established and applied to the description of dimensional crossover phenomena.
The classical{to{quantum dimensional crossover in a wide class of systems with quenched impurities and inhomogeneities is investigated within the framework of the renormalization group.The one{loop renormalization group equations are derived with the help of a double (; Æ){expansion, where the small parameter Æ describes the quantum eects and = 2 d is the usual expansion parameter (0 < 2; d is the spatial dimensionality).The classical{to{quantum crossover phenomenon is established and the critical exponents are calculated to rst order in and Æ.The analogy between this approach and that of double (; Æ){expansion in treating the extended impurities problem is discussed.
The current state of Monte Carlo work on phase transitions and critical phenomena is reviewed. Both classical and quantum Monte Carlo results are discussed with emphasis on statistical lattice-model studies involving the highly accurate calculation of critical exponents. It is shown that finite-size scaling is an effective method whereby not only simple lattice models but also those of complex practical magnetic materials can be treated. For models allowing crossover phenomena, it is shown that both finite-size scaling theory and conventional power-law functions must be used in carrying out a Monte Carlo analysis.
In this work, we extend the analysis of interacting bosons at 2D-1D dimensional crossover for finite size and temperature by using the field theory approach ( b o s o n i z a t i o n ) and quantum Monte Carlo simulations. Stemming from the fact that finite size low-dimensional systems are allowed only to have quasiordered phase, we consider the self-consistent harmonic approximation to compute the fraction of quasicondensate and its scaling with the system size. It allows us to understand the important role played by finite size and temperature across the dimensional crossover for deciding the condensate nature. Furthermore, we consider a mean-field approximation to compute the finite size effect on the crossover temperature for both weak and strong interactions. All the physical quantities we discuss here provide essential information for quantum gas at dimensional crossover and are directly detectable in cold atom experiments.
The critical behaviour of thin films containing quenched random impurities and inhomogeneities is investigated by the renormalization-group method to the one-loop order within the framework of the n-component φ4-model. The finite-size crossover in impure films has been considered on the basis of the fundamental relationship between the effective dimensionality Deff and the characteristic lengths of the system. The fixed points, their stability properties and the critical exponents are obtained and analyzed, using an ε̃=(4−Deff)-expansion near the effective spatial dimensionality Deff of the fluctuation modes in D-dimensional hyperslabs with two types of quenched impurities: point-like impurities with short-range random correlations and extended (linear) impurities with infinite-range random correlations along the small-size spatial direction. The difference between the critical properties of infinite systems and films is demonstrated and investigated. A new critical exponent, describing the scaling properties of the thickness of films with extended impurities has been derived and calculated. A special attention is paid to the critical behaviour of real impure films (D=3).
A quantum system can undergo a continuous phase transition at the absolute zero of temperature as some parameter entering its Hamiltonian is varied. These transitions are particularly interesting for, in contrast to their classical finite-temperature counterparts, their dynamic and static critical behaviors are intimately intertwined. Considerable insight is gained by considering the path-integral description of the quantum statistical mechanics of such systems, which takes the form of the classical statistical mechanics of a system in which time appears as an extra dimension. In particular, this allows the deduction of scaling forms for the finite-temperature behavior, which turns out to be described by the theory of finite-size scaling. It also leads naturally to the notion of a temperature-dependent dephasing length that governs the crossover between quantum and classical fluctuations. Using these ideas, a scaling analysis of experiments on Josephson-junction arrays and quantum-Hall-effect systems is presented.
Quasi-one-dimensional (Q1D) systems, i.e., three- and two-dimensional (3D/2D) arrays composed of weakly coupled one-dimensional lattices of interacting quantum particles, exhibit rich and fascinating physics. They are studied across various areas of condensed matter and ultracold atomic lattice-gas physics, and are often marked by dimensional crossover as the coupling between one-dimensional systems is increased or temperature decreased, i.e., the Q1D system goes from appearing largely 1D to largely 3D. Phase transitions occurring along the crossover can strongly enhance this effect. Understanding these crossovers and associated phase transitions can be challenging due to the very different elementary excitations of 1D systems compared to higher-dimensional ones. In the present work, we combine numerical matrix product state (MPS) methods with mean-field (MF) theory to study paradigmatic cases of dimensional crossovers and the associated phase transitions in systems of both hard-core and soft-core lattice bosons, with relevance to both condensed matter physics and ultracold atomic gases. We show that the superfluid-to-insulator transition is a first order one, as opposed to the isotropic cases, and calculate transition temperatures for the superfluid states, finding excellent agreement with analytical theory. At the same time, our MPS+MF\mathrm{MPS}+\mathrm{MF}MPS+MF approach keeps functioning well where the current analytical framework cannot be applied. We further confirm the qualitative and quantitative reliability of our approach by comparison to exact quantum Monte Carlo calculations for the full 3D arrays.