  Reformulation of a combinatorial problem into optimization of a
statistical-mechanics system, enables finding a better solution using
heuristics derived from a physical process, such as by the SA (Simulated
Annealing). In this paper, we present a Hadamard matrix (H-matrix) searching
method based on the SA on an Ising model. By equivalence, an H-matrix can be
converted into an SH (Seminormalized Hadamard) matrix; whose first columns are
unity vector and the rest ones are vectors with equal number of -1 and +1
called SH-vectors. We define SH spin-vectors to represent the SH vectors, which
play a similar role to the spins on an Ising model. The topology of the lattice
is generalized into a graph, whose edges represent orthogonality relationship
among the SH spin-vectors. Started from a randomly generated quasi H-matrix Q,
which is a matrix similar to the SH-matrix without imposing orthogonality, we
perform the SA. The transitions of Q are conducted by random exchange of {+,-}
spin-pair within the SH-spin vectors which follow the Metropolis update rule.
Upon transition toward zero-energy, the Q-matrix is evolved following a Markov
chain toward an orthogonal matrix, at which point the H-matrix is said to be
found. We demonstrate the capability of the proposed method to find some low
order H-matrices, including the ones that cannot trivially be constructed by
the Sylvester method.
