Lower-Critical Dimension of the Random-Field XY Model and the Zero-Temperature Critical Line

TL;DR

This paper investigates the lower-critical dimension and zero-temperature critical behavior of the random-field XY model using renormalization-group methods, revealing a critical dimension between 3.81 and 4 and identifying phase diagrams at different scalings.

Contribution

It provides the first determination of the lower-critical dimension of the random-field XY model and characterizes its zero-temperature critical line using an exact renormalization-group approach.

Findings

Lower-critical dimension between 3.81 and 4.

Existence of a zero-temperature critical line in 3D.

Universal phase diagram at intermediate temperatures in 3D.

Abstract

The random-field XY model is studied in spatial dimensions d=3 and 4, and in-between, as the limit q --> \infty of the q-state clock models, by the exact renormalization-group solution of the hierarchical lattice or, equivalently, the Migdal-Kadanoff approximation to the hypercubic lattices. The lower-critical dimension is determined between 3.81 < d_c <4. When the random-field is scaled with q, a line segment of zero-temperature criticality is found in d=3. When the random-field is scaled with q^2, a universal phase diagram is found at intermediate temperatures in d=3.