1,720,972 research outputs found
On the localization dichotomy for gapped quantum systems
Full text linkSince their introduction by G. Wannier in 1937, Wannier functions have been extensively used in solid state physics to analyze and understand the physical properties of perfect crystalline quantum systems. In 2016, D. Monaco, G. Panati, A. Pisante and S. Teufel proved a localization dichotomy result for periodic Schrödinger operators, namely that the localization properties of Wannier functions are deeply connected to the topological properties of the quantum system. The original results presented in this thesis concern the possibility of extending such localization dichotomy to generic gapped quantum systems. First of all, by reviewing and analyzing the different few existing results about generalized Wannier functions, we give a precise definition of generalized Wannier functions for generic gapped quantum systems. Moreover, we prove the existence of Parseval frames of exponentially localized generalized Wannier functions for a large class of magnetic systems, as a byproduct we show the existence of a generalized Wannier basis for magnetic Hamiltonians. Furthermore, we analyze the Chern number in position space, namely the Chern character, by proving a gap labelling theorem for Bloch-Landau Hamiltonians using gauge covariant magnetic perturbation theory and investigating the validity of the gap labelling theorem in a non-covariant setting. We also explicitly show how to connect the Chern character to the Středa formula. Finally, we show that an ultra generalized type of Wannier basis is not capable to encode the physical properties of the systems and we prove that the existence of a well-localized localized generalized Wannier basis implies the vanishing of the Chern character
Parseval Frames of Exponentially Localized Magnetic Wannier Functions
No full textMotivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension d≤3 , we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states. When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model m occupied energy bands by a real-analytic and Zd -periodic family P(k)k∈Rd of orthogonal projections of rank m. A moving orthonormal basis of RanP(k) consisting of real-analytic and Zd -periodic Bloch vectors can be constructed if and only if the first Chern number(s) of P vanish(es). Here we are mainly interested in the topologically obstructed case. First, by dropping the generating condition, we show how to algorithmically construct a collection of m−1 orthonormal, real-analytic, and Zd -periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of m+1 real-analytic and Zd -periodic Bloch vectors which generate RanP(k) . Both algorithms are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. A moving Parseval frame of analytic, Zd -periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of magnetic Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting. Our results are illustrated in crystalline insulators modelled by 2d discrete Hofstadter-like Hamiltonians, but apply to certain continuous models of magnetic Schrödinger operators as well
Symmetry and localization for magnetic Schrödinger operators: Landau levels, Gabor frames and all that
Full text linkWe investigate the relation between broken time-reversal symmetry and localization of the electronic states, in the explicitly tractable case of the Landau model.
We first review, for the reader's convenience, the symmetries of the Landau Hamiltonian and the relation of the latter with the Segal-Bargmann representation of Quantum Mechanics. We then study the localization properties of the Landau eigenstates by applying an abstract version of the Balian-Low Theorem to the operators corresponding to the coordinates of the centre of the cyclotron orbit in the classical theory. Our proof of the Balian-Low Theorem, although based on Battle's main argument, has the advantage of being representation-independent
Parseval Frames of Exponentially Localized Magnetic Wannier Functions
No full textMotivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension d≤ 3 , we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states. When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model m occupied energy bands by a real-analytic and Z d-periodic family {P(k)}k∈Rd of orthogonal projections of rank m. A moving orthonormal basis of RanP(k) consisting of real-analytic and Z d-periodic Bloch vectors can be constructed if and only if the first Chern number(s) of P vanish(es). Here we are mainly interested in the topologically obstructed case. First, by dropping the generating condition, we show how to algorithmically construct a collection of m- 1 orthonormal, real-analytic, and Z d-periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of m+ 1 real-analytic and Z d-periodic Bloch vectors which generate RanP(k). Both algorithms are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. A moving Parseval frame of analytic, Z d-periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of magnetic Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting. Our results are illustrated in crystalline insulators modelled by 2d discrete Hofstadter-like Hamiltonians, but apply to certain continuous models of magnetic Schrödinger operators as well. </p
Beyond Diophantine Wannier diagrams:Gap labelling for Bloch–Landau Hamiltonians
Full text linkIt is well known that, given a 2d purely magnetic Landau Hamiltonian with a constant magnetic field b which generates a magnetic flux ' per unit area, then any spectral island σb consisting of M infinitely degenerate Landau levels carries an integrated density of states Ib D M'. Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any 2d Bloch–Landau operator Hb which also has a bounded Z2-periodic electric potential. Assume that Hb has a spectral island σb which remains isolated from the rest of the spectrum as long as ' lies in a compact interval Œ'1; '2]. Then Ib D c0 C c1' on such intervals, where the constant c0 2 Q while c1 2 Z. The integer c1 is the Chern marker of the spectral projection onto the spectral island σb. This result also implies that the Fermi projection on σb, albeit continuous in b in the strong topology, is nowhere continuous in the norm topology if either c1 ¤ 0 or c1 D 0 and ' is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields.</p
Středa formula for charge and spin currents
Full text linkWe consider a 2-dimensional Bloch-Landau-Pauli Hamiltonian for a spinful electron in a constant magnetic field subject to a periodic background potential. Assuming that the z-component of the spin operator is conserved, we compute the linear response of the associated spin density of states to a small change in the magnetic field, and identify it with the spin Hall conductivity. This response is in the form of a spin Chern marker, which is in general quantized to a half-integer, and to an integer under the further assumption of time-reversal symmetry. Our result is thus a generalization to the context of the quantum spin Hall effect of the well-known formula by Středa, which is formulated instead for charge transport
From Decay of Correlations to Locality and Stability of the Gibbs State
No full textWe show that whenever the Gibbs state of a quantum spin system satisfies decay of correlations, then it is stable, in the sense that local perturbations affect the Gibbs state only locally, and it satisfies local indistinguishability, i.e. it exhibits local insensitivity to system size. These implications hold in any dimension, require only locality of the Hamiltonian, and are based on Lieb-Robinson bounds and on a detailed analysis of the locality properties of the quantum belief propagation for Gibbs states. To demonstrate the versatility of our approach, we explicitly apply our results to several physically relevant models in which the decay of correlations is either known to hold or is proved by us. These include Gibbs states of one-dimensional spin chains with polynomially decaying interactions at any temperature, and high-temperature Gibbs states of quantum spin systems with finite-range interactions in any dimension. We also prove exponential decay of correlations above a threshold temperature for Gibbs states of one-dimensional finite spin chains with translation-invariant and exponentially decaying interactions, and then apply our general results
The Haldane model and its localization dichotomy
Full text linkGapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of |x| 2 diverges. Intermediate regimes are forbidden. Following the lesson of our Maestro, to whom this contribution is gratefully dedicated, we find useful to explain this subtle mathematical phenomenon in the simplest possible model, namely the discrete model proposed by Haldane [10]. We include a pedagogical introduction to the model and we explain its Localization Dichotomy by explicit analytical arguments. We then introduce the reader to the more general, model-independent version of the dichotomy proved in [19]
Localization of generalized Wannier bases implies Chern triviality in non-periodic insulators
Full text linkWe investigate the relation between the localization of generalized Wannier
bases and the topological properties of two-dimensional gapped quantum systems
of independent electrons in a disordered background, including magnetic fields,
as in the case of Chern insulators and quantum Hall systems. We prove that the
existence of a well-localized generalized Wannier basis for the Fermi
projection implies the vanishing of the Chern character, which is proportional
to the Hall conductivity in the linear response regime. Moreover, we state a
localization dichotomy conjecture for general non-periodic gapped quantum
systems.Comment: 37 pages, no figures. The text in this version, which corresponds to
the paper published in Annales Henri Poincar\'e, is the updated version of
the preprint in v1 entitled "Localization implies Chern triviality in
non-periodic insulators". Comparison with v1: title changed, presentation and
main result improved, typos fixed, appendix adde
Finding spectral gaps in quasicrystals
Full text linkWe present an algorithm for reliably and systematically proving the existence
of spectral gaps in Hamiltonians with quasicrystalline order, based on
numerical calculations on finite domains. We apply this algorithm to prove that
the Hofstadter model on the Ammann-Beenker tiling of the plane has spectral
gaps at certain energies, and we are able to prove the existence of a spectral
gap where previous numerical results were inconclusive. Our algorithm is
applicable to more general systems with finite local complexity and eventually
finds all gaps, circumventing an earlier no-go theorem regarding the
computability of spectral gaps for general Hamiltonians.Comment: paper + supplemental materia
- …
