Adiabatic Tracking of a State: A New Route to Nonequilibrium Physics

We present a novel numerical approach to track the response of a quantum system to an external perturbation that is progressively switched on. The method is applied, within the framework of the density matrix renormalization group technique, to track current-carrying states of interacting fermions in one dimension and in the presence of an Aharonov-Bohm magnetic flux. This protocol allows us to access highly excited states. We also discuss the connection with the entanglement entropy of these excited states.

Figure 1

Left panel: Landau-Zener effect. At an avoided level crossing, a diabatic evolution allows a transition to an excited state while in an adiabatic evolution the system remains in its GS. Right panel: Without a gap, the adiabatic tracking allows reaching excited states.

Figure 2

50 lowest energy levels and currents of Eq. (1) versus the flux $\Phi$ computed with a standard DMRG procedure in a $L=30$ ring. Different colors correspond to different levels. Left panel: The current of the GS is highlighted with a black line. Right panel: The red curve shows the tracked states, and the triangles correspond to values discussed in the text (${\Phi }_{\max }=0.4$, 0.95, 1.15, 1.9, 2.15, and 3.15).

Figure 3

Current versus interaction for excited states taken at ${\Phi }_{\max }=2.15$ and 3.15 in rings of various lengths.

Figure 4

Comparison between the adiabatic tracking (red dots) and standard DMRG (crosses) keeping the 5 lowest levels at ${\Phi }_{\max }=1.15$ in $L=50$ rings. Top panel: Energy differences between excited states and the GS ${\Delta }_i=E_i-E_0$. Inset: The adiabatic tracking (red dot) comes through the energy level crossing between the fourth and the second levels. Bottom panel: Currents of the corresponding states.

Figure 5

Fits of $c$ and $b$ from Eq. (3) versus interaction at different values of ${\Phi }_{\max }$ (see Fig. 2) in $L=50$ rings, except $L=30$ for ${\Phi }_{\max }\ge 1.90$. Right (a) panel: ${\varphi }_{\max }=0.4$, the system is still in its GS. The error bars are not shown since they are smaller than the symbols.