Nontrivial dynamics of a two-site system: Transient crystals

We analyze theoretically quench dynamics of a two-site system abruptly driven from its equilibrium state focusing on the time dependent spectral density function. In the presence of electron reservoirs this function reveals in time a nontrivial regular pattern of peaks corresponding to the stationary quantum chain structure. Such dynamical system with periodic structure of the spectral density stands for a new transient crystal material. We investigate here the role of the Coulomb repulsion between the sites, nontrivial substrates and different system geometries on the transient crystal pattern which can be measured in the tunneling conductance experiments. We also propose the transient crystal system between unbiased leads as an effective monoparametric pump.

Figure 1

Spectral density function ${\rho }_2(E,t)$ of the two-site system after the quench at $t=0$ (for $t<0$: ${\mathrm{\Gamma }}_{L/R}=0$ and $V=0$). For $t>0$ the site-site coupling is $V=0.4,2,4$, and 8 (from upper to bottom panels, respectively) and ${\mathrm{\Gamma }}_L=1$. The other parameters are ${\mathrm{\Gamma }}_R=0$, $V_{1,kL}=4$, $V_{2,kL}=V_{1,kR}=V_{2,kR}=0$, ${\varepsilon }_0=0$, $U^C=0$. The units of all energies and time are ${\mathrm{\Gamma }}_L=\mathrm{\Gamma }$ and $\hslash /\mathrm{\Gamma }$, respectively.

Figure 2

Time of the first maximum in ${\rho }_i(E,t)$ as a function of the coupling parameter $V$ for $i=1$ (solid curve) and $i=2$ (broken curve). The inset shows time evolution of the spectral density function at the Fermi level for $V=2$ and $i=1$ and 2, respectively. The other parameters are the same as in Fig. 1; ${\tau }_{\text{max}}$ is expressed in the time units, $\hslash /\mathrm{\Gamma }$.

Figure 3

(Upper panel) Effective chain length, $M$, as a function of time for the system described in Fig. 1 for $V=4$ (dots) and $V=2$ (crosses). The inset figure shows the spectral density function (normalized DOS) obtained for $M=3$ linear sites within the Green function method with the effective coupling, $V_M=2.83$, ${\varepsilon }_i=0$ (solid curve) and for the dimer taken at time $t=2.3$ after the quench for $V=4$ (dotted curve). The bottom panel shows time dependent spectral density function for $V=4$ with indicated effective chain lengths, $M$. The circles correspond to maxima of equilibrium DOS obtained for $M$-site linear chain for $V_M=V/2.3cos\left(\frac{\pi }{M+1}\right)$. The other parameters are the same as in Fig. 1.

Figure 4

Electrode density of states considered in the calculations: rectangular DOS (curve A), 2D square lattice DOS with the van Hove singularity (curve B), and 2D honeycomb (hexagonal) lattice DOS with two van Hove singularities (curve C) [38, 39]. The energy unit is $\mathrm{\Gamma }$, all curves are normalized to 1, and each DOS width is $w=30\mathrm{\Gamma }$.

Figure 5

Spectral density function ${\rho }_2(E,t)$ of the two-site system coupled vertically with the lead characterized by different DOS shown in Fig. 4, curves A, B, and C (upper, middle, and bottom panel, respectively). The other parameters are the same as in Fig. 1, $V=4$.

Figure 6

Spectral density function ${\rho }_2(E,t)$ for linear change of $V$ from $V=0$ (at $t=10$) up to $V=4$ (for $t\ge 15$). The other parameters are the same as in Fig. 1.

Figure 7

Spectral density function of the two-site system in the horizontal geometry ${\rho }_2(E,t)={\rho }_1(E,t)$ after the quench with the substrate characterized by 2D tight-binding DOS with a van Hove peak in the middle of the band. The other parameters are the same as in Fig. 1, $V=4$, $V_{1,kL}=V_{2,kR}=V_k=2.82$.

Figure 8

Time dependent currents flowing from the left (red solid curve) and from the right (blue broken curve) electrodes for ${\varepsilon }_0=-2$. The coupling $V=4$ was switched off at $t=10$ for a period of $\mathrm{\Delta }=5$ time units (black dotted line). The left (right) electrode is characterized by 2D-TB DOS with one (two) van Hove singularity (see Fig. 4), both DOS are symmetrical and their widths are $w=50$ energy units, the current unit is $e\mathrm{\Gamma }/\hslash$.

Figure 9

Pumped charge through the two-site system between the unbiased leads during the changing of $V$ parameter (shown in Fig. 8) as a function of $\mathrm{\Delta }$ for ${\varepsilon }_0=0,2,4$ (upper panel) and as a function of ${\varepsilon }_0$ for $\mathrm{\Delta }=0,2,5,10,20$ (bottom panel). The unit of $\mathrm{\Delta }$ is $\hslash /\mathrm{\Gamma }$.

Figure 10

Time dependent spectral density function of the two-site system after the quench at $t=0$ for small Coulomb repulsion between sites $U^C=1$ (upper panel) and for the stronger one, $U^C=4$, (bottom panel). The arrows indicate the energy shifting of the stationary peaks of the local DOS due to the Coulomb repulsion. The other parameters are the same as in Fig. 1, $V=4$.