Square Ice in Two Dimensions

Setup

Square ice is a two-dimensional analog of spin ice. Here Ising spins \(\sigma = \pm 1\) live on checkerboard lattice. At each fully connected plaquette, \(I\), of lattice we define a charge $$ Q = (-1)^I\sum_{i \in I} \sigma_i $$ as the sum of the four spins at its corners with a staggered sign, taking values \(0,\,\pm 2,\,\pm 4\). The energy penalises non-zero charge with $$ H = \tfrac{J}{2}\sum_I Q_I^2 = J\sum_{\langle ij\rangle} \sigma_i \sigma_j + \text{const.}, $$ Each state with \(Q_I=0\) i.e. two up and two down spins on each tetrahedron, is a ground state. There are six configurations per vertex, with a residual entropy of \(S/N = 3\log(4/3)/4 \approx 0.432\) for the full lattice. Excitations about these ground states have \(Q_I=\pm 2\), which behave as emergent magnetic monopoles or \(Q_I=\pm 4\) which are double monopoles.

Method

Metropolis single-spin flips using a four-colour checkerboard decomposition are used to equilibriate. Every tetrahedron has one corner of each colour and the GPU updates them in parallel using four passes per sweep. There is an optional plaquette update that flips all four spins around an inactive plaquette. Curves in plots use the single-tetrahedron approximation. Arctic circle mode imposes domain wall boundary conditions within a central diamond region, enables loop flips and set \(T=0\).

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