Time Propagation and Spectroscopy of Fermionic Systems Using a Stochastic Technique

We present a stochastic method for solving the time-dependent Schrödinger equation, generalizing a ground state full configuration interaction quantum Monte Carlo method. By performing the time integration in the complex plane close to the real-time axis, the numerical effort is kept manageable and the analytic continuation to real frequencies is efficient. This allows us to perform ab initio calculation of electron spectra for strongly correlated systems. The method can be used as a cluster solver for embedding schemes.

Figure 1

(a) Time evolution of $\mathrm{Re}⟨\mathrm{\Psi }(0)|\mathrm{\Psi }(t)⟩$ and contour in complex time and (b) corresponding photoemission spectra (for $\mu =0$) for the time evolution using 70 000, $1.6\times {10}^6$, and $1.7\times {10}^7$ walkers for the 18-site Hubbard model at $U/t=2$, $k=(0,0)$, and half filling. All calculations start from the same initial state with 350 000 walkers, and three different time contours were used leading to 70 000, $1.6\times {10}^6$, and $1.7\times {10}^7$ walkers for longer times. The utilized time step is ${10}^{-3}$. Both the Lanczos and FCIQMC spectra were convoluted with a Lorentzian of FWHM of 0.02 to simplify visual comparison of the FCIQMC spectrum to the discrete eigenvalues obtained in the Lanczos method. The integrated weights of the peaks of the FCIQMC spectra are indicated and agree well with the weights of the discrete Lanczos spectrum, which are given in the first graph of (b). The bracketed numbers indicate the weights of not fully resolved peaks. (c) Photoemission and inverse photoemission spectra for a 24-site cluster with lattice vectors (3,3) and $(-5,3)$ with 22 electrons at $U/t=4$ for $k=(0,0)$ obtained using $\sim 1.5\times {10}^8$ and $\sim 3\times {10}^7$ walkers respectively. The inverse photoemission part carries very low weight and is also shown in the inset. For comparison, the same spectrum computed by means of the Hirsch-Fye [29] auxiliary-field quantum Monte Carlo (AFQMC) calculation is displayed.

Figure 2

Atomic multiplet of the carbon atom, obtained from two distinct initial states created by adding a $2p$ electron to the cation ground state. One of the states is prepared as a singlet (red). The second state (blue) is a mixture of singlet and triplet but with $L_z\ne 0$. The time evolution is carried out for 1600 a.u. of time and the zero of the frequency axis corresponds to $-37.3706\mathrm{H}$, which is the ground state energy of the cation computed using the projective FCIQMC algorithm. The VTZ basis is used in this example. The experimental values are according to Ref. [33]. The frequency resolution is 3.9 mH. The inset shows a portion of the computed Green’s function in real time.

Figure 3

Photoabsorption spectra for the carbon dimer for a single excitation from the $2{\sigma }_u$ to the $3{\sigma }_g$ (blue) and from the $1{\pi }_u$ to the $3{\sigma }_g$ (red) orbital using $\mathrm{V}X\mathrm{Z}$ basis sets. Also shown is spectral decomposition of the FCIQMC ground state (green) as a reference. The spectra are not normalized for better display. For the VTZ basis set, we also computed ${\mathrm{\Sigma }}_u$ photoemission (PE) and inverse photoemission (IPE) spectra for the ${\mathrm{C}}_2^{-}$ and ${\mathrm{C}}_2^{+}$, respectively. All energies and spectra are obtained with ME analytic continuation from 44–48 independent calculations. The experimental values are taken from Ref. [39]; these are also used to attribute singlet and triplet states and the $\pm$ symmetry of the $\mathrm{\Sigma }$ states. The zero of the frequency axis is set to $-75.649\mathrm{H}$. The time step used is $\mathrm{\Delta }t={10}^{-3}$ for the VTZ basis set and $\mathrm{\Delta }t={10}^{-3}$ (green) and $\mathrm{\Delta }t=5\times {10}^{-4}$ (red and blue) for the VQZ basis set.