224 research outputs found
Non-equilibrium almost-stationary states and linear response for gapped quantum systems
We prove the validity of linear response theory at zero temperature for
perturbations of gapped Hamiltonians describing interacting fermions on a
lattice. As an essential innovation, our result requires the spectral gap
assumption only for the unperturbed Hamiltonian and applies to a large class of
perturbations that close the spectral gap. Moreover, we prove formulas also for
higher order response coefficients.
Our justification of linear response theory is based on a novel extension of
the adiabatic theorem to situations where a time-dependent perturbation closes
the gap. According to the standard version of the adiabatic theorem, when the
perturbation is switched on adiabatically and as long as the gap does not
close, the initial ground state evolves into the ground state of the perturbed
operator. The new adiabatic theorem states that for perturbations that are
either slowly varying potentials or small quasi-local operators, once the
perturbation closes the gap, the adiabatic evolution follows non-equilibrium
almost-stationary states (NEASS) that we construct explicitly.Comment: v1->v2 section 4 on linear response added, presentation partly
reworked. v2->v3 slightly stronger statements for "fast" switching. Final
version as to appear in CM
Semiclassical approximations for adiabatic slow-fast systems
In this letter we give a systematic derivation and justification of the
semiclassical model for the slow degrees of freedom in adiabatic slow-fast
systems first found by Littlejohn and Flynn [5]. The classical Hamiltonian
obtains a correction due to the variation of the adiabatic subspaces and the
symplectic form is modified by the curvature of the Berry connection. We show
that this classical system can be used to approximate quantum mechanical
expectations and the time-evolution of operators also in sub-leading order in
the combined adiabatic and semiclassical limit. In solid state physics the
corresponding semiclassical description of Bloch electrons has led to
substantial progress during the recent years, see [1]. Here, as an
illustration, we show how to compute the Piezo-current arising from a slow
deformation of a crystal in the presence of a constant magnetic field
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
Adiabatic Decoupling and Time-Dependent Born-Oppenheimer Theory
We reconsider the time-dependent Born-Oppenheimer theory with the goal to
carefully separate between the adiabatic decoupling of a given group of energy
bands from their orthogonal subspace and the semiclassics within the energy
bands. Band crossings are allowed and our results are local in the sense that
they hold up to the first time when a band crossing is encountered. The
adiabatic decoupling leads to an effective Schroedinger equation for the
nuclei, including contributions from the Berry connection.Comment: Revised version. 19 pages, 2 figure
Generalised Quantum Waveguides
We study general quantum waveguides and establish explicit effective
Hamiltonians for the Laplacian on these spaces. A conventional quantum
waveguide is an -tubular neighbourhood of a curve in
and the object of interest is the Dirichlet Laplacian on this
tube in the asymptotic limit . We generalise this by
considering fibre bundles over a -dimensional submanifold
with fibres diffeomorphic to ,
whose total space is embedded into an -neighbourhood of . From
this point of view takes the role of the curve and that of the
disc-shaped cross-section of a conventional quantum waveguide. Our approach
allows, among other things, for waveguides whose cross-sections are
deformed along and also the study of the Laplacian on the boundaries of
such waveguides. By applying recent results on the adiabatic limit of
Schr\"odinger operators on fibre bundles we show, in particular, that for small
energies the dynamics and the spectrum of the Laplacian on are reflected by
the adiabatic approximation associated to the ground state band of the normal
Laplacian. We give explicit formulas for the according effective operator on
in various scenarios, thereby improving and extending many of the
known results on quantum waveguides and quantum layers in
Constrained Quantum Systems as an Adiabatic Problem
We derive the effective Hamiltonian for a quantum system constrained to a
submanifold (the constraint manifold) of configuration space (the ambient
space) in the asymptotic limit where the restoring forces tend to infinity. In
contrast to earlier works we consider at the same time the effects of
variations in the constraining potential and the effects of interior and
exterior geometry which appear at different energy scales and thus provide, for
the first time, a complete picture ranging over all interesting energy scales.
We show that the leading order contribution to the effective Hamiltonian is the
adiabatic potential given by an eigenvalue of the confining potential
well-known in the context of adiabatic quantum wave guides. At next to leading
order we see effects from the variation of the normal eigenfunctions in form of
a Berry connection. We apply our results to quantum wave guides and provide an
example for the occurrence of a topological phase due to the geometry of a
quantum wave circuit, i.e. a closed quantum wave guide.Comment: 19 pages, 4 figure
Peierls substitution for magnetic Bloch bands
We consider the Schr\"odinger operator in two dimensions with a periodic
potential and a strong constant magnetic field perturbed by slowly varying
non-periodic scalar and vector potentials, and , for . For each isolated family of magnetic Bloch bands we
derive an effective Hamiltonian that is unitarily equivalent to the restriction
of the Schr\"odinger operator to a corresponding almost invariant subspace. At
leading order, our effective Hamiltonian can be interpreted as the Peierls
substitution Hamiltonian widely used in physics for non-magnetic Bloch bands.
However, while for non-magnetic Bloch bands the corresponding result is well
understood, for magnetic Bloch bands it is not clear how to even define a
Peierls substitution Hamiltonian beyond a formal expression. The source of the
difficulty is a topological obstruction: magnetic Bloch bundles are generically
not trivializable. As a consequence, Peierls substitution Hamiltonians for
magnetic Bloch bands turn out to be pseudodifferential operators acting on
sections of non-trivial vector bundles over a two-torus, the reduced Brillouin
zone. Part of our contribution is the construction of a suitable Weyl calculus
for such pseudos. As an application of our results we construct a new family of
canonical one-band Hamiltonians for magnetic Bloch bands with
Chern number that generalizes the Hofstadter model
for a single non-magnetic Bloch band. It turns out
that is isospectral to for any
and all spectra agree with the Hofstadter spectrum depicted in his famous black
and white butterfly. However, the resulting Chern numbers of subbands,
corresponding to Hall conductivities, depend on and , and thus the
models lead to different colored butterflies.Comment: 39 pages, 4 figures. Final version to appear in Analysis & PD
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