Download From Fractional Quantum Hall Effect To Fractional Chern Insulator

From Fractional Quantum Hall Effect
To Fractional Chern Insulator
N. Regnault
Laboratoire Pierre Aigrain,
Ecole Normale Supérieure, Paris
Acknowledgment
Y.L. Wu (PhD, Princeton)
A.B. Bernevig (Princeton University)
Motivations : Topological insulators
An insulator has a (large) gap separating a fully filled valence band
and an empty conduction band
Atomic insulator : solid
argon
Semiconductor : Si
How to define equivalent insulators ? Find a continuous
transformation from one Bloch Hamiltonian H0 (~k) to another
H1 (~k) without closing the gap
Vacuum is the same kind of insulator than solid argon with a
gap 2me c 2
Are all insulators equivalent to the vacuum ? No
Motivations : Topological insulators
What is topological order ?
cannot be described by symmetry breaking (cannot use
Ginzburg-Landau theory)
some physical quantities are given by a “topological invariant”
(think about the surface genus)
a bulk gapped system (i.e. insulator) system feeling the
topology (degenerate ground state, cannot be lifted by local
measurement).
naive picture : normal strip versus Moebius strip
a famous example : Quantum Hall Effect (QHE)
TI theoretically predicted and experimentally observed in the past
4 years missed by decades of band theory
2D TI :
3D TI :
Motivations : FTI
A rich physics emerge when turning on strong interaction in QHE
What about Topological insulators ?
no interaction
Time-reversal
breaking
Time reversal
invariant
QHE (B field)
Chern Insulator
strong
interaction
FQHE
FCI
QSHE
FQSHE ?
3D TI
FTI ?
Outline
Fractional Quantum Hall Effect
Fractional Chern Insulator
FQHE vs FCI
Fractional Quantum Hall Effect
Landau level
Cyclotron frequency : ωc =
Filling factor : ν =
Rxx
hn
eB
=
eB
m
N
Nφ
At ν = n, n completely filled levels
and a energy gap ~ωc
Rxy
Integer filling : a (Z) topological
insulator with a perfectly flat band
/ perfectly flat Berry curvature !
B
N=2
N=1
h wc
N=0
h wc
Partial filling + interaction →
FQHE
Lowest Landau level
(ν < 1) :
z m exp −|z|2 /4l 2
N-body wave function : P
Ψ = P(z1 , ..., zN ) exp(− |zi |2 /4)
The Laughlin wave function
A (very) good approximation of the ground state at ν =
ΨL (z1 , ...zN ) =
P
Y
(zi − zj )3 e − i
1
3
|zi |2
4l 2
i<j
ρ
x
The Laughlin state is the unique (on genus zero surface)
densest state that screens the short range (p-wave) repulsive
interaction.
Topological state : the degeneracy of the densest state
depends on the surface genus (sphere, torus, ...)
The Laughlin wave function : quasihole
Add one flux quantum at z0 = one quasi-hole
Y
Ψqh (z1 , ...zN ) =
(z0 − zi ) ΨL (z1 , ...zN )
i
ρ
x
Locally, create one quasi-hole with fractional charge +e
3
“Wilczek” approach : quasi-holes obey fractional statistics
Adding quasiholes/flux quanta increases the size of the droplet
For given number of particles and flux quanta, there is a
specific number of qh states that one can write
These numbers/degeneracies can be classified with respect
some quantum number (angular momentum Lz ) and are a
fingerprint of the phase (related to the statistics of the
excitations).
Fractional Chern Insulator
Interacting Chern insulators
A Chern insulator is a zero magnetic field version of the QHE
(Haldane, 88)
Basic building block of 2D Z2 topological insulator (half of it)
Is there a zero magnetic field equivalent of the FQHE ? →
Fractional Chern Insulator
Several proposals for a CI with nearly flat band (K. Sun et al.,
Neupert et al., E. Tang et al.) that may lead to FCI
But “nearly” flat band is not crucial for FCI like flat band is
not crucial for FQHE (think about disorder)
The checkerboard lattice model
-t 2
t2
t1
-t 2
t2
K. Sun, Z.C. Gu, H. Katsura, S. Das
Sarma, Phys. Rev. Lett. 106, 236803
(2011).
P † †
H1 = k (ckA
, ckB )h1 (k)(ckA , ckB )T
P
h1 (k) = i di (k)σi
dx (k) =
4t1 cos(φ) cos(kx /2) cos(ky /2)
dy (k) = 4t1 sin(φ) sin(kx /2) sin(ky /2)
dz (k) = 2t2 (cos(kx ) − cos(ky ))
φ, t1 and t2 parameters (plus second
NNN) can be optimized to get a
nearly flat band
The flat band limit
δ Ec ∆ (Ec being the
interaction energy scale)
1
0.8
0
0.6
0.2
0.4
kx Lx / 2 π
0.4
0.6
E
E/t1
The checkerboard lattice model
has a nearly flat valence band.
4
3
2
1
0
-1
-2
0.2
0.8
1 0
D
d
k
ky Ly / 2 π
We can deform continuously the
band structure to have a
perfectly flat valence band
and project the system onto the
lowest band, similar to the
projection onto the lowest
Landau level
Two body interaction and the checkerboard lattice
Our goal : stabilize a Laughlin-like state at ν = 1/3.
A key property : the Laughlin state is the unique densest state
that screens the short range repulsive interaction.
A nearest neighbor repulsion
should mimic the FQH
interaction.
We give the same energy
penalty to any of these
Red-Blue clusters
Hint =
X
<i,j>
ni nj
Sheng et al. Nature
Communications 2, 389 (2011),
Neupert et al. Phys. Rev. Lett.
106, 236804 (2011)
The ν = 1/3 filling factor
(E - E1)/ t1
An almost threefold degenerate ground state as you expect for the
Laughlin state on a torus (here lattice with periodic BC)
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
N=6
N=10
N=12
0
5
10
15
20
Kx + Nx Ky
25
30
35
But 3fold degeneracy is not enough to prove that you have
Laughlin-like physics there.
FCI vs FQHE
0.4
1
0.36
0.8
0.34
0.6
Energy
Energy
0.38
0.32
0.3
0.4
0.2
0.28
0.26
0
0.24
0
5
10
15
kx * Ny + ky
20
25
80
90
100
110
120
130
kx * NΦ + ky
FQHE on torus
FCI (ν = 1/3, N = 8)
The degeneracy is not perfect. The Berry curvature density is not
flat and thus we don’t have an exact magnetic translation algebra
(see S. A. Parameswaran, R. Roy, S. L. Sondhi, arXiv :1106.4025)
140
Gap
Many-body gap can actually increase with the number of
particles due to aspect ratio issues.
Finite size scaling not and not monotonic reliable because of
aspect ratio in the thermodynamic limit.
The 3-fold degeneracy at filling 1/3 in the continuum exists
for any potential and is not a hallmark of the FQH state. On
the lattice, 3-fold degeneracy at filling 1/3 means more than
in the continuum, but still not much
0.1
0.1
0.06
δ/∆
∆/t1
0.08
0.04
Ny=3
Ny=6
Ny=9
0.02
0
0
0.01
Ny=3
Ny=6
Ny=9
0.001
0.05
0.1
1/N
Gap vs size
0.15
0.2
0
0.05
0.1
0.15
0.2
1/N
spread of the GS manifold
Quasihole excitations
The form of the groundstate of the Chern insulator at filling
1/3 is not exactly Laughlin-like. However, the universal
properties SHOULD be.
The hallmark of FQH effect is the existence of fractional
statistics quasiholes.
In the continuum FQH, Quasiholes are zero modes of a model
Hamiltonians - they are really groundstates but at lower filling.
In our case, for generic Hamiltonian, we have a gap from a low
energy manifold (quasihole states) to higher generic states.
0.3
0.28
E / t1
0.26
N = 9, Nx = 5, Ny = 6
The number of states below
the gap matches the one of
the FQHE !
0.24
0.22
0.2
0.18
0.16
0
5
10
15
Kx + Nx * Ky
20
25
30
Direct comparison
Question : “Why not using overlaps ?”
Possible answers :
high overlaps do not guarantee that you are in the same phase
we don’t know how to write the Laughlin state on such a
lattice
we don’t know if there is an exact hamiltonian for the
Laughlin state on such a lattice
we cannot compute an overlap but we can compute an
entanglement spectrum...
Particle entanglement spectrum
Particle cut : start with the ground state Ψ for N particles,
remove N − NA , keep NA
ρA (x1 , ..., xNA ; x 0 1 , ..., x 0 NA )
Z
Z
= ... dxNA +1 ...dxN
Ψ∗ (x1 , ..., xNA , xNA +1 , ..., xN )
× Ψ(x 0 1 , ..., x 0 NA , xNA +1 , ..., xN )
“Textbook expression” for the reduced density matrix.
The particle ES reveals the
fingerprint of the phase.
18
16
ξ
14
This information that comes from
the bulk excitations is encoded
within the groundstate !
12
10
8
6
-25
-20
-15
-10
-5
0
5
10
15
20
25
LzA
Laughlin ν = 1/3 state N = 8,
NA = 4
If the FCI at ν = 1/3 is a
Laughlin-like phase, we should
observe the Laughlin counting.
Particle entanglement spectrum
24
22
20
ξ
18
16
14
12
10
8
-5
0
5
10
15
20
25
30
35
40
Kx + Nx * Ky
PES for N = 12, NA = 4, 58905 states below the gap as expected
From topological to trivial insulator
One can go to a trivial insulator, adding a +M potential on A sites
and −M potential on B sites. A perfect (atomic) insulator if
M → ∞.
2.5
0.1
0.08
0.06
∆PES
∆/t1
N=8, NA=3
N=10, NA=3
N=10, NA=4
N=10, NA=5
2
0.04
0.02
N=8
N=10
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
M/4t2
Energy gap
1
1.2
1.4
0
0
0.5
1
1.5
M/4t2
Entanglement gap
Sudden change in both gaps at the transition (M = 4t2 )
2
Emergent Symmetries in the Chern Insulator
In FCIs, there is in principle no exact
degeneracy (apart from the lattice
symmetries).
But both the low energy part of the
energy and entanglement spectra exhibit
an emergent translational symmetry.
FQHE BZ
Kx=(0,...,m-1)
FOLDING
The momentum quantum numbers of the
FCI can be deduced by folding the FQH
Brillouin zone.
FQH : m = GCD(N, Nφ = Nx × Ny ),
FCI : nx = GCD(N, Nx ),
ny = GCD(N, Ny )
Ky=(0,...,m-1)
FCI BZ
Ky=(0,...,ny -1)
Kx=(0,...,nx -1)
FQHE vs FCI
Some questions about the FCIs
What is specific about the checkerboard lattice model ?
What are the important ingredients ?
Any evidence for other fractions ?
p/2p + 1 series ? Composite fermions ?
Moore-Read / Read-Rezayi states
Is there any FQHE feature missing on the FCI side ?
Four other models
b2
a2
b1
a3
+M
e2
t1
iφ
t2 e
e1
−M
a1
Kagome (3 bands)
Haldane model (2 bands)
b2
t1
y
it1
−t1
6 1 2
b1
−it1
V
5 4 3
t2
+M
x
−t2
−M
1/2 HgTe (2 bands)
Ruby lattice (6 bands)
FCI on the Haldane model
ν = 1/3 on the Haldane model with short range repulsion.
b2
0.05
a2
+M
t2 eiφ
a1
0.04
t1
−M
0.03
E−E0
b1
a3
0.02
0.01
(N,Nx ,Ny ) =(8,6,4)
(N,Nx ,Ny ) =(10,6,5)
0.00
0
We can tune t1 , t2 and Φ to make a
flat band model (Neupert et al.)
5
10
15
20
kx + Nx ky
Exactly the same features than ν = 1/3 on the checkerboard
lattice (but ...)
25
FCI on the Haldane model
How does the flatness of the Berry curvature affects the FCI
spectrum ?
Look at N = 8, ν = 1/3, tuning Φ tracking σB , the deviation to
the average Berry curvature (1/2π)
0.025
1.4
0.025
1.2
0.8
0.010
0.6
0.4
0.005
0.000
0.1
0.015
1.0
0.015
0.2
0.2
0.3
0.4
σB
0.5
0.6
Energy gap
0.7
W
∆E
0.020
3.0
1.35
1.20
1.05
0.90
0.75
0.60
0.45
0.30
0.15
0.020
0.010
0.005
0.000
-0.005
0.0
0.2
0.4
0.6
σB
0.8
1.0
1.2
Energy spread
1.4
1.35
1.20
1.05
0.90
0.75
0.60
0.45
0.30
0.15
2.5
2.0
∆ξ
0.030
1.5
1.0
0.5
0.0
0.1
0.2
0.3
0.4
σB
0.5
Ent. gap
The flatness of the Berry curvature helps but not as much as
expected
0.6
0.7
Benchmarking the FCIs
both the Haldane and 1/2 HgTe models
show a weak FCI phase at ν = 1/3
0.03
Checkerboard, Kagome and ruby models
exhibit a clear Laughlin like phase.
0.02
∆E
0.01
0.00
-0.01
-0.02
-0.03
-0.04
0.00
Ny =3
Ny =4
Ny =5
Ny =6
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
1/N
Gaps for the Haldane model
0.07
0.06
Two versions of the Kagome model : the
one with the flattest band (with NNN) is
the weakest FCI.
A flat band model is not a guarantee to
get a good FCI
0.03
0.02
0.01
0.00
0.00
20
Ny =3
Ny =4
Ny =5
Ny =6
0.02
0.04
18
16
ξ
∆E
0.05
0.04
0.06
0.08
0.10
0.12
0.14
0.16
1/N
Gaps for the Kagome model
14
12
10
0
5
10
15
20
25
30
35
kx + Nx ky
PES Kagome N = 12, NA = 5
The Moore-Read state
Prototype of the non-abelian state that may describe the
incompressible fraction ν = 5/2. In the FQHE, the MR state can
be exactly produced using a three-body interaction.
What about the FCI ?
Two types of nearest neighbor
3 particle cluster :
Blue-Blue-Red
Red-Red-Blue
We give the same energy
penalty to any of these clusters
The Moore-Read state
Energy spectrum at ν = 1/2 for N = 12, Nx = 6, Ny = 4
0.21
Energy
0.2
0.19
0.18
0.17
0.16
0.15
0
5
10
15
20
25
kx * Ny + ky
6-fold almost degenerate groundstate (2 at kx = ky = 0, 4 at
kx = 0, ky = 4), as predicted by the FQHE to FCI mapping !
The Moore-Read state
Particle entanglement spectrum at ν = 1/2 for
N = 12, Nx = 6, Ny = 4, NA = 5
22
20
18
ξ
16
14
12
10
8
0
5
10
15
20
25
Kx + Nx * Ky
1277 states in each momentum sector below the dotted line.
The Moore-Read state
Energy spectrum at ν = 1/2 + one flux quantum for
N = 12, Nx = 5, Ny = 5
0.165
0.16
Energy
0.155
0.15
0.145
0.14
0.135
0.13
0.125
0
5
10
15
20
25
kx * Ny + ky
7 states in each momentum sector below the dotted line.
Can we stabilize the MR state with a two-body interaction ? Work
in progress...
The Read-Rezayi states
We can cook-up an interaction
for each RR states.
works like a charm for the
RR k = 3.
Lg
RRk=4
RRk=3
MR
not completely clear for
the RR k = 4. Finite size
effect ?
MR do not show up in the
Haldane and 1/2 HgTe
models
Strong MR phase in the
Kagome and ruby models.
Particle-hole symmetry
An important feature of the FQHE (with a two body interaction) :
ν = 1/3 ↔ ν = 2/3
0.5
5
0.45
4.95
Energy
Energy
0.4
0.35
0.3
4.9
4.85
4.8
0.25
4.75
0.2
0.15
4.7
-2
0
2
4
6
8
10
12
14
16
18
kx * Ny + ky
N = 6, Nx = 6, Ny = 3ν = 1/3
-2
0
2
4
6
8
10
12
14
16
18
kx * Ny + ky
N = 12, Nx = 6, Ny = 3ν = 2/3
The p-h symmetry is absent from the FCI model, it might be
restored at large N
FCI are not as boring as expected
Some questions about the FCI
What is specific about the checkerboard lattice model ? not
really
What are the important ingredients ? Don’t know
Any evidence for other fractions ?
p/2p + 1 series ? Composite fermions ? TODO
Moore-Read / Read-Rezayi states yes !
Is there any FQHE feature missing on the FCI side ?
particle-hole symmetry
Conclusion
Fractional topological insulator at zero magnetic field exists as
a proof of principle.
There is a counting principles for the groundstate and the
excitations.
Beyond ν = 1/3 : ν = 1/5, “ν = 5/2” (at least with three
body interaction). Other fractions ?
What are the good ingredients for an FCI ?
First time entanglement spectrum is used to find information
about a new state of matter whose ground-state wavefunction
is not known.
Entanglement spectrum powerful tool to understand strongly
interacting phases of matter.
Roadmap : find one such insulator experimentally, 2D-3D
fractional topological insulators ?