Krylov-Projected Quantum Monte Carlo Method

We present an approach to the calculation of arbitrary spectral, thermal, and excited state properties within the full configuration interaction quzantum Monte Carlo framework. This is achieved via an unbiased projection of the Hamiltonian eigenvalue problem into a space of stochastically sampled Krylov vectors, thus, enabling the calculation of real-frequency spectral and thermal properties and avoiding explicit analytic continuation. We use this approach to calculate temperature-dependent properties and one- and two-body spectral functions for various Hubbard models, as well as isolated excited states in ab initio systems.

Figure 1

$E(\beta )$ for the 12-site 1D Hubbard model at $U/t=1$ sampled with $\sim 2\times {10}^3$ walkers, with comparison FTLM. Error bars show standard deviation (note: not standard error) over ten independent calculations to demonstrate the spread of results. High and low temperature results are almost exact, while at intermediate temperatures, the variance in the stochastic sampling as well as systematic errors (such as from the nonlinear diagonalization step, and finite $R$) increases the variation between runs. Simulation parameters were $\tau =0.01$, $n_{\mathrm{a}}=2.0$, and a deterministic space of double excitations [36, 37].

Figure 2

$E(\beta )$ for the 18-site 2D Hubbard model at $U/t=1$, with ground-state FCIQMC energy for comparison. Ten independent simulations were used to create the standard deviations shown as error bars. Simulation parameters were $\tau =0.01$, $n_{\mathrm{a}}=2.0$, and a deterministic space of double excitations.

Figure 3

(a) $A_1(k,\omega )$ from $k=-\frac{6}{7}\pi$ (bottom) to $k=\pi$ (top) for the 1D 14-site Hubbard model at $U/t=2$, compared to dynamical Lanczos method. Poles coming from the ground state or low-lying excited states with large transition amplitudes are captured accurately. (b) The local density of states. The low-energy results are reproduced accurately by the KP-FCIQMC method while the qualitative behavior is captured at high energies. Simulation parameters were $\tau =0.01$, $n_{\mathrm{a}}=3.0$, and a deterministic space of 50 000 determinants [36].

Figure 4

$A_2(\omega )$ calculated for the 10-site Hubbard model with $U/t=1$, and compared to the near-exact dynamical Lanczos method. Inset shows integrated weight, ${\int }_0^{\omega }{A}_2({\omega }^{\prime })d{\omega }^{\prime }$. Simulation parameters were $\tau =0.01$, with a deterministic space of double excitations.