Quantum Monte Carlo with directed loops

Abstract

We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The "algorithmic discontinuities" between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic S=1/2 Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the XY model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For LxL lattices with L up to 64, we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: chi( perpendicular )=0.0659+/-0.0002.

A two-dimensional spin liquid in quantum kagome ice

“…We implement a finitetemperature Stochastic Series Expansion [34][35][36] (SSE) QMC algorithm with directed loop updates in a 2 þ 1 dimensional simulation cell, designed specifically to study the Hamiltonian equation (1) with J ± ¼ 0 (for details, see the Methods Summary). Note, this Hamiltonian explicitly breaks U(1) invariance, retaining global Z 2 symmetries.…”

“…Within the SSE, the Hamiltonian was implemented with a triangular plaquette breakup 36 , which helps ergodicity in the regime where J z /J ±± is large. Using this Hamiltonian breakup, the standard SSE-directed loop equations 35 were modified to include sampling of off-diagonal operators of the type S þ r S þ r 0 þ h:c. The resulting algorithm is highly efficient, scaling linearly in the number of lattice sites V and the inverse temperature b. This scaling is modified to V 2 b in the cases where a full q-dependent structure factor measurement is required.…”

Phase diagram of the one-dimensional Hubbard-Holstein model at half and quarter filling

“…[17][18][19][20][21] SSE provides statistically exact results ͑no Trotter discretization of imaginary time is used͒ and has been adapted for many different quantum lattice models. Although this method has been described in detail elsewhere, we briefly describe here our treatment of the Holstein phonon interaction.…”

“…For the electrons, we use the directed loop algorithm. 21 We note that the operators ͓Eqs. ͑6͒-͑10͔͒ are not changed during the electron loop update.…”

Antiferromagnetism in a Family of S = 1 Square Lattice Coordination Polymers NiX2(pyz)2 (X = Cl, Br, I, NCS; pyz = Pyrazine)

“…Numerical calculations of the spin-1 antiferromagnetic Heisenberg model in an applied magnetic field were performed using the stochastic series expansion quantum Monte Carlo (QMC) method with directed loop updates. 43 For antiferromagnetic exchange interactions, sublattice rotation is required to avoid the sign problem in QMC. By taking the direction of the applied magnetic field as the discretization axis, sublattice rotation on a bipartite lattice leads to a sign problem free Hamiltonian as long as the applied field is parallel or perpendicular to the axis of exchange anisotropy.…”