Spectral Properties of Schrödinger Operators Arising in the Study of Quasicrystals

Damanik, David; Embree, Mark; Gorodetski, Anton

doi:10.1007/978-3-0348-0903-0_9

Cited by 61 publications

(70 citation statements)

References 174 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Even for the simplified model of non-interacting electrons in quasicrystalline potentials, nothing is known which would look like as a generalization of the band theory for crystals. Moreover, there is a striking difference between the one-dimensional case, where considerable progress has been achieved and the more physical case of higher dimensions, where the results are quite scant (see a survey [1]). After an initial enthusiasm, the problem was nearly abandoned, despite its obvious importance for the physics of quasicrystals.…”

Section: Introductionmentioning

confidence: 99%

Electrons in deterministic quasicrystalline potentials and hidden conserved quantities

Kalugin¹,

Katz²

2014

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Abstract. We propose an ansatz for the wave function of a non-interacting quantum particle in a deterministic quasicrystalline potential. It is applicable to both continuous and discrete models and includes Sutherland's hierarchical wave function as a special case. The ansatz is parameterized by a first cohomology class of the hull of the structure. The structure of the ansatz and the values of its parameters are preserved by the time evolution. Numerical results suggest that the ground states of the standard vertex models on Ammann-Beenker and Penrose tilings belong to this class of functions. This property remains valid for the models perturbed within their MLD class, e.g. by adding links along diagonals of rhombi. The convergence of the numerical simulations in a finite patch of the tiling critically depends on the boundary conditions, and can be significantly improved when the choice of the latter respects the structure of the ansatz.

show abstract

Section: Introductionmentioning

confidence: 99%

Electrons in deterministic quasicrystalline potentials and hidden conserved quantities

Kalugin¹,

Katz²

2014

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…Thanks to extensive study, there exist both rigorous and numerical upper and lower bounds on α þ u [10][11][12][13][14][15][16]. We mention that quasiperiodic sequences serve as models for one-dimensional quasicrystals and their sometimes exotic transport properties.…”

mentioning

confidence: 99%

“…We mention that quasiperiodic sequences serve as models for one-dimensional quasicrystals and their sometimes exotic transport properties. Especially, the discrete one-body Schrödinger operator with Fibonacci potential, see (5), has been considered [10,[12][13][14][15][16][17][18][19][20][21][22][23][24]. Quasiperiodic spin chains (in particular with Fibonacci disorder) have also been studied extensively, with a focus on spectral properties and critical phenomena [25][26][27][28][29][30][31][32].…”

mentioning

confidence: 99%

See 1 more Smart Citation

New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

et al. 2014

Self Cite

View full text Add to dashboard Cite

We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson (LR) bound for an isotropic XY chain in a quasiperiodic transversal magnetic field. Instead of the usual effective light cone jxj ≤ vjtj, we obtain jxj ≤ vjtj α for some 0 < α < 1. We can characterize the allowed values of α exactly as those exceeding the upper transport exponent α þ u of a one-body Schrödinger operator. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. We also discuss anomalous LR bounds with power-law tails for a random dimer field. Introduction.-Relativistic systems are local in the sense that information propagates at most at the speed of light. In their seminal paper [1], Lieb and Robinson found that nonrelativistic quantum spin systems described by local Hamiltonians satisfy a similar "quasilocality" under the Heisenberg dynamics. Their Lieb-Robinson (LR) bound and its recent generalizations [2,3] implies the existence of a "light cone" jxj ≤ vjtj in space-time, outside of which quantum correlations (concretely: commutators of local observables) are exponentially small. In other words, the LR bound shows that, to a good approximation, quantum correlations propagate at most ballistically, with a systemdependent "Lieb-Robinson velocity" v.About ten years ago, the general interest in LR bounds resurged when Hastings and coworkers realized that they are the key tool for deriving exponential clustering, a higher-dimensional Lieb-Schultz-Mattis theorem, and the celebrated area law for the entanglement entropy in onedimensional systems with a spectral gap [2,4,5]. These results highlight the role of entanglement in constraining the structure of ground states in gapped systems and yield many applications to quantum information theory, e.g., in developing algorithms to simulate quantum systems on a classical computer [6,7].In this Letter, we announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for an isotropic XY chain in a quasiperiodic transversal magnetic field. The LR bound is anomalous in the sense that the forward half of the ordinary light cone is changed to the region jxj ≤ vjtj α for some 0 < α < 1.Previous study has focused on the dependence of the Lieb-Robinson velocity v on the system details [3], with particular interest in the case v ¼ 0, since it may be interpreted as dynamical localization [8]. In a very recent paper [9], a logarithmic light cone was obtained for long-range, i.e., power-law decaying, interactions. The

show abstract

“…On the other hand, the absence of actual physical systems exhibiting the singular continuous component relegated this measure as a merely mathematical issue for some time. From this perspective, the discovery of QCs bridged the long standing gap between the theory of spectral operators in Hilbert spaces and condensed matter theory [16,17]. Now, from the viewpoint of condensed matter physics there are two different (though closely related) measures one can consider when studying the properties of solid materials.…”

Section: Spectral Measure Classificationmentioning

confidence: 99%

On the Nature of Electronic Wave Functions in One-Dimensional Self-Similar and Quasiperiodic Systems

Maciá

2014

ISRN Condensed Matter Physics

View full text Add to dashboard Cite

The interest in the precise nature of critical states and their role in the physics of aperiodic systems has witnessed a renewed interest in the last few years. In this work we present a review on the notion of critical wave functions and, in the light of the obtained results, we suggest the convenience of some conceptual revisions in order to properly describe the relationship between the transport properties and the wave functions distribution amplitudes for eigen functions belonging to singular continuous spectra related to both fractal and quasiperiodic distribution of atoms through the space.

show abstract

Spectral Properties of Schrödinger Operators Arising in the Study of Quasicrystals

Cited by 61 publications

References 174 publications

Electrons in deterministic quasicrystalline potentials and hidden conserved quantities

Electrons in deterministic quasicrystalline potentials and hidden conserved quantities

New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

On the Nature of Electronic Wave Functions in One-Dimensional Self-Similar and Quasiperiodic Systems

Contact Info

Product

Resources

About