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Entanglement in Fractional Quantum Hall states
Emil Bergholtz1, Andreas Läuchli1, Masud Haque1 and Juha Suorsa2
1 Max
Planck Institute for the Physics of Complex Systems, Dresden, Germany
2 Department of Physics, University of Oslo, Norway
Entanglement spectrum
Introduction
|<L|C>|=0.995
Entanglement in CM systems
The description of condensed matter (CM) phases using entanglement
measures, borrowed from the field of quantum information (QI) theory,
has led to an explosive growth of interdisciplinary work. However, there
are only a few cases where entanglement concepts provide physical
information that is not obtainable through more conventional quantities,
such as correlation functions. A prominent example where these new
concepts are indeed valuable is ...
Bz
The QH system:
The quantum Hall (QH) system - cold electrons
in two dimensions in a perpendicular magnetic
field - is a striking example of a system where
unexpected phenomena emerge at low energies
due to subtle correlations [1,2].
Emergent CFT structure in Coulomb GS
|ψ! =
A
|ψi !
e
i
⊗
B
|ψi !,
|ψiA !, |ψiB !
where |ψ! is the ground state and
"
Vy
! ! ! ! ! ! ! ! ! ! ! ! ! ! !
• Only experimentally realized system
|<L|C>|=0.985
L1=10
L1=12
L1=14
Laughlin
Coulomb
4
label states in the two partitions.
0
L1=6
-4
analyze the ES of Laughlin states on the torus and show that it is arranged in
towers, each of which is generated by modes of two spatially separated
chiral edges. This structure is present for all torus circumferences, which
allows for a microscopic identification of the prominent features of the
spectrum by perturbing around the thin torus limit. [6]
+ + + + + + + + + + + + + + + + + + +
|<L|C>|=0.986
8
Recently it was shown [5] that the structure of the ES is related to the edge theory
5
of FQH states
and provides a promising route to identifying topological order.
We...
Ix
−ξi /2
|<L|C>|=0.987
12
The entanglement spectrum (ES), {ξi }, is defined in terms of the Schmidt decomposition
!
|<L|C>|=0.978
L1=8
0
4
-4
!KA
0
4
-4
!KA
0
4
-4
0
!KA
4
-4
!KA
0
4
!KA
Comparison of the ES (and overlaps) between the Laughlin
wave function and the Coulomb ground state for various L1.
The ES become very similar for sufficiently large L1. The
appearance of "generic levels" in the Coulomb state lead to
a tentative notion of "Entanglement gap".
FIG. 1: Sketch of the Hall experiment. The 2DEG is exposed to a strong perpendicular magnetic field Bz . A current Ix is
ES as two chiral edges
passed through the sample along the x-direction, and the resulting transverse voltage Vy measured for varying values of the
2
1
1
B! -D !
3
2
1
20
1
!
C -A
-3 -2 -1 0 1 2 3
investigate two probes of bi-partite
entanglement
- the
FIG. 2: Sketch of
the (integer) quantum Hall effect.
The Hall resistance as function of the magnetic field is quantized, i.e. A-B
exhibits and
plateaux. von
(The result
predicted by the classical
Hall effect corresponds
entanglement spectrum (ES)
Neumann
entropy
- in to a straight line through the centres
0 of the
plateaux.) The longitudinal resistivity is zero except at transitions between plateaux. Courtesy of D.R. Leadley, Warwick
fractional quantum Hall (FQH)
states on the torus [7-8]. The torus
University 1997.
geometry provides new natural continuos parameter; its
(bulk) to
excitations. The integer effect was discovered in 1980 by von Klitzing et al[15]; two years later, using even
circumference. We exploit this
20
!(NA=6)
2
"K
B-A
!
A -B
!
10
!
0
20
cleaner samples, Tsui and collaborators reported the discovery of the fractional effect[16] at ν = 1/3. Since then, with
• Disentangle the ES originating from two spatially separated cuts.
• Obtain a physical understanding of the ES starting from the
the fabrication of ever-higher mobility samples, a large number of fractions have been observed[17]. The quantization
10
10
of the Hall resistance turned out to be extremely exact (to at least ten parts in a billion), which has led to the
introduction of a new standard of resistance, with the so-called von Klitzing constant RK = h/e2 , roughly equal to
B-B
microscopic Hamiltonian. 25812.8 ohms, as the fundamental unit.
entanglement entropy, and extract its
• Determine the scaling of the
0
Physically, the number ν in (1) corresponds to the Landau level filling fraction at the center of the corresponding
-18
-12
topological part, to an accuracy
superior
totheprevious
studies
[8] on
plateau[18].
In other words,
IQHE occurs when
an integer number
of Landau levels is filled, while the FQHE
the sphere.
3
B
A
x
6
A
x
BL
B
x
1ν 0=0p/q,
1 1 0q 0 0 1 0 0 0 0 1 0
ν =: p/(2mp
Hamiltonian
Hee =+ 1)
Vkm c†n+m c†n+k cn+m+k cn
1.2
No kinetic energy!
0
100 = 102 , ν = 1/3
• Following Ref. [8] we divide the lattice sites into two disjoint
groups A and B. This is very convenient for numerical studies and
amounts to a similar, but not identical, to a spatial partitioning.
1
• Fractional quantum Hall states have degenerate ground states on
the torus geometry. We label the ground states by their
corresponding thin torus (or Tao-Thouless, TT) ground states [9]
which are adiabatically connected to the bulk ground states. For
example, for 1/3 filling there are three degenerate states, which
correspond to the following TT configurations
!
!
!
!
100100100!100100100100100100!100100100
!
!
!
!
010010010!010010010010010010!010010010
!
!
!
!
001001001!001001001001001001
!001001001
#$
%
"
2
A
• Thinking about the entanglement in terms of the FQH states as
dressed TT states turn out to be very convenient as we explain
below.
3
L1=12
(a)
6
9
12
15
2
lA
0.1
k+m
e! = ±e/q
L1=10
0.6
(all ee-terms that preserve position of CM)
L1=10
1
(b)
Ns=18
1
Ns=24
Ns=24
0.0
ΔS
0.0
L1=8
Ns=12
Ns=30
1
Ns=30
(c)
Ns=36
00
0.0
k-m
1......1
• Exact mapping of a single Landau level!
• The matrix elements
k,m depend on the real space interaction
ν =V1/3
and on the torus dimensions, in particular the lattice constant 2π/L1
0.8
3
L1=14
wave function 1
wave function 2
wave function 3
averaged S(lA)
0
5
10
15
20
0
3
6
L1
9
12
We...
0
15
a) SA in different
degenerate sectors for the
1/3 Laughlin state, and
their average for fixed L1.
b) Difference between
degeneracy-averaged SA
and largest individualsector SA, as a function of
L1. They differ significantly
only for intermediate L1.
c) Degeneracy-averaged
SA versus lA, for different
L 1.
-0.5
-Ln(3)
-Ln(3)
-Ln(5)
-1.5
ν=1/3, Laughlin
(a)
ν=1/5, Laughlin
(c)
0
-0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
4
2
-Ln(3)
-1.5
-Ln(3)
-2
-Ln(5)
0
-2
-4
-2.5
-3
ν=1/3, Coulomb
(b)
-3.5
0
3
6
L1
18
0
6
12
L1
0
a) Chiral edge levels identified from ES, as a
function of torus thickness L1. b) `Aspect
ratio' of the diamond formed by the four
lowest ES levels.
9
L1
12
15
18 0
ν=1/5, Coulomb
(d)
3
Ns=12
4
Ns=18
Ns=24
3
Ns=30
2
1
Ns=36
5
Ns=39
4
0.2
0
Ns=18
3
(c)
Ns=24
Ns=30
2
0
5
10
L1
Ns=36
1
(a)
0.1
Ns=12
15
-0.1
-0.2
ν=1/3, Laughlin
ν=1/3, Coulomb
Ns=36 (Laughlin)
(a)
0
0
(b)
0.6
0.5
0.5
0.4
0.3
0.2
0.1
0.4
0
0
2
4
6
8
10
12
14
16
18
20
L1
0.3
0.2
0.1
0
0
(b)
2
4
6
8
10
12
14
16
18
20
L1
Conclusions and outlook
0
-1
12
Degeneracy-averaged SA (a) and its derivative (b) for the
1/3 Laughlin state as a function of L1. The inset shows
results for the Coulomb interaction.
Extraction of the topological part of SA
-2
6
Exploit the possibility of continuously varying L=2L1 in the torus
setting to obtain significantly improved control of the scaling
form of SA.
We find that
the scaling
regime is
indeed
reached (at
least for
simple FQH
states) in a
sizable
window of
numerically
accessible
parameter
space.
lA
-1
0
0.5
Sphere
5
Conclusion: Degeneracy-averaged SA have much less finite size
oscillations. Moreover, lA should be chosen large (close to lB).
A way of determining the
topological part, −2γ , is to
fit to the scaling for each
L1 as is done here. The
sought value should be
extracted at very large L1.
Torus
1
Entanglement scaling in action
1.4
n k>m
Vk,m
B
There are several sources of finite sizes corrections that has to be
considered; such as finite torus dimensions (L1 and L2) and finite
block size (lA). Moreover the degenerate states do in general lead to
different SA.
1
10
for n large enough block boundaries of total length L [3,4]. For the
torus setup we have n=2 and L=2L1. The constant term is related to
the topological field theory and thus contains information of the
universality class of the state. The leading term gives the entropy
density and is of relevance for the simulation cost of the state.
6
9
12
L1
15
• In the ideal case (1/3 Laughlin) our estimates are accurate to about
2-3% which is much better than earlier results (10-30% at best).
• For other states the performance is worse albeit still better than
previous results.
18
-6
-8
SA(L1) - L1 dSA/dL1
2π/L1
L1 , L 2
!!
y
1.5
12
18
24
30
36
SA = αL − nγ + O(1/L),
L2
y
2
The entanglement entropy of topologically ordered states in two
dimensions is expected to fulfill an ”area law” and scale as
A
L2
y
2.5
ξ
x
SA(lA)
y
ν = p/q
18
(b)
20
Finite size considerations
01
ν = 1/2, 1/4, 3/8, . . .
SA(lA)
ν = 1/3
ν≤1
A
x
y
BB
L1
B
1
SA(L1) - L1 dSA/dL1
e = e/(2m + 1)
12
30
KA
ν=1
!
Partitioning:
0
SA = −tr[ρA lnρA ]
2 4 10
ν= , ,
...
5 11 21
1..0.....0..1 ↔ 0..1.....1..0
-6
The von Neumann entropy measure of entanglement, SA, measure the
entanglement between a block (A) and the rest (B) of a many-particle
system in a pure state |ψ! in terms of the reduced density matrix
ρA = trB |ψ!"ψ| as
....
1
ν=
2m + 1
N-particle
states:
-12
Entanglement entropy
ν = 2/5, 4/11, 10/21, . . .
A
-18
0
Δk=0 (1)
Δk=1 (1)
Δk=2 (2)
Δk=3 (3)
Δk=4 (5)
dSA/dL1
2
18
KA
SA(L1) - L1 dSA/dL1
1
12
3
(a)
SA(L1)
(! = 1)
Note: The y-position is given by the momentum in the x-direction!
ν = 1/(2m + 1)
k=
6
SA(L1) - L1 dSA/dL1
1
0
A!-A!
dSA/dL1
A Landau level (LL) is 1D! With periodic b.c. using Landau
gauge this is explicit:
1-particle ψk ∼ e 1 e
ν =states:
1/(2m + 1) = 1/3, 1/5, . . .
-6
B! -B!
Entanglement spectrum of the 12-particle 1/3 Laughlin state for L1 =10. Left panels show
the symmetric cut and right panels show one of the asymmetric cuts (see above). The blue
squares represent numerically obtained data. The assigned edge modes are labeled by
green dots while the combinations of those edges are marked by red crosses. The script
letters are microscopic identifiers for the two edges combining to form each tower. The
inset shows a CFT tower formed by two ideal (linear dispersion) chiral edges and the
degeneracies of each such level.
Setup and partitioning
2π
2π 2
ik L
x −(y−k L
) /2
A-A
Diamond Aspect Ratio
10
2
S(C)-S(L)
We...
3
3
2
1
0
precision, be reproduced from two chiral
edges with split degeneracies.
• There is a lot of non-trivial structure
such as identical edges appearing in
several sectors.
• The ES structure is present for all L1,
enabling us to understand many features
by studying the problem close to the TT
limit: The TT state determines the
position of the lowest ES level, and the
positions and energetics of the other
prominent towers can be understood by
studying the leading quantum
fluctuations.
SA(L1)
!(NA=5)
20
[001001001001001001]A
!(NA=5)
[010010010010010010]A
• The ES is arranged in towers.
• Each tower can, to a remarkably high
!(NA=6)
exhibiting topological order. magnetic field.
• Extremely rich system: precision and
universality, fractional charge and statistics,
topological q-bits (?), ...
• Ideal setting for applying ideas from
quantum information/entanglement theory
[3-5].
• Investigating the ES on the torus gave us the opportunity
to study the interplay of two chiral edges, and to reach a
microscopic understanding thereof [6].
• The continuous variation of the block boundary
significantly improves on the performance of numerically
determining the entanglement scaling of FQH states [7].
• Similar studies of non-abelian states are under way...
References
1. D.C. Tsui, H.L. Störmer, and A.C. Gossard, Phys. Rev. Lett. 48, 1599 (1982).
2. R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
3. A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
4. M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).
5. H. Li and F.D.M. Haldane, Phys. Rev. Lett. 101, 010504 (2008).
6. A.M. Läuchli, E.J. Bergholtz, J. Suorsa and M. Haque, Phys. Rev. Lett. 104,
156404 (2010).
7. A.M. Läuchli, E.J. Bergholtz and M. Haque, New J. Phys. 12, 075004 (2010).
8. M. Haque, O. Zozulya, and K. Schoutens, Phys. Rev. Lett. 98, 060401 (2007).
9. See eg E.J. Bergholtz and A. Karlhede, Phys. Rev. B 77, 155308 (2008).