Criticality in correlated quantum matter

Abstract

At quantum critical points (QCP) [1,2,3,4,5,6,7] there are quantum fluctuations on all length scales, from microscopic to macroscopic lengths, which, remarkably, can be observed at finite temperatures, the regime to which all experiments are necessarily confined. A fundamental question is how high in temperature can the effects of quantum criticality persist? That is, can physical observables be described in terms of universal scaling functions originating from the QCPs? Here we answer these questions by examining exact solutions of models of correlated systems and find that the temperature can be surprisingly high. As a powerful illustration of quantum criticality, we predict that the zero temperature superfluid density, ρs(0), and the transition temperature, Tc, of the cuprates are related by Tc ∝ ρs(0) y , where the exponent y is different at the two edges of the superconducting dome, signifying the respective QCPs. This relationship can be tested in high quality crystals.Do quantum critical points (QCPs) provide a powerful framework for understanding complex correlated many body problems? Do they shed new light on quantum mechanics of macroscopic systems, providing, for example, a deeper understanding of entanglement? Answers to many such questions require a precise recognition of the experimentally observable regime of quantum criticality. An important class of QCPs are analogs of classical critical points, but in a dimension higher than the actual spatial dimension of the system; that is, a quantum critical point in d spatial dimensions is equivalent to a classical critical point in (d + z), dimensions, where z is the dynamic critical exponent. We shall be primarily concerned with z = 1, although an example involving z = 1 is given below. This extra dimension (imaginary time) is the sine qua non of the effect of quantum mechanics.It seems paradoxical that often the region of classical critical fluctuations is small and tends to zero as the temperature, T , tends to zero, while the region of quantum critical fluctuations fans out as T increases. The reason is that the quantum critical region is determined by the dominant microscopic energy scale in the Hamiltonian, whereas the classical critical region is determined by the ratio of the transition temperature, T c , to the same scale raised to a positive power (the dimensionality). Since this ratio is usually small compared to unity (it is 10 −5 in a conventional superonductor), the classical critical region is small as well. As a classical critical point is driven to zero by tuning a parameter to reach the QCP, T c itself vanishes as ξ −z , provided the T = 0 spatial correlation length, ξ, is large compared to the lattice spacing, further amplifying the contrast between classical and quantum criticalities.How high in temperature does the effect of a QCP persist? Consider perhaps the simplest possible example of a quantum phase transition described by the Hamiltonian of an Ising model in a transverse field [8] in one dimension,where the usual Pauli matrices,...

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