One-dimensional transient crystals

A phase of solid state matter which exists in time dimension is investigated theoretically for a linear chain of quantum dots or atomic sites using the evolution operator technique and the Laplace transform technique.. The studies focus on the spectral density of states (DOS) function and its modifications after a sudden quench in the system. It is shown that this function reveals a very regular structure which oscillates in time and which corresponds to the stationary DOS of one-dimensional crystals. Thus, such a system stands for a transient crystal material which exists in time dimension but appears only for a short period and suddenly vanishes. Moreover, we observe different impulse propagation velocities due to the local DOS inertia, i.e., it is more difficult to rebuild a many-peaked structure of DOS related to longer chains. We also show that the transient crystal pattern in a linear chain can be observed for real electrodes characterized by the van Hove singularities or by a flat spectral density.

Figure 1

Spectral density function ${\rho }_N(E,t)$ of a linear chain composed of $N=10$ (upper panel) and $N=20$ sites (bottom panel) after the quench at $t=0$ (for $t<0$: $\mathrm{\Gamma }=0, V_i=V=0$ and for $t>0$: ${\mathrm{\Gamma }}_L=1, V=4$), ${\varepsilon }_0=0$. A and B indicate the flat density area and the transient crystal region, respectively. The reference energy point is the lead Fermi energy, $E_F=0$, and energy is expressed in $\mathrm{\Gamma }$ units and time in $\hslash /\mathrm{\Gamma }$ units.

Figure 2

Time evolution of the spectral density function at the Fermi level for $V=4, N=20$ at the first and last chain sites, $i=1$ and $i=20$ (upper panel)—first maxima at these curves appear for time ${\tau }_1=0.45$ and ${\tau }_N=3.31$. The bottom panel shows the time of first maxima in ${\rho }_i(E,t)$ for every chain site $i$ for $N=20$ (blue points) and $N=40$ (red lines) and for different couplings $V=1,2$, and 4. The other parameters are the same as in Fig. 1.

Figure 3

Time of the first maximum in the spectral density function ${\rho }_1(E,t)$, i.e., at the first site (${\tau }_1$, upper panel) and ${\rho }_N(E,t)$ at the last site (${\tau }_N$, bottom panel), as a function of the coupling parameter $V$ for $N=5,10,20$, and 30 sites. The black line in the bottom panel corresponds to the velocity of the signal propagation through area A, $v_A=\frac{N}{{\tau }_N-{\tau }_1}$, as a function of the site-site coupling $V$. The values of $v_A$ correspond the same scale as time values on the $y$ axis. The other parameters are the same as in Fig. 1.

Figure 4

Time-dependent spectral density function ${\rho }_N(E,t)$ (upper panel) for $V=4$ and $N=20$ with indicated effective chain lengths of the transient crystal $N_{\mathrm{eff}}$ form 1–10. The circles correspond to maxima of the stationary DOS obtained for a linear chain composed of the $N_{\mathrm{eff}}$ site, and for the effective coupling $V_{\mathrm{eff}}=V/3.2cos\left(\frac{\pi }{N_{\mathrm{eff}}+1}\right)$ and ${\varepsilon }_i=0$. The bottom panel shows the time difference between the first and next maxima in ${\rho }_N(E_F,t)$ in the transient crystal area as a function of the effective chain length $N_{\mathrm{eff}}$ for $N=20,40,60$ (red crosses, orange triangles, and blue circles, respectively) and for $V=1,2$, and 4 (as indicated in the panel). The lines are plotted for better visualization. The other parameters are the same as in Fig. 1.

Figure 5

Time-dependent spectral density function ${\rho }_N(E,t)$ after the quench at $t=0$ for chain length $N=10$ and coupling strength $V=4$, for different lead DOS: (a) rectangular DOS, (${\mathrm{DOS}}_A$), (b) 2D square lattice DOS with one van Hove singularity for $E=0$ (${\mathrm{DOS}}_B$), and (c) 2D honeycomb (hexagonal) lattice DOS with two van Hove singularities for $E=\pm 5$ (${\mathrm{DOS}}_C$) [48, 49]. All DOS curves have the width $w=30\mathrm{\Gamma }$ and they are all normalized. The other parameters are the same as in Fig. 1.