Local symmetries in one-dimensional quantum scattering

We introduce the concept of parity symmetry in restricted spatial domains—local parity—and explore its impact on the stationary transport properties of generic, one-dimensional aperiodic potentials of compact support. It is shown that, in each domain of local parity symmetry of the potential, there exists an invariant quantity in the form of a nonlocal current, in addition to the globally invariant probability current. For symmetrically incoming states, both invariant currents vanish if weak commutation of the total local parity operator with the Hamiltonian is established, leading to local parity eigenstates. For asymmetrically incoming states which resonate within locally symmetric potential units, the complete local parity symmetry of the probability density is shown to be necessary and sufficient for the occurrence of perfect transmission. We connect the presence of local parity symmetries on different spatial scales to the occurrence of multiple perfectly transmitting resonances, and we propose a construction scheme for the design of resonant transparent aperiodic potentials. Our findings are illustrated through application to the analytically tractable case of piecewise constant potentials.

Figure 1

Local symmetries in one dimension. (a) Schematic of a completely local parity symmetric potential, composed of three different arbitrary mirror-symmetric scatterers, labeled $A,B,C$, and intervening potential-free regions (gaps). The circular arcs above the scatterer array provide all possible spatial decompositions into local symmetry domains ${\mathcal{D}}_n$, of lengths $L_n/2$ and center positions ${\alpha }_n$, which completely cover the potential region (up to variations including part of the intervening gaps). Two selected decompositions are shown below the array (solid red and dashed green lines), demonstrating the presence of nested local symmetries within the same system. In a scattering setup, unit amplitude plane waves incident on the left (and on the right) are considered (see Sec. 3), leading to transmitted $t$ (and $\tilde{t}$) and reflected $r$ (and $\tilde{r}$) amplitudes. (b) In one dimension, any translation of isolated constituents or parts of the potential can be reduced to combined overlapping local parity transformations: here, subsequent inversion through the centers of two subdomains, first ${\mathcal{D}}_1$, then ${\mathcal{D}}_2$.

Figure 2

Local parity eigenstates (their real parts) schematically shown for a single subdomain $\mathcal{D}$. Eigenstates of ${\widehat{\Pi }}_{+}^{\mathcal{D}}$ (${\widehat{\Pi }}_{-}^{\mathcal{D}}$) with eigenvalue $\lambda =+1$ ($-1$) are even (odd) within $\mathcal{D}$ and arbitrary outside. Eigenstates of ${\widehat{\Pi }}_{+}^{\mathcal{D}}$ (${\widehat{\Pi }}_{-}^{\mathcal{D}}$) with eigenvalue $\lambda =-1$ ($+1$) are odd (even) within $\mathcal{D}$ and zero outside.

Figure 3

Transmission spectrum of the simplest LP-symmetric barrier setup, with ${\psi }_{k_c}\left(x\right)$ and ${\rho }_{k_r}\left(x\right)$ shown (in arbitrary units) for a zero-current state ($E_c=8.70\epsilon$) under SAC and a PTR ($E_r=4.60\epsilon$) under AAC, respectively.

Figure 4

Potential with two resonators ${\mathcal{R}}_1$ (with $n_1=5$ barriers) and ${\mathcal{R}}_2$ ($n_2=9$), exhibiting two close (tunneling) PTRs at $E_1=7.00\epsilon$ and $E_2=7.46\epsilon$, with corresponding probability densities shown (in arbitrary units) for AAC. Their reducibility within the resonators is indicated by vertical dotted lines.

Figure 5

Potential decomposed into $N_{\mathcal{R}}=5$ resonators, exhibiting a zero-current state [with shown $\mathrm{Re}\left({\psi }_{k_r}\right)$ for SAC and ${\rho }_{k_r}$ for AAC, in arbitrary units] at energy $E_r=k_r^2/2=7.354\epsilon$. If ${\mathcal{R}}_1$ is removed (dashed lines), the state becomes a PTR (with shown ${\rho }_{k_r}$ for AAC, in arbitrary units) at the same energy.

Figure 6

Finite array of $N_{\mathcal{S}}=12$ rectangular barriers, two of which are deformed (defects). The array is decomposed into $N_{\mathcal{R}}^{\left(1\right)}=5$ and $N_{\mathcal{R}}^{\left(2\right)}=12$ resonators with respect to supported irreducible PTRs at energies $E_1=5.00\epsilon$ and $E_2=8.47\epsilon$ (${\rho }_{k_1}$ and ${\rho }_{k_2}$ are plotted in arbitrary units).

Figure 7

Part of the potential $V\left(x\right)$ of around the $n$th scatterer, with corresponding wave-function modulus $u_k\left(x\right)$. Either (i) a subdomain ${\mathcal{D}}_l$ is itself perfectly transmitting with LP symmetric $u_k\left(x\right)$ (solid lines), which is shown by considering a BVP in each of its halves (see the Appendix), or (ii) it must be augmented by a subsequent subdomain, so that $u_k\left(x\right)$ is LP symmetric in the resulting subdomain ${\mathcal{D}}_{\tilde{m}}$ (dashed lines).