We investigate the functional form of the order-parameter (two-point) correlation function in quantum critical phenomena. Contrary to the common lore, when there is no particle-hole symmetry we find that the equal-time correlation function at criticality does not display a diverging correlation length. We illustrate our conclusions by Monte Carlo calculations of the quantum rotor model in
d=2 space dimensions
We study the role of particle-hole symmetry on the universality class of various quantum phase transitions corresponding to the onset of superfluidity at zero temperature of bosons in a quenched random medium. To obtain a model with an exact particle-hole symmetry it is necessary to use the Josephson junction array, or quantum rotor, Hamiltonian, which may include disorder in both the site energies and the Josephson couplings between wave function phase operators at different sites. The functional integral formulation of this problem in
d spatial dimensions yields a
(d+1)-dimensional classical
XY model with extended disorder, constant along the extra imaginary time dimension---the so-called random rod problem. Particle-hole symmetry may then be broken by adding nonzero site energies, which may be uniform or site dependent. We may distinguish three cases: (i) exact particle-hole symmetry, in which the site energies all vanish; (ii) statistical particle-hole symmetry, in which the site energy distribution is symmetric about zero, vanishing on average; and (iii) complete absence of particle-hole symmetry in which the distribution is generic. We explore in each case the nature of the excitations in the nonsuperfluid Mott insulating and Bose glass phases. We show, in particular, that, since the boundary of the Mott phase can be derived exactly in terms of that for the pure, nondisordered system, there can be no direct Mott-superfluid transition. Recent Monte Carlo data to the contrary can be explained in terms of rare region effects that are inaccessible to finite systems. We find also that the Bose glass compressibility, which has the interpretation of a temporal spin stiffness or superfluid density, is positive in cases (ii) and (iii), but that it vanishes with an essential singularity as full particle-hole symmetry is restored. We then focus on the critical point and discuss the relevance of type (ii) particle-hole symmetry-breaking perturbations to the random rod critical behavior, identifying a nontrivial crossover exponent. This exponent cannot be calculated exactly but is argued to be positive and the perturbation therefore relevant. We argue next that a perturbation of type (iii) is irrelevant to the resulting type (ii) critical behavior: The statistical symmetry is restored on large scales close to the critical point, and case (ii) therefore describes the dirty boson fixed point. Using various duality transformations we verify all of these ideas in one dimension. To study higher dimensions, we attempt, with partial success, to generalize the Dorogovtsev--Cardy--Boyanovsky double-epsilon expansion technique to this problem. We find that when the dimension of time ${ϵ}_{\ensuremath{\tau}}<{ϵ}_{\ensuremath{\tau}}^{c}\ensuremath{\simeq}\frac{8}{29}$ is sufficiently small a type (ii) symmetry-breaking perturbation is irrelevant, but that for sufficiently large ${ϵ}_{\ensuremath{\tau}}>{ϵ}_{\ensuremath{\tau}}^{c}$ particle-hole asymmetry is a relevant perturbation and a new stable fixed point appears. Furthermore, for ${ϵ}_{\ensuremath{\tau}}>{ϵ}_{\ensuremath{\tau}}^{c2}\ensuremath{\approx}\frac{2}{3}$, this fixed point is stable also to perturbations of type (iii): at $ϵ={ϵ}_{\ensuremath{\tau}}^{c2}$ the generic type (iii) fixed point merges with the new fixed point. We speculate, therefore, that this new fixed point becomes the dirty boson fixed point when ${ϵ}_{\ensuremath{\tau}}=1$. We point out, however, that ${ϵ}_{\ensuremath{\tau}}=1$ may be quite special. Thus, although the qualitative renormalization group flow picture the double-epsilon expansion technique provides is quite compelling, one should remain wary of applying it quantitatively to the dirty boson problem.
We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one-dimension: the dynamical exponent is infinite and, at the critical point, the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point, there may be different exponents for the divergence of the average and typical correlation lengths, again as in one-dimension, but the evidence for this is less strong.
We determine the dynamical critical exponent z appearing at the Bose glass to superfluid transition in two dimensions by performing large scale numerical studies of two microscopically different quantum models within the universality class: The hard-core boson model and the quantum rotor (soft core) model, both subject to strong on-site disorder. By performing many simulations at different system size L and inverse temperature β close to the quantum critical point, the position of the critical point and the critical exponents, z, ν, and η can be determined independently of any implicit assumptions of the numerical value of z, in contrast to most prior studies. This is done by a careful scaling analysis close to the critical point with a particular focus on the temperature dependence of the scaling functions. For the hard-core boson model we find z=1.88(8), ν=0.99(3), and η=-0.16(8) with a critical field of h(c)=4.79(3), while for the quantum rotor model we find z=1.99(5), ν=1.00(2), and η=-0.3(1) with a critical hopping parameter of t(c)=0.0760(5). In both cases do we find a correlation length exponent consistent with ν=1, saturating the bound ν≥2/d as well as a value of z significantly larger than previous studies, and for the quantum rotor model consistent with z=d.
Earlier Monte Carlo calculations on the dissipative two-dimensional XY model are extended in several directions. We study the phase diagram and the correlation functions when dissipation is very small, where it has properties of the classical 3D-XY transition, i.e., one with a dynamical critical exponent
z=1. The transition changes from
z=1 to the class of criticality with $z\ensuremath{\rightarrow}\ensuremath{\infty}$ driven by topological defects, discovered earlier, beyond a critical dissipation. We also find that the critical correlations have power-law singularities as a function of tuning the ratio of the kinetic energy to the potential energy for fixed large dissipation, as opposed to essential singularities on tuning dissipation keeping the former fixed. A phase with temporal disorder but spatial order of the Kosterlitz-Thouless form is also further investigated. We also present results for the transition when the allowed Caldeira-Leggett form of dissipation and the allowed form of dissipation coupling to the compact rotor variables are both included. The nature of the transition is then determined by the Caldeira-Leggett form.
The two-dimensional quantum XY model, with a Caldeira-Leggett form of dissipation, is applicable to the quantum-critical properties of diverse experimental systems, ranging from superconductor to insulator transitions, ferromagnetic and antiferromagnetic transitions in metals, to the loop-current order transition in cuprates. We solve the reexpression of this model in terms of orthogonal topological excitations, vortices, and a variety of instantons, by renormalization group methods. The calculations explain the extraordinary properties of the model discovered in Monte Carlo calculations: the product form of the quantum-critical fluctuations in space and time, a spatial correlation length proportional to the logarithm of the temporal correlation length near the transition from a disordered to a fully ordered state, and the occurrence of a phase with spatial order without temporal order. They are intimately related to the flow of the metric of time in relation to the metric of space, i.e., of the dynamical critical exponent
z. These properties appear to be essential in understanding the strange metallic phase found in a variety of quantum-critical transitions as well as the accompanying high-temperature superconductivity.