Analog of Hamilton-Jacobi theory for the time-evolution operator

In this paper we develop an analog of Hamilton-Jacobi theory for the time-evolution operator of a quantum many-particle system. The theory offers a useful approach to develop approximations to the time-evolution operator, and also provides a unified framework and starting point for many well-known approximations to the time-evolution operator. In the important special case of periodically driven systems at stroboscopic times, we find relatively simple equations for the coupling constants of the Floquet Hamiltonian, where a straightforward truncation of the couplings leads to a powerful class of approximations. Using our theory, we construct a flow chart that illustrates the connection between various common approximations, which also highlights some missing connections and associated approximation schemes. These missing connections turn out to imply an analytically accessible approximation that is the “inverse” of a rotating frame approximation and thus has a range of validity complementary to it. We numerically test the various methods on the one-dimensional Ising model to confirm the ranges of validity that one would expect from the approximations used. The theory provides a map of the relations between the growing number of approximations for the time-evolution operator. We describe these relations in a table showing the limitations and advantages of many common approximations, as well as the approximations introduced in this paper.

Figure 1

Illustration shows the relation between the different approximations discussed in the text. An approximation that is implied by symmetry is shown in the dashed box on the left side of the figure. We supply these in later sections of the manuscript. In the downward direction the approximations become progressively worse. To the left the approximations are expected to work better for larger constant parts of the Hamiltonian $\overline{H}$, and on the right for larger time-dependent parts of the Hamiltonian $V\left(t\right)$. The crossed out arrows signify that the result cannot be recovered without going to higher order in the approximate Wilcox series.

Figure 2

Plot of the $l_2$ distance, Eq. (64), between the different approximations and the exact time-evolution operator for strong driving $B=4,J_z=1$, and $L=14$ sites. Evidently, all approximations perform better as the large frequency limit is approached.

Figure 3

Plot of the $l_2$ distance between the different approximations and the exact time-evolution operator for weak driving $B=1$, and strong static interaction $J_z=4$, for a chain of $L=14$ sites.

Figure 4

Plot of the $l_2$ distance between the different approximations and the exact time-evolution operator as a function of the couplings for a chain with $L=16$ sites. The upper row has $J_z=1$ fixed with $B$ being varied. The lower row has $B=1$ fixed and $J_z$ is varied. For both cases the left column has $\omega =5$, the middle column $\omega =15$, and the right column $\omega =25$.