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Physica A Statistical Mechanics and its Applications

On the finite-size scaling in quantum critical phenomena

N. S. Tonchev

Published Feb 1, 1991• 5 citations
Publisher
DOI
10.1016/0378-4371(91)90284-j

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Singapore perspectives•Jan 1, 2000•137 citations
The validity of modified finite-size scaling above the upper critical dimension is demonstrated for the quantum phase transition whose dynamical critical exponent is z=2. We consider the N -component Bose-Hubbard model, which is exactly solvable and exhibits mean-field type critical phenomena in the large- N limit. The modified finite-size scaling holds exactly in that limit. However, the usual procedure, taking the large system-size limit with fixed temperature, does not lead to the expected (and correct) mean-field critical behavior because of the limited range of applicability of the finite-size scaling form. By quantum Monte Carlo simulation, it is shown that the modified finite-size scaling holds in the case of N=1.
Abstract Interacting many-particle systems may undergo phase transitions and exhibit critical phenomena in the limit of infinite system size, while the precursors of these phenomena are studied in the theory of finite-size scaling. After surveying the basic notions of phases, phase diagrams, and phase transitions, this chapter focuses on critical behaviour at a second-order phase transition. The Landau-Ginzburg theory and the concept of scaling prepare readers for an elementary introduction to the concepts of the renormalization group, followed by an introduction into the field of quantum phase transitions where quantum fluctuations take over the role of thermal fluctuations.
The aim of this book is to familiarise the reader with the rich collection of ideas, methods and results available in the theory of critical phenomena in systems with confined geometry. The existence of universal features of the finite-size effects arising due to highly correlated classical or quantum fluctuations is explained by the finite-size scaling theory. This theory (1) offers an interpretation of experimental results on finite-size effects in real systems; (2) gives the most reliable tool for extrapolation to the thermodynamic limit of data obtained by computer simulations; (3) reveals