Local symmetries in one-dimensional quantum scattering

Abstract

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

Dynamics of local symmetry correlators for interacting many-particle systems

“…quantum systems with local symmetries. It is only recently that it has been shown [11][12][13][14][15][16][17][18][19][20] that the unique feature of the presence of local symmetries are the existence of invariant two-point correlator currents. However, all of these works address the case of one-dimensional stationary and non-interacting wave mechanical setups ranging from acoustics, optics to quantum mechanics and make no statement how a more general concept of local symmetries could look like, in particular in the case of dynamically interacting quantum systems.…”

“…Reducing to a single particle in one dimension which is the case previously treated extensively [11][12][13][14][15][16][17][18][19][20] we obtain…”

“…Recently several steps in the direction of a theory of local symmetries have been performed indeed [11][12][13][14][15][16][17][18][19][20]. The focus was hereby on the case of non-interacting stationary one-dimensional single particle or wave mechanical (acoustics, optics) problems which are described by the corresponding Schrödinger or Helmholtz equation Ψ ′′ (x) + S(x)Ψ(x) = 0 where the prime denotes differentiation with respect to the spatial variable x.…”

Quantum Network Transfer and Storage with Compact Localized States Induced by Local Symmetries

“…Furthermore, there is a class of CLSs which are protected by spatial local symmetry of the Hamiltonian against any perturbations preserving this local symmetry and the geometry within their localization domain 10 . As a side note, the more general study of local symmetries has recently been put on new grounds by introducing a framework of nonlocal currents [29][30][31][32][33][34] by means of which the parity and Blochtheorem are generalized to locally symmetric systems 35 . Recently, it has been shown that CLSs may result from particular local symmetries which commute with the Hamiltonian in discrete systems 10 .…”

Spectral Classification of One‐Dimensional Binary Aperiodic Crystals: An Algebraic Approach

“…In this way, the vanishing commutator condition satis…ed by the renormalized blocks is directly connected to the local mirror symmetry of the underlying atomic clusters, in agreement with the presence of a number of perfect transmission resonant states in general aperiodic systems exhibiting translation or inversion local symmetries. [57][58][59] Furthermore, from the graphical summary displayed in Fig.11 we realize that those BACs sharing the resonance energy E in their spectra (i.e, Fibonacci, precious mean and period doubling) have in common the presence of both T AAB and T BAA local transfer matrices in their renormalization block matrices, whereas those BACs lacking the resonance energy E in their spectra (i.e, metallic and mixed based means) include the conjugated local transfer matrices T BBA and T ABB as well due to the presence of B B bonds in addition to A A and A B ones. This is illustrated in the last column of Fig.11, where we compare the spectra obtained for the period doubling and copper limit-periodic crystals.…”