xiao-gang wen- ‘complete’ characterization of topological order: modular trans., pattern of...
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Complete characterization of topological order:
Modular trans., Pattern of zeros, Tensor net-work
Xiao-Gang Wen, MIT
July, 2009; Dresden
Z.-C. Gu M. Levin M. Barkeshli Y.-M. Lu Z.-H. Wang
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How to characterize it? F. Wilczek
We used to believe that all phases and phases transition aredescribed by Landau theory in terms of symmetry breaking, order
parameters, long-range-correlations. People proposed chiral spin liquid (CSL) to explain high Tc
superconductor, and we found the CSL is characterized by T and Pbreaking S1 (S2 S3) = 0. Kalmeyer &Laughlin, 87; Wen & Wilczek & Zee, 89
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How to characterize it? F. Wilczek
We used to believe that all phases and phases transition aredescribed by Landau theory in terms of symmetry breaking, order
parameters, long-range-correlations. People proposed chiral spin liquid (CSL) to explain high Tc
superconductor, and we found the CSL is characterized by T and Pbreaking S1 (S2 S3) = 0. Kalmeyer &Laughlin, 87; Wen & Wilczek & Zee, 89
But this is not the end of story, but a begining.Many CSLs with the same symmetry new kind of order Wen, 89
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How to characterize it? F. Wilczek
We used to believe that all phases and phases transition aredescribed by Landau theory in terms of symmetry breaking, order
parameters, long-range-correlations. People proposed chiral spin liquid (CSL) to explain high Tc
superconductor, and we found the CSL is characterized by T and Pbreaking S1 (S2 S3) = 0. Kalmeyer &Laughlin, 87; Wen & Wilczek & Zee, 89
But this is not the end of story, but a begining.Many CSLs with the same symmetry new kind of order Wen, 89
To describe/define the new order, we need to identifymeasurable universal quantities
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How to characterize it? F. Wilczek
We used to believe that all phases and phases transition aredescribed by Landau theory in terms of symmetry breaking, order
parameters, long-range-correlations. People proposed chiral spin liquid (CSL) to explain high Tc
superconductor, and we found the CSL is characterized by T and Pbreaking S1 (S2 S3) = 0. Kalmeyer &Laughlin, 87; Wen & Wilczek & Zee, 89
But this is not the end of story, but a begining.Many CSLs with the same symmetry new kind of order Wen, 89
To describe/define the new order, we need to identifymeasurable universal quantities
Topology-dependent and topologically stable ground statedegeneracy can (partially) describe the new order. Wen 89, Wen & Niu 90(Motivate us to name the new order as topological order Wen 89)
Topological entanglement entropy and spectrum can describe(partially) the topological order. Kitaev & Preskill 06, Levin & Wen 06, Haldane 08
(Can be probed by quantum noise Klich & Levitov 08)Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra
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Topological order = pattern of long range entanglement
eiHlocal |topo. ordered = |direct product state = |1 |2 ...
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Topological order = pattern of long range entanglement
eiHlocal |topo. ordered = |direct product state = |1 |2 ...
Lord Kelvins atoms (knotted tubes of ether) Kelvin, 1867 cannot befermions. (Knots are not topo. enough against local destruction.)
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Topological order = pattern of long range entanglement
eiHlocal |topo. ordered = |direct product state = |1 |2 ...
Lord Kelvins atoms (knotted tubes of ether) Kelvin, 1867 cannot befermions. (Knots are not topo. enough against local destruction.)
|long range entangled =
|loops or string-nets
A modern picture of atoms (which unifies fermions and light): Electons/Quarks = ends of (long) tubes of ether.
Light = fluctuations of tubes of ether (density wave of tubes).
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How to find a complete description of topological orders?
Two issues:
What symbols/notions should we use to completely describetopological orders?
Knowing the symbols, how to calculate the symbol that describethe topological order in the ground state of a generic Hamiltonian.
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How to find a complete description of topological orders?
Two issues:
What symbols/notions should we use to completely describetopological orders?
Knowing the symbols, how to calculate the symbol that describethe topological order in the ground state of a generic Hamiltonian.
In this talk, I will describe three attempts to completely describe
topological orders
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An old attempt: Non-Abelian Berrys phases
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Projective representation of modular group
Consider a family of FQH Hamiltonian H(), = x + iy, with the
following inverse mass matrix: (m1)ij() =
y+
2x
y x
y xy
1y
.
+1 H( ) H( ) H( ) H( )
(x, y) (x + y, y) : + 1; (x, y) (y, x) : 1/.
Non-Abelian Berry phase of deg. ground states: U( ) = e
Non-Abelian Berry phase T = U( + 1) and
S = U( 1/) of the degenerate ground state projective representation of modular group, which maycompletely (?) describe the topological order. Wen 89
Eigenvalues of T quasiparticle statistics. Checked for Abelian FQH states described by the K-matrix.
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Topological order in FQH states and Pattern of zeros
We can use FQH wavefunction ({zi}) to completely characterizetopological order in ({zi}).
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Topological order in FQH states and Pattern of zeros
We can use FQH wavefunction ({zi}) to completely characterizetopological order in ({zi}).
({zi}) contain to too much useless information and the label ismany-to-one.
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Topological order in FQH states and Pattern of zeros
We can use FQH wavefunction ({zi}) to completely characterizetopological order in ({zi}).
({zi}) contain to too much useless information and the label ismany-to-one.
The key is to extract universal information from ({zi}).
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Topological order in FQH states and Pattern of zeros
We can use FQH wavefunction ({zi}) to completely characterizetopological order in ({zi}).
({zi}) contain to too much useless information and the label ismany-to-one.
The key is to extract universal information from ({zi}).
Filling fraction = 1/m Laughlin state 1/m = (zi zj)m
is characterized by the mth
zero as we bring two electrons together. Generalizing that, we bring a electrons together in a wave function:
Let zi = i + z(a), i = 1, 2, , a
({zi}) = SaP(1,...,a; z(a), za+1, za+2, ) + O(
Sa+1)
The sequence of integers {Sa} characterizes the FQH states in thefirst Landau level and is called the pattern of zeros (POZ).
The pattern of zeros {Sa} is a quantitative symbol todescribe and characterize the (chiral) topological order in
FQH states Wen & Wang 08, Barkeshli & Wen 08Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra
T d l ifi i f T l i l d i FQH
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Towards a classification of Topological order in FQH states
Not all sequences of integers {Sa} can correspond to a FQHwavefunction. Only those that satisfy, for any a, b, c,
Sa+b Sa Sb 0
Sa+b+c Sa+b Sb+c Sa+c + Sa + Sb + Sc = even 0
can correspond to FQH wavefunctions.Finding all those sequences may lead to a classification oftopological orders in FQH states
Relation to 1D CDW picture: Seidel & Lee 06, Bergholtz etal 06, Bernevig & Haldane 07,Seidel & Yang 08, Ardonne etal 08
view la = Sa Sa1 as the orbital occupied by ath electron
occupation distribution nl=number of la = l.Z2 : (S2, S3,...) = (0, 2, 4, 8,...), (nl) = (20|20|20|...), n = 2
Z(2)5 : (S2, S3,...) = (0, 2, 6,...), (nl) = (20102000|20102000|...), n = 5
n-cluster structure: n = number of particles in one period of nl.Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra
A li i f {S } h i i
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An application of {Sa} characterization
A quasiparticle in a FQH state can also be quan-titatively characterized by pattern of zeros {S;a}:
S ;a
Let (, zi) be a FQH wavefunction with a quasiparticle at ,then S;a is the order of zero of (, zi) when we bring a electronsto . ((, zi) =
(zi )1/m for a Laughlin quasiparticle.)
{S;a} must satisfies:
S;a+b S;a Sb 0,
S;a+b+c S;a+b S;a+c Sb+c + S;a + Sb + Sc 0
The above equations have many solutions, and each solution
correspond to a type of quasiparticle.{Sa} and {S;a} are quantitative characterizations of FQHstate and their quasiparticles. Such quantitativecharacterizations allow us to calculation topologicalproperties quantitatively.
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T l i l i f h f
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Topological properties from the pattern of zeros
From {Sa} we can get (with the help of vertex algebra): n-cluster structure: wave function for n-electron bound states
reduce to Laughlin wavefunction ODLRO for n-electron clusters.ODLRO only contains cluster condition very limited info.
filling fraction = lima2a2
Sa= n
Sn+1Sn ground state degeneracy on genus g surfaces number of quasiparticle types quasiparticle charges Q = S,nSn
Sn+1Sn quasiparticle statistics/scaling-dim, and quantum dimensions the fusion algebra of quasiparticles the central charge of edge excitations
;a 1 2 3S ;a S ;a S
Quasiparticle fusionXiao-Gang Wen, MIT Complete characterization of topological order: Modular tra
Ch t i t d th h fi d i t L i
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Characterize topo. orders through fixed-point Lagrangian
We may want to use Lagrangian to characterize phases, but- some times similar Lagrangian correspond to the same phase- some times similar Lagrangian correspond to the different phases
cgg
Again the Lagrangian is a many-to-one label of topological orders(and symmetry breaking orders)
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Ch t i t d th h fi d i t L i
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Characterize topo. orders through fixed-point Lagrangian
We may want to use Lagrangian to characterize phases, but- some times similar Lagrangian correspond to the same phase- some times similar Lagrangian correspond to the different phases
cgg
Again the Lagrangian is a many-to-one label of topological orders(and symmetry breaking orders)
A RG idea: under the RG transformation, a Lagrangian flows tofixed-point Lagrangian. It is the fixed-point Lagrangian that
characterizes phase.
cgg
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Limitations of standard RG approach
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Limitations of standard RG approach
But the RG approach appear not to apply to topological phases:If a bosonic system is in a topological phase, then its low energyeffective Lagrangian (the fixed-point Lagrangian) can be
A pure gauge theory with G = Z2, Zn, U(1), SU(2),... A Chern-Simons gauge theory with any G A QED (U(1) gauge theory + massless fermions) A QCD (SU(2) gauge theory + massless fermions) A gravity theory with gapless gravitions
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Limitations of standard RG approach
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Limitations of standard RG approach
But the RG approach appear not to apply to topological phases:If a bosonic system is in a topological phase, then its low energyeffective Lagrangian (the fixed-point Lagrangian) can be
A pure gauge theory with G = Z2, Zn, U(1), SU(2),... A Chern-Simons gauge theory with any G A QED (U(1) gauge theory + massless fermions) A QCD (SU(2) gauge theory + massless fermions) A gravity theory with gapless gravitions
How can the RG flow of a bosonic Lagrangian L() with a scaler
field produces such rich class of fixed-point Lagrangian withgauge fields and fermionic fields?
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Tensor renormalization group: a new RG approach
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Tensor renormalization group: a new RG approach
After a discretization, we can rewrite any space-time path integralat a tensor-trace over a tensor network
D(x, t)eL() ={i}
Tijkl tTr T
T
T
T
T J
T
T
T
T
T
T
T
TT
T
T
T
J
If we know how to calculate tensor-trace, we can solve any thing.
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Tensor renormalization group: a new RG approach
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Tensor renormalization group: a new RG approach
After a discretization, we can rewrite any space-time path integralat a tensor-trace over a tensor network
D(x, t)eL() ={i}
Tijkl tTr T
T
T
T
T J
T
T
T
T
T
T
T
TT
T
T
T
J
If we know how to calculate tensor-trace, we can solve any thing.
Unfortunately, according to the principle of no-free lunch,calculating the tensor-trace is an NP hard problem. Schuch etc 07
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Tensor renormalization group: a new RG approach
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Tensor renormalization group: a new RG approach
After a discretization, we can rewrite any space-time path integralat a tensor-trace over a tensor network
D(x, t)eL() ={i}
Tijkl tTr T
T
T
T
T J
T
T
T
T
T
T
T
TT
T
T
T
J
If we know how to calculate tensor-trace, we can solve any thing.
Unfortunately, according to the principle of no-free lunch,calculating the tensor-trace is an NP hard problem. Schuch etc 07
But Levin and Nave discovered a principle of free lousy lunch: ifyou are willing to accept some errors, calculating the tensor-tracehas only polynomial complexity.
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Filtering out local entanglements
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Filtering out local entanglements
T
T
t T
T T
T
T
T
T
T
T T T T
TTTT
SS
SS
T
TT
TT
T
4
3
2
1
T
T
repeat the above iteration to reduce the number of tensors to one,then take the trace of that tensor to calculate the path integral.
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Filtering out local entanglements
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Filtering out local entanglements
T
T
t T
T T
T
T
T
T
T
T T T T
TTTT
SS
SS
T
TT
TT
T
4
3
2
1
T
T
repeat the above iteration to reduce the number of tensors to one,then take the trace of that tensor to calculate the path integral.
Levin and Naves implementation of TRG flow T T
has a smallproblem: the resulting fixed-point tensor is not isolated.Local non-universal entanglements are not completely removed
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Filtering out local entanglements
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Filtering out local entanglements
T
T
t T
T T
T
T
T
T
T
T T T T
TTTT
SS
SS
T
TT
TT
T
4
3
2
1
T
T
repeat the above iteration to reduce the number of tensors to one,then take the trace of that tensor to calculate the path integral.
Levin and Naves implementation of TRG flow T T
has a smallproblem: the resulting fixed-point tensor is not isolated.Local non-universal entanglements are not completely removed
T
TTTT
T
TT
TT
T
T
T
T
T T
T T
T
T T T T
TTT
T
T
T
T
T T
S2
S4
S4S2
2S
4S
2S4S
S
S
3S
1S
3
1
S
S
S
S
1
3
1
3
Tensor entanglement filtering renormalization (TEFR): Gu & Wen 09Filter out local entanglement but keep long range entanglement
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Application of TEFR to 2D statistical Ising model
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Application of TEFR to 2D statistical Ising model
Using TRG/TEFR and the resulting fixed-point tensor, we cancalculate free energy, ground state energy, low energy spectrum,
central charge and scaling dimensions, entanglement entropy,correlation functions, etc Zhengcheng Gu & Wen 09
2
T
FF"
X1X2
=(b)/(a)2X1
T
c X =(c)/(a)2
(c)
T TTTT
(a) (b)
0.2
0.4
0.6
0.8
1
2 2.4 2.6 2.82.20
1.2
2.82.62.42.221.8
1
1.2
1.4
0.5
1
1.5
2
2.5
1.8
Fixed-point tensors: Thigh1111 = 1 and Tlow1111 = T
low2222 = 1.
Symmetry breaking as direct sum: Tlow = Thigh Thigh
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Application of TEFR to 2D statistical Ising model
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Application of TEFR to 2D statistical Ising model
Using TRG/TEFR and the resulting fixed-point tensor, we cancalculate free energy, ground state energy, low energy spectrum,
central charge and scaling dimensions, entanglement entropy,correlation functions, etc Zhengcheng Gu & Wen 09
2
T
FF"
X1X2
=(b)/(a)2X1
T
c X =(c)/(a)2
(c)
T TTTT
(a) (b)
0.2
0.4
0.6
0.8
1
2 2.4 2.6 2.82.20
1.2
2.82.62.42.221.8
1
1.2
1.4
0.5
1
1.5
2
2.5
1.8
Fixed-point tensors: Thigh1111 = 1 and Tlow1111 = T
low2222 = 1.
Symmetry breaking as direct sum: Tlow = Thigh Thigh
c h1 h2 h3 h4
0.49942 0.12504 0.99996 1.12256 1.12403
1/2 1/8 1 9/8 9/8
Computation time: 10 hours on a desktop.Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra
Application of TEFR to spin-1 chain
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pp p
H =i
Si Si+1 + U(S
zi )
2
+ Bi
Sxi
y
2z
B
U
2.0
0
Haldane
TRI
Z
0.6
Z 2
y2
TRI
Haldane
0
2.0
Z
Zz
0.6
2
B
U
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Application of TEFR to spin-1 chain
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pp p
H =i
Si Si+1 + U(S
zi )
2
+ Bi
Sxi
y
2z
B
U
2.0
0
Haldane
TRI
Z
0.6
Z 2
y2
TRI
Haldane
0
2.0
Z
Zz
0.6
2
B
U
H =i
Si Si+1 + U(S
zi )
2
+B
2
i
(Sxi (Szi+1)
2 + Sxi+1(Szi )
2 + 2Szi )
0.5
z
y
B
U
2.0
0
Haldane
TRI
Z
Z 2
2
0.5
y
TRI
Haldane
0
2.0
Z
Zz2
2
B
U
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Haldane phase is a symmetry protected topological phase
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p y y p p g p
Fixed-point tensor for Haldane phase:HT
2 2
22
Tabcd = Ta1a2,b1b2,c1c2,d1d2 = 2a1b22b1c22c1d22d1a2a = 1, 2, 3, 4; a1, a2 = 1, 2.
TH + T TH if T has time-reversal, parity and translationsymmetry.
Haldane phase is a symmetry protected topological phase. Fixed-point wavefunction:
)1 2a a1 2 1 2c c( b b
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Haldane phase is a symmetry protected topological phase
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p y y p p g p
Fixed-point tensor for Haldane phase:HT
2 2
22
Tabcd = Ta1a2,b1b2,c1c2,d1d2 = 2a1b22b1c22c1d22d1a2a = 1, 2, 3, 4; a1, a2 = 1, 2.
TH + T TH if T has time-reversal, parity and translationsymmetry.
Haldane phase is a symmetry protected topological phase. Fixed-point wavefunction:
)1 2a a1 2 1 2c c( b b
The boundary spin-1/2 and string order parameter are not goodways to characterize the Haldane phase.
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Theory of topological order
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Topological
Order of zerosNetworkPatternTensor
TransformationModular
Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra
Theory of topological order
http://goforward/http://find/http://goback/ -
8/3/2019 Xiao-Gang Wen- Complete characterization of topological order: Modular trans., Pattern of zeros, Tensor net-work
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Emergent
photons & electrons
Spin
liquid
Long range
entanglement
Lattice
gauge theory
Statistics
NonAbelian
Vertex Algebra
(CFT)
Topological
quantum computation
Classification
of knots
Classification
of 3manifoldsADS/CFT
Order
Stringnet
condensation
Herbertsmithite
Topological
High Tcsuperconductor
Emergent
gravity
of zeros
symm. breaking
Pattern
Network
Tensor
Category
Tensor
Numerical
Approach FQH
Edge state
TransformationModular
Cont. trans. without
Xiao-Gang Wen, MIT Complete characterization of topological order: Modular tra
http://goforward/http://find/http://goback/