Benchmarking the accuracy of superconducting pair-pair correlations within Constrained Path Quantum Monte Carlo

Abstract

Ground state properties of the Hubbard model are of fundamental importance to understand the mechanism of unconventional superconductivity in the high-T_c cuprates and other materials. One of the most powerful numerical methods for strongly interacting models is quantum Monte Carlo, which however faces a fundamental limitation, the Fermion sign problem. The sign problem can be mitigated using approximate methods such as Constrained Path Monte Carlo, but additional approximations must be made in order to measure different observables, particularly for operators that do not commute with the Hamiltonian. We examine critically the most commonly used approximation, back propagation, as well as a recently proposed constraint release measurement technique. In comparisons with a variety of systems that can be solved numerically exactly by other methods, we find that back propagation tends to underestimate superconducting pair-pair correlations. The constraint release technique can provide accurate results, with the disadvantages that it much more computationally expensive and reintroduces the sign problem.

Figures (5)

• Figure 1: (color online) Pair-pair correlation for nearest-neighbor pairs, $P(r)$, versus distance $r$ for a 32-site one-dimensional lattice with open boundary conditions, 28 particles, and $U$=8. The x's are exact results calculated using DMRG; circles are calculated using CPMC with BP. The inset shows the convergence of the CPMC mixed energy estimator versus the imaginary time step $\Delta\tau$. The line is a quadratic fit to the CPMC data; the x at $\Delta\tau$=0 is the energy calculated by DMRG.
• Figure 2: (color online) Pair-pair correlation $P(r)$ versus distance $r$ for a 32-leg ladder with open boundary conditions, 56 particles, and $U$=4. The x's are numerically exact DMRG results, open circles are CPMC with BP, and filled squares are CPMC with CR. Both BP and CR used $\beta$=3. The inset shows $\beta$ dependence of the average sign in the CR measurements.
• Figure 3: (color online) Percent relative error in the average long-range $d_{x^2-y^2}$ pair-pair correlation $\bar{P}$ versus $\beta$ for the 4$\times$4 lattice with 10 electrons, $U$=4, and $\Delta\tau$=0.01. Circles and filled squares are BP and CR estimates with $\tau=\beta/2$, respectively. Both used the free electron trial function. The inset shows the average sign for the CR estimate.
• Figure 4: (color online) $\bar{P}$ versus $\beta$ for for the 4$\times$4 lattice with 10 electrons, $U$=8, and $\Delta\tau$=0.01. Open (filled) symbols are BP (CR) estimates. Filled squares are CR estimates with $\tau=\beta/2$, while filled diamonds used $\tau=\beta/4$. Dashed lines indicate the QP-PIRG rather than free electron trial function. The inset shows the average sign for the CR estimate using the free electron trial function.
• Figure 5: (color online) (a) Variance extrapolation of the energy using the QP-PIRG method for the half-filled 6$\times$6 anisotropic lattice with $t^\prime=0.5$ and $U$=5. The line is a quadratic least-squares fit. (b) Average long-range $d_{x^2-y^2}$ pair-pair correlation versus $U$ on the 6$\times$6 anisotropic triangular lattice with $t^\prime$=0.5. X's, circles, and filled squares were calculated using extrapolated QP-PIRG, CPMC with BP, and CPMC with CR, respectively. CPMC results used $\Delta\tau=$0.05.