A dedicated algorithm for calculating ground states for the triangular random bond Ising model

Triggered by the exchange of ideas between computer science and theoretical physics, several disordered systems with complex energy landscapes can now be analyzed numerically exact through computer simulations [1] by using fast combinatorial optimization algorithms. E.g., the ground state problem for the planar 2d random bond Ising model (RBIM) can be mapped to an auxiliary minimum-weight perfect matching problem, solvable in polynomial time. The corresponding mapping uses the relation between perfect matchings and paths on a lattice. In effect, these paths can be used to partition the graph into domains of up and down spins that comprise a GS spin configuration, see Fig. 1(a),(b). Consequently, the GS properties as well as minimum-energy domain wall (MEDW) excitations, see Fig. 1(c), can be analyzed very fast [2]. Here, we introduce a dedicated algorithm for the 2d RBIM on planar triangular lattices that improves on the running time of existing algorithms and allow us to study systems with up to N =512×512 spins. Further, we investigate the critical behavior of the corresponding T = 0 ferromagnet to spin-glass transition, signaled by a breakdown of the magnetization, using finite-size scaling analyses of the MEDW excitation energy. Finally, we contrast our numerical results with previous simulations and presumably exact results [3]. In this regard, we obtain a highly precise estimate of the critical point for the triangular lattice geometry and we verify the critical exponents obtained earlier for the RBIM on the planar square lattice [2].