Fermionic mean-field dynamics for spin systems beyond free fermions

Abstract

We introduce the fermionized time-dependent Hartree-Fock (fTDHF), a real-time quantum dynamics method for spin-1/2 Hamiltonians following their mapping to fermions via the Jordan-Wigner transformation. fTDHF is formally equivalent to exact dynamics in the case of free fermions and can efficiently handle non-local string operators arising from long-range interactions via transition matrix elements between non-orthogonal Slater determinants. We show that the fTDHF method can be implemented on a classical computer with a cost that scales polynomially with system size, and linearly with the time steps. We benchmark fTDHF against exact dynamics on three separate spin-1/2 models, representing adiabatic preparation of states with long-range correlations, disorder-driven observation of many-body localization, and particle production in the Schwinger model. For each of these systems, fTDHF is shown to reproduce the qualitative dynamics generated by the exact evolutions, while maintaining a simple physical picture due to its mean-field nature.

Figures (4)

• Figure 1: Spin-spin correlation matrices $\Xi_{pq}$ obtained using the fTDHF method (left) and with exact time evolution (right) for the state that belongs to the CSB phase, shown in panels (a) and (b), and for the state that belongs to the XY phase, shown in panels (c) and (d).
• Figure 2: Spatially averaged spin-spin correlations $\Xi_M(l)$ as a function of distance $l$ obtained using the fTDHF method (open markers) and with exact time evolution (filled markers) for the states that belong to the CSB and XY phases.
• Figure 3: Expectation value of the $z$-spin alignment as a function of time, $\langle S^z_p(t)\rangle$, with (a) small disorder strength, $W=J_{max}$, and (b) large disorder strength, $W=10\cdot J_{max}$. The fTDHF values are shown with solid lines, while exact time evolution results are shown with dashed lines.
• Figure 4: Evolution of the particle density as a function of time for different values of $x$, starting from the bare vacuum. The fTDHF values are shown with solid lines, while exact time evolution results are shown with dashed lines.