Quantum critical phenomena influence the finite temperature behavior of condensed matter systems through quantum critical fans whose extents are determined by the exponents of the zero temperature criticality. Here we emphasize the aspects of quantum critical lines, as discussed previously, and study an exactly solved model involving a transverse field Ising model with added three-spin interaction. This model has three critical lines. We compute the spin-spin correlation function and extract the correlation length, and identify the crossovers: quantum critical to quantum disordered, or renormalized classical regimes. We construct the quantum critical fans along one of the critical lines. In addition, we also construct finite temperature dynamic structure factors. We hope this model will become experimentally realizable in the future, and our results could stimulate studies in many similar models.
We face the problem of detecting and featuring footprints of quantum criticality in the finite-temperature behavior of quantum many-body systems. Our strategy is that of comparing the phase diagram of a system displaying a
T=0 quantum phase transition with that of its classical limit, in order to single out the genuinely quantum effects. To this aim, we consider the one-dimensional Ising model in a transverse field: while the quantum
S=1∕2 Ising chain is exactly solvable and extensively studied, results for the classical limit $(S\ensuremath{\rightarrow}\ensuremath{\infty})$ of such model are lacking, and we supply them here. They are obtained numerically, via the transfer-matrix method, and their asymptotic low-temperature behavior is also derived analytically by self-consistent spin-wave theory. We draw the classical phase diagram according to the same procedure followed in the quantum analysis, and the two phase diagrams are found unexpectedly similar: Three regimes are detected also in the classical case, each characterized by a functional dependence of the correlation length on temperature and field analogous to that of the quantum model. What discriminates the classical from the quantum case are the different values of the exponents entering such dependencies, a consequence of the different nature of zero-temperature quantum fluctuations with respect to the thermal ones.
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.
A zero-temperature real-space renormalization-group method is applied to the one-dimensional quantum spin-\textonehalf{} isotropic $X\ensuremath{-}Y$ model in a transverse field. A line of unstable fixed points is found and is attributed to the oscillatory behavior of the correlation functions. Magnetization components and critical indices are in good agreement with the exact results. The behavior of the
xx correlation function is studied near the critical field.
We introduce and analyze a class of model systems to study transitions in the integer quantum Hall effect (IQHE). Even without disorder our model exhibits an IQHE transition as a control parameter is varied. We find that the transition is in the two-dimensional Ising universality class and compute all associated exponents and critical transport properties. The fixed point has time-reversal, particle-hole, and parity invariance. We then consider the effect of quenched disorder on the IQHE transition and find the following. (i) Randomness in the control parameter (which breaks all the above symmetries) translates into bond randomness in the Ising model and is hence marginally irrelevant. The transition may equally well be viewed as a quantum percolation of edge states localized on equipotentials. The absence of random-phase factors for the edge states is responsible for the nongeneric (Ising) critical properties. (ii) For a random magnetic field (which preserves particle-hole symmetry in every realization) the model exhibits an exactly solvable fixed line, described in terms of a product of a Luttinger liquid and an SU(n) spin chain. While exponents vary continuously along the fixed line, the longitudinal conductivity is constant due to a general conformal sum rule for Kac-Moody algebras (derived here), and is computed exactly. We also obtain a closed expression for the extended zero-energy wave function for every realization of disorder and compute its exact multifractal spectrum f(\ensuremath{\alpha}) and the exponents of all participation ratios. One point on the fixed line corresponds to a recently proposed model by Gade and Wegner. (iii) The model in the presence of a random on-site potential scales to a strong disorder regime, which is argued to be described by a symplectic nonlinear-sigma-model fixed point. (iv) We find a plausible global phase diagram in which all forms of disorder are simultaneously considered. In this generic case, the presence of random-phase factors in the edge-state description indicates that the transition is described by a Chalker-Coddington model, with a so far analytically inaccessible fixed point.
Critical properties of quantum spin chains with varying degrees of disorder are studied at zero temperature by analytical and extensive density matrix renormalization methods. Generally the phase diagram is found to contain three phases. The weak disorder regime, where the critical behavior is controlled by the fixed points of the pure system, and the strong disorder regime, which is attracted by an infinite randomness fixed point, are separated by an intermediate disorder regime, where dynamical scaling is anisotropic and the static and dynamical exponents are disorder dependent.