Hopping Modulation in a One-Dimensional Fermi-Hubbard Hamiltonian

We consider a strongly repulsive two-component Fermi gas in a one-dimensional optical lattice described in terms of a Hubbard Hamiltonian. We analyze the response of the system to a periodic modulation of the hopping amplitude in the presence of a large two-body interaction. By (essentially) the exact simulations of the time evolution, we find a nontrivial double occupancy frequency dependence. We show how the dependence relates to the spectral features of the system given by the Bethe ansatz. The discrete nature of the spectrum is clearly reflected in the double occupancy after a long enough modulation time. We also discuss the implications of the 1D results to experiments in higher dimensional systems.

Figure 1

Double occupancy $\sum _i⟨n_{i,\uparrow }{n}_{i,\downarrow }⟩$ as a function of frequency for long ($t=10$, main panel) and short ($t=0.5$, inset) times. A single broad peak appears for $\omega /U\simeq 1$ ($L=20$, $U/J=60$, $\Omega /J=0.1$).

Figure 2

Double occupancy $\sum _i⟨n_{i,\uparrow }{n}_{i,\downarrow }⟩$ as a function of time, for two different frequency values corresponding to a minimum ($\omega /U=0.978$) and a maximum ($\omega /U=0.988$) of the plot in Fig. 1 (same parameters as Fig. 1).

Figure 3

DO spectrum for $U/J=500$ and $L=4,6,8,10$ (top left to bottom right). Vertical lines represent the value $\tilde{\Delta }E$ for various $k_p$ and $k_h$. Red continuous lines represent the values listed in Table . Blue dashed lines represent transitions between states of even parity (i.e., $i_p$ and $i_h$ even), which do not seem to correspond to a peak in the DO spectrum. Gray dotted lines represent states with $i_p=2n+1$ or $i_h=2n+1$, with $n\in \mathbb{Z}$. Green dashed-dotted lines represent a particular condition of coincidence of the frequency between a “red line” transition and a “gray line” transition with $i_p=2n+1$, $i_h=2n^{\prime }+1$, and $i_p-i_h=2n^{\prime \prime }$ (see Table ). For ease of reading, the right part of the spectrum has been omitted due to its symmetry around $\omega =1$ (cf. the slight asymmetry in the trapped case, Fig. 1). $x$ axis: $\omega /U$; $y$ axis: $⟨n_{i\uparrow }{n}_{i\downarrow }⟩$.

Figure 4

Trapped system: energy levels (continuous red lines correspond to even values of $i_p$ and $i_h$) for $L=20$, $N_{\mathrm{tot}}=12$, $U/J=60$, and $\Omega /J=0.1$. The peaks at the edge of the diagram do not correspond to any peak due to the approximate description of the energy levels.

Figure 5

Trapped system: energy levels (same definitions as in Fig. 4) $L=40$, $N=24$, $U=20$, and $\Omega /J=0.1$ (right), when values of $i_h$ and $i_p$ up to 20 are considered.