Worm-algorithm-type simulation of the 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 nonintersecting 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 ${h}_{c}=3.044\phantom{\rule{0.16em}{0ex}}330(6)$, significantly improving over the existing results and nearly excluding the central value of the previous result ${h}_{c}=3.044\phantom{\rule{0.16em}{0ex}}38(2)$. At criticality, the fractal dimensions of the loops are estimated as ${d}_{\ensuremath{\ell}\ensuremath{\downarrow}}(1\mathrm{D})=1.37(1)\ensuremath{\approx}\frac{11}{8}$ and ${d}_{\ensuremath{\ell}\ensuremath{\downarrow}}(2\mathrm{D})=1.75(3)$, 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 ($0\ensuremath{\le}h<{h}_{c}$), having a fractal dimension ${d}_{\ensuremath{\ell}}=1.750(7)$ that is consistent with the hull dimension ${d}_{\mathrm{H}}=\frac{7}{4}$ 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.