Contents - Max-Planck-Institut für Physik komplexer Systeme

2.12 Entanglement Analysis of Fractional Quantum Hall States on Torus Geometries
EMIL J. BERGHOLTZ, MASUDUL HAQUE, ANDREAS M. LÄUCHLI
Introduction. The description of condensed matter
phases using entanglement measures, borrowed from the
field of quantum information theory, has led to a large
body of work [1]. One situation where this interdisciplinary
work has been particularly fruitful is in the
study of topologically ordered states [2, 3]. Fractional
quantum Hall (FQH) states of two-dimensional electrons
in a magnetic field stand out as the only experimentally
realized topologically ordered phases. These
states have recently received renewed intense attention
due to quantum computation proposals based on their
topological properties. An intriguing feature of FQH
states is that their edges have gapless modes, described
by chiral Luttinger liquids.
Here we summarize studies of the entanglement entropy
and the entanglement sepctrum in the most prominent
FQH states, namely the Laughlin states, placed on a
toroidal geometry [4, 5].
Entanglement measures, topological order, and edge
modes. A prominent measure of entanglement is the
von Neumann entropy of entanglement, SA, measuring
the entanglement between a block (A) and the rest
(B) of a many-particle system in a pure state. The entanglement
entropy SA = −tr [ρA lnρA] is defined in
terms of the reduced density matrix, ρA = trB ρ, obtained
by tracing out B degrees of freedom from the
system density matrix ρ = |ψ〉〈ψ|, with |ψ〉 denoting a
ground state wave function.
In one dimension the scaling behavior of the block entanglement
entropy is well understood. In two dimensions
(2D), no such generic classification exists. However,
for topologically ordered states in two dimensions,
the scaling of SA contains topological information
about the state:
SA = αL − nγ + O(1/L), (1)
where L is the block boundary length, γ characterizes
the topological field theory describing the state [2],
while n counts the number of disconnected components
of the boundary. The value of γ is related to the
“quantum dimensions” of the quasiparticle types of the
theory, γ = ln D, where D is the total quantum dimension.
For Laughlin states at filling ν = 1/m, γ = 1
2 lnm.
It is also useful to study the complete spectrum of the
reduced density matrix, obtained by a Schmidt decomposition
of the ground state:
|ψ〉 =
i
e −ξi/2 |ψ A i 〉 ⊗ |ψ B i 〉.
The {ξi} form the entanglement spectrum (ES). The states
|ψA i 〉 (|ψB i 〉) form an orthonormal basis for the subsystem
A (B).
Very recently, ES studies have been used [3] to probe
edge modes of FQH states. The entanglement between
two partitions of an edgeless wavefunction may seem
at first sight unrelated to edge physics. However, the
entanglement spectrum can be regarded as the spectrum
of an effective “entanglement Hamiltonian” confined
to the A region of space, which typically contains
similar physics as the original physical Hamiltonian,
but is dominated by the boundary region between A
and B. Since the region A does have a boundary, the
low-lying spectrum would then show an edge structure,
even though the total system has no edge. This
construction of studying edge physics in ES works particularly
well in situations where the bulk is gapped
but edges are gapless, which is the case for FQH states.
Refs. [3] analyzed the ES of FQH states on the sphere
with hemispheric partitioning. The Virasoro multiplet
structure of the conformal field theory (CFT) describing
the edge appears in the low-lying part of the ES.
A B
x
y
Figure 1: Torus setup for block entanglement computations. The
lowest Landau level is spanned by orbitals which in Landau gauge
are centered along the circles shown. The arrows indicate the chiralities
of the virtual ‘edges’ created by the block partitioning.
The torus geometry. The torus geometry (Fig. 1)
gives us access to new physics and new analysis tools,
compared to the spherical geometry used, e.g., in [3,7].
The two circumferences of the torus (L1, L2 in the x,y
directions) are continously variable parameters which
allow a sensitive analysis of the scaling behaviors.
The natural partitions of the torus are cylinder-like segments
with two disjoint edges. The ES thus contains
the physics of a combination of two separate conformal
edges. This leads to ‘towers’ in the ES spectrum,
when plotted against appropriate quantum numbers.
Even in cases where the two edges have different spectra,
the two spectra combine to form towers. The twoedge
picture provides enormous predictive power, as
the assignment of only a few edge mode energies enables
us to construct the remaining ES.
64 Selection of Research Results