Lower critical dimension of the random-field $XY$ model and the zero-temperature critical line

The random-field $XY$ model is studied in spatial dimensions $d=3$ and 4, and in between, as the limit $q\to \infty$ 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 $d=3$.

Figure 1

From Ref. [24]: (a) Migdal-Kadanoff approximate renormalization-group transformation for the $d=3$ cubic lattice with the length-rescaling factor of $b=2$. (b) Construction of the $d=3,b=2$ hierarchical lattice for which the Migdal-Kadanoff recursion relation is exact. For general spatial dimension $d$, the bond moving is $\left(b^{d-1}\right)$-fold. The renormalization-group solution of a hierarchical lattice proceeds in the opposite direction of its construction.

Figure 2

Phase diagrams for $(q=7,10,20,50,100,150)$-state random-field clock models in $d=3$, occurring in the figure, respectively, from high field to low field. Disordered and ferromagnetic phases occur at high temperature-high field and low temperature-low field, respectively. It is seen that the ferromagnetic phase, in random field, disappears as $q\to \infty$, indicating that no ferromagnetic phase occurs in the random-field $XY$ model at nonzero temperature in $d=3$. However, for high $q$, the ordered phase extends to $qH/J=5.1$ at zero temperature, as also seen in the left inset. For high $q$, the zero-field ferromagnetic transition temperature saturates, as also seen in Ref. [24] and in the right inset in this figure.

Figure 3

Phase diagrams for $(q=7,10,20,50,100,150)$-state random-field clock models in $d=3$. At low temperature, the curves are, from low field to high field, $q=7,10$ and indistinguishably $q=20,50,100,150$. It is seen that when the random field is scaled with $q^2$, a universal phase diagram is found above low temperature for high $q$.

Figure 4

Phase diagrams for $(q=3,4,5,6,7,10)$-state random-field clock models in $d=3.32$, occurring in the figure, respectively, from high field to low field.

Figure 5

The critical line segment, at zero temperature, is between $qH/J=0$ and the $qH/J$ values shown in this figure for each dimension $d$. The values are consistent with a divergence as $d=4$ is approached.

Figure 6

Phase diagrams for $(q=7,10,20)$-state random-field clock models in $d=4$, occurring in the figure, respectively, from low field to high field.