We study the nite temperature crossovers in the vicinity of a zero temperature quantum phase transition. The universal crossover functions are observables of a continuum quantum eld theory. Particular attention is focussed on the high temperature limit of the continuum eld theory, the socalled \quantum-critical region. Basic features of crossovers are illustrated by a simple solvable model of dilute spinless fermions, and a partially solvable model of dilute bosons. The low frequency relaxational behavior of the quantum-critical region is displayed in the solution of the transverse-eld Ising model. The insights from these simple models lead to a fairly complete understanding of the system of primary interest: the two-dimensional quantum rotor model, whose phase transition is expected to be in the same universality class as those in antiferromagnetic Heisenberg spin models. Recent work on the experimental implications of these results for the cuprate compounds is briey reviewed.
A systematic method for the computation of finite-temperature (T) crossover functions near quantum-critical points close to, or above, their upper-critical dimension is devised. We describe the physics of the various regions in the T and critical tuning parameter (t) plane. The quantum-critical point is at T=0, t=0, and in many cases there is a line of finite-temperature transitions at T=
Tc(t), t0, with
Tc(0)=0. For the relativistic, n-component ${\mathrm{\ensuremath{\varphi}}}^{4}$ continuum quantum field theory [which describes lattice quantum rotor (n\ensuremath{\geqslant}2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d3, \ensuremath{\epsilon}=3-d is the control parameter over the entire phase diagram. In the region |T-
Tc(t)|\ensuremath{\ll}
Tc(t), we obtain an \ensuremath{\epsilon} expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of $\sqrt{\ensuremath{\epsilon}}$, modulo powers of ln \ensuremath{\epsilon}, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (\ensuremath{\omega}), the perturbative $\sqrt{\ensuremath{\epsilon}}$ expansion describes quantum relaxation at \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}
kBT or larger, but fails for \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}$\sqrt{\ensuremath{\epsilon}}$
kBT or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t=0, for T>0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.
The purpose of this work is to understand the zero temperature phases and the phase transitions of Heisenberg spin systems which can have an extensive, spontaneous magnetic moment; this entails a study of quantum transitions with an order parameter which is also a non-abelian conserved charge. To this end, we introduce and study a new class of lattice models of quantum rotors. We compute their mean-field phase diagrams and present continuum, quantum field-theoretic descriptions of their low energy properties in different regimes. We argue that, in spatial dimensiond=1, the phase transitions in itinerant Fermi systems are in the same universality class as the corresponding transitions in certain rotor models. We discuss implications of our results for itinerant fermions systems in higherdand for other physical systems.
We investigate the zero-temperature and finite-temperature phase transitions of quantum Ising and quantum rotor models. We here assume a long-range (falling off as ${1/r}^{d+\ensuremath{\sigma}},$ where r is the distance between two spins/rotors in units of lattice spacing) ferromagnetic interaction among the spins or rotors. We find that the long-range behavior of the interaction drastically modifies the universal critical behavior of the system. The corresponding upper critical dimension and the hyperscaling relation and exponents associated with the quantum transition are modified and, as expected, they attain values of short-range system when $\ensuremath{\sigma}=2.$ The dynamical exponent varies continuously as the parameter \ensuremath{\sigma} and is unity for $\ensuremath{\sigma}=2.$ The one-dimensional long-range quantum Ising system shows a phase transition at
T=0 for all values of \ensuremath{\sigma}. The most interesting observation is that the phase diagram for $\ensuremath{\sigma}=d=1$ shows a line of Kosterlitz-Thouless transition at finite temperature even though the
T=0 transition is a simple order-disorder transition. These finite temperature transitions are studied near the phase boundary using renormalisation group equations and a region with diverging susceptibility is located. We have also studied one-dimensional quantum rotor model which exhibits a rich and interesting transition behavior depending upon the parameter \ensuremath{\sigma}. We explore the phase diagram extending the short-range quantum nonlinear \ensuremath{\sigma} model renormalisation group equations to the present case.
This paper presents a unified perspective on the results of two recent works (C. Buragohain and S. Sachdev cond-mat/9811083 and S. Sachdev cond-mat/9810399) along with additional background. We describe the low frequency, non-zero temperature, order parameter relaxational dynamics of a number of systems in the vicinity of a quantum critical point. The dynamical correlations are properties of the high temperature limit of renormalizable quantum field theories in spatial dimensions d=1,2. We study, as a function of d and the number of order parameter components, n, the crossover from the finite frequency, "amplitude fluctuation", gapped quasiparticle mode in the quantum paramagnet (or Mott insulator), to the zero frequency "phase" (n>=2) or "domain wall" (n=1) relaxation mode near the ordered phase. Implications for dynamic measurements on the high temperature superconductors and antiferromagnetic spin chains are noted.
We review the non-zero temperature relaxational dynamics of quantum systems near a zero temperature, second-order phase transition. We begin with the quantum Ising chain, for which universal and exact results for the relaxation rates can be obtained in all the distinct limiting regimes of the phase diagram. Next, we describe the crossovers in the electron spectral function near a transition involving a change in the pairing symmetry of BCS superconductors in two dimensions. Finally, we consider dynamic spin models which may provide a mean-field description of magnetic ordering transitions in the heavy fermion compounds.