arXiv subjects

Domenico Monaco

Publications and source records attributed to Domenico Monaco.

30 records · Page 2Linked to original sources

arXiv2019

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 $|\mathbf{x}|^2$ 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 (Phys. Rev. Lett. 61, 2017 (1988)). 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 (Commun. Math. Phys. 359, 61-100 (2018)), and finally we announce further generalizations to non-periodic models.

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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.

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arXiv2018

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.

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arXiv2017

Localised Wannier functions in metallic systems

The existence and construction of exponentially localised Wannier functions for insulators is a well-studied problem. In comparison, the case of metallic systems has been much less explored, even though localised Wannier functions constitute an important and widely used tool for the numerical band interpolation of metallic condensed matter systems. In this paper we prove that, under generic conditions, $N$ energy bands of a metal can be exactly represented by $N+1$ Wannier functions decaying faster than any polynomial. We also show that, in general, the lack of a spectral gap does not allow for exponential decay.

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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.

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arXiv2017

Chern and Fu-Kane-Mele invariants as topological obstructions

The use of topological invariants to describe geometric phases of quantum matter has become an essential tool in modern solid state physics. The first instance of this paradigmatic trend can be traced to the study of the quantum Hall effect, in which the Chern number underlies the quantization of the transverse Hall conductivity. More recently, in the framework of time-reversal symmetric topological insulators and quantum spin Hall systems, a new topological classification has been proposed by Fu, Kane and Mele, where the label takes value in $\mathbb Z_2$. We illustrate how both the Chern number $c \in \mathbb Z$ and the Fu-Kane-Mele invariant $δ\in \mathbb Z_2$ of 2-dimensional topological insulators can be characterized as topological obstructions. Indeed, $c$ quantifies the obstruction to the existence of a frame of Bloch states for the crystal which is both continuous and periodic with respect to the crystal momentum. Instead, $δ$ measures the possibility to impose a further time-reversal symmetry constraint on the Bloch frame.

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arXiv2017

Gauge-theoretic invariants for topological insulators: A bridge between Berry, Wess-Zumino, and Fu-Kane-Mele

We establish a connection between two recently-proposed approaches to the understanding of the geometric origin of the Fu-Kane-Mele invariant $\mathrm{FKM} \in \mathbb{Z}_2$, arising in the context of 2-dimensional time-reversal symmetric topological insulators. On the one hand, the $\mathbb{Z}_2$ invariant can be formulated in terms of the Berry connection and the Berry curvature of the Bloch bundle of occupied states over the Brillouin torus. On the other, using techniques from the theory of bundle gerbes it is possible to provide an expression for $\mathrm{FKM}$ containing the square root of the Wess-Zumino amplitude for a certain $U(N)$-valued field over the Brillouin torus. We link the two formulas by showing directly the equality between the above mentioned Wess-Zumino amplitude and the Berry phase, as well as between their square roots. An essential tool of independent interest is an equivariant version of the adjoint Polyakov-Wiegmann formula for fields $\mathbb{T}^2 \to U(N)$, of which we provide a proof employing only basic homotopy theory and circumventing the language of bundle gerbes.

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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.

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arXiv2016

$\mathbb{Z}_2$ invariants of topological insulators as geometric obstructions

We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to -1. We investigate the existence of periodic and time-reversal invariant Bloch frames in dimensions 2 and 3. In 2d, the obstruction to the existence of such a frame is shown to be encoded in a $\mathbb{Z}_2$-valued topological invariant, which can be computed by a simple algorithm. We prove that the latter agrees with the Fu-Kane index. In 3d, instead, four $\mathbb{Z}_2$ invariants emerge from the construction, again related to the Fu-Kane-Mele indices. When no topological obstruction is present, we provide a constructive algorithm yielding explicitly a periodic and time-reversal invariant Bloch frame. The result is formulated in an abstract setting, so that it applies both to discrete models and to continuous ones.

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arXiv2016

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.

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arXiv2015

Construction of real-valued localized composite Wannier functions for insulators

We consider a real periodic Schrödinger operator and a physically relevant family of $m \geq 1$ Bloch bands, separated by a gap from the rest of the spectrum, and we investigate the localization properties of the corresponding composite Wannier functions. To this aim, we show that in dimension $d \leq 3$ there exists a global frame consisting of smooth quasi-Bloch functions which are both periodic and time-reversal symmetric. Aiming to applications in computational physics, we provide a constructive algorithm to obtain such a Bloch frame. The construction yields the existence of a basis of composite Wannier functions which are real-valued and almost-exponentially localized. The proof of the main result exploits only the fundamental symmetries of the projector on the relevant bands, allowing applications, beyond the model specified above, to a broad range of gapped periodic quantum systems with a time-reversal symmetry of bosonic type.

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arXiv2014

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.

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