Adiabatic currents for interacting electrons on a lattice

We prove an adiabatic theorem for general densities of observables that are sums of local terms in finite systems of interacting fermions, without periodicity assumptions on the Hamiltonian and with error estimates that are uniform in the size of the system. Our result provides an adiabatic expansion to all orders, in particular, also for initial data that lie in eigenspaces of degenerate eigenvalues. Our proof is based on ideas from a recent work of Bachmann et al. who proved an adiabatic theorem for interacting spin systems. As one important application of this adiabatic theorem, we provide the first rigorous derivation of the so-called linear response formula for the current density induced by an adiabatic change of the Hamiltonian of a system of interacting fermions in a ground state, with error estimates uniform in the system size. We also discuss the application to quantum Hall systems.Comment: 46 pages; v1->v2: typos corrected, references added, Remark 4 after Thm 2 slightly reworded, v2->v3: major revision of the presentation of the result, 3 figures adde

Topological invariants of eigenvalue intersections and decrease of Wannier functions in graphene

We investigate the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the multilayer case. Since the decrease of the Wannier functions is characterised by the structure of the Bloch eigenspaces around the Dirac points, we introduce a geometric invariant of the family of eigenspaces, baptised eigenspace vorticity. We compare it with the pseudospin winding number. For every value $n \in Z$ of the eigenspace vorticity, we exhibit a canonical model for the local topology of the eigenspaces. With the help of these canonical models, we show that the single band Wannier function $w$ satisfies $|w(x)| \leq \mathrm{const} |x|^{- 2}$ as $|x| \rightarrow \infty$, both in monolayer and bilayer graphene.Comment: 54 pages, 4 figures. Version 2: Section 1.0 added; improved results on the decay rate of Wannier functions in graphene (Th. 4.3 and Prop. 4.6). Version 3: final version, to appear in JSP. New in V3: previous Sections 3.1 and 3.2 are now Section 2.2; Lemma 2.4 modified (previous statement was not correct); major modifications to Section 2.3; Assumption 4.1(v) on the Hamiltonian change

Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry

We describe some applications of group- and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e.g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, called the Bloch bundle. The role of additional $\mathbb{Z}_2$-symmetries, as time-reversal and space-reflection symmetry, is discussed, showing how time-reversal symmetry implies the triviality of the Bloch bundle, both in the bosonic and in the fermionic case. Moreover, the same $\mathbb{Z}_2$-symmetry allows to define a finer notion of isomorphism and, consequently, to define new topological invariants, which agree with the indices introduced by Fu, Kane and Mele in the context of topological insulators.Comment: Contribution to the proceedings of the conference "SPT2014 - Symmetry and Perturbation Theory", Cala Gonone, Italy (2014). Keywords: Periodic Schr\"{o}dinger operators, composite Wannier functions, Bloch bundle, Bloch frames, time-reversal symmetry, space-reflection symmetry, invariants of topological insulator

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.Comment: 29 pages. Version 2: minor corrections of misprints, corrected proofs of Theorems 2.4 and 2.9, added references. Accepted for publication in Annales Henri Poicar\'

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 $\varphi$ per unit area, then any spectral island $\sigma_b$ consisting of $M$ infinitely degenerate Landau levels carries an integrated density of states $\mathcal{I}_b=M \varphi$. 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 $\sigma_b$ which remains isolated from the rest of the spectrum as long as $\varphi$ lies in a compact interval $[\varphi_1,\varphi_2]$. Then $\mathcal{I}_b=c_0+c_1\varphi$ 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 $\sigma_b$. This result also implies that the Fermi projection on $\sigma_b$, 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 $\varphi$ 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.Comment: 20 pages, no figures. Appendix C added. Final version accepted for publication in Journal of the European Mathematical Societ

Geometric phases in graphene and topological insulators

This thesis collects three of the publications that the candidate produced during his Ph.D. studies. They all focus on geometric phases in solid state physics. We first study topological phases of 2-dimensional periodic quantum systems, in absence of a spectral gap, like e.g. (multilayer) graphene. A topological invariant n_v in Z, baptized eigenspace vorticity, is attached to any intersection of the energy bands, and characterizes the local topology of the eigenprojectors around that intersection. With the help of explicit models, each associated to a value of n_v in Z, we are able to extract the decay at infinity of the single-band Wannier function w in mono- and bilayer graphene, obtaining |w(x)| <= const |x|^{-2} as |x| tends to infinity. Next, we investigate gapped periodic quantum systems, in presence of time-reversal symmetry. When the time-reversal operator Theta is of bosonic type, i.e. it satisfies Theta^2 = 1, we provide an explicit algorithm to construct a frame of smooth, periodic and time-reversal symmetric (quasi-)Bloch functions, or equivalently a frame of almost-exponentially localized, real-valued (composite) Wannier functions, in dimension d <= 3. In the case instead of a fermionic time-reversal operator, satisfying Theta^2 = -1, we show that the existence of such a Bloch frame is in general topologically obstructed in dimension d=2 and d=3. This obstruction is encoded in Z_2-valued topological invariants, which agree with the ones proposed in the solid state literature by Fu, Kane and Mele

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 Schr\"{o}dinger operators as well.Comment: 40 pages. Improved exposition and minor corrections. Final version matches published paper on Commun. Math. Phy

The Haldane model and its localization dichotomy

Gapped 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]

A Z2 invariant for chiral and particle–hole symmetric topological chains

We define a Z2-valued topological and gauge invariant associated with any one-dimensional, translation-invariant topological insulator that satisfies either particle–hole symmetry or chiral symmetry. The invariant can be computed from the Berry phase associated with a suitable basis of Bloch functions that is compatible with the symmetries. We compute the invariant in the Su–Schrieffer–Heeger model for chiral symmetric insulators and in the Kitaev model for particle–hole symmetric insulators. We show that in both cases, the Z2 invariant predicts the existence of zero-energy boundary states for the corresponding truncated models