Exposing local symmetries in distorted driven lattices via time-averaged invariants

Time-averaged two-point currents are derived and shown to be spatially invariant within domains of local translation or inversion symmetry for arbitrary time-periodic quantum systems in one dimension. These currents are shown to provide a valuable tool for detecting deformations of a spatial symmetry in static and driven lattices. In the static case the invariance of the two-point currents is related to the presence of time-reversal invariance and/or probability current conservation. The obtained insights into the wave functions are further exploited for a symmetry-based convergence check which is applicable for globally broken but locally retained potential symmetries.

Figure 1

Illustrative sketches of static potential landscapes possessing a local symmetry for $x,\overline{x}\in \mathcal{D}$ under (a) translation by $L$ and (b) inversion through position $\alpha$. (c) Sketch of a time-dependent potential with $V(x,t)=V(\overline{x},t)$ within localized domains of translation (${\mathcal{D}}_1$) and inversion (${\mathcal{D}}_2$) symmetry.

Figure 2

(a1) Ground state probability densities $|{\mathrm{\Psi }}_0{|}^2$ with corresponding (b1) nonlocal current $|Q_{{\mathrm{\Psi }}_0}{|}^2$ (for translation $\overline{x}=x+L$ with $L=5$) as a function of $x$ and (c1) ${log}_{10}\left[\right|Q_{{\mathrm{\Psi }}_0}^{\mathrm{\Delta }L}{|}^2]$ for varying $x$ and shift $\mathrm{\Delta }L$ (see text), for a $N=5$-barrier lattice with $V_n=1$ for $n\ne 0$ and $V_0=0.8$. The potential is shown in arbitrary units in the top panel. (a2, b2, c3) Same as above but for $V_0=1.2$. (d) Density $|{\mathrm{\Psi }}_0{\left(x\right)|}^2$ and (e) current $|Q_{{\mathrm{\Psi }}_0}{|}^2$ for a lattice containing $N=25$ barriers with five defects with $V_i=1.1$ at positions $X_i=-55,-30,-20,10,x=40$. In all cases ${\left|Q(x-L/2)\right|}^2$ is shown so that the deviations from $Q_{{\mathrm{\Psi }}_0}^{\prime }=0$ are centered around the defect locations).

Figure 3

(a) Magnitude squared of the locally invariant current ${\overline{Q}}_{{\mathrm{\Psi }}_0}$ (again shifted by $+L/2$ in $x$) of the Floquet mode ${\mathrm{\Psi }}_0$ for the setup in Fig. 2(a1), but now oscillating with amplitude $A=1$ and frequency $\omega =0.5$. (b) Density plot, $|{\mathrm{\Psi }}_0{|}^2$, of the Floquet mode used in (a). (c) Integral kernel $|Q_{{\mathrm{\Psi }}_0}{(x,t)|}^2$ [cf. Eq. (8)] for one period of the driving.

Figure 4

Convergence measure ${\epsilon }_{\mathrm{\Psi }}$ on a logarithmic scale for the ground state of the setup in Fig. 2(a1) in dependence of $k_{\text{max}}$ (with the plane wave basis size being $2k_{\text{max}}+1$) for three different spacings $\mathrm{\Delta }x$ of the numerical grid on which $\mathrm{\Psi }\left(x\right)$ is represented.

Figure 5

Ratios of the absolute squares of the three different values $Q_{\mathcal{L}}, Q_{\mathcal{C}}$, and $Q_{\mathcal{R}}$ that the current $Q\left(x\right)$ takes for a single point scatterer located at $X_c$ [see Eq. (A2)] as functions of the momentum $k$ of the incoming wave. Shown are the analytic results stated in Eq. (A3) for two different strengths (a) ${\mathrm{\Lambda }}_c=2.5$ and (b) ${\mathrm{\Lambda }}_c=5$ of the delta potential.