It from qubits (not bits): Quantum information unify ...research.kek.jp/group/riron/workshop/theory2015Dec/theory2015wint… · y i@ i ˙ i ; form the 16-dim. spinor rep. of SO(10)

It from qubits (not bits):Quantum information unify everything

Xiao-Gang Wen, MIT, Dec, 2015

Xiao-Gang Wen, MIT, Dec, 2015 It from qubits (not bits): Quantum information unify everything

It from qubits: All elementary particles come from qubits

If elementary particles were spin-0 bosons, then they couldcome from qubits

• The space = an ocean of qubits.• The 0-state = the vacuum.• The 1-state = a spin-0 boson.

Space = A collection of qubits

1−state

0−state

bosonspin−0

• Ground state of the space-forming qubits = vacuumExcitations above the ground state = elementary particlesXiao-Gang Wen, MIT, Dec, 2015 It from qubits (not bits): Quantum information unify everything

But our elementary particles are very complicated

Seven wonders of our universe:

1. Identical particles2. Spin-1 bosons with only two-components (gauge bosons)3. Particles with Fermi statistics4. Fractional angular momentum (spin-1/2)5. Only left-hand fermions couple the SU(2)-gauge-bosons6. Lorentz symmetry7. Spin-2 bosons with only two-components (gravitons?)

Can simple qubits produce the above seven wonders?

• Yes for 1-6. Such a magic is possible if the qubits areLong-range entangledChen-Gu-Wen 10 (also refer as topologicallyorderedWen 89 in condensed matter systems)

• A great unification: Qubits unify gauge boson and fermion• A new view: Our world is made of quantum information!























Make long range entanglement (topo. order)

• The product states | ↑↓↓↑ ...〉 are not entangled• To make topological order, we need to sum over many different

product states, but we should not sum over everything.∑all spin configurations | ↑↓↓↑ ...〉 = | →→→→ ...〉

• Sum over a subset of spin configurations: sum over all the “stringstates”, where the up-spins form strings:

|Φclosed string〉 =∑

all loops Ψ∣∣∣ ⟩

|Φstring-net〉 =∑

all string-nets Ψ

∣∣∣∣∣⟩

→ string-net condensation Levin-Wen 05 (string-net liquid).

→ |Φstring〉 has long-range entanglement anda non-trivial topological order


Make long range entanglement (topo. order)

• The product states | ↑↓↓↑ ...〉 are not entangled• To make topological order, we need to sum over many different

product states, but we should not sum over everything.∑all spin configurations | ↑↓↓↑ ...〉 = | →→→→ ...〉

• Sum over a subset of spin configurations: sum over all the “stringstates”, where the up-spins form strings:

|Φclosed string〉 =∑

all loops Ψ∣∣∣ ⟩

|Φstring-net〉 =∑

all string-nets Ψ

∣∣∣∣∣⟩

→ string-net condensation Levin-Wen 05 (string-net liquid).

→ |Φstring〉 has long-range entanglement anda non-trivial topological order


The magic of long-range entanglements→ emergence of electromagnetic waves (photons)

• Wave in superfluid state |ΦSF〉 =∑

all position conf.

∣∣∣ ⟩:

density fluctuations:∂2t ρ− ∂2

xρ = 0→ Longitudinal wave

• Wave in closed-string liquid |Φstring〉 =∑

closed strings

∣∣∣ ⟩:

String density E (x) fluctuations → waves in string liquid. Closedstrings → ∂ · E = 0 → only two transverse modes. →E − ∂ × B = B + ∂ × E = ∂ · B = ∂ · E = 0. (E electric field)


The magic of long-range entanglement→ Emergence of Yang-Mills theory (gluons)

• If string has different types and can branch→ string-net liquid → Yang-Mills theory• Different ways that strings join → different gauge groups• String types →

representations ofgauge group.

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

Closed strings →U(1) gauge theoryString-nets → SU(2)× SU(3) Yang-Mills gauge theory


The magic of long-range entanglement→ Emergence of Fermi statistics

• In string liquids, the ends of string behavelike point particles (gauge charges).• String attached to the particle does not cost energy, but can

change the statistics of the particle.

• For string liquid state |Φ〉 =∑

all conf.

∣∣∣∣ ⟩, = +1

→ End of strings = boson (Higgs boson).

• For string liquid state |Φ〉 =∑

all conf.±∣∣∣∣ ⟩

, = −1

→ End of string = fermion (electron & quark). Levin-Wen 03

• A unification of gauge interactions and Fermi statistics


Chiral fermion problem: the qubit theory is wrong?

• Qubit theory is a lattice theory.

• For a long time, we thought:Lattice theory cannot produce chiral fermion/gauge theory, such asthe standard model.Qubit theory cannot produce an observed property: onlyleft-hand fermions couple the SU(2)-gauge-bosonsThis seems rule out the string-net theory of our world

• Recently, this long standing problem was solved:qubit lattice theory can produce a modified standard modelwith 16 Weyl fermions per family (SO(10) GUT). Wen 13

In other words, the SO(10) GUT:

H = ψ†α i∂iσ

iψα, ψα form the 16-dim. spinor rep. of SO(10)can be put on on a 3D spatial lattice, if we just allow directfermion-fermion interaction.


Chiral fermion problem: the qubit theory is wrong?

• Qubit theory is a lattice theory.

• For a long time, we thought:Lattice theory cannot produce chiral fermion/gauge theory, such asthe standard model.Qubit theory cannot produce an observed property: onlyleft-hand fermions couple the SU(2)-gauge-bosonsThis seems rule out the string-net theory of our world

• Recently, this long standing problem was solved:qubit lattice theory can produce a modified standard modelwith 16 Weyl fermions per family (SO(10) GUT). Wen 13

In other words, the SO(10) GUT:

H = ψ†α i∂iσ

iψα, ψα form the 16-dim. spinor rep. of SO(10)can be put on on a 3D spatial lattice, if we just allow directfermion-fermion interaction.


A solution of chiral fermion problem

A chiral fermion theory in d-dimensional space-time with agauge group G is can be put on lattice without breaking thegauge symmetry if Wen arXiv:1305.1045

(1) there exist (possibly symmetry breaking) mass terms thatmake all the fermions massive, and(2) πn(G/Ggrnd) = 0 for n ≤ d + 1, where Ggrnd is theunbroken symmetry group.

→The above SO(10) chiral fermion theory can be put on lattice withthe SO(10) realized as an on-site symmetry, if we allow directfermion-fermion interaction on lattice.

→The U(1)× SU(2)× SU(3) standard model can be put on latticeif we allow direct fermion-fermion interaction on lattice.

This is the first non-perturbative definition of the standard model


How to obtain a non-perturbative definition ofthe SO(10) chiral fermion model

Try to put 16 gapless chiral fermions (Weyl fermions) in 3+1D

H = ψ†α i∂iσ

iψα, ψα form the 16-dim. spinor rep. of SO(10)on a 3D spatial lattice.

• There is a 4+1D lattice free fermion model that is fully gapped inthe 4+1D bulk, but produces a gapless chiral fermion (Weyl)fermion at boundary H = ψ† i∂iσ

iψ

• We consider 16 copies of the above 4+1D lattice model. At theboundary, we obtain H = ψ†

α i∂iσiψα, α = 1, · · · , 16, which has a

SU(16) symmetry. It also has SO(10) ⊂ SU(16) symmetry withψα as the 16-dim. spinor representation.

• 4+1D → 3+1D lattice: The low energyeffective theory of such a 3+1D latticemodel is the SO(10) chiral fermion+ the mirror of the SO(10) chiral fermion.

the mirrorof chiralfermionfermion

chiral

theory theorystate

gapped


How to obtain a non-perturbative definition ofthe SO(10) chiral fermion model

We gap out the mirror sectorwithout breaking the SO(10) symmetryto obtain the SO(10) chiral fermionas the low energy effective theory

the mirrorof chiralfermionfermion

chiral

theory theory

gapping

gappedstate

How to gapped out the mirror sector:• We first, introduce a Higgs field φa in 10-dim. representation of

the SO(10). The Higgs field φa live only on the boundary of themirror sector.• We add the following SO(10) symmetric Higgs coupling for the

mirror sector δH = ψTα φaΓa

αβψβ + h.c . where Γaαβ is the SO(10)

invariant tensor. Γa has 8 +1 eigenvalues and 8 −1 eigenvalues.• In the Higgs condensed phases 〈φa〉 6= 0, the mirror sector is fully

gapped at the cost of SO(10) symmetry breaking.• We disorder the Higgs condensation via smooth fluctuations〈φa〉 = 0→ gap the mirror sector without SO(10) symm. breaking.


Are fermions really gapped in the disordered phase?

• All the fermions in the mirror sector are gappedin the ordered phase 〈φa〉 = const.

• If φa has long-wave-length fluctuations→ the disordered phase 〈φa〉 = 0, canall the fermions in the mirror sectorstill be gapped?

• Even when the fluctuations of φa(xµ)are long-wave-length, there are may bepoint/line/... defects where φa(xµ) = 0.

So the fermions in the mirror sector may not be gapped (bemassive) in the disordered symmetric phase.


Fermions gapped (massive) for SO(10) disordered phase

• The SO(10) vector field 〈φa〉 = const. breaks SO(10) down toSO(9). The long-wave-length fluctuations of φa are described by aSO(10)/SO(9) = S9 non-linear σ-model.

• In 3+1D boundary theory, S9 non-linear σ-model has nopoint/line/... defects, since πd(S9) = 0 for d = 0, · · · , 4. → forfixed random φa(xµ), there no fermion zero-model, gapless linemodes, etc. → the spectrum of D + φa(xµ)Γa has a finite gap.

• S9 non-linear σ-model does not even has induced Wess-Zuminoterm after integrating out the massive (gapped) fermions on theboudary and in the bulk, since πd(S9) = 0 for d = 5.→ the 3+1D effective S9 non-linear σ-model on the boundary hasa gapped/massive symmetric disordered phase.

We can define the SO(10) GUT and U(1)× SU(2)× SU(3)standard model as a 3+1D lattice gauge theory withinteracting fermions on lattice.




















A very conservative sufficient conditionto put chiral fermion/gauge theory on lattice

A chiral fermion theory in d-dimensional space-time with agauge group G is can be put on lattice without breaking thegauge symmetry if Wen arXiv:1305.1045

(1) there exist (possibly symmetry breaking) mass terms thatmake all the fermions massive, and(2) πn(G/Ggrnd) = 0 for n ≤ d + 1, where Ggrnd is theunbroken symmetry group.

A more general sufficient condition:Any anomaly-free chiral fermion/gauge theory can be put onlattice without breaking the gauge symmetry, if the latticefermions have a proper direct fermion-fermion interaction.


What is anomaly-free? Need a classification of anomalyTopological orders and SPT orders

2g

1g

2g

SY−SRE 1

SB−SRE 1

SB−LRE 2

SY−LRE 2

SB−LRE 1

SY−LRE 1

g1

SRE

SB−SRE 2

SY−SRE 2

symmetry breaking

(group theory)

(tensor category

(group cohomology

theory)

LRE 1 LRE 2

SET orders

w/ symmetry)

SPT orderes

intrinsic topo. order

topological order(tensor category)

• Topological orders = equivalent classes of gapped ground states

Two gapped ground states are equivalent if they can smoothlydeform into each other without closing energy gap

- Trivial topological order: equivalent to a product state.

• Symmetry protected trivial (SPT) order = equivalent classes ofsymmetric gapped ground states with trivial topological order.Two symmetric gapped states are equivalent if they can smoothlydeform into each other without closing energy gap and withoutbreaking the symmetry



2g

1g

2g

SY−SRE 1

SB−SRE 1

SB−LRE 2

SY−LRE 2

SB−LRE 1

SY−LRE 1

g1

SRE

SB−SRE 2

SY−SRE 2

symmetry breaking

(group theory)

(tensor category

(group cohomology

theory)

LRE 1 LRE 2

SET orders

w/ symmetry)

SPT orderes



• Topological orders = equivalent classes of gapped ground statesTwo gapped ground states are equivalent if they can smoothlydeform into each other without closing energy gap





2g

1g

2g

SY−SRE 1

SB−SRE 1

SB−LRE 2

SY−LRE 2

SB−LRE 1

SY−LRE 1

g1

SRE

SB−SRE 2

SY−SRE 2

symmetry breaking

(group theory)

(tensor category

(group cohomology

theory)

LRE 1 LRE 2

SET orders

w/ symmetry)

SPT orderes








2g

1g

2g

SY−SRE 1

SB−SRE 1

SB−LRE 2

SY−LRE 2

SB−LRE 1

SY−LRE 1

g1

SRE

SB−SRE 2

SY−SRE 2

symmetry breaking

(group theory)

(tensor category

(group cohomology

theory)

LRE 1 LRE 2

SET orders

w/ symmetry)

SPT orderes





• Symmetry protected trivial (SPT) order = equivalent classes ofsymmetric gapped ground states with trivial topological order.

Two symmetric gapped states are equivalent if they can smoothlydeform into each other without closing energy gap and withoutbreaking the symmetry



2g

1g

2g

SY−SRE 1

SB−SRE 1

SB−LRE 2

SY−LRE 2

SB−LRE 1

SY−LRE 1

g1

SRE

SB−SRE 2

SY−SRE 2

symmetry breaking

(group theory)

(tensor category

(group cohomology

theory)

LRE 1 LRE 2

SET orders

w/ symmetry)

SPT orderes







Topological orders and SPT orders classifyanomalies in one lower dimensions Wen arXiv:1303.1803

2g

1g

2g

SY−SRE 1

SB−SRE 1

SB−LRE 2

SY−LRE 2

SB−LRE 1

SY−LRE 1

g1

SRE

SB−SRE 2

SY−SRE 2

symmetry breaking

(group theory)

(tensor category

(group cohomology

theory)

LRE 1 LRE 2

SET orders

w/ symmetry)

SPT orderes



theorywith

Topolocallyordered

stateanomaly

grav.

SPTstate

symmetryon−sitewith

anomaly(symm.)

gaugewiththeory

SPTstate

symmetryon−sitewith

mixedwiththeory

anomalygauge−grav.


A lattice definition of any anomaly-free chiral theories

Wen arXiv:1303.1803

chiralgaugetheory

the mirrorof chiralgaugetheory

theorychiralanomaly−free

gaugeSPTstate

ν

statesSPTν

ν

ν

1

2

3

(a) (b)

gaugetheorychiral

the mirror ofanomaly−free

l

gapping

Anomaly-free boundary → the bulk phase has trivialtopological/SPT order → the boundary (the mirror sector) can befully gapped without breaking the symmetry.

Any anomaly-free chiral fermion/gauge theory can be put onlattice without breaking the gauge symmetry, if the latticefermions have a proper direct fermion-fermion interaction.


Documents

It from qubits (not bits): Quantum information unify ...research.kek.jp/group/riron/workshop/theory2015Dec/theory2015wint… · y i@ i ˙ i ; form the 16-dim. spinor rep. of SO(10)