トポロジカル秩序とクォーク閉じ込め問題

University of Tokyo, ISSP, Masatoshi SATO

@ 九州, Dec.25,‘081

z Part 1
What is topological order ?
Fractionalization and topological orders.
MS, M. Kohmoto and Y.-S.Wu, Phys. Rev. Lett. 97, 010601 (2006)z Part 2
Quark (De)confinement as a Topological Order
MS, Phys. Rev. D77, 045013 (2008) + alpha2Physics of topological orders
z Bulk-edge correspondence ・・・ holographic relationindex theorem
z fractionalization

・・・ fractional charge
fractional statistics(anyon, non-abelion)
fractional QHE

z topology-changing phase transitionz (topological) entanglement entropy ・・・quantum computer3http://www.issp.u-tokyo.ac.jp/public/TASSP

4

① What is topological order ?
Orders: conventionally understood in terms of SSB

spontaneous symmetry breaking

But, recently it is recognized that there exist orders whichcannot be described by spontaneous symmetry breaking

5

Examplefractional quantum Hall system

R.Willett et al. ‘87

z Phase transitions (jumps of Hall conductance) occur by changing
the magnetic fieldz The FQH states do not break the symmetry of the Hamiltonian(quantum liquid state).
z They have excitations with different fractional charges. e* = eν

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(Wen ‘04)
OrdersSymmetry-breaking orders‘Particle’ condensation

Non-symmetry-breaking orders
Quantum system

Classical system

Symmetry group
Nambu-Goldstone mode
Quantum orders

GappedTopological orders

Fermi liquids

7Topological order Topological orders are conventionally characterized by theground-state degeneracy depending on the topology of thespace (topological degeneracy) Wen ‘90

fractional quantum Hall systemsex.) Laughlin statedegeneracy

Wen-Niu ‘90

8The ground state degeneracy is useful even forsymmetry breaking ordersordered
2-fold degeneracy

not ordered
no degeneracy

But, the degeneracy is independent of the topology !

9222

At present, several different systems are known to exhibit
topological orders.
 boson system fermion system
 in the presence or in the absence of magnetic filed(without or with time-reversal invariance) 2+1 and 3+1 dim. system
X.G.Wen ‘91,
Read-Sachdev ‘91,
Senthil-Fisher ‘00,
Moessner-Sondhi ‘01,
Misguichi-Serban-Pesquir ‘02,
Balents-Fisher-Girvin ‘02,
Motrunich-Senthil ‘02,
Kitaev, ‘97,…….

 fractional quantum Hall systems
 insulators
 (frustrated) spin systems
 unconventional superconductors Sr2RuO4
 superfluid 3He A-phase10Common characteristics① All the known models have an excitation with fractionalcharge.
Oshikawa-Senthil ‘06

In addition, some models have the following otherfractionalization② Some models have an excitation with fractional statistics
or non-abelian statistics.
③ Some models show fractional quantum Hall effects.

11All these fractionalizations induce hidden symmery(topological discrete algebra).

Fractionalization = Topological order12Plan of part1① Topological discrete algebra② Ground state degeneracy③ Fractionalization and topological invariant(Fractional Quantum Hall effect)13

② Topological Discrete Algebra (d=2+1)
Our assumptions are the following

definition of charge① The system is on a torus.
② There exists an exact U(1) symmetry.
③ The system is gapped.
④ Charge fractionalization occurs.
In other words, we assume that
there exist a quasi-particle with fractional charge

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We start with the following non-trivial processes on a torus(Wu-Hatsugai-Kohmoto ’91)

a. adiabatic unit flux insertions through holes of torus

a unit flux

b. translations of i-th quasi-particle along loops of torus

15Note that the unit flux insertions are unitary equivalent tothe following adiabatic changes of the boundary conditions.
Consider the twisted boundary condition on the torus
phase

phase

By large gauge transformation

1. the flux is reduced as
2. the boundary conditions are changed as
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Using the large gauge transformation, we can delete
the inserted unit flux completely, but the boundary condition
parameters θx and θy change by one period.
 Thus Ux (Uy) is unitary equivalent to adiabatic changeof θx (θy) by 2π

The spectrum should be invariant under Ux and Uy(“gauge invariance due to large gauge transformation”)
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We also consider the exchange of quasi-particles.
c. exchange between i-th and (i+1)-th quasi-particles

The reason why we take into account this operation is thatthe translation along the loops are not independent to theexchange of quasi-particles. Indeed they form the braid
group algebra.
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Braid Group on torus

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The commutation relations between the braid group operatorsand the flux insertions are determined by AB effect.
From AB effect, we have

AB phasetranslation after flux insertion

In a similar manner, we obtain

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On the other hand, the commutation relation between the flux
insertion operators is determined by Schur’s lemma
commutes with all the braid group operators
Schur’s lemma

To have a finite dimensional representation, we have21

Thus, our tool to examine the topological order isthe following.
① Braid Group② AB effect③ Schur’s lemma

Topological discrete algebra (1)22If the quasi-particle obeys abelian statistics, the algebra issimplified.

The solution of braid group is

Topological discrete algebra (2)23The statistical property is naturally combined with the charge
fractionalization in terms of the topological algebra.

Questions:
1. Can the topological algebra explain the ground statedegeneracy ?
2. (What is the physical meaning of the parameter λ ?)24

③ Ground state degeneracy
To count ground state degeneracy, we define Tx on theground statepair-creation

pair-annihilation

In a similar manner, we define Ty on the ground state25

Let us take the basis of the ground state to be aneigenstate of Txthen, we have n different eigenstates

n-fold degeneracy26

We can also take the basis to be an eigenstate of Tx and Tyn

In this case, we have qQ different eigenstatesqQ eigenvalues

qQ-fold degeneracy
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The minimal ground state degeneracy is the least common
multiple of n and qQ=nQ2/N

In general¾ Ground state degeneracy is obtained from charge
fractionalization and fractional statistics¾ It depends on the topology of the space28

Test of our result(torus)
1) Laughlin state (fractional Hall effect)
D.Arovas, J.Schrieffer, F.Wilczek, ‘84

Agreement with numerical simulation andthe Chern-Simon effective theory
W.P.Su ’84, Wen-Niu ‘902) Kitaev model (topological computer, RVB state)

Agreement with the exact result
Kitaev ‘9729

④ The physical meaning of λTo consider the physical meaning of λ, we calculate theHall conductance by using the linear response theory

average over the boundary conditions
Niu-Thouless-Wu ‘84

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31

Using this, we have

Fractional λ implies the fractional quantum Hall effect32

Part 1.

Summary Using the braid group formulation, we found a closed
algebraic structure which characterizes topological orders.
chargeTDA
statistics

Hall conductance Topological degeneracy is due to the topologicaldiscrete algebra.
cf.) For symmetry breaking ordersσx
The degeneracy is due to broken generators.

The fractionalization implies non-trivial topologicalinvariants

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Part 2.

⑤ Quark deconfinement as a topological
order
Idea

The Quark has fractional charges

The deconfinement implies the topological order ?

yes
But.. the quark is an elementary particle, not a collectiveexcitation.
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Nevertheless, non-trivial topological discrete algebra can beconstructed in a similar manner ..

An important difference
Due to the existence of gluons, the unit flux is
reduced to Φa= 2π /3ee: the minimal charge of the constitute particle(charge of down quark )cf.) For electron system, Φa= 2π /e35Due to gluon fluctuations, there exist center vortices of SU(3)in each holes of three dimensional torus.

Thus, the physics is the same after the flux insertion by 2π/3e(not 2π/e)
The unit flux is reduced to Φa= 2π /3e36

permutation group on T337The only allowed representation of the permutation group isboson or fermion.
The unique solution38

If quarks are deconfined, the physical states are classified
with the permutation group of quarks.

Quarks have fractional charges, so we have non-trivial AB
effects with inserted flux

Topological Discrete Algebra

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From the topological discrete algebra, the groundstate degeneracy is found to be 33.

The quark deconfinement phase is topologicallyordered !

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On the other hand, …
If quarks are confined, the physical states are classified withthe permutation group of hadrons.

Hadrons have integer charges, so we have no non-trivial AB
effect with the inserted flux.

Topological discrete algebra is trivial

No topological degeneracy !
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The confinement and deconfinement phases in QCD
are discriminated by the topological ground statedegeneracy !

For SU(N) QCD on Tn ×R4-n• deconfinement:

Nn –fold ground state degeneracy• confinement:

No topological degeneracy42

We check this in the following manners
z comparison with the heavy quark limit
z perturbative calculation of the topological groundstate degeneracy
z consistency check with Fradkin-Shenker’s phase
diagramz comparison with Witten index43

1. comparison with the heavy quark limit
SU(3) YM

QCDheavy quark limit

The pure SU(3) YM has an additional symmetry known as
center symmetry
center symmetry

τ44

confinement phaseτ① area law

temporal gauge② cluster property

The center symmetry is not broken
No ground state degeneracy

45deconfinement phase① perimeter law

breaking of the center symmetry

33 degeneracy

The degeneracy reproduces the one obtained from thetopological discrete algebra46

In the static limit, our criterion for confinement coincides withthe Wilson’s.

remarkIn this limit, topological discrete algebra becomes asfollows.

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2. perturbative calculation of topological degeneracyz We can calculate the topological degeneracy in theperturbation theory.
z Since the QCD is deconfined in the weak coupling,
there must be the topological degeneracy.
SU(2) gauge theory on a S1 x R1,2S1ΦR22-fold degeneracy
should arises.
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The Result

N.Weiss 81,
Hosotani 83

Again, the result is consistent with the topological degeneracy49

3. comparison with Fradkin-Shenker’s phase diagram
Fradkin-Shenker’s result (79)z Higgs and the confinement phase are smoothly connected whenthe Higgs fields transform like fundamental rep (complementarity).
z They are separated by a phase boundary when the Higgs fields
transform like other than fundamental rep.

Our topological argument implies that no ground state degeneracy
exists when Higgs and the confinement phase are smoothly connected.

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4. N=1 SU(N) SYM
z Witten index has information on the ground statedegeneracy on the torus T3
z N=1 SU(N) SYM is in confinement phase, so theremust be no topological degeneracy.
We know that there exists degeneracy on T3
Witten index

at least N-degeneracy

51But …
zThe theory has a discrete chiral symmetry, which isexpected to be spontaneously broken.

z The rotation symmetry is expected not to broken.

N-fold degeneracy is a consequence of SSBof chiral symmetry

No topological degeneracy = quark confinement52 Summary of Part 2
 We can generalize the topological discrete algebra to thenon-Abelian gauge theories.
 Topological degeneracy gives a gauge-invariant “orderparameter” which distinguishes the quark confinement phaseand deconfinement one.
33S
T−β H
Tr e( )lim
=1
Tr ( e )T3−β Hβ →∞S3 Topological criterion of the quark confinement is available
even in the presence of the dynamical quarks.
 We can check our idea in various ways.
 Our topological discrete algebra can be generalized into
other systems, (i.e. standard model, SUSY )53