Entanglement Hamiltonian of Interacting Fermionic Models

Recent numerical advances in the field of strongly correlated electron systems allow the calculation of the entanglement spectrum and entropies for interacting fermionic systems. An explicit determination of the entanglement (modular) Hamiltonian has proven to be a considerably more difficult problem, and only a few results are available. We introduce a technique to directly determine the entanglement Hamiltonian of interacting fermionic models by means of auxiliary field quantum Monte Carlo simulations. We implement our method on the one-dimensional Hubbard chain partitioned into two segments and on the Hubbard two-leg ladder partitioned into two chains. In both cases, we study the evolution of the entanglement Hamiltonian as a function of the physical temperature.

Figure 1

Nearest-neighbor hopping terms of the entanglement Hamiltonian for a segment of $L_a=8$ sites in a Hubbard chain of total length $L=32$, and parameters $t=1$ and $U=4$, as a function of the temperature. The dashed line indicates the expected values far from the boundary; see the main text.

Figure 2

The same as Fig. 1, for a Hubbard two-leg ladder of linear length $L=8$, with fixed parameters $t=1$ and $U=4$, as a function of the temperature for two values of $t_{\perp }$. The dashed line indicates the expected high-temperature limit $t_{i,i+1}\simeq \beta t$. The inset shows a magnification of the plot for $\beta \ge 1$ in a semilogarithmic scale.

Figure 3

The same as Fig. 2 for the on-site Hubbard repulsion $U$.

Figure 4

The same as Fig. 2 for the nearest-neighbor spin-spin interaction $J$.

Figure 5

The same as Fig. 2 for the next-nearest-neighbor spin-spin interaction $J^{\prime }$.