Some Examples of Quantum Phase Transitions

Diana V Shopova, Dimo I Uzunov

G. G. Batrouni, Richard T. Scalettar

This chapter discusses quantum phase transitions (QPT). It starts with a brief review of thermal phase transitions, critical exponents and scaling laws. The scaling laws are then generalized to the QPT case which is also illustrated with two specific examples. The first example that of the one-dimensional Ising model in a transverse magnetic field; the second is that of the bosonic Hubbard model. Quantum Monte Carlo is described briefly and mean field theory is introduced with the help of several examples and exercises.

K. Sengupta

Igor F. Herbut

The dynamical critical exponent is introduced. The phase diagram and the phase transitions in the Bose–Hubbard model of interacting bosons on a lattice are determined. The concept of quantum fluctuations is introduced on the example of an interacting superfluid, and finally the special scaling of conductivity is discussed. Dynamical critical exponent The finite temperature phase transitions studied in previous chapters are the result of the competition between the entropy and the energy terms in the free energy: the weight of entropy increases with temperature, and ultimately destroys the order that may be existing in the system. A sharp phase transition between two phases exhibiting qualitatively different correlations may, however, occur even at zero temperature, by varying a coupling constant in the Hamiltonian. The transition then corresponds to a non-analyticity of the energy of the ground state. A simple example is provided by the interacting bosons, where the superfluid transition may be brought about by tuning the chemical potential at T = 0. Such T = 0 phase transitions are called quantum phase transitions , and will be the subject of the present chapter. In general, a quantum phase transition may lie at the end of a line of thermal phase transitions, as in the bosonic example mentioned above. It is possible, however, that the system may not even have an ordered state at finite temperatures, but still exhibits a quantum critical point. Two different situations are depicted in Fig. 8.1. Examples of such phase diagrams are provided by the system of interacting bosons which will be discussed in Sections 8.2 and 8.3.

FRANCESCO IACHELLO

Shape phase transitions in nuclei, an example of quantum phase transitions in mesoscopic systems, are briefly reviewed. Spectral signatures of critical points in finite quantal systems are discussed and experimental examples are shown.