1,720,995 research outputs found
Geometric phases in graphene and topological insulators
Full text linkThis 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
Adiabatic currents for interacting fermions on a lattice
Full text linkWe 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 [6], where Bachmann et al. proved an adiabatic theorem for interacting spin systems. As one important application of this adiabatic theorem, we provide the first rigorous derivation of the adiabatic 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
KK-theory, gauge theory and topological phases
Full text linkThe history of science is full of moments when physics and mathematics both benefited
from mutual interaction, with the early twentieth century providing us with two big examples. On one hand, Einstein’s theory of gravitation could not have been developed without
the work of Riemann on the geometry of manifolds. On the other hand, the advent of
quantum mechanics fostered the development of new mathematics especially in operator
algebras. In line with this spirit of interdisciplinarity, the school and workshop ‘KK-theory,
Gauge Theory and Topological Phases’ took place from 27 February to 10 March 2017 at
the Lorentz Center in Leiden. Francesca Arici and Domenico Monaco report about this event
Chern and Fu–Kane–Mele Invariants as Topological Obstructions
Full text linkThe 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 Z2.
We illustrate how both the Chern number c 2 Z and the Fu–Kane–Mele invariant
ı 2 Z2 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, ı measures the possibility to impose a further time-reversal
symmetry constraint on the Bloch frame
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
From charge to spin: analogies and differences in quantum transport coefficients
Full text linkWe review some recent results from the mathematical theory of transport of
charge and spin in gapped crystalline quantum systems. The emphasis will be in
transport coefficients like conductivities and conductances. As for the former,
those are computed as appropriate expectations of current operators in a
non-equilibrium almost-stationary state (NEASS), which arises from the
perturbation of an equilibrium state by an external electric field. While for
charge transport the usual double-commutator Kubo formula is recovered (also
beyond linear response), we obtain formulas for appropriately-defined spin
conductivities which are still explicit but more involved. Certain "Kubo-like"
terms in these formulas are also shown to agree with corresponding
contributions to the spin conductance. In addition to that, we employ similar
techniques to show a new result, namely that even in systems with non-conserved
spin there is no generation of spin torque, that is the spin torque operator
has an expectation in the NEASS which vanishes faster than any power of the
intensity of the perturbing field.Comment: 12 pages; minor changes, updated references; Contribution to the
proceedings of ICMP 202
Topology vs localization in synthetic dimensions
Full text linkMotivated by recent developments in quantum simulation of synthetic
dimensions, e.g. in optical lattices of ultracold atoms, we discuss here
-dimensional periodic, gapped quantum systems for , with focus on
the topology of the occupied energy states. We perform this analysis by asking
whether the spectral subspace below the gap can be spanned by smooth and
periodic Bloch functions, corresponding to localized Wannier functions in
position space. By constructing these Bloch functions inductively in the
dimension, we show that if they are required to be orthonormal then in general
their existence is obstructed by the first two Chern classes of the underlying
Bloch bundle, with the second Chern class characterizing in particular the
4-dimensional situation. If the orthonormality constraint is relaxed, we show
how occupied energy bands can be spanned by a Parseval frame comprising at
most Bloch functions.Comment: 26 pages, 3 figures. Submission for the JMP special topic
'Mathematical aspects of topological phases'. Proceedings volume for the
conference 'Topological phases of matter', Leysin (CH), July 25-28, 2021.
v1-->v2: small changes, matches published versio
Purely linear response of the quantum Hall current to space-adiabatic perturbations
Full text linkUsing recently developed tools from space-adiabatic perturbation theory, in
particular the construction of a non-equilibrium almost stationary state, we
give a new proof that the Kubo formula for the Hall conductivity remains valid
beyond the linear response regime. In particular, we prove that, in quantum
Hall systems and Chern insulators, the transverse response current is quantized
up to any order in the strength of the inducing electric field. The latter is
introduced as a perturbation to a periodic, spectrally gapped equilibrium
Hamiltonian by means of a linear potential; existing proofs of the exactness of
Kubo formula rely instead on a time-dependent magnetic potential. The result
applies to both continuum and discrete crystalline systems modelling the
quantum (anomalous) Hall effect.Comment: 18 page
A invariant for chiral and particle-hole symmetric topological chains
Full text linkWe define a -valued topological and gauge invariant associated
to any 1-dimensional, translation-invariant topological insulator which
satisfies either particle-hole symmetry or chiral symmetry. The invariant can
be computed from the Berry phase associated to a suitable basis of Bloch
functions which 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 invariant predicts the existence of zero-energy boundary
states for the corresponding truncated models.Comment: REVTeX class, 20 pages, no figures; to appear in the proceedings
volume for the workshop "Learning from Insulators: New Trends in the Study of
Conductivity of Metals", 9-13 August 2021, Lorentz Center (Leiden, NL
On the construction of Wannier functions in topological insulators: the 3D case
Full text linkWe investigate the possibility of constructing exponentially localized
composite Wannier bases, or equivalently smooth periodic
Bloch frames, for three-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 Z2-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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