Local Quantum Criticality in the Two-dimensional Dissipative Quantum XY Model

TL;DR

This paper uses quantum Monte Carlo simulations to explore the phase diagram and critical fluctuations of the two-dimensional dissipative XY model, revealing scale-invariant temporal correlations and an infinite dynamic exponent, relevant to cuprate strange metals.

Contribution

It provides the first numerical verification of an analytic topological excitation-based solution for the dissipative XY model, elucidating its critical behavior and correlation structure.

Findings

Critical fluctuations are scale-invariant in imaginary time.

Spatial correlations grow logarithmically with temporal correlation length.

Dynamic exponent z is infinite, indicating extreme anisotropy.

Abstract

We use quantum Monte-Carlo simulations to calculate the phase diagram and the correlation functions for the quantum phase transitions in the two-dimensional dissipative XY model with and without four-fold anisotropy. Without anisotropy, the model describes the superconductor to insulator transition in two-dimensional dirty superconductors. With anisotropy, the model represents the loop-current order observed in the under-doped cuprates and its fluctuations, as well as the fluctuations near the ordering vector in simple models of itinerant antiferromagnets. These calculations test an analytic solution of the model which re-expressed it in terms of topological excitations - the vortices with interactions only in space but none in time, and warps with leading interactions only in time but none in space, as well as sub-leading interactions which are both space and time-dependent. For parameters for which the proliferation of warps dominates the phase transition, the critical fluctuations as a function of the deviation of the dissipation parameter $\alpha$ from its critical value $\alpha_c$ are scale-invariant in imaginary time $\tau$ as the correlation length in time $\xi_{\tau} = \tau_c e^{|\alpha_c/(\alpha_c-\alpha)|^{1/2}}$ diverges. On the other hand, the spatial correlations develop with a correlation length $\xi_x \approx \xi_0 \log{(\xi_{\tau})}$, with $\xi_0$ of the order of a lattice constant. The dynamic correlation exponent $z$ is therefore $\infty$. These results are consistent with the phenomenology proposed for the strange metal properties of the cuprates. The Monte-Carlo calculations also directly show warps and vortices. Their density and correlations across the various transitions in the model are calculated and related to those of the order parameter in the original model.