Michael Levin and Xiao-Gang Wen- An origin of light and electrons: a unification of gauge interaction and Fermi statistics

8/3/2019 Michael Levin and Xiao-Gang Wen- An origin of light and electrons: a unification of gauge interaction and Fermi statistics

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An origin of light and electrons

– a unification of gauge interaction and Fermi statistics

Michael Levin and Xiao-Gang Wen

http://dao.mit.edu/˜wen

• Artificial light and quantum orders ...

PRB 68 115413 (2003)

• Fermions, strings, and gauge fields ...

PRB 67 245316 (2003)

• Strings-net condensation ...PRB 71 045110 (2005)

• Quantum field theory of many-body systems

(Oxford Univ. Press, 2004)



Deep mysteries of nature

• Identical particles (Why two hydrogen atoms are exactly the same?)

• Gauge interactions (long range, massless gauge bosons)

• Fermi Statistics (Who ordered it?)

•Massless fermions (nearly, M f /M P

∼10−20)

• Chiral fermions (Are we edge excitations?)

• Gravity (The correct physical theory allows only integers)

A great-grand unification:

a single structure that explains all the mysteries

We will discuss a baby-grand unification that explains the first four

mysteries from a single structure – local bosonic model.



Where do Maxwell equation and Dirac equation come from?

• Eular equation: ∂ 2t ρ − v2∂ 2i ρ = 0 → massless scalar identical bosons

superfluid → density fluctuations

• Navier equation: ∂ 2t ui − T ijkm ∂ j∂ kum = 0 → phonons (identical bosons)

crystal → lattice fluctuations

Identical particles → vacuum is not empty

• Maxwell equation: ∂ × E + ∂ tB = ∂ ×B− ∂ tE = 0 → photons

???

Dirac equation: (γ µ∂ µ m)ψ = 0 fermions



• Both Maxwell equation and Dirac equation can come from local

bosonic models or lattice spin models if bosons/spin (a) form Long

strings and (b) strings from a quantum liquid (string-net condensed

state):

Gauge bosons and fermions can emerge as low energy collective

modes of the condensed string-nets

String-net condensation provides a way to unify gauge interac-tions and Fermi statistics

The appearance of the gauge interaction and Fermi statistics in our

nature is not an accident.



A local bosonic model on cubic lattice

3

4 1

2

iI+z

I+y I+x

I

I+z

I+y

I

I+x

A rotor θi on every link of the cubic lattice:

H = U I

Q2I − g

p

(Bp + h.c.) + J i

(Lzi )2

QI =

i next to I

Lzi , Bp = L+

1 L−2 L+3 L−4

Lz = i∂ θ: the angular momentum of the rotor

L± = e±iθ: the raising/lowering operators of Lz



What is string-net

zL =0

ClosedStrings

StringsOpen

U g,J

+

+ + +

+

+

+

+

+ +

+

String−net

zL =1

zL =−1

Physical meaning of the three terms:

•the U -term

→closed strings. Open ends cost energy.

• J -term → string tension

• the g-term → strings can fluctuate



What is string-net condensation

U / g

J/g

String−net condensed

Higgs phase

Confined phase

|string-net condensed =

all closed-string-nets

|string-net



Fluctuations of condensed string-nets = U (1) gauge bosons

• When U = ∞, the rotor model can be mapped to U (1) lattice gauge

model, with θi on link IJ as the U (1) gauge potential:

θi = aIJ,

i I J

IJ

• For finite U and with other perturbations:

Leff = LU (1)(aIJ) + 1a20 + 2 cos(aIJ)

Since aIJ = θi is compact, the -terms do not generate a mass for theU (1) gauge bosons, if 1,2 are small enough.

Gauge bosons and “gauge symmetry” can be emergent



Ends of open strings = gauge charges

• Strings are unobservable in string condensed state.

• Ends of strings behave like independent particles.

•Lzi

∼θi = aIJ correspond to electric field/flux.

Ends of condensed strings are gauge chargesThey also carry fractional rotor-angular momentum (Lz = ±1/2)



Can get fermions for free (almost) Levin & Wen 04

Just add some legs

6

5

34 1

2

a crossed leg a leg

• Dressed-string model:

H = U I

QI − gp

(Bp + h.c.) + J i

(Lzi )2

Bp = L+1 L−2 L+

3 L−4 (−1)Lz5+Lz

6

•Different ground state wave function for the string-net condensed

state

|string-net condensed =

all closed-string-nets

±|string-net

which leads to different statistics for the ends of condensed strings.



String operators – creation operators of gauge charges

• A pair of gauge charges is created by an open string operator whichcommute with the Hamiltonian except at its two ends.Strings cost no energy and is unobservable.

leg

crossed legi

i+x

i+z

i+y

i

i+x

i+z

i+y dressed string

C

• In simple-string model – simple-string operator

L+i1

L−i2

L+i3

L−i4

...

• In dressed-string model – dressed-string operator

(L+i1

L−i2

L+i3

L−i4

...)

i on crossed legs of C

(−1)Lzi



A dressed-string operator creates a pair of fermions

• The statistics is determined by particle hopping operators Levin & Wen

02:

a

b

c

d

b

c

a d

12

3

4

5a d

b

a d

c

b

c

t bd t cb t ba

t cbt ba t bd

t cb

t ba

t bd

• An open string operator is a hopping operator of the gauge charges.

Open string operator determine the statistics.

• For simple-string model:ˆtba = L

+

2 ,ˆtcb = L−3 ,

ˆtbd = L

+

1We find tbdtcbtba = tbatcbtbdThe ends of simple-string are bosons.

• For dressed-string model: tba = (−)Lz4+Lz

1L+2 , tcb = (−)Lz

5L−3 , tbd = L+1

We find tbdtcbtba = −tbatcbtbd

The ends of dressed-string are fermions.



What make fermions massless?

• Consider the hopping Hamiltonian for a single end of string

H = ij

(tij

+ h.c.)

H may realize translation symmetry only projectively.

• The translation T (2)a of the two ends of a string satisfies the translation

algebra

T (2)a T

(2)b

= T (2)b

T (2)a , a,b = x,y, z

The translation T a of the one ends of a string satisfies

T aT b = ηT bT a, η = ±1

• η = −1 → π-flux through each square → massless fermions

• The string-net wave function Φ(X) = (−1)N X given rise to the π-flux,

where N X=number of spares enclosed by the closed string X.



Comparison with superstring theory

Superstring theory

•gauge boson = small open string of size lP .

• fermion comes from “super world sheet” (σ1, σ2, θα).

• graviton = small closed string of size lP .

Fermions do not have to carry gauge charges.

String-net theory

Every thing comes from local bosonic model — locality principle

1. Htot = Hi ⊗H j ⊗ ....

2. Local operator = operators acting within Hi.

3. Hamiltonian = sum of local operators.

• gauge boson = fluctuations of large string-nets that fill the space.

• fermion = one end of open string.

• graviton = ???.

Fermions (including composite fermions) must carry gauge charges.



• 123 standard model is inconsistent with the locality principle.

• SU (5) GUT is inconsistent with the locality principle.

But can be fixed by including additional discrete (say Z 2) gauge theory.

Prediction, cosmic string associated with the discrete gauge theory.

• SO(10) GUT can be consistent with the locality principle.



Summary

• Gauge interaction and Fermi statistics are just phenomena of quan-

tum interference in infinity dimension – many-body quantum entan-

glements.

• No need to introduce gauge bosons and fermions by hand. They just

emerge if our vacuum has a string-net condensation.

• Constructed spin model on cubic lattice that reproduce QED and

QCD Wen 03.

They are the U (1) and the SU (3) in the U (1) × SU (2) × SU (3) stan-dard model.

But ... have trouble to get the chiral coupling of the SU (2).

Six fascinating properties of nature:

•Identical particles

•Gauge interaction

• Fermi statistics • Massless fermions• Chiral fermions • Gravity

The string-net condensation picture can explain four of them.

Four down and two more to go!



A picture of our vacuum

- a recipe for making an artificial vacuum in condensed matter

A picture of our vacuum A string−net theory of light and electrons



General string-net condensed wave functions Levin & Wen 04

Too hard to describe Φ(X) = const. directly.

Indirect description:• Types of strings: 1, 2,...,N . 0 represents no string.

• Branching rule: Kogut & Susskind 75

δijk = 1 → (ijk) branching is allowed in ground state.

δijk = 0 → (ijk) branching is not allowed in ground state.

• Topological: Φ(X) = Φ(X ) if two string-nets X and X has the sametopology. Freedman etal 03

• Rebranching relation and 6j-symbol:

Φ

i

k

lm

j

=

N

n=0

F ijmkln Φ

i j n k

l

Topological string-net condensation is described by a set of data

(N, δijk , F ijmkln )



• Not all sets (N, δijk, F ijmkln ) describe consistent string-net condensation.

Moore & Seiberg 89

l

m

q p

m m

j lk

j lk

m

j l j l

m

q

sr

p

nn

(a) (b) (c)

(d) (e)

r

k

k k

i

s

iii

i

•Pentagon identity

nF

mlqkp∗nF

jipmns∗F

js∗nlkr∗ = F

jipq∗kr∗F

riq∗mls∗

• The solutions of the above non-linear equations (called tensor cate-

gories) describe all the string-net condensed state.



• All string-net condensed states characterized by (N, δijk, F

ijm

kln ) can berealized by exactly soluble lattice models with 12 spin interactions.

• The low energy effective theories are topological theories, and almost

all the topological theories can be realized this way.

• The 6-j symbol of a group G, satisfy the pentagon identity.The fluctuations of the corresponding string-net condensation

→ gauge boson with gauge group G.

• The 6-j symbol of a quantum group G, satisfy the pentagon identity.

The corresponding string-net condensation

→ doubled Chern-Simons theory.



• Spins on Kagome lattice: Li = 0, 1, 2, · · ·• No string state = |Li = 0. Type-s string: string of Li = s spins

•Exactly soluble Hamiltonian is obtained from the data (N, δijk , F

ijmkln )

i

p

I

H strnet = gp

(1 − Bp) + U I

(1 − QI), Bp =N

s=0

asBsp

QI ac

b = δabc a

c

b Bsp

ab c

e

d

h

l

i

k

g

j

=

m,...,r

Bs,ghijklp,ghi jkl(abcdef )

a

b c

e

d

k’l’

i’h’

g’

j’



Bs,ghijklp,ghi jkl(abcdef ) = F

bg∗hs∗hg∗F ch∗i

s∗ih∗F di∗ js∗ ji∗F

ej∗ks∗k j∗F

f k∗ls∗lk∗F

al∗gs∗gl∗

• Bsp create a small loop of type-s string around hexagon p

Bsp

a

g

bh

c

i

d

j

ekf

l =

f k e

j

d

i

ch

b

g

a

l

s

=

ghi jkl

F bg∗hs∗hg∗F ch∗i

s∗ih∗F di∗ js∗ ji∗F

ej∗ks∗k j∗F

f k∗ls∗lk∗F

al∗gs∗gl∗

ab

h’c

i’

d

j’

ek’f

l’

g’

• (

s asBsp, QI) is a commuting set of operators. H strnet is exactlysoluble.

• Bsp term generates string hopping. QI term enforce the branching rule

in ground state.

Some examples from the solutions of the pentagon identity



Some examples – from the solutions of the pentagon identity

Z 2 gauge theory

• N = 1, δ000 = δ110 = 1, δ100 = 0 (only closed strings), F ijmkln leads to

Φ

=Φ

, Φ

=Φ

• The Hamiltonian

edge

leg

σ z

σ z

σ z

σ x

σ x

σ x

σ x

σ x

σ x

I

i

p

H strnet = gp

edges of p

σxi + U

I

legs of I

σzi

• Ground state wave function Φ(X) = const.• Effective theory: Z 2 gauge theory = U (1) × U (1) Chern-Simons theory

L =1

4πK IJ aIµ∂ ν aJλµνλ, K =

0 22 0

Doubled semion theory



Doubled semion theory

• N = 1, δ000 = δ110 = 1, δ100 = 0 (only closed strings), F ijmkln leads to

Φ

= − Φ

, Φ

= − Φ

•The Hamiltonian

edge

leg

legσ

z

σ z

σ z

σ x

σ x

σ x

σ x

σ x

σ z

σ z

σ z

σ z

σ zσ

x

σ z

I

i

p

H strnet = −I

legs of I

σzi +

p

(

edges of p

σx j )(

legs of p

(−)1−σz

j4 )

• Ground state wave function Φ(X) = (−)Xc, where Xc is the number

of loops in the string configuration X

• Effective theory: U (1) × U (1) Chern-Simons theory

L =1

4πK IJ aIµ∂ ν aJλµνλ, K =

2 00 −2

• Ends of open strings → Semions with θ = ±π/2



Doubled Yang-Lee theory

• N = 1, δ000 = δ110 = δ111 = 1, δ100 = 0 (branched string-nets),

F ijmkln leads to

Φ

=γ · Φ

Φ

=γ −1 · Φ

+ γ −1/2 · Φ

Φ =γ −1/2

·Φ −

γ −1

·Φ

where γ = 1+√

52

• Ground state has a string-net condensation• Effective theory: SO3(3) × SO3(3) Chern-Simons theory

• Ends of open strings → particles with non-Abelian statistics

Documents

Michael Levin and Xiao-Gang Wen- An origin of light and electrons: a unification of gauge interaction and Fermi statistics