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Horia D. Cornean

Publications and source records attributed to Horia D. Cornean.

At least 37 records · Page 2Linked to original sources
arXiv2021

Spectral analysis near a Dirac type crossing in a weak non-constant magnetic field

This is the last paper in a series of three in which we have studied the Peierls substitution in the case of a weak magnetic field. Here we deal with two $2d$ Bloch eigenvalues which have a conical crossing. It turns out that in the presence of an almost constant weak magnetic field, the spectrum near the crossing develops gaps which remind of the Landau levels of an effective mass-less magnetic Dirac operator.

math-ph↗
arXiv2020

Beyond Diophantine Wannier diagrams: Gap labelling for Bloch-Landau Hamiltonians

It is well known that, given a $2d$ purely magnetic Landau Hamiltonian with a constant magnetic field $b$ which generates a magnetic flux $φ$ per unit area, then any spectral island $σ_b$ consisting of $M$ infinitely degenerate Landau levels carries an integrated density of states $\mathcal{I}_b=M φ$. 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 $H_b$ which also has a bounded $\mathbb{Z}^2$-periodic electric potential. Assume that $H_b$ has a spectral island $σ_b$ which remains isolated from the rest of the spectrum as long as $φ$ lies in a compact interval $[φ_1,φ_2]$. Then $\mathcal{I}_b=c_0+c_1φ$ on such intervals, where the constant $c_0\in \mathbb{Q}$ while $c_1\in \mathbb{Z}$. The integer $c_1$ is the Chern marker of the spectral projection onto the spectral island $σ_b$. This result also implies that the Fermi projection on $σ_b$, albeit continuous in $b$ in the strong topology, is nowhere continuous in the norm topology if either $c_1\ne0$ or $c_1=0$ and $φ$ 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.

math-ph↗
arXiv2019

Parseval frames of exponentially localized magnetic Wannier functions

Motivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension $d \le 3$, 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 $\mathbb Z^{d}$-periodic family $\{P({\bf k})\}_{{\bf k} \in \mathbb R^{d}}$ of orthogonal projections of rank $m$. A moving orthonormal basis of $\mathrm{Ran} P({\bf k})$ consisting of real-analytic and $\mathbb Z^d$-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 periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of $m+1$ real-analytic and periodic Bloch vectors which generate $\mathrm{Ran} P({\bf k})$. 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, 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.

math-ph↗
arXiv2018

Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

First, we reconsider the magnetic pseudodifferential calculus and show that for a large class of non-decaying symbols, their corresponding magnetic pseudodifferential operators can be represented, up to a global gauge transform, as generalized Hofstadter-like, bounded matrices. As a by-product, we prove a Calderón-Vaillancourt type result. Second, we make use of this matrix representation and prove sharp results on the spectrum location when the magnetic field strength $b$ varies. Namely, when the operators are self-adjoint, we show that their spectrum (as a set) is at least $1/2$-Hölder continuous with respect to $b$ in the Hausdorff distance. Third, when the magnetic perturbation comes from a constant magnetic field we show that their spectral edges are Lipschitz continuous in $b$. The same Lipschitz continuity holds true for spectral gap edges as long as the gaps do not close.

math.AP↗
arXiv2018

Localization for gapped Dirac Hamiltionians with random perturbations: Application to graphene antidot lattices

In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator $H_0 := D_S + V_0$ is the sum of a Dirac-like operator $D_S$ plus a periodic matrix valued potential $V_0$, and is assumed to have an open gap. The random potential $V_ω$ is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family $H_ω:= H_0 + V_ω$ is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.

math-ph↗
arXiv2018

Resolvent convergence to Dirac operators on planar domains

Consider a Dirac operator defined on the whole plane with a mass term of size m supported outside a domain Omega. We give a simple proof for the norm resolvent convergence, as m goes to infinity, of this operator to a Dirac operator defined on Omega with infinite mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials.

math-ph↗
arXiv2018

A local directional growth estimate of the resolvent norm

We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm grows at least quadratically with the distance from this point. This provides a new proof not using the maximum principle that the resolvent norm of the considered class cannot have local maxima. Finally, we give new criteria for the existence of local non-degenerate minima of the resolvent norm and provide examples of (un)bounded non-normal operators having this property.

math.SP↗
arXiv2018

Two dimensional Schr{\" o}dinger operators with point interactions: threshold expansions, zero modes and $L^p$-boundedness of wave operators

We study the threshold behaviour of two dimensional Schr{\" o}dinger operators with finitely many local point interactions. We show that the resolvent can either be continuously extended up to the threshold, in which case we say that the operator is of regular type, or it has singularities associated with $s$ or p-wave resonances or even with an embedded eigenvalue at zero, for whose existence we give necessary and sufficient conditions. An embedded eigenvalue at zero may appear only if we have at least three centres. When the operator is of regular type we prove that the wave operators are bounded in $L^p(\R^2)$ for all $1<p<\infty$. With a single center we always are in the regular type case.

math.SP↗
arXiv2017

Peierls' substitution for low lying spectral energy windows

We consider a $2d$ magnetic Schrödinger operator perturbed by a weak magnetic field which slowly varies around a positive mean. In a previous paper we proved the appearance of a `Landau type' structure of spectral islands at the bottom of the spectrum, under the hypothesis that the lowest Bloch eigenvalue of the unperturbed operator remained simple on the whole Brillouin zone, even though its range may overlap with the range of the second eigenvalue. We also assumed that the first Bloch spectral projection was smooth and had a zero Chern number. In this paper we extend our previous results to the only two remaining possibilities: either the first Bloch eigenvalue remains isolated while its corresponding spectral projection has a non-zero Chern number, or the first two Bloch eigenvalues cross each other.

math-ph↗
arXiv2017

On the construction of Wannier functions in topological insulators: the 3D case

We investigate the possibility of constructing exponentially localized composite Wannier bases, or equivalently smooth periodic Bloch frames, for 3-dimensional time-reversal symmetric topological insulators, both of bosonic and of fermionic type, so that the bases in question are also compatible with time-reversal symmetry. This problem is translated in the study, of independent interest, of homotopy classes of continuous, periodic, and time-reversal symmetric families of unitary matrices. We identify three $\mathbb{Z}_2$-valued complete invariants for these homotopy classes. When these invariants vanish, we provide an algorithm which constructs a "multi-step" logarithm that is employed to continuously deform the given family into a constant one, identically equal to the identity matrix. This algorithm leads to a constructive procedure to produce the composite Wannier bases mentioned above.

math-ph↗
arXiv2017

On the adiabatic theorem when eigenvalues dive into the continuum

We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.

math-ph↗
arXiv2017

On the existence of impurity bound excitons in one-dimensional systems with zero range interactions

We consider a three-body one-dimensional Schrödinger operator with zero range potentials, which models a positive impurity with charge $κ> 0$ interacting with an exciton. We study the existence of discrete eigenvalues as $κ$ is varied. On one hand, we show that for sufficiently small $κ$ there exists a unique bound state whose binding energy behaves like $κ^4$, and we explicitly compute its leading coefficient. On the other hand, if $κ$ is larger than some critical value then the system has no bound states.

math-ph↗
arXiv2017

Low lying spectral gaps induced by slowly varying magnetic fields

We consider a periodic Schrödinger operator in two dimensions perturbed by a weak magnetic field whose intensity slowly varies around a positive mean. We show in great generality that the bottom of the spectrum of the corresponding magnetic Schrödinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure. First, we construct an effective Hofstadter-like magnetic matrix which accurately describes the low lying spectrum of the full operator. The construction of this effective magnetic matrix does not require a gap in the spectrum of the non-magnetic operator, only that the first and the second Bloch eigenvalues do not cross but their ranges might overlap. The crossing case is more difficult and will be considered elsewhere. Second, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted to slowly varying magnetic fields. As an application, we prove in the overlapping case the appearance of spectral islands separated by gaps.

math.SP↗
arXiv2017

Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators

We provide a constructive proof of exponentially localized Wannier functions and related Bloch frames in 1- and 2-dimensional time-reversal symmetric (TRS) topological insulators. The construction is formulated in terms of periodic TRS families of projectors (corresponding, in applications, to the eigenprojectors on an arbitrary number of relevant energy bands), and is thus model-independent. The possibility to enforce also a TRS constraint on the frame is investigated. This leads to a topological obstruction in dimension 2, related to $\mathbb{Z}_2$ topological phases. We review several proposals for $\mathbb{Z}_2$ indices that distinguish these topological phases, including the ones by Fu--Kane [Phys. Rev. B 74 (2006), 195312], Prodan [Phys. Rev. B 83 (2011), 235115], Graf--Porta [Commun. Math. Phys. 324 (2013), 851] and Fiorenza--Monaco--Panati [Commun. Math. Phys., in press]. We show that all these formulations are equivalent. In particular, this allows to prove a geometric formula for the the $\mathbb{Z}_2$ invariant of 2-dimensional TRS topological insulators, originally indicated in [Phys. Rev. B 74 (2006), 195312], which expresses it in terms of the Berry connection and the Berry curvature.

math-ph↗
arXiv2016

On the construction of composite Wannier functions

We give a constructive proof for the existence of an $N$-dimensional Bloch basis which is both smooth (real analytic) and periodic with respect to its $d$-dimensional quasi-momenta, when $1\leq d\leq 2$ and $N\geq 1$. The constructed Bloch basis is conjugation symmetric when the underlying projection has this symmetry, hence the corresponding exponentially localized composite Wannier functions are real. In the second part of the paper we show that by adding a weak, globally bounded but not necessarily constant magnetic field, the existence of a localized basis is preserved.

math-ph↗
arXiv2015

Peierls substitution and magnetic pseudo-differential calculus

We revisit the celebrated Peierls-Onsager substitution employing the magnetic pseudo-differential calculus for weak magnetic fields with no spatial decay conditions, when the non-magnetic symbols have a certain spatial periodicity. We show in great generality that the symbol of the magnetic band Hamiltonian admits a convergent expansion. Moreover, if the non-magnetic band Hamiltonian admits a localized composite Wannier basis, we show that the magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix. In addition, if the magnetic field perturbation is slowly variable, then the spectrum of this matrix is close to the spectrum of a Weyl quantized, minimally coupled symbol.

math-ph↗
arXiv2014

Spectral edge regularity of magnetic Hamiltonians

We analyse the spectral edge regularity of a large class of magnetic Hamiltonians when the perturbation is generated by a globally bounded magnetic field. We can prove Lipschitz regularity of spectral edges if the magnetic field perturbation is either constant or slowly variable. We also recover an older result by G. Nenciu who proved Lipschitz regularity up to a logarithmic factor for general globally bounded magnetic field perturbations.

math.SP↗