Clock Spin Glass

The clock model (and corresponding clock spin glass) describes interacting spins which can take discrete values on the 2D planar unit circle.

\mathcal{H} = -\sum_{ij} J_{ij}\, s_i\!\cdot\!s_j,\quad s_i=(\cos\varphi_i, \sin\varphi_i)

where \(`s_i\!\cdot\!s_j`\) is a scalar product and each \(`\varphi_i`\) can take one of \(`q`\) values:

\varphi_i = v_i \frac{2\pi}{q},\quad v_i\in\{0,\ldots,q-1\}

Or, as a visual representation for \(`q=5`\):

\begin{tikzpicture} \draw (0,0) circle(2); \draw[->,line width=1.5pt] (0,0) -- (30:2); \draw[->,line width=1.5pt] (0,0) -- (102:2); \draw[->,line width=1.5pt] (0,0) -- (174:2); \draw[->,line width=1.5pt] (0,0) -- (246:2); \draw[->,line width=1.5pt] (0,0) -- (318:2); \end{tikzpicture}

NOTE: For the case discretized to two values, \(`q=2`\), this model corresponds to the Ising spin glass.

Hypergraph Generalization

The scalar product in the standard clock Hamiltonian does not lend itself to a generalization for multi-spin interactions. Instead, the qiotoolkit implementation uses a Hamiltonian based on the normalized sum of vectors \(`\vec v_e`\):

\mathcal{H} = -\sum_e J_e (2v_e^2-1),\quad
\vec v_e = \frac{1}{|e|}\sum_{i\in e} \vec{s_i}\,

where the term in brackets reduces to the scalar product for the case of 2-spin interactions and covers the range \(`[-1,1]`\) for any \(`q`\):

• The minimal value is realized when the clock spins "cancel out" (i.e., sum up to \(`\vec 0`\))
• The maximal value is realized when all spins are aligned (have the same value).

Example Input Data

{
  "type": "clock",
  "version": "0.1",
  "q": 5,
  "terms": [
    {"c": 1.0, "ids": [0, 1]},
    {"c": 1.0, "ids": [1, 2]},
    {"c": 1.0, "ids": [2, 3]},
    {"c": -2.0, "ids": [3, 0]}
  ]
}