Adiabatic Preparation of a Heisenberg Antiferromagnet Using an Optical Superlattice

We analyze the possibility to prepare a Heisenberg antiferromagnet with cold fermions in optical lattices, starting from a band insulator and adiabatically changing the lattice potential. The numerical simulation of the dynamics in 1D allows us to identify the conditions for success, and to study the influence that the presence of holes in the initial state may have on the protocol. We also extend our results to two-dimensional systems.

Figure 1

The proposed adiabatic protocol. First, a BI in a lattice with depth $V_1$ (a) is transformed to a product of singlets $|\varphi ⟩$ (b) by slowly switching on a second lattice with depth $V_2$ and half the original wave length. Then, by lowering the barrier $V_1\to 0$, the system turns into an AFM (c).

Figure 2

Absence of holes. (a), (b) $m^2$, $p_0$, and $e_{\mathrm{spin}}$ as functions of the ramping time $T$, for chain length $N=22$ (solid blue lines), 42 (dashed red lines), 62 (dash-dotted green lines), and 82 (dash double-dotted brown lines). The inset of (a) shows the squared size $N^2$ of the longest chain reaching a fixed $m^2=0.85$ at ramping time $T$, and reveals the scaling $T\propto N^2$. (c), (d) Same quantities as above, evaluated on sublattices of length $L=22$ (solid blue lines), 42 (dashed red lines), and 62 (dash-dotted green lines) for $N=82$. Now, the inset of (c) shows the squared size $L^2$ of the largest sublattice reaching $m^2=0.85$ at ramping time $T$, and reveals the scaling $T\propto L^2$. All results were obtained with MPS of bond dimension $D=60$ and Trotter step $\delta t=0.02$ [29].

Figure 3

Effect of holes and harmonic trap. (a) $m^2$ as a function of the ramping time $T$, evaluated on a sublattice of length $L=42$ for $N=86$, without holes (thin solid line), and with initially 2 holes at each boundary (thick solid line), and $t=2$. The inset shows the largest sublattice size $L$ reaching $m^2=0.85$ at ramping time $T$, for the two cases of the main plot, and the size of the hole-free region (dashed red line). (b) $m^2$ evaluated on a middle sublattice of length $L=82$ for $N=102$, with no holes (thin solid line), and with initially 10 holes at each boundary and a harmonic trap of strength $V_t=0.004$ (dash-dotted green line), $0.006$ (dashed red line), and $0.02$ (solid blue line), and $t=3$. For $V_t=0.2$ (not shown) the exact behavior of the ideal case is recovered. Again, the inset shows the largest sublattice size $L$ reaching $m^2=0.85$ at ramping time $T$, for the cases of the main plot. All results were obtained with MPS of bond dimension $D=60$ and Trotter step $\delta t=0.02$ [29].

Figure 4

Performance in 2D. (a) $m^2$ and $e_{\mathrm{spin}}$ as functions of the ramping time $T$, for systems of size $N=4\times 4$ (solid blue line), $6\times 6$ (dashed red line), $8\times 8$ (dash-dotted green line), and $10\times 10$ (dash double-dotted brown line). (b) $m^2$ for $N=8\times 8$, with initially 4 holes at the boundary and no harmonic trap (dash-dotted green line), and with a trap of strength $V_t=0.25$ (dashed red line), and 2.5 (solid blue line), and $t=2.5$. The results were obtained with PEPS of bond dimension $D=4$ (a) and $D=2$ (b) and Trotter step $\delta t=0.03$ [29].