Transitionless quantum driving for spin systems

We apply the method of transitionless quantum driving for time-dependent quantum systems to spin systems. For a given Hamiltonian, the driving Hamiltonian is constructed so that the adiabatic states of the original system obey the Schrödinger equation. For several typical systems such as the $XY$ spin chain and the Lipkin-Meshkov-Glick model, the driving Hamiltonian is constructed explicitly. We discuss possible interesting situations when the driving Hamiltonian becomes time independent and when the driving Hamiltonian is equivalent to the original one. For many-body systems, a crucial problem occurs at the quantum phase transition point where the energy gap between the ground and first excited states becomes zero. We discuss how the defect can be circumvented in the present method.

Figure 1

Protocols for $J_x\left(t\right)$, $J_y\left(t\right)$, and $h\left(t\right)$ to be examined in the numerical analysis.

Figure 2

Fidelity for the driving of the ground state in the Lipkin-Meshkov-Glick model. The solid lines use the fixed-point protocol and dotted lines the non-fixed-point one. The upper figure is for the symmetric phase and the lower for the broken one.

Figure 3

The size dependence of the fidelity at the final time $t=1$. The upper figure is for the symmetric phase and the lower for the broken one.