xiao-gang wen- an introduction of topological order

8/3/2019 Xiao-Gang Wen- An introduction of topological order

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An introduction of topological order

Xiao-Gang Wen, MIT

December 22, 2009

Xiao-Gang Wen, MIT An introduction of topological order
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What are phases?



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What are phases?

Phases are dened through phase transitions.



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What are phases?

Phases are dened through phase transitions.What are phase transitions?As we change a parameter g in Hamilto-nian H (g ), the ground state energy den-sity or average of some other local oper-ators may have a singularity at g c thesystem has a phase transition at g c .

B

A

C

g1

g2



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What are phases?


B

A

C

g1

g2

The Hamiltonian H (g ) is a smooth function of g . How can theground state energy E g be singular at a certain g c ?



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What are phases?


B

A

C

g1

g2

The Hamiltonian H (g ) is a smooth function of g . How can theground state energy E g be singular at a certain g c ?Spontaneous symmetry breaking is a mechanism to cause asingularity in ground state energy E g . Spontaneous symmetry breaking causes phase transition.



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Symmetry breaking theory of phase transition

Ground state is obtained by minimizing an energy function E g ()against the internal variable .E g () is a smooth function of and g . How can it minimal valueE g E g (min ) have singularity as a function of g ?

Minimum splitting singularity in ground state energy E g at g c State-B has less symmetry than state-A.

State-A State-B: spontaneous symmetry breaking. For a long time, we believe that

phase transition = change of symmetry the different phases = different symmetry .

A A

BB

E E E

g g c gA



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FQH a new chapter of condensed matter physics

Different FQH states have the same symmetry Tsui & Stormer & Gossard, 82

So actually different phases can have the same symmetry and sametype of correlations notion of topological order for gapped states Wen, 89

algebraic order for gapless states .Xiao-Gang Wen, MIT An introduction of topological order


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Topological order = pattern of long-range entanglement E

g

E

g

E

g

A basic assumption: Singularity of E g at g c = gap closing at g c Ground states |(g ) of H (g ), 0 g 1 are in the same phase if the gaps of H (g ), 0 g 1 are nite.

Since the gap does not close, we have, for largeT

|(1) = P e i T 10 dg H (g ) |(0)We rewrite H as H = H A + H B where the terms in H A act on

non-overlapping sites, and the terms in H B also act on

non-overlapping sites. For example, for H = i S i S i +1 , we haveH = i =even S i S i +1 and H = i =od S i S i +1 .

e i TH (g ) = e i TH A(g )e i TH B (g )

e i TH A(g ) and e i TH B (g ) generate the local unitary transformations .



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A

i T H B

i T H

e

eU i

1 2 l...

|(1) = P e i T 1

0 dg H (g ) |(0)

= (local unitary transformation) |(0)

The local unitary transformations dene an equivalence relation:Two states related by a local unitary transformation are in the same phase.A quantum phase is an equivalence class of local unitarytransformations.

A state that can be transformed into a direct-product state througha local unitary transformation has a trivial topological order.

Non-trivial topological order cannot be transformed into adirect-product state and has long-range quantum entanglement



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How to study long-range entanglement (topological order)?

Topological order/long-range entanglement

cannot be described by symmetry breaking,cannot be described by order parameters,cannot be described by long range correlations,cannot be described by Ginzberg-Landau theory.But can we describe topological order in terms what it is, not in terms what it is not?



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cannot be described by symmetry breaking,cannot be described by order parameters,cannot be described by long range correlations,cannot be described by Ginzberg-Landau theory.But can we describe topological order in terms what it is, not in terms what it is not? Some basic issues for a theory of topological order

Characterize topo. order through experimental/numerical probe(Dene topological order via physical characterization)

Calculate topological orders (ie their physical characterizations)from ideal/generic ground state wave functions.

Calculate topological orders from Hamiltonian (or Lagrangian) Classify and nd mathematical frame work of topological orders.



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cannot be described by symmetry breaking,cannot be described by order parameters,cannot be described by long range correlations,cannot be described by Ginzberg-Landau theory.But can we describe topological order in terms what it is, not in terms what it is not? Some basic issues for a theory of topological order

Characterize topo. order through experimental/numerical probe(Dene topological order via physical characterization)

Calculate topological orders (ie their physical characterizations)from ideal/generic ground state wave functions.

Calculate topological orders from Hamiltonian (or Lagrangian) Classify and nd mathematical frame work of topological orders.



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Physical characterizations/denition of topological order

Topo. ordered states are gapped. Trivial low energy dynamics.



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Topo. ordered states are gapped. Trivial low energy dynamics.Low energy dynamics can be non-trivial if ground state degeneracydepend on topology

The ground state degeneracy on sphere D sph = 1 D tor = D sph , and is robust against any perturbations

D tor is a physical characterization/denition of topo. orders Wen 89



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D tor is a physical characterization/denition of topo. orders Wen 89 1) When D disk = 1 non-chiral topological order.

2) When D disk = chiral topological order and a low energyspectrum on disk E diskn . Spectrum E diskn = spectrum of a CFT.Edge CFT is a more complete characterization/denition of topological orders Wen 90



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D tor is a physical characterization/denition of topo. orders Wen 89 1) When D disk = 1 non-chiral topological order.

2) When D disk = chiral topological order and a low energyspectrum on disk E diskn . Spectrum E diskn = spectrum of a CFT.Edge CFT is a more complete characterization/denition of topological orders Wen 90

For example, = 1 / 2 FQH state has D sph , D tor , D disk = 1 , 2, :E n

k 1

12

3

5 7



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Non-Abelian Berry phase of deg. ground statesWen 89

Try squeeze more information from the deg. ground states

Consider a family of FQH HamiltonianH ( ) on a torus. Thefamily of degenerate ground states | ( ) can give raise tonon-Abelian Berry phase

a ( ) = i |d

d | , U ( 1 2) = P e

i 2 1 d a

which may contain more information than ground statedegeneracy.

But U ( ) = e is Abelian for topologically ordered states.

If U ( ) is non-Abelian, then there will exist localperturbations to the Hamiltonian that will lift the ground statedegeneracy on torus.U ( ) = e Hall viscosity. Read 08,Haldane 09How to get a non-Abelian Berrys phase?


b l h d d l f


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Non-Abelian Berry phase and modular transformation

Now we assume = x + i y in H ( ) parameterize

the following inverse mass matrix: (m1)ij ( ) =

y +

2x

y x

y x y 1 y

.+1 H( ) H( ) H( ) H( )

(x , y ) (x + y , y ) : + 1; ( x , y ) ( y , x ) : 1/. Non-Abelian Berry phase of deg. ground states | ( ) :

T = U ( + 1) , and S = U ( 1/ ) has non-Abelian part. The non-Abelian part is path independent and universal:

U path 1 = e i

U path 2 S and T projective representation of modular group, which maycompletely (?) characterize the topological order. Wen 89

Eigenvalues of T quasiparticle statistics.Checked for Abelian FQH states described by the K -matrix.


A i d i f H ll ff


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An introduction of quantum Hall effect

Classical Hall effect:V H = R xy I , R xy = B ec Quantum Hall effect: For 2D electron gas Appears for strong magnetic eld,when lling fraction

density of electronsdensity of ux quanta

1

Hall coefficients are quantized:

R xy = nm he 2 = 1 he 2 = mn

Observed lling fractions = 1 , 2,..., 1/ 3, 2/ 3, 2/ 5...

xy

B

HI R

R

V = Ixy

2D electrons gas at thosedensities (with rational ) form an incompressiblestate. The electron density

and are quantized exactly.Xiao-Gang Wen, MIT An introduction of topological order

Wh i ibl i lli f i


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Why incompressible state at integer lling fraction = n

Magnetic eld discrete Landau levels (LL). Number of states in a LL = number of ux quanta. n LLs are lled if = n incompressible states .

Filled

Empty

Empty

Filled

Wave functions:Single electron wave function (in rst LL) has a ring shape.

m = z m e 1

4l 2B |z |2

, z = x + iy

Fill the orbits m = 0 ,..., N 1 N -electron droplet of uniformdensity = 1 HQ state:

(z 1,..., z N ) = ( z 1)0(z 2)1... + ... =i < j

(z i z j )i

e 1

4l 2B |z i |2


I ibl t t t f ti l lli f ti 1 / 3


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Incompressible states at fractional lling fractions = 1 / 3

First Landau level is partially lled Huge degeneracy fornon-interacting electron

Interaction lift degeneracy incompressibility at fractional mustcome from interactionLaughlins theory for FQH effect:Every electron want to stay away from every other electron try

1/ 3(z 1,..., z N ) =i < j

(z i z j )3i

e 14l 2B |z i |2

Every electron in rst Landau level. Third order zero between any pair of electrons. Good for energy Low density. Filling fraction = 1 / 3. 1/ 3 is very rigid.

Compression create rst order zero nite energy costincompressibility.


C l l t t d FQH t t ff ti th


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Calculate topo. order FQH state: effective theory

Calculate topological order = Calculate D tor , E diskn ,...

An effective theory that can calculate D tor

will be a complete anduseful effective theory. Consider an electron system in a magnetic eld:

L(J , A) = eAi J i + kinetic/potential energy

where J (x) = i vi (x xi ), J 0(x) = i (x xi ) are electroncurrent and density. The kinetic/potential energy is given by

i 12 mv

2i + i < j V (xi x j ).

In a hydrodynamical approach, we assume that the low-energycollective modes can be described by the density and the currentuctuation J , and the low-energy effective theory has the formL(A , J ) = eAi J i + eA0J 0 + L (J ). For small J , we mayassume L (J ) to be quadratic in J .


As an incompressible uid, the density of FQH state is tied to the


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p , y Qmagnetic eld, i.e. J 0 = B hc / e . When combined with the nite

Hall conductance xy = e 2

hc , we nd that J satises

eJ = xy A = e 2

hc A

We choose the effective Lagrangian L(A , J ) in such a way thatit produces the above equation of motion.

It is convenient to introduce a U (1) gauge eld a to describe theelectron number current:

J =1

2 a , a a + f

The current dened in this way automatically satises theconservation law.

Then the effective Lagrangian that produces the above equationtakes the following form:

L = 1

4 a a +

e hc

A a + ...



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Is L = 14 a a + e hc A a

+ 12g f 2

a correct andcomplete effective theory?

Can we calculate D tor

? k = 0 modes all have a nite energy gap. So to calculate D tor weconcentrate on the k = 0 mode.

Take the a0 = 0 gauge, and rewrite (ax , ay ) = ( 2L X ,2L Y ) we nd

L =

(X Y Y X ) 12g

(X 2 + Y 2)

which describes a mass-g 1 particle in (X , Y ) plane with amagnetic eld B = 2 .



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+ 12g f 2





L =

(X Y Y X ) 12g

(X 2 + Y 2)


The states in the rst Landau level form the degenerate groundstates and D tor = !



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+ 12g f 2





L =

(X Y Y X ) 12g

(X 2 + Y 2)


The states in the rst Landau level form the degenerate groundstates and D tor = which is a wrong result.


How to x this problem?


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We note that d 2x J 0 = d

2x 12ij i a j is quantized as a integer.

So the ux of a is quantized as multiple of 2. This implies that the charge of a is quantized as integer.

The gauge transformation of a is really generated by U (x ):

a a + i U 1 U

or

a a + f , U = e i f , f f + 2 .

Using the gauge transformation on torus U = e i 2 (mx + ny )/ L, wene that

(ax , ay ) (ax +2L

m, ay +2L

n), (X , Y ) (X + n, Y + m)

are gauge equivalent points.Xiao-Gang Wen, MIT An introduction of topological order


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L =

(X Y Y X ) 1

2g (X 2 + Y 2)

actually describes a mass-g 1 charge-1 particle on a torusX X + 1 , Y Y + 1 with a magnetic eld B = 2 . The totalmagnetic ux is 2 and total ux quantum is 1/ .

But the new consideration causes an even bigger problem: Anconsistent theory must have integer ux quantum. Now we do noteven have a consistent theory!



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L =

(X Y Y X ) 1

2g (X 2 + Y 2)

actually describes a mass-g 1 charge-1 particle on a torusX X + 1 , Y Y + 1 with a magnetic eld B = 2 . The totalmagnetic ux is 2 and total ux quantum is 1/ .

But the new consideration causes an even bigger problem: Anconsistent theory must have integer ux quantum. Now we do noteven have a consistent theory!

This is not a bug but a feature. 1 must quantized as an integer m: = 1 , 1/ 2, 1/ 3, 1/ 4,... .So is quantized to = 1 , 1/ 3,... , which is right.But = 1 / 2, 1/ 4,... is also allowed, which is wrong. = 2 / 5, 2/ 3,... is not allowed, which is wrong.


Quasiparticle excitations


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Q pThe a charge- l excitation is described by

L = m

4a a +

e

hc A a + la0(x x0)

From the equation of motion L/ a0 = 0

J 0 =1

2ij i a j =

1m

B e / hc

+l m

(x x0)

a charge- l excitation carries electric charge Q = e l m , and aux 2 l m . Exchange two a charge- l excitation generate a phase = 12 l 2

l m = l

2/ m.

a charge-1 excitation: Q = e / m, = / m.a charge-m excitation: Q = e , = m. Only when m=odd, the charge- e excitations are fermions, which

are electrons.When m=odd, the charge- e excitations are bosons.


The = 1 / m Laughlin state is described by the following compact


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g y g pU (1) Chern-Simons effective theory

L = m

4

a a +e

2

A a + ......

where the a charge is quantized as integers. When m=odd, the electrons have Fermi statistics.

When m=even, the electrons have Bose statistics.

The ground state sector is described by

L = m(X Y Y X ) 1

2g (X 2 + Y 2)

which describes a mass-g 1 charge-1 particle on a torusX X + 1 , Y Y + 1 with a magnetic eld B = 2 m. Thetotal ux quanta is m. So D tor = m.It is interesting to see that just trying to calculate one physical quantity D tor allow us to understand so much.


Topological order from ideal wave function: effective theory


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opo og ca o de o dea wave u ct o : e ect ve t eo y





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p g y


Projective (parton) construction




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p g y

Calculate topological order = Calculate D tor , E diskn ,... Topo. order for the = 1 / 2 FQH state 1/ 2 = (z i z j )2

1/ 2(z i ) = [ 1(z i )]2 = 1(z (1)i ) 1(z

(2)i ) z i = z (1)i = z (2)i

Before the z i = z (1)i = z

(2)i projection, 1(z

(1)i ) 1(z

(2)i ) is the

ground state of 2 kinds of partons, each kind of parton z (a )i form = 1 IQH 1 = (z

(a )i z

(a ) j )

Effective theory of independent partons

L = i I t I 12m I ( iA)2I

The electron wave function 1/ 2(z i ) = 0| e (z i )| 1 1The electron operator e = 12 is SU (2) singlet.



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Introduce SU (2) gauge eld aIJ to do projection (glue partons


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( ) g g IJ p j (g pback to electrons) at Lagrangian level:

L = i I t I I ( iAIJ iaIJ )2J

Low energy effective theory is obtained by integrating out thegapped parton elds: L = 14 Tr (a a +

i 3 a

3) SU 1(2) CS theory.

More general states k / n = [ k (z i )]n : effective theory is

L =k

4Tr (a a +

i 3

a3), a = SU (n) gauge eld

SU k (n) CS theory.

D tor , E diskn , S , T (ie topological order) can be calculated from theeffective CS theory.1/ n states are Abelian FQH statesk / n states are non-Abelian FQH states Wen 91


Quantum spin liquids


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Herbertsmithsite : spin-1/2 on Kagome lattice H = J Si S j .

J 200K , but no phase transition down to 50mK spin liquidHelton etal 06


Organics -(ET) 2X:


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t / t = 0 .5 1.1 X=Cu[N(CN) 2]Cl, Cu2(CN)3,...

Cu[N(CN)2]Cl t / t = .75 Cu2(CN)3 t / t = 1 .06

Spin int.J = 250 K But no AF

order downto 35mK


Calculate topo. order from ideal wave function: Spin liquid


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How to construct a many-boson/spin wave function (i1, i2, )for quantum spin liquids? Zou & Baskaran & Anderson, 87

Enlarge Hilbert space: Start with two fermions 1 and 2SU (2) singlets |0 , 12 |0 | = |empty , | = |one bosonSU (2) doublet 1 |0 , 2|0 unphysical states

Choose a trial Hamiltonian

H trial =ij

i u ij j, i = 1i2i , u

ij = i 0ij + l ij l

Spin liquid |u ij = P| u ijmean or u ij(i1, i2, ..) = 0|b i1 b i2 ..|u ijmean

where |u ijmean is the ground state of H trial and b i = 1(i)2(i) .

P : projection to local SU (2) singlet; b i is SU (2) singlet. u ij: variational parameters SU (2) gauge structure many-to-one label:

|u ij = |u ij , u ij = W iu ijW j , W i SU (2)


Mean-eld phase diagram of J 1-J 2 model


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The trial wave function |u ij

is more general than the tra-ditional trial wave functioni(u i| + v i| .

|u ij can have long range en-tanglements and we may get a

richer phase diagram.

H = J 1nn

Si S j + J 2nnn

Si S j

where J 1

+ J 2

= 1 .

A

ID

B

C

GE

H

J 2

F

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 .9 10.5

0.45

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 0.2 0.4 0.6 0.8 1

Obtain trial ground state by minimizing E (u ij) = u ij |H |u ijWe nd many local minima.


Mean-eld states


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state-A ( -ux): u i,i+ x = 3 1, u i,i+ y = 3 + 1

state-D (chiral-spin): u i,i+ x = 3 1, u i,i+ y = 3 + 1u i,i+ x+ y = 2, u i,i+ xy = +

2.

state-I (uRVB): u i,i+ x = 3, u i,i+ y = 3

state-G: u i,i+ x = 3

1

, u i,i+ y = 3

+ 1

u i,i+ x+ y = 3, u i,ix+ y = + 3.

state-H: u i,i+ x = 3 1, u i,i+ y = 3 + 1u i,i+ x+ y = + 3, u i,ix+ y = +

3.

All those states are spin rotation invariant and translation invariantThey are spin liquids with one spin-1/2 per unit cell.

How to tell if two ansatz u ij and u ij belong to the same phase ornot?


Projective symmetry group (PSG)Wen, 01


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Landau symmetry breaking theory for minimizing energy :1) The energy function E (u ij) = u ij |H |u ij has a symmetry.

2) u ij that minimize E (u ij) has a lower symmetry.3) If we change the energy function E (u ij), u ij will change.a) If the symmetry of u ij does not change no singularity in theminimal energy and no phase transition.b) If the symmetry of u ij does change a singularity in theminimal energy and a phase transition.


Projective symmetry group (PSG)Wen, 01


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Landau symmetry breaking theory for minimizing energy :1) The energy function E (u ij) = u ij |H |u ij has a symmetry.

2) u ij that minimize E (u ij) has a lower symmetry.3) If we change the energy function E (u ij), u ij will change.a) If the symmetry of u ij does not change no singularity in theminimal energy and no phase transition.b) If the symmetry of u ij does change a singularity in theminimal energy and a phase transition.

The symmetry of the energy function E (u ij)1) Hamiltonian symmetry group (SG H ), eg T x : u i, j u i+ x, j+ x2) SU (2) -gauge symmetry W i: u i, j W iu i, jW j

The symmetry group of u ij is PSG SG H SU (2)-gauge :The PSG of u ij is the group formed by all the combined symmetry and gauge transformations that leave u ij unchanged.

PSG of the ansatz characterize the quantum phases.PSG is a new label to characterize quantum p has e s.


Physical symmetry of ground state and PSG


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Relation to physical symmetry of the ground state |u ij :u i,i+ x = 3 + ( )i y 1, u i,i+ y = 3 ( ) i y 1:

PSG = {W 0i , T x , W T y i T y ,...} SG H SU (2)-gauge symmetry .

W 0i = 1 pure gauge trans. that leave u ij unchanged invariant gauge group (IGG): IGG PSG

The ansatz with the above PSG has physical symmetry T x , T

y The ansatz is translation inv. up to gauge transformations.In general physical symmetry group (SG) = PSG / IGG

PSG IGG SGState-G: Z2Azz13 Z 2 T x , T y , P x , P y , P xy , T

State-H: Z2A0013 Z 2 T x , T y , P x , P y , P xy , T 1) State-G and state-H have the same physical symmetry, but are distinct quantum phases since there PSGs are different.2) If Hamiltonian breaks P x : x x and P y : y y , thenstate-G and state-H will belong to the same p has e .


PSG and symmetry protected topological/algebraic order


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Systems with no symmetry can have rich topological orderSystems with symmetry can have even richer symmetry protectedtopological order (ie with long range entanglement)

PSG IGG SGState-A: SU2Bn0 SU (2) T x , T y , P x , P y , P xy , T State-D: SU2???? SU (2) T x , T y , P x , P y , P xy State-I: SU2An0 SU (2) T x , T y , P x , P y , P xy , T State-G: Z2Azz13 Z 2 T x , T y , P x , P y , P xy , T State-H: Z2A0013 Z 2 T x , T y , P x , P y , P xy , T

The distinction between state-G and state-H requires P x and P y symmetry. Wen, 01

The distinction between state-A ( -ux) and state-I (uRVB)requires T x and T y symmetry. Wen, 01

The distinction between band insulator and topological insulatorrequires T symmetry. Kane & Mele, 05; Bernevig & Zhang, 06


PSG in topological/algebraic order plays a role of i b ki d


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symmetry group in symmetry-breaking order

PSG characterizes different symm. protected quant./topo. orderSymmetry group characterizes different symmetry breaking orders

Phase trans. = a change in PSGPhase trans. = a change insymm. of ground state

Both PSG and symm.-breakingcharacterizations require H tohave some symm.

A G B continuous trans.without any change of physicalsymm. (PSG changes). Wen, 01

A D continuous trans. thatbreaks T , but not in 3D Isingclass. Ran & Wen, 06

G: Z2Azz13linear

D: SU2gapped(chiral spin, break T)

A: SU2Bn0linear(piflux)

E: SU2gapless(break 90)

I: SU2An0gapless

J

(uRVB)

C: SU2xSU2linear

B: SU2xSU2gapless

F: U1Cn00xgappedH: Z2A0013linear

20.50.5

0.3

0.1

0 10.50

0.1

0.3

1


Low energy effective theory of spin liquids


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Starting from H trial = ij i u ij j |u ijmean

Ground state: |u ij = P| u ijmean1) Collective uctuations |u ij+ u ij SU (2) gauge elds.2) Topological excitations spinons at i1 and i2:

|i1, i2 = P 1i1 2i2 |

u ijmean

whose dynamics is described byH = ij i u ij j. Lattice effective theory = spinons coupled to SU (2) -gauge theory:

H = ij i (u ij + u ij) j.Lattice gauge group is SU (2)

Low effective theory = spinons coupled to IGG -gauge theory :H = ij i (u ij + u ij) j, restrict u ij to IGG gauge mode.Low energy gauge group isIGG The ansatz u ij is not SU (2) invariant. The Higgs mechanismbreaks the SU (2) gauge group to the IGG gauge group.


Spinon spectrum:1 1 1


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1 0 0.5 11

0.5

0

0.5

4

0.3

321

0.5

234

1 0.5 0 0.5 11

0.5

0

0.51

0.3

0.5 11

0.5

0

0.51234

1 0.5 0

0.3

State-A ( -ux) State-G (Z2Azz13) State-H (Z2A0013)

Low energy effective theories :State-A ( -ux): SU (2) -gauge + 2 massless Dirac fermionsState-D (chiral-spin): SU (2)2 CS gauge theory (massive spinons)

Topological order with semions D tor = 2 , E diskn , S , T State-I (uRVB): SU (2) -gauge + gapless spinon with Fermi surfaceState-G (Z2Azz13): Z 2-gauge + 2 massless Dirac fermionsState-H (Z2A0013): Z 2-gauge + 2 massless Dirac fermions


The measurable characters of PSG spin liquids


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Spectrum of physical spin-1 excitations (2 spinons):

0.4

10.3

0.2

0

0.2

0.434

0.4 0.2 0 0.2 0.4

2

0.2

32

10.3

0 0.2 0.4

0.4

0.2

0

0.2

0.4

0.4

4

0.4

32

10.3

0.2

0

0.2

0.4

0.4 0.2 0 0.2 0.4

4

State-A ( -ux) State-G (Z2Azz13) State-H (Z2A0013) The spin-1 spectrum of state-A (-ux/SU2Bn0) is periodic in 1/4

of B.Z.!

The spin-1 nodal points near (0, ), (, 0) split along the zoneboundary for the state-G (Z2Azz13). The spin-1 nodal points near (0, ), (, 0) split perpendicular to

the zone boundary for the state-H (Z2A0013).


PSG does not describe all symm. protected topo. orders


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How many Z 2 topological orders with translation symmetry?(How many translation symmetry protected Z 2 topological orders?)

1) Z 2 topological order = fully gapped state described low energyZ 2 gauge theory (IGG = Z 2)2) We have only translation symmetry T x , T y . No spin rotation,no time reversal T , no any other symmetries.




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According to PSG, there are only two classes:Z2A: the spinon hopping on lattice see 0 ux per unit cellZ2B: the spinon hopping on lattice see ux per unit cell




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According to PSG, there are only two classes:Z2A: the spinon hopping on lattice see 0 ux per unit cellZ2B: the spinon hopping on lattice see ux per unit cell

At least 16 trans. symm. protected Z2A topo. orders Kou & Wen, 09The deg. D tor on torus depend on Lx , Ly = even/odd , even/odd

Lx Ly \ ind. 15 14 13 11 7 12 3 9 6 10 5 8 4 2 1 0ee 4 3 3 3 3 4 4 4 4 4 4 3 3 3 3 4eo 4 3 3 3 3 4 4 2 2 2 2 3 3 3 3 4oe 4 3 3 3 3 2 2 2 2 4 4 3 3 3 3 4oo 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4

States 1,2,4,8, 7,11,13,14 are actually non-Abeli an stat es.Xiao-Gang Wen, MIT An introduction of topological order

Translation symmetry protected topo. superconductorsN


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Spinons form a paired state in the Z2A states Trans. symm. protected Z2A topo. orders

are closely related to trans. symm. protectedtopo. superconductors .

How many fully gapped and distinct super-conducting states with only trans. symm.?(no parity, no time-reversal, no spin, ...)2D: 16, 3D: 256, d -dimension: 2(2d )

Even/odd electrons in ground state dependon Lx , Ly = even/odd , even/odd Kou & Wen, 09

T protected topo. superconductorsRoy, 06; Qi & Hughes & Raghu & Zhang, 09; Sato & Fujimoto, 09;

T x , T y protected topo. superconductorsKou & Wen, 09

T symm. is not important to have topo.superconductors.

( )N e ee eo oe oo0000 : + + + +1111 : + + + 0101 : + + +1010 : + + 0011 : + + +1100 : + + 0110 : + +

1001 : + 0001 : + + +1110 : + + 0100 : + +1011 : +

0010 : + +1101 : + 0111 : +1000 :


Towards a systematic theory of topological orders


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Ideal wave function for particle condensation (| = |1 , | = |0 )

|part = all conf. = i(|0 + |1 ) = i| x

1) Foundation for symmetry breaking order2) No long range entanglement



f f | | | |


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|part = all conf. = i(|0 + |1 ) = i| x


Ideal wave function for loop condensation |loop =



Id l f i f i l d i (| |1 | |0 )


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|part = all conf. = i(|0 + |1 ) = i| x


Ideal wave function for loop condensation |loop =

1) Z 2 topological order: D sph , D tor , D disk = 1 , 4, 12) Exact ground state of Kitaev, 97H Z 2 = U I Q I g p B p , B p edges of p

x i , Q I legs of I

z i

=4

x

x

x

x

z z

z z e o

e e

e

o o

op

Dtor

iI

B

Q


String-net wave function for generic topological orders


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Ideal wave function for string-net condensation Levin & Wen, 04

|strnet = all conf.

1) string has types a = 0 , 1,..., N (spins on links haveN + 1 states)2) Branching rule:ijk = 1 (ijk ) branching is allowed in ground state.ijk = 0 (ijk ) branching is not allowed in ground state.

3) Topological: (X ) = ( X ) if two string-nets X and X has thesame topology. Freedman etal 034) Recoupling relation and 6j-symbol:

ji k

ml =

N

n=0F

ijmkln

ji k

ln

Topological string-net condensation is described by a set of data(N , ijk , F

ijmkln )


Pentagon identityijm


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Not all sets (N , ijk , F ijmkln ) describe consistent string-net

condensation. Moore & Seiberg 89

ji k l

m

pn =

q

F mknlpq

ji k l

m

p

q =q ,s

F mknlpq F ijmqps

ji k l

p

qs

ji k l

m

pn =

t

F ijmknt

ji k l

pn

t =t ,s

F ijmknt F itnlps

ji k l

p

st

=t ,s ,q

F ijmknt F

itnlps F

jkt lsq

ji k l

p

qs

Pentagon identity (non-lin. alg. equ.): t F ijmk nt F

itnlp s F

jkt ls q = F

ijmk nt F

itnlp s


Tensor category and topological orders

A l ti f t g id tit (t t g )


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A solution of pentagon identity (tensor category)

An ideal string-net condensed state

which is the exact ground state of a local Hamiltonian H . The pentagon identity has many solutions a large class of 2D topological ordersF ijmkln D

tor , S , T , ..[classify all the non-chiral topological ordersD disk = 1 ]

Excitations in string-net condensed states Charge excitations = ends of strings Vortex excitations = modied string-net wave functions:

Z 2 state: ground = 1 , vortex pairx1 ,x2 (X ) = ( )# of crossings .


Z 2 topological order N = 1 , 000 = 110 = 1 , 100 = 0 (only closed strings), F

ijmkln


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=

,

=

, (X ) = 1

D tor = 4 , 4 types of quasiparticles: boson,boson,boson,fermion

Effective theory: U (1) U (1) CS theory with K = 0 22 0Doubled semion theory

N = 1 , 000 = 110 = 1 , 100 = 0 (only closed strings), F ijmkln

= , = ,(X ) = ( )# of loops

D tor = 4 , 4 types of quasiparticles: boson,semion,semion,boson

Effective theory: U (1) U (1) CS theory with K = 2 00 2


Doubled Fibonacci theory N = 1 , 000 = 110 = 111 = 1 , 100 = 0 (branched string-nets),

ijm


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F ijmkln leads to

= = 1 + 1/ 2

= 1/ 2

1

where = 1+ 52 Effective theory: SO 3(3) SO 3(3) Chern-Simons theory with

D tor = 4

Ends of open strings particles with non-Abelian statisticsDifferent string-net condensations different low energy gauge theories and different statistics:A unication of gauge theory and quantum statistics .



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Topological order a rich world


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(CFT)

Classificationof 3manifolds

High Tcsuperconductor

Vertex Algebra

Herbertsmithite

ADS/CFT

TopologicalOrder =Long rangeentanglement

Latticegauge theory

Topologicalquantum fieldtheory

Topologicalquantum comp.

NonAbelianStatistics

Spin

Category

liquid

Emergent

of zeros

gravity

Emergent

PatternNetwork Tensor

photons & electrons

Edge state

Stringnetcondensation

ModularTransf ormati on

FQHNumericalApproach

Cont. trans. withoutsymm. breaking

Tensor


xiao-gang wen- an introduction of topological order

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