Download Classification of Topologically ordered Phases

Classification of Topologically
ordered Phases
Ching-Yu Huang (⿈黃靜瑜)
C.N. Yang Institute for Theoretical Physics
State University of New York at Stony Brook, Stony Brook
!
"Topological phases of matter, quantum field theory, and quantum
information school/ workshop’’
Jan,02, 2015
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Outline
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Introduction: phase transition, topological order…
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Symmetry-protected quantum state renormalization
Detection of symmetry enriched topological order
Summary
•
•
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Matter occurs in different phases
•
•
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Most phases of matter can
be understood through the
lens of spontaneous
symmetry-breaking. For example:
Crystals: translation and
rotation symmetry broken
Magnets: spin rotation and
time-reversal symmetry
broken
Local order parameters
distinguish different phases
[Landau]
ordered
[Gu, Levin,& Wen’ 08]
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disordered
Topological order: Beyond the
Landau Paradigm
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•
•
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Fractional quantum Hall effect
No symmetry breaking
No local order parameters
Characterized by
- Ground state degeneracy
[Tsui, Stormer, &
- Fractional statistics of
quasiparticles (anyons)
- Topological entanglement entropy
- Long-range entanglement (LRE)
[Chen, Gu, & Wen’ 10]
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Gossard ’82]
Local unitary transformation and
quantum state
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For gapped system, a quantum phase transition can
happen only when energy gap closes
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Two gapped ground states, |Ψ1> and |Ψ2> , belong
to the same phase
- if they can connected by an adiabatic evolution that
does not close energy gap
- if they are related by a local unitary transformation. |Ψ1> = U |Ψ2>
[Chen, Gu & Wen ’10]
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Quantum states classification
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Local unitary (LU) transformation Quantum information view – Entanglement
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Short-range entanglement(SRE)
Long-range entanglement (LRE)
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Symmetric local unitary transformation
(I)
(II)
LRE
topological order
(III)
with symmetry
(I)
(a)
LRE
SRE
SET order
(I)
(III)
(II)
breaking symmetry
(I)
SRE
(III)
(II)
(II)
(III)
SPT order
(I)
(II)
(III)
breaking symmetry
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(b)
[Chen, Gu & Wen ’10]
Matrix/Tensor product states
•
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Question: how to represent efficiently quantum
states of a quantum lattice system
Solution: divide the coefficient of the quantum state
in a “Tensor Network” that only depends on a small
number of parameters
| Ψ〉 =
!
•
d
d
∑ ...∑ c
i1 =1
in =1
i1 ...in
| i1...in 〉
The numerical implementation for finding the
quantum states of 1D/2D spin systems are based
on the matrix/tensor product states.
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PART II : Symmetry-protected
quantum state renormalization
work with Xie Chen and Feng-Li Lin
[Huang, Chen, & Lin
PRB. 88,205124 (2013) ]
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Symmetry-protected topological
order phases
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•
•
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Symmetry-protected topological order phases (SPT)
Example: Topological insulator, Haldane phase in
spin-1 chains
SPT ground states have short-range entanglement
which is protected by symmetry
The local unitary transformations preserve some
particular symmetry
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Motivation
•
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Tensor Product State
and quantum state RG
provide a useful
numerical tool to identify
or even classify the
topological order
How to identify the
symmetry-protected
topological phases with
this ideas?
trivial phase
Topologically ordered phase
[ Chen, Gu & Wen ’10]
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Quantum state renormalization
group
The usual renormalization
group : effective Hamiltonian
The wave function
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•
!
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Quantum state RG is to remove SRE and coarse grain
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Symmetry-protected
Quantum state
renormalization group
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1-Dimensional AKLT phase
[I. Affleck, T. Kennedy, E. H. Lieb, & H. Tasaki . 1987]
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The 1-dimensional AKLT state
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Merging neighboring sites : the valence bond solid
The fixed-point state of the QSRG
[Verstraete et al. ’05]
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Preserving symmetry transformations
trivial (product) state
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non-trivial state
Symmetry-protected Quantum state
renormalization group (SP-QSRG)
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Schematic procedure
!
!
!
!
[Huang, Chen, & Lin’13] !
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The key point is to make sure the physical leg be in the same projective
class (half integer and integer)
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1D AKLT and 1D dimer state flow to the same fixed point
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2-Dimensional AKLT phase
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Spin 3/2 particle on the honeycomb lattice
The Hamiltonian of the system is
!
!
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AKLT point
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2-Dimensional (SP)- QSRG
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Remove SRE - the dimer on
the middle bond gets removed
!
!
!
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coarse grain
!
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Ensure the physical
indices j1 and j2 are in the same class of projective representation
[Huang, Chen, & Lin’13] 15
2-Dimensional AKLT-like model
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Phase diagram ! !
! ! 2
! ! 3
H = ∑ Si ⋅ S j + J 2 ∑ ( Si ⋅ S j ) + J 3 ∑ ( Si ⋅ S j )
<ij >
<ij >
<ij >
!
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TPS method
Tensor renormalization group
1.4
1.2
1.5
1.0
1.0
0.8
0.5
0.6
0.4
0.2
0.0
-1.0
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AKLT phase
0.0
J2=0.5
-0.1
0.1
0.2
J3
MZ
MZs
λ1−λ2
-0.5
0.0
AKLT phase
0.0
J3
0.5
1.0
2-Dimensional AKLT-like model
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Phase diagram , fix J2=0.5
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(SP)-QSRG results
√
!
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Summary
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We try to promote the QSRG as a useful numerical tool
in finding and classifying the SPT phase via MPS/TPS.
A key observation is that preserving symmetry in the
QSRG allows us to study fix points of symmetry
protected topological phases
We have consider the 1- and 2-dimensional AKLT
phases as the examples.
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outlook
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Generalize QSRG procedure to SET phases
Generalize this QSRG procedure to quantum dimer models
Quantum Dimer Model on the Triangular Lattice has Z2 fixed
point
| i = (1
g)| icol + g| irvb
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PART III : Detection of symmetry
enriched topological order
work with Xie Chen and Frank Pollmann
[Huang, Chen, & Pollmann. PRB, 90,045142 (2014)}
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Topological order: Topological
degeneracy
•
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Topological degeneracy depends on the topology of
space
g
Ground state degeneracy (N ):
!
!
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g=0
g=1
g=2
For example:
υ=1/3 FQH - Degeneracy on torus= 3 Z2 liquid
- Degeneracy on torus= 4
[Wen & Niu ‘90 ]
[Laughlin ‘83]
Degeneracy sector: winding
number ( Even and odd)
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Topological order: anyons
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Topological order characterized by its quasiparticle
excitations “anyons” (with nontrivial braiding
statistics)
Particle Exchange
iθ
e
Boson/ Fermion ψ (x1 ,x 2 ) = ± ψ (x 2 ,x1 )
Abelian anyons ψ (x ,x ) = e ψ (x ,x )
Non-Abelian anyons ψ a (x1 ,x 2 ) = M ab ψ b (x 2 ,x1 )
iθ
1
2
2
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Topological entanglement entropy
• Topological entanglement entropy (TEE) with smooth
boundary, circumference L
S = cL - γ
!
!
B
A
!
• ϒ >0, ϒ=log D (D: total quantum dimension )
KP
S
=
S
+
S
+
S
+
S
−
S
topo
A
B
C
D
AB
• Z2 gauge theory: ϒ=Log 2
− S BC − SCD − S AD + S ABCD
• General partition
[Kitaev &Preskill ’06; Levin & Wen ’06; Li & Haldane ’08]
Minimally entangled states
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A bipartition with “non-contractible” boundary, where the
TEE depends on the ground state
!
A
S = cL - γ '
B
Ψ = ∑ α λα φα ,a
A
φα ,a
B
!
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There is a special basis of ground state for an
“Nontrivial bipartition”- Minimally Entangled States
(MES)
The minimally entangled state can be identified with the
quasiparticles of the topological phase
[Zhang et al. ’12; Grover et al. ’12; Cincio & Vidal ’12; Wen ’90]
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Motivation
Topologically ordered systems have a richer structure
when symmetries are present
- Symmetry enriched topological order (SET) phase
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Trivial phases
Topological
order B
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Topological
order A
Topological
order A2
Topological
order A3
Topological
order A1
Toric code :
Z2 liquid
How to detect the SET phase?
Order parameter, some quantities ?
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RVB (resonating
valence bond
state) liquid:
Z2 liquid +
SU(2) symmetry
Fractionalization
Symmetry fractionalization
• Laughlin’s liquid
- When an electron is
added to a FQH state
• RVB state on the kagome
lattice
- a pair of excitations
carrying a spin ½ each
Electron
charge= e
Quasiparticles
(charge = e/3 for υ=1/3 )
Quasiparticles carry
“fractional “symmetry representation !
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Symmetry enriched topological
order
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Hamiltonian and ground state have
the same symmetry
!
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U
G H =Gψ
For abelian group, fractionalization of symmetry
operators
- Quasiparticles carry projective
representations
!
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Classify symmetry fractionzation with Second cohomology group
[Chen,Gu & Wen ’10; Pollmann & Turner ‘12 ]
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Ug
g ∈ Hψ
+
g
a
a∗
Z2 topologically ordered phase
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Z2 topologically ordered phase has four quasiparticle types
Four minimally entangled state
Tensor product state representation of four anyons can
be obtained from the complete set of ground states
!
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Consider “spin 1/2 representations on the Z2 gauge
charges”
How to detect symmetry enriched topological ordered
phases?
- projective representation and non-local order parameter
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Spin-1 Bosons on the hexagonal
lattice
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Z2 loop state of S = 1 AKLT
chains on the honeycomb
(AKLT string state)
An excited state with two
defects which carry a spin
1/2 each.
It can be represented by
χ=3 tensor product state
ω = 01 − 10 + 22
(S=1/2) ⊕ ( S = 0)
d = 4 (ϕ , s = 1, 0, −1)
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Spin-1 Bosons on the hexagonal
lattice
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The non-local order parameter
Projective representations of
quasiparticles
! a (g, X L , X R ) =
O
!
!
!
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•
!
• Projective representations
from the minimally
entangled states
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lim n→∞
$ n −1
% R
Ξ X (1) ' ∏ g (k ) ( X (n) Ξ a
) k =2
*
a
L
Selection rule forces the nonlocal order to vanish if edge
spins are fractionalized
RVB phase on kagome lattice
Z2 topological order
A singlet can fractionalize
into two spinons which
carry a spin ½
Topological entanglement
entropy
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Projective representation
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!
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Non-local parameter
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In the same SET phase as
the spin-1 model above
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Conclusion and outlook
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The symmetries can enrich the structure of topological phases.
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Propose methods to detect the symmetry enriched topological order phases
by measuring projective representations and the non-local parameters
!
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Generalize this method to non-abelian group
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If we have a Hamiltonian (with topological order), numerically, how to classify all
possible state?
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The applications : Quantum computation? 1D AKLT state: [Chen, Duan, Ji, & Zeng’ 10 , Else, Schwarz, Bartlett, &
Doherty’ 12]
2D AKLT state [Wei, Affleck, & Raussendorf ’11 ; Wei, Affleck, &
Raussendorf ’12]
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Thank You
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