Worm-algorithm-type Simulation of Quantum Transverse-Field Ising Model

Abstract

We apply a worm algorithm to simulate the quantum transverse-field Ising model in a path-integral representation of which the expansion basis is taken as the spin component along the external-field direction. In such a representation, a configuration can be regarded as a set of non-intersecting loops constructed by "kinks" for pairwise interactions and spin-down (or -up) imaginary-time segments. The wrapping probability for spindown loops, a dimensionless quantity characterizing the loop topology on a torus, is observed to exhibit small finite-size corrections and yields a high-precision critical point in two dimensions (2D) as h c = 3.044 330(6), significantly improving over the existing results and nearly excluding the best one h c = 3.044 38(2). At criticality, the fractal dimensions of the loops are estimated as d ↓ (1D) = 1.37(1) ≈ 11/8 and d ↓ (2D) = 1.75(3), consistent with those for the classical 2D and 3D O(1) loop model, respectively. An interesting feature is that in 1D, both the spin-down and -up loops display the critical behavior in the whole disordered phase (0 ≤ h < h c ), having a fractal dimension d = 1.750(7) that is consistent with the hull dimension d H = 7/4 for critical 2D percolation clusters. The current worm algorithm can be applied to simulate other quantum systems like hard-core boson models with pairing interactions.