Partition Functions of Non-Abelian Quantum Hall States

122 Pages·2010·1.51 MB·English

DIPARTIMENTO DI FISICA E ASTRONOMIA
UNIVERSITA` DEGLI STUDI DI FIRENZE
Scuola di Dottorato in Scienze
Dottorato di Ricerca in Fisica - XXIII ciclo
SSD FIS/02
Dissertation in Physics
to Obtain the Degree of
Dottore Ricerca
di
in Fisica
Title:
Partition Functions of Non-Abelian
Quantum Hall States
presented by
Giovanni Viola
Supervisor Andrea Cappelli
Coordinator Alessandro Cuccoli
Referees Eddy Ardonne
Domenico Seminara
December 2010

To my Parents

Overview
ThefractionalquantumHalleffectisacollectivequantumphenomenonoftwo-dimensio-
nal electrons placed in a strong perpendicular magnetic field (B 1 10 Tesla) and
∼ −
at very low temperature (T 50 500 mK). It manifests itself in the measure of
∼ −
the transverse (σ ) and longitudinal (σ ) conductances as functions of the magnetic
xy xx
field. TheHallconductanceσ showsthecharacteristic step-like behavior: ateach step
xy
(plateau), itis quantized in rational multiples of thequantum unitof conductance e2/h:
σ = νe2; furthermore, at the plateaus centers the longitudinal resistance vanishes.
xy h
These features are independent on the microscopic details of the samples; in particular,
the quantization of σ is very accurate, with experimental errors of the order 10 8Ω.
xy −
The rational number ν is the filling fraction of occupied one-particle states at the given
value of magnetic field.
The fractional Hall effect is characterized by a non-perturbative gap due to the
Coulomb interaction among electrons in strong fields. The microscopic theory is im-
practicable, besides numerical analysis; thus, effective theories, effective field theories in
particular, are employed to study these systems. The first step was made by Laughlin,
who proposed the trial ground state wave function for fillings ν = 1 = 1, 1, de-
2s+1 3 5 ···
scribing the main physical properties of the Hall effect. One general feature is that the
electrons form a fluid with gapped excitations in the bulk, the so-called incompressible
fluid. In a finite sample, the incompressible fluid gives rise to edge excitations, that
are gapless and responsible for the conduction properties. Therefore the low-energy
effective field theory should describe these edge degrees of freedom.
The Laughlin theory revealed that the excitations are ”anyons”, i.e. quasiparticles
with fractional values of charge and exchange statistics. The latter is a possibility in
bidimensional quantum systems, where the statistics of particles is described by the
braid group instead of the permutation group of higher-dimensional systems. Each fill-
ing fraction corresponds to a different state of matter that is characterized by specific
fractional values of charge and statistics. The braiding of quasiparticles can also occur
for multiplets of degenerate excitations, by means of multi-dimensional unitary trans-
formations; in this case, called non-Abelian fractional statistics, two different braidings
donotcommute. Thecorrespondingnon-Abelianquasiparticles aredegenerate forfixed
positions and given quantum numbers.
Non-Abelian anyons are candidate for implementing topological quantum computa-
tionsaccordingtotheproposalbyKitaevandothers. SinceAnyonsarenon-perturbative
collective excitations, they are less affected by decoherence due to local disturbances.
III
IV
Experimental observation of non-Abelian statistics is a present challenge.
Thetopological fluids, suchasthequantumHallfluidcanbedescribedbytheChern-
Simoneffective fieldtheoryinthelow-energy long-rangelimit. TheChern-Simontheory
is a gauge theory with symmetry algebra g. Witten’s work showed the connection
between the Chern-Simon theory and the rational conformal field theory (RCFT). For
the applications to quantum Hall physics, this gives two results: i) the wave functions
are correlators of the rational conformal field theory with Affine algebras g, and ii) on
spaces with boundaries, e.g. a disc or annulus, the Hilbert space of the edge modes is
given by the same conformal theory. Therefore, the low-energy effective fibeld theories
for Hallfluidsarerational CFT,thatlive onthetwo-dimensional spaceor, inthecase of
edge modes, on the space-time boundary (e.g. the cylinder for a droplet of Hall fluid).
Once the rational CFT has been proposed for wave functions, it also describes the low
energy excitations on the edge.
Conformal field theories describe each anyon by an appropriate field, the fusion be-
tween anyons andtheirbraidingpropertiescoincidewiththefusionrulesoftheoperator
algebra and the monodromy properties of the conformal blocks, respectively. The basic
quasiparticles correspond to the sectors in the theory. In the conformal description the
Verlinde’s formula implies that the braiding matrix, the fusion rules and the quantum
dimensions are function of the modular transformation of the partition function; the
S
latter is a fundamental quantity in the theory.
[
While the Laughlin states are described by the Abelian CFT with U(1) symmetry
(the chiral Luttinger liquid), more involved theories describe the non-Abelian anyons.
[
Moore and Read proposed the U(1) Ising theory for the plateaus at ν = 5/2. In
×
the literature, this approach has been generalized to other states in the second Landau
level, that are supposed to be non-Abelian fluids: their wave functions are conformal
[ [
blocks of rational conformal field theories U(1) g. Here U(1) describes the charge part
×
of the theory and the other factor g is the neutral part of the edge modes characterized
by an Affine symmetry algebra g or a coset g/bh. The most relevant proposals are:
the Read-Rezayi and anti-Read-Rebzayi states, the Bonderson-Slingerland hierarchy, the
Wen non-Abelian fluid and the nobn-Abelian spbin-bsinglet state. These theories describe
different filling fractions ν = 2+ p′ and have been supported by numerical simulations.
pˆ
This thesis concerns the study of edge excitations of Hall states, in particular their
modular invariant partition functions. This quantity provides a complete definition of
the Hilbert space and its decomposition into sectors; moreover, the fusion rules, the
selection rules for the fields, are built in. In this thesis, we obtain a straightforward
method to derive the modular invariant partition functions for edge excitations in the
annulus geometry. Starting from the two choices of: i) the conformal field theory g of
theneutralpartofexcitations, andii)Abelianfieldrepresentingtheelectron, thecharge
and statistics of all excitations can be self-consistently found without further physbical
input. The annulus geometry is chosen because it enjoys the modular symmetry where
the edge modesliving on two circles realize the holomorphicand anti-holomorphic parts
of the theory. The disk and Hall bar geometries can also be obtained by taking limits
of the annulus.
V
The partition function is derived by considering the role of the electron field in the
Hall fluid: this is a spin one-half Abelian field, that must possesses integer exchange
statisticswithalltheotherexcitationsinthefluid. Oncetheseconditionsareconsidered,
the derivation of the partition function is unique. We observe that these conditions are
also required by invariance under modular transformations of the partition function,
that have clear physical meaning for Hall systems.
In the last part of the thesis, we compute experimental signatures of non-Abelian
statistics by using the partition function. We describe the Coulomb Blockade current
peaks, both at zero and non-zero temperatures (determining the low-lying energy spec-
trum)andstudythethermopower, theratio ofelectric andthermoelectric conductances
(measuring the so-called quantum dimension). We show that the derivation of the peak
patterns from the partition function is very simple, both at zero and finite tempera-
tures. All these experimental predictions are worked out in details for the prominent
non-Abelian states in the second Landau level. These experimental quantities are al-
ternative to the interference effects that would directly test the braiding properties of
quasiparticles.
The thesis is organized as follows: in chapter 1 we recall some general aspects of
planar physics: quantum Hall effect, Chern-Simons theory and quantum statistics in
two dimensions. The chapter 2 introduces the reader to the conformal field theory
and its connections with Chern-Simons theory, anyons model and the Hall fluids. In
chapter 3 we discussthe derivation of the partition function for Hall fluidsin thesecond
Landau level, the technical details are reported in appendix B. In the chapter 4, the
physical applications of the partition function are analyzed. Appendix A also reports
the derivation of the partition function and the Coulomb peak patterns for the Jain
states, that are Abelian states in the first Landau level.

The work of this thesis is based on the papers:
A. Cappelli, G. Viola and G. R. Zemba,
Chiral Partition Functions of Quantum Hall Droplets,
Annals Phys, 325 (2010) 465,
arXiv:0909.3588v1 [cond-mat.mes-hall]
A. Cappelli and G. Viola
Partition Functions of Non-Abelian Quantum Hall States,
arXiv:1007.1732v2 [cond-mat.mes-hall];
submitted to J. Phys. A.
Other author’s publications:
S. Caracciolo, F. Palumbo and G. Viola,
Bogoliubov transformations and fermion condensates in lattice field theories,
Annals Phys. 324, (2009) 584,
[arXiv:0808.1110 [hep-lat]].