Multi-reference Trial State for Lattice Quantum Monte Carlo Simulations

TL;DR

This work tackles the slow convergence and extrapolation uncertainties in Nuclear Lattice EFT (NLEFT) by introducing a multi-reference trial-state framework built from shell-model configurations. It develops an efficient sampling algorithm (MultSlat-SamplerII) for computing correlation functions between multiple Slater determinants within AFQMC, enabling practical inclusion of multi-determinant trial states. By optimizing a deformed-harmonic-oscillator basis and linearly combining several shell-distributions, the method significantly reduces excited-state contamination for $^7$Li and $^8$Li, yielding faster imaginary-time convergence and more accurate observables, including electromagnetic transitions. The approach lays a robust foundation for accurate multi-channel calculations of nuclear spectra and transitions in NLEFT, with potential extensions to alpha-clustering and other complex shell structures.

Abstract

Nuclear lattice effective field theory (NLEFT) is an efficient \textit{ab initio} tool for solving nuclear many-body systems using the imaginary-time projection technique, where the preparation of trial states is essential for substantially reducing the computational cost required to achieve the desired numerical precision. It has been challenging in forming optimal multi-reference trial states using multiple Slater determinants within auxiliary-field based quantum Monte Carlo frameworks like NLEFT. In this work, we develop a novel sampling method for efficiently incorporating such multi-reference trial states into NLEFT calculations. We applied the optimized trial state to $^7$Li and $^8$Li, finding overall improvements in calculated energies, electromagnetic properties, and transitions compared to results obtained without these optimizations. Our approach provides a reliable foundation for accurately simulating nuclear ground and low-lying excited states within the NLEFT framework.

Figures (11)

• Figure 1: The three different shell distributions, $d_1$, $d_2$ and $d_3$, for constructing the trial state of $^7$Li(3/2$^-$). The red (blue) circle represents the occupied orbit of the proton (neutron). The number close to the circle denotes the magnetic quantum number of the $p$-shell orbit. $d_1$ consists of two Slater determinants shown in the first row. $d_2$ and $d_3$ comprises two and one Slater determinants, respectively.
• Figure 2: The brute-force way to calculate the matrix element $\langle \Phi^{\lambda, L}_{J_z, P}(\xi)|O|\Phi^{\lambda, R}_{J_z, P}(\xi)\rangle$. The $N_S$ Slater determinants are first independently evolved to the middle of the projection time under the operation of the transfer matrices $M$, the $N_S^2$ matrix elements are then calculated individually.
• Figure 3: The efficient way to calculate the matrix element $\langle \Phi^{\lambda, L}_{J_z, P}(\xi)|O|\Phi^{\lambda, R}_{J_z, P}(\xi)\rangle$. The $N_O$ single-nucleon states are first evolved to the middle of the projection time, which are then employed to reconstruct the $N^2_S$ matrix elements.
• Figure 4: The ratio $E^{ii}(\tau)=H^{ii}(\tau)/C^{ii}(\tau)$ of $^{7}$Li(3/2$^-$) versus the projection time $\tau$. The red, blue and green points denote the result of the three different trial states $d_1$, $d_2$ and $d_3$. The circles and the crosses represent the two methods, MultSlat-SamplerII and MultSlat-SamplerI, respectively. The error bar stands for statistical errors.
• Figure 5: The energy of $^{7}$Li(3/2$^-_1$), $^{7}$Li(3/2$^-_2$) and $^{7}$Li(3/2$^-_3$) versus the projection time $\tau$, which are represented by the red, blue and green points. The circles and the crosses represent MultSlat-SamplerII and MultSlat-SamplerI, respectively.