Universal Monopole Scaling near Transitions from the Coulomb Phase

Certain frustrated systems, including spin ice and dimer models, exhibit a Coulomb phase at low temperatures, with power-law correlations and fractionalized monopole excitations. Transitions out of this phase, at which the effective gauge theory becomes confining, provide examples of unconventional criticality. This Letter studies the behavior at nonzero monopole density near such transitions, using scaling theory to arrive at universal expressions for the crossover phenomena. For a particular transition in spin ice, quantitative predictions are made by mapping to the $XY$ model and confirmed using Monte Carlo simulations.

Figure 1

Schematic phase diagram for a system with a continuous transition from the Coulomb phase. The latter exists only at $z=0$ (blue line), and the confinement transition is therefore at an isolated point in the phase diagram. It nonetheless influences properties in a broad region and leads to universal scaling forms such as Eq. (2). The dotted line ($T>T_{\mathrm{C}}$) indicates a crossover from Coulomb-like behavior to a conventional paramagnet, while the dash-dotted line ($T<T_{\mathrm{C}}$) is either a transition (in a conventional universality class) or a crossover. Both have the form $z\sim |T-T_{\mathrm{C}}{|}^{\varphi }$, where $\varphi$ is a crossover exponent. The arrow shows an example path, as $T$ is reduced at fixed $\Delta$ and $V$.

Figure 2

$W(T,z=0,L)$ versus $T/h$ for various system sizes $L$, showing a crossing at $T_{\mathrm{C}}/h\simeq 3.252$, indicative of a continuous transition. (The crossing becomes sharper for larger $L$, omitted for clarity.) Left inset: The applied field causing the transition and the configuration of the spins for $T\ll T_{\mathrm{C}}$. Right inset: Collapse of $W(T,0,L)$ versus $t{L}^{1/\nu }$ using $\nu =0.6717$ (3D $XY$).

Figure 3

Scaling at nonzero monopole fugacity $z$. Main figure: Plot of $W(T_{\mathrm{C}},z,L)$ (empty symbols, left scale) and $\partial W(T,z,L)/\partial T$ at $T=T_{\mathrm{C}}$ (filled symbols, right scale) versus $z^{\nu }{L}^{\varphi }$. The data lie on a single curve in each case, with $\nu$ and $\varphi =1.6665$ taking values for the 3D $XY$ universality class. Inset: Monopole density ${\rho }_m$ divided by $|t{|}^{2-\alpha }$, using the 3D $XY$ exponent $\alpha =-0.015$ [35]. Plotted against $z|t{|}^{-\varphi }$, data for various $T$ and $z$ (and fixed $L=16$) collapse separately for each sign of $t$.