Implementation of Spin Hamiltonians in Optical Lattices

We propose an optical lattice setup to investigate spin chains and ladders. Electric and magnetic fields allow us to vary at will the coupling constants, producing a variety of quantum phases including the Haldane phase, critical phases, quantum dimers, etc. Numerical simulations are presented showing how ground states can be prepared adiabatically. We also propose ways to measure a number of observables, like energy gap, staggered magnetization, end-chain spins effects, spin correlations, and the string-order parameter.

Figure 1

(a) Different phases of the ground state of Eq. (1). (b) Type of Hamiltonian (3) as a function of the gradient of the electric field, $\Delta$, for $U_0=0.75U_2$. The solid line is obtained naturally, the dashed line when working with the upper part of the spectrum. The conjectured nematic phase (dotted line) is achieved for $U_0=U_2$ or $\Delta \to \infty$.

Figure 2

(a) A ladder is the combination of two spin chains that interact with each other. Interactions along a chain and between legs can be different. (b) We can build a ladder combining a 3D lattice, $I_2(x)$, with an additional 1D optical lattice, $I_1(x)$, that has twice the period of the first one. This induces a tunable hopping $J^{\prime }$, different from the longitudinal one, $J$, and suppresses hopping between neighboring ladders.

Figure 3

(a) Procedure for the adiabatic construction of the ground state of the $s=1$ AF with open boundary conditions. (b) Infidelity of the final state for the AKLT (dashed line), Heisenberg (solid line), and $s=\frac{1}{2}$ spin ladder (dotted line), versus duration of the process, for either nine spins (chains) or rungs (ladders).

Figure 4

Maximum staggered magnetization per spin acquired by the ground state of the (a) AKLT and (b) Heisenberg models, under an oscillating magnetic field, $B_j=(-1{)}^{j}0.025\alpha \cos (2\pi \nu t)$, for five, seven, and nine spins (right to left). Vertical lines mark the finite excitation gap for the given size.