Resolution of the sign problem for a frustrated triplet of spins

We propose a mechanism for solving the “negative sign problem”—the inability to assign non-negative weights to quantum Monte Carlo configurations—for a toy model consisting of a frustrated triplet of spin-1/2 particles interacting antiferromagnetically. The introduced technique is based on the systematic grouping of the weights of the recently developed off-diagonal series expansion of the canonical partition function [Phys. Rev. E 96, 063309 (2017)]. We show that, although the examined model is easily diagonalizable, the sign problem it encounters can nonetheless be very pronounced, and we offer a systematic mechanism to resolve it. We discuss the prospects of generalizing the suggested scheme and the steps required to extend it to more general and larger spin models.

Figure 1

A triplet of antiferromagnetically coupled spin-1/2 particles. The two-body interactions are antiferromagnetic in the $x$ direction with coupling strength $\mathrm{\Gamma }$ and in the $z$ direction with strength $J$.

Figure 2

Diagrammatic representation of weights in the partition function expansion as closed paths on the hypercube of basis states. Each configuration is represented by a closed cycle whose nodes are basis states and is assigned a Boltzmann-like weight of the form ${\mathrm{\Gamma }}^q{e}^{-\beta [E_{z_0},...,E_{z_q}]}$ calculated from the classical energies $E_{z_i}$ of classical states $|z_i\rangle$. The boldfaced sequence of digits appearing next to each path corresponds to the sequence of visited basis states. (a) A zeroth-order classical term. These terms appear in the decomposition of the classical partition function. Their weights are standard Boltzmann weights. (b) A second-order term containing three basis states and two edges. (c) A third-order term generating a negative weight (for positive $\mathrm{\Gamma }$). (d) A fourth-order term, whose weight is positive regardless of the sign of $\mathrm{\Gamma }$.

Figure 3

The action of the off-diagonal operators on the even-parity basis states of the spin triplet. Similar relations hold for the odd-parity states; these are obtained via the substitution $|0\rangle \leftrightarrow |1\rangle$.

Figure 4

Top: Average QMC weight as a function of expansion order $q$ in the sign-problem-free case (black) and the sign problematic case (red). The latter distribution has negative weights for odd values of the expansion order $q$ (here $\beta J=5$ and $\mathrm{\Gamma }/J=\pm 1/2$). Bottom: Severity of the sign problem as measured by the weighted sign $\langle \mathrm{sgn}\rangle$ as a function of $\beta J$ on a logarithmic-linear scale for different values of the coupling parameter $\mathrm{\Gamma }/J$. For negative $\mathrm{\Gamma }$ values, $\langle \mathrm{sgn}\rangle =1$, whereas for positive values, the quantity decays exponentially fast. (The lines are to guide the eye.)

Figure 5

QMC thermal averages of the diagonal energy $\langle H_{\mathrm{c}}/J\rangle$ as a function of $\beta J$ for different values of $\mathrm{\Gamma }/J$. The solid lines are exact-diagonalization results. (a) Grouped ODE averages for negative values of $\mathrm{\Gamma }$ (no sign problem). (b) Standard ODE for negative values of $\mathrm{\Gamma }$ (no sign problem). (c) Grouped ODE averages for positive values of $\mathrm{\Gamma }$ (sign problem). (d) Standard ODE for positive values of $\mathrm{\Gamma }$ (sign problem). Whereas for $\mathrm{\Gamma }<0$ [(a) and (b)] both standard ODE and grouped ODE perform similarly well, the performances of the two algorithms differ considerably in the sign problematic case [(c) and (d)]. Whereas grouped ODE encounters no problems, in contrast, standard ODE diverges (note the difference in the vertical scales between the two cases).