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 spin-down 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 $h_c=3.044330(6)$, significantly improving over the existing results and nearly excluding the best one $h_c=3.04438(2)$. At criticality, the fractal dimensions of the loops are estimated as $d_{\ell \downarrow }(1\mathrm{D})=1.37(1)\approx 11/8$ and $d_{\ell \downarrow }(2\mathrm{D})=1.75(3)$, consistent with those for the classical 2D and 3D O(1) loop model, respectively. An interesting feature is that in one dimension (1D), both the spin-down and -up loops display the critical behavior in the whole disordered phase ($0\le h<h_c$), having a fractal dimension $d_{\ell }=1.750(7)$ that is consistent with the hull dimension $d_{\mathrm{H}}=7/4$ 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.

11 pages, 11 figures

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