Absence of mobility edges in mosaic Wannier-Stark lattices

Abstract

Mobility edges, separating localized from extended states, are known to arise in the single-particle energy spectrum of certain one-dimensional models with quasiperiodic disorder. Recently, some works claimed rather unexpectedly that mobility edges can exist even in disorder-free one-dimensional models, suggesting as an example the so-called mosaic Wannier-Stark lattice where a Stark potential is applied on every M site of the lattice. Here, we present an exact spectral analysis of the mosaic Wannier-Stark Hamiltonian and prove that, strictly speaking, there are not mobility edges, separating extended and localized states. Specifically, we prove that the energy spectrum is almost pure point with all the wave functions displaying a higher than exponential localization with the exception of (M − 1)-isolated extended states at energies around which a countably infinite number of localized states with a diverging localization size condense.

Accelerating Quantum Decay by Multiple Tunneling Barriers

“…However, an infinitely countable set of eigenenergies, with weakly extended eigenstates, accumulate toward the zero energy point E = 0 of the extended state, with a diverging localization length of corresponding eigenstates. Such a property can be proven rigorously in some special potential models, such as the Stark potential model [53] or other integrable models such as the Maryland model [54]. More generally, for small energies |E/κ| → 0, the potential entering in Equation (A5) is almost vanishing and thus we expect that any eigenstate, if localized, should have a large localization length, diverging as E → 0.…”

Accelerating Quantum Decay by Multiple Tunneling Barriers

“…Curve 4 in Figure 3c shows the numerically computed behavior of the survival probability P(t) = |a 0 (t)| 2 for V 0 /κ = 10 and F = 1, clearly showing the acceleration of the decay in the early stage of the decay. This is a rather striking and unexpected result, given that the added barriers have a monotonously increasing and unbounded height and the corresponding Hamiltonian ( 18) has an almost pure point spectrum with localized eigenstates (see Appendix B and [53]).…”

“…(iii) The third example of decay acceleration concerns deterministic potential barriers with continuously increasing and unbounded heights, namely we assume symmetric Stark potential barriers with and Curve 4 in Figure 3 c shows the numerically computed behavior of the survival probability for and , clearly showing the acceleration of the decay in the early stage of the decay. This is a rather striking and unexpected result, given that the added barriers have a monotonously increasing and unbounded height and the corresponding Hamiltonian (18) has an almost pure point spectrum with localized eigenstates (see Appendix B and [ 53 ]).…”

“…However, an infinitely countable set of eigenenergies, with weakly extended eigenstates, accumulate toward the zero energy point of the extended state, with a diverging localization length of corresponding eigenstates. Such a property can be proven rigorously in some special potential models, such as the Stark potential model [ 53 ] or other integrable models such as the Maryland model [ 54 ]. More generally, for small energies , the potential entering in Equation (A5) is almost vanishing and thus we expect that any eigenstate, if localized, should have a large localization length, diverging as .…”

Spectral Structure and Doublon Dissociation in the Two‐Particle Non‐Hermitian Hubbard Model

“…impact of non-Hermiticity on the spectral and dynamical features of two strongly-correlated particles, and should stimulate further theroertical and experimental studies on an emergent area of research. [87] Appendix A: Energy Spectrum on the Infinite Lattice…”