Nonequilibrium phase transition in transport through a driven quantum point contact

We study the transport of noninteracting fermions through a periodically driven quantum point contact (QPC) connecting two tight-binding chains. Initially, each chain is prepared in its own equilibrium state, generally with a bias in chemical potentials and temperatures. We examine the heating rate (or, alternatively, energy increase per cycle) in the nonequilibrium time-periodic steady state established after initial transient dynamics. We find that the heating rate vanishes identically when the driving frequency exceeds the bandwidth of the chain. We first establish this fact for a particular type of QPCs where the heating rate can be calculated analytically. Then we verify numerically that this nonequilibrium phase transition is present for a generic QPC. Finally, we derive this effect perturbatively in leading order for cases when the QPC Hamiltonian can be considered a small perturbation. Strikingly, we discover that for certain QPCs the current averaged over the driving cycle also vanishes above the critical frequency, despite a persistent bias. This shows that a driven QPC can act as a frequency-controlled quantum switch.

Figure 1

(a) Driven QPC connecting two tight-binding chains. On-site potentials indicated by double lines can be present on two QPC sites. Wiggly lines indicate the time dependence of the QPC. The whole system is described by the Hamiltonian (1). (b) The Floquet Hamiltonian (7) of the system with the conformal QPC (4).

Figure 2

The single-particle spectrum of the Floquet Hamiltonian $H_{\mathrm{F}}^{\mathrm{c}}$ of the conformal QPC as a function of the driving frequency.

Figure 3

The transmission coefficient (left) and the heating function (right) for the conformal QPC for different driving frequencies.

Figure 4

(a) and (c) Average heating rate $\overline{\mathcal{W}}$ and (b) and (d) average current through the QPC $\overline{\mathcal{J}}$ vs driving frequency. Initially, the left chain is filled by fermions in the ground state with particle density $N/L=0.4$ fermion per site, while the right chain is empty. (a) and (b) Conformal QPC (4). The heating rate and the current are obtained analytically from Eqs. (11) and (12), respectively. The heating rate vanishes for $\omega \ge 2$, while the current remains finite. (c) and (d) Tunneling QPC (16). Plots are obtained numerically for a system with $L=50$ sites in each chain and $N=20$ fermions. Both the heating rate and the current vanish for $\omega \ge 2$. Insets show the real-time dynamics of the total energy $\mathcal{E}-{\mathcal{E}}_0$ [(a) and (c)] and the particle number in the right chain $N_R$ [(b) and (d)].

Figure 5

Averaged heating rate $\overline{\mathcal{W}}$ (left) and current $\overline{\mathcal{J}}$ (right) for a QPC with $J_t=0.3sin\omega t,U_t^L=-U_t^R=0.3cos\omega t$ calculated numerically (solid lines; the system size and initial state are the same as in Fig. 4) and perturbatively [dashed lines; see Eqs. (19) and (20), respectively]. Both the average heating rate and current vanish for $\omega \ge 2$ in the leading order. However, the actual current remains finite due to the higher-order corrections, as illustrated in the inset in the right panel.