The world of long-range entanglement– from new states of quantum matter to
a unification of gauge inter. and Fermi statistics
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013
.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Quantum phases ← Symmetry
there are much more than four phases:different phases = different symmetry breaking
(a) (b) (b)(a)
A A A
BB’φ φ φ
g g c g’ε ε ε
(a) (b)
From 230 ways of translation symmetry breaking, we obtain the230 crystal orders in 3D. The math foundation is group theory
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Quantum phases ← q-Topology
there are much more than symmetry-breaking phases:• Example:
- Quantum Hallstates σxy = m
ne2
h- Spin liquid states
• FQH states and spin-liquid states haveno symmetry breaking, no crystal order, no spin order, ...so they must have a new order – topological order Wen 89
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
What is topological order?
Two aspects:• Macroscopic definition• Microscopic picture
For example,• crystal order is defined/probed by X-ray diffraction:
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Macroscopic definition of topological orders
Topological order can be defined “experimentally” through twounusual topological probes (at least in 2D)
(1) Topology-dependent ground state degeneracy Dg Wen 89
Deg.=D Deg.=D1 2Deg.=1
g=0
g=1
g=2
(2) Non-Abelian geometric’s phases of the degenerate groundstate from deforming the torus: Wen 90
- Shear deformation T : |Ψα〉 → |Ψ′α〉 = Tαβ|Ψβ〉
- 90◦ rotation S : |Ψα〉 → |Ψ′′α〉 = Sαβ|Ψβ〉
• T ,S , define topological order “experimentally”.
• T ,S is a universal probe for any 2D topological orders, just likeX-ray is a universal probe for any crystal orders.Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Symmetry-breaking/topological orders through experiments
Order Experiment
Crystal order X-ray diffraction
Ferromagnetic order Magnetization
Anti-ferromagnetic order Neutron scattering
Superconducting order Zero-resistance & Meissner effect
Topological order Topological degeneracy,(Global dancing pattern) non-Abelian geometric phase
• The linear-response probe Zero-resistance and Meissner effectdefine superconducting order. Treating the EM fields as non-dynamical fields
• The topological probe Topological degeneracy and non-Abeliangeometric phases T ,S define a completely new class of order –topologically order.
• T ,S determines the quasiparticle statistics. Keski-Vakkuri & Wen 93;
Zhang-Grover-Turner-Oshikawa-Vishwanath 12; Cincio-Vidal 12
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
What is the microscopic picture of topological order?
Deg.=D Deg.=D1 2Deg.=1
g=0
g=1
g=2
represent an experimental definition of topological order.
• But what is the microscopic understanding of topological order?
• Zero-resistance and Meissner effect → experimental definition ofsuperconducting order.
• A microscopic picture ofsuperconducting order:electron-pair condensationBardeen-Cooper-Schrieffer 57
• A microscopic picture of topological order:long-range entanglements Chen-Gu-Wen 10
(defined by local unitary trans. andmotivated by topological entanglemententropy). Kitaev-Preskill 06,Levin-Wen 06
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
What is the microscopic picture of topological order?
Deg.=D Deg.=D1 2Deg.=1
g=0
g=1
g=2
represent an experimental definition of topological order.
• But what is the microscopic understanding of topological order?
• Zero-resistance and Meissner effect → experimental definition ofsuperconducting order.
• A microscopic picture ofsuperconducting order:electron-pair condensationBardeen-Cooper-Schrieffer 57
• A microscopic picture of topological order:long-range entanglements Chen-Gu-Wen 10
(defined by local unitary trans. andmotivated by topological entanglemententropy). Kitaev-Preskill 06,Levin-Wen 06
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Quantum entanglements through examples
• | ↑〉 ⊗ | ↓〉 = direct-product state → unentangled (classical)
• | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉 → entangled (quantum)• | ↑〉 ⊗ | ↑〉+ | ↓〉 ⊗ | ↓〉+ | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉
→ more entangled
= (| ↑〉+ | ↓〉)⊗ (| ↑〉+ | ↓〉) = |x〉 ⊗ |x〉 → unentangled
• = | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉... → unentangled
• = (| ↓↑〉 − | ↑↓〉)⊗ (| ↓↑〉 − | ↑↓〉)⊗ ... →short-range entangled (SRE) entangled
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Quantum entanglements through examples
• | ↑〉 ⊗ | ↓〉 = direct-product state → unentangled (classical)• | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉 → entangled (quantum)
• | ↑〉 ⊗ | ↑〉+ | ↓〉 ⊗ | ↓〉+ | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉
→ more entangled
= (| ↑〉+ | ↓〉)⊗ (| ↑〉+ | ↓〉) = |x〉 ⊗ |x〉 → unentangled
• = | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉... → unentangled
• = (| ↓↑〉 − | ↑↓〉)⊗ (| ↓↑〉 − | ↑↓〉)⊗ ... →short-range entangled (SRE) entangled
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Quantum entanglements through examples
• | ↑〉 ⊗ | ↓〉 = direct-product state → unentangled (classical)• | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉 → entangled (quantum)• | ↑〉 ⊗ | ↑〉+ | ↓〉 ⊗ | ↓〉+ | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉 → more entangled
= (| ↑〉+ | ↓〉)⊗ (| ↑〉+ | ↓〉) = |x〉 ⊗ |x〉 → unentangled
• = | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉... → unentangled
• = (| ↓↑〉 − | ↑↓〉)⊗ (| ↓↑〉 − | ↑↓〉)⊗ ... →short-range entangled (SRE) entangled
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Quantum entanglements through examples
• | ↑〉 ⊗ | ↓〉 = direct-product state → unentangled (classical)• | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉 → entangled (quantum)• | ↑〉 ⊗ | ↑〉+ | ↓〉 ⊗ | ↓〉+ | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉
→ more entangled
= (| ↑〉+ | ↓〉)⊗ (| ↑〉+ | ↓〉) = |x〉 ⊗ |x〉 → unentangled
• = | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉... → unentangled
• = (| ↓↑〉 − | ↑↓〉)⊗ (| ↓↑〉 − | ↑↓〉)⊗ ... →short-range entangled (SRE) entangled
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Quantum entanglements through examples
• | ↑〉 ⊗ | ↓〉 = direct-product state → unentangled (classical)• | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉 → entangled (quantum)• | ↑〉 ⊗ | ↑〉+ | ↓〉 ⊗ | ↓〉+ | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉
→ more entangled
= (| ↑〉+ | ↓〉)⊗ (| ↑〉+ | ↓〉) = |x〉 ⊗ |x〉 → unentangled
• = | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉... → unentangled
• = (| ↓↑〉 − | ↑↓〉)⊗ (| ↓↑〉 − | ↑↓〉)⊗ ... →short-range entangled (SRE) entangled
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Quantum entanglements through examples
• | ↑〉 ⊗ | ↓〉 = direct-product state → unentangled (classical)• | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉 → entangled (quantum)• | ↑〉 ⊗ | ↑〉+ | ↓〉 ⊗ | ↓〉+ | ↑〉 ⊗ | ↓〉+ | ↓〉 ⊗ | ↑〉
→ more entangled
= (| ↑〉+ | ↓〉)⊗ (| ↑〉+ | ↓〉) = |x〉 ⊗ |x〉 → unentangled
• = | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉 ⊗ | ↑〉 ⊗ | ↓〉... → unentangled
• = (| ↓↑〉 − | ↑↓〉)⊗ (| ↓↑〉 − | ↑↓〉)⊗ ... →short-range entangled (SRE) entangled
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
How to make long range entanglements (topo. orders)
To make topological order, we need to sum over different productstates, but we should not sum over everything.
• Sum over a subset of the particle configurations, by first join theparticles into strings, then sum over the loop states
→ string-net condensation (string liquid):Kogut-Susskind 75, Kitaev 97, Wen 03, Levin-Wen 05
|Φloop〉 =∑
all loop conf.
∣∣∣∣ ⟩which is not a direct-product
state and not a local deformation of direct-product states→ non-trivial topological orders (long-range entanglements)
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Topological orders through pictures
FQH state String liquid (spin liquid)
• Global dance:All spins/particles dance following a local dancing “rules”
→ The spins/particles dance collectively→ a global dancing pattern.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Local dancing rule → global dancing pattern
• Local dancing rules of string-net liquid:- Up-spins from closed strings with no ends.- Strings can fluctuate and re-connect freely.
Φstr
( )= Φstr
( ), Φstr
( )= Φstr
( )→ Global dancing pattern Φstr
( )= 1
• Local dancing rules of another string liquid:(1) Dance while holding hands (no open ends)
(2) Φstr
( )= Φstr
( ), Φstr
( )= −Φstr
( )→ Global dancing pattern Φstr
( )= (−)# of loops
• Two string-net condensations → two topological orders Levin-Wen 05
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Local dancing rule → global dancing pattern
• Local dancing rules of string-net liquid:- Up-spins from closed strings with no ends.- Strings can fluctuate and re-connect freely.
Φstr
( )= Φstr
( ), Φstr
( )= Φstr
( )→ Global dancing pattern Φstr
( )= 1
• Local dancing rules of another string liquid:(1) Dance while holding hands (no open ends)
(2) Φstr
( )= Φstr
( ), Φstr
( )= −Φstr
( )→ Global dancing pattern Φstr
( )= (−)# of loops
• Two string-net condensations → two topological orders Levin-Wen 05
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Emergence of fractional spin/statistics
• Why electron carry spin-1/2 and Fermi statistics?
• Ends of strings are point-like excitations,which can carry spin-1/2 and Fermi statistics?Fidkowski-Freedman-Nayak-Walker-Wang 06
• Φstr
( )= 1 string liquid Φstr
( )= Φstr
( )360◦ rotation: → and = → : R360◦ =
(0 11 0
)+ has a spin 0 mod 1. − has a spin 1/2 mod 1.
• Φstr
( )= (−)# of loops string liquid Φstr
( )= −Φstr
( )360◦ rotation: → and = − → − : R360◦ =
(0 −11 0
)+ i has a spin −1/4 mod 1. − i has a spin 1/4 mod 1.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Emergence of fractional spin/statistics
• Why electron carry spin-1/2 and Fermi statistics?• Ends of strings are point-like excitations,
which can carry spin-1/2 and Fermi statistics?Fidkowski-Freedman-Nayak-Walker-Wang 06
• Φstr
( )= 1 string liquid Φstr
( )= Φstr
( )360◦ rotation: → and = → : R360◦ =
(0 11 0
)+ has a spin 0 mod 1. − has a spin 1/2 mod 1.
• Φstr
( )= (−)# of loops string liquid Φstr
( )= −Φstr
( )360◦ rotation: → and = − → − : R360◦ =
(0 −11 0
)+ i has a spin −1/4 mod 1. − i has a spin 1/4 mod 1.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Emergence of fractional spin/statistics
• Why electron carry spin-1/2 and Fermi statistics?• Ends of strings are point-like excitations,
which can carry spin-1/2 and Fermi statistics?Fidkowski-Freedman-Nayak-Walker-Wang 06
• Φstr
( )= 1 string liquid Φstr
( )= Φstr
( )360◦ rotation: → and = → : R360◦ =
(0 11 0
)+ has a spin 0 mod 1. − has a spin 1/2 mod 1.
• Φstr
( )= (−)# of loops string liquid Φstr
( )= −Φstr
( )360◦ rotation: → and = − → − : R360◦ =
(0 −11 0
)+ i has a spin −1/4 mod 1. − i has a spin 1/4 mod 1.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Emergence of fractional spin/statistics
• Why electron carry spin-1/2 and Fermi statistics?• Ends of strings are point-like excitations,
which can carry spin-1/2 and Fermi statistics?Fidkowski-Freedman-Nayak-Walker-Wang 06
• Φstr
( )= 1 string liquid Φstr
( )= Φstr
( )360◦ rotation: → and = → : R360◦ =
(0 11 0
)+ has a spin 0 mod 1. − has a spin 1/2 mod 1.
• Φstr
( )= (−)# of loops string liquid Φstr
( )= −Φstr
( )360◦ rotation: → and = − → − : R360◦ =
(0 −11 0
)+ i has a spin −1/4 mod 1. − i has a spin 1/4 mod 1.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Spin-statistics theorem
(a) (b) (c) (d) (e)
• (a) → (b) = exchange two string-ends.
• (d) → (e) = 360◦ rotation of a string-end.
• Amplitude (a) = Amplitude (e)
• Exchange two string-ends plus a 360◦ rotation of one of thestring-end generate no phase.
→ Spin-statistics theorem
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
New math for entanglements → Tensor category theory
Levin-Wen 05
Generalize the local dancing rule
Φ
( )= Φ
( )to
Φ
(kj
i lm
)=
N∑n=0
F ijmkln Φ
(ij n k
l
)which must satisfy
N∑n=0
Fmlqkpn F
jipmnsF
jsnlkr = F jip
qkrFriqmls
The theory about the solutions = tensor category theory→ classify 2D gapped phases with no symmetry (topological order)→ produce non-Abelian statistics in 2D and fermions in any dim.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Long range entanglements (closed oriented strings)→ emergence of electromagnetic waves (photons)
• Wave in superfluid state |ΦSF〉 =∑
all position conf.
∣∣∣ ⟩:
density fluctuations:Euler eq.: ∂2
t ρ− ∂2xρ = 0
→ Longitudinal wave
• Wave in closed-string liquid |Φstring〉 =∑
closed strings
∣∣∣ ⟩:
String density E(x) fluctuations → waves in string condensed state.Strings have no ends → ∂ · E = 0 → only two transverse modes.Equation of motion for string density → Maxwell equation:E− ∂ × B = B + ∂ × E = ∂ · B = ∂ · E = 0. (E electric field)Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Long range entanglements (string nets)→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
A picture of our vacuum A string−net theory of light and electrons
Closed strings → Maxwell gauge theoryString-nets → Yang-Mills gauge theoryXiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
A unification of gauge interaction and Fermi statistics
• Previous understanding of gauge interaction:- Connection of a fiber bundle.Weyl 20
- Geometric phases of quantum spins.- “Glue” to glue partons into physical spins. (Dynamically generated . gauge field). D’Adda & Di Vecchia & Luscher 78, Witten 79, Baskaran & Anderson 88
- Collective modes of long-range entanglement. (of strings) Foerster & Nielsen & Ninomiya 80, Kitaev 97, Wen 02
. Senthil & Motrunich 02, Levin-Wen 05
• Previous understanding of emergent fermions:- Bound states of charge and flux (works only in 2+1D).
Leinaas &Myrheim 77, Wilczek 82
- Bound states of charge and monople (only for U(1))Goldhaber 82, Wilczek 82
- End of strings (for any dim. and any gauge groups) Levin-Wen 03
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Can long-range entangled qubits produce everything?
At moment, the fundamental properties in our universe can besummarized by the following short list:. (1) Locality.. (2) Identical particles.. (3) Gauge interactions.. (4) Fermi statistics.. (5) Tiny masses of fermions (∼ 10−20 Planck mass).. (6) Chiral fermions.. (7) Lorentz invariance.. (8) Gravity.
Long-range entangled qubits can produce (1–6)→ (an extended) standard model.An experimental prediction: All composite fermions must carrynon-trivial gauge charge. There are additional discrete gaugetheories beyond U(1)× SU(2)× SU(3). There are new cosmicstrings (Z2 gauge flux) at high energies.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Can long-range entangled qubits produce everything?
At moment, the fundamental properties in our universe can besummarized by the following short list:. (1) Locality.. (2) Identical particles.. (3) Gauge interactions.. (4) Fermi statistics.. (5) Tiny masses of fermions (∼ 10−20 Planck mass).. (6) Chiral fermions.. (7) Lorentz invariance.. (8) Gravity.
Long-range entangled qubits can produce (1–6)→ (an extended) standard model.
An experimental prediction: All composite fermions must carrynon-trivial gauge charge. There are additional discrete gaugetheories beyond U(1)× SU(2)× SU(3). There are new cosmicstrings (Z2 gauge flux) at high energies.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Can long-range entangled qubits produce everything?
At moment, the fundamental properties in our universe can besummarized by the following short list:. (1) Locality.. (2) Identical particles.. (3) Gauge interactions.. (4) Fermi statistics.. (5) Tiny masses of fermions (∼ 10−20 Planck mass).. (6) Chiral fermions.. (7) Lorentz invariance.. (8) Gravity.
Long-range entangled qubits can produce (1–6)→ (an extended) standard model.An experimental prediction: All composite fermions must carrynon-trivial gauge charge. There are additional discrete gaugetheories beyond U(1)× SU(2)× SU(3). There are new cosmicstrings (Z2 gauge flux) at high energies.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
What is long-range entanglement
For gapped systems with no symmetry:• According to Landau theory, no symmetry to break→ all systems belong to one trivial phase
• Thinking about entanglement: Chen-Gu-Wen 2010
- There are long range entangled (LRE) states
→ many phases
- There are short range entangled (SRE) states
→ one phase
|LRE〉 6= |product state〉 = |SRE〉
local unitarytransformation
LREproduct
SREstate
state
local unitarytransformation
LRE 1 LRE 2
local unitarytransformation
productstate
productstate
SRE SRE
g1
2g
SRE
LRE 1 LRE 2
phase
transition
topological order
• All SRE states belong to the same trivial phase
• LRE states can belong to many different phases= different patterns of long-range entanglements defined by the LU trans.
= different topological orders→ A category theory of topological order Levin-Wen 05, Chen-Gu-Wen 2010
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
What is long-range entanglement
For gapped systems with no symmetry:• According to Landau theory, no symmetry to break→ all systems belong to one trivial phase• Thinking about entanglement: Chen-Gu-Wen 2010
- There are long range entangled (LRE) states
→ many phases
- There are short range entangled (SRE) states
→ one phase
|LRE〉 6= |product state〉 = |SRE〉
local unitarytransformation
LREproduct
SREstate
state
local unitarytransformation
LRE 1 LRE 2
local unitarytransformation
productstate
productstate
SRE SRE
g1
2g
SRE
LRE 1 LRE 2
phase
transition
topological order
• All SRE states belong to the same trivial phase
• LRE states can belong to many different phases= different patterns of long-range entanglements defined by the LU trans.
= different topological orders→ A category theory of topological order Levin-Wen 05, Chen-Gu-Wen 2010
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
What is long-range entanglement
For gapped systems with no symmetry:• According to Landau theory, no symmetry to break→ all systems belong to one trivial phase• Thinking about entanglement: Chen-Gu-Wen 2010
- There are long range entangled (LRE) states → many phases
- There are short range entangled (SRE) states → one phase
|LRE〉 6= |product state〉 = |SRE〉
local unitarytransformation
LREproduct
SREstate
state
local unitarytransformation
LRE 1 LRE 2
local unitarytransformation
productstate
productstate
SRE SRE
g1
2g
SRE
LRE 1 LRE 2
phase
transition
topological order
• All SRE states belong to the same trivial phase
• LRE states can belong to many different phases= different patterns of long-range entanglements defined by the LU trans.
= different topological orders→ A category theory of topological order Levin-Wen 05, Chen-Gu-Wen 2010
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Short-range entanglements w/ symmetry → SPT phases
For gapped systems with a symmetry G (no symmetry breaking):
• there are LRE symmetric states → many different phases
• there are SRE symmetric states → one phase
many different phases
We may call them symmetry protected trivial (SPT) phase
or symmetry protected topological (SPT) phase
1g
2g
2g
SY−SRE 1
SB−SRE 1
SB−LRE 2
SY−LRE 2
SB−LRE 1
SY−LRE 1
SB−SRE 2
SY−SRE 2
g1
LRE 2LRE 1
SRESPT phases
symmetry breaking
(group theory)
topological orders
( ??? )
( ??? )
topological ordertopological order
symmetrypreserve
no symmetry
phase
transition
SPT 1 SPT 2
- Haldane phase of 1D spin-1 chain w/ SO(3) symm. Haldane 83
- Topo. insulators w/ U(1)× T symm.: 2D Kane-Mele 05; Bernevig-Zhang 06
and 3D Moore-Balents 07; Fu-Kane-Mele 07
1D 2D 3D
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Short-range entanglements w/ symmetry → SPT phases
For gapped systems with a symmetry G (no symmetry breaking):
• there are LRE symmetric states → many different phases
• there are SRE symmetric states →
one phase
many different phases
We may call them symmetry protected trivial (SPT) phase
or symmetry protected topological (SPT) phase
1g
2g
2g
SY−SRE 1
SB−SRE 1
SB−LRE 2
SY−LRE 2
SB−LRE 1
SY−LRE 1
SB−SRE 2
SY−SRE 2
g1
LRE 2LRE 1
SRESPT phases
symmetry breaking
(group theory)
topological orders
( ??? )
( ??? )
topological ordertopological order
symmetrypreserve
no symmetry
phase
transition
SPT 1 SPT 2
- Haldane phase of 1D spin-1 chain w/ SO(3) symm. Haldane 83
- Topo. insulators w/ U(1)× T symm.: 2D Kane-Mele 05; Bernevig-Zhang 06
and 3D Moore-Balents 07; Fu-Kane-Mele 07
1D 2D 3D
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Short-range entanglements w/ symmetry → SPT phases
For gapped systems with a symmetry G (no symmetry breaking):
• there are LRE symmetric states → many different phases
• there are SRE symmetric states →
one phase
many different phases
We may call them symmetry protected trivial (SPT) phase
or symmetry protected topological (SPT) phase
1g
2g
2g
SY−SRE 1
SB−SRE 1
SB−LRE 2
SY−LRE 2
SB−LRE 1
SY−LRE 1
SB−SRE 2
SY−SRE 2
g1
LRE 2LRE 1
SRESPT phases
symmetry breaking
(group theory)
topological orders
( ??? )
( ??? )
topological ordertopological order
symmetrypreserve
no symmetry
phase
transition
SPT 1 SPT 2
- Haldane phase of 1D spin-1 chain w/ SO(3) symm. Haldane 83
- Topo. insulators w/ U(1)× T symm.: 2D Kane-Mele 05; Bernevig-Zhang 06
and 3D Moore-Balents 07; Fu-Kane-Mele 07
1D
2D 3D
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Short-range entanglements w/ symmetry → SPT phases
For gapped systems with a symmetry G (no symmetry breaking):
• there are LRE symmetric states → many different phases
• there are SRE symmetric states →
one phase
many different phases
We may call them symmetry protected trivial (SPT) phaseor symmetry protected topological (SPT) phase
1g
2g
2g
SY−SRE 1
SB−SRE 1
SB−LRE 2
SY−LRE 2
SB−LRE 1
SY−LRE 1
SB−SRE 2
SY−SRE 2
g1
LRE 2LRE 1
SRESPT phases
symmetry breaking
(group theory)
topological orders
( ??? )
( ??? )
topological ordertopological order
symmetrypreserve
no symmetry
phase
transition
SPT 1 SPT 2
- Haldane phase of 1D spin-1 chain w/ SO(3) symm. Haldane 83
- Topo. insulators w/ U(1)× T symm.: 2D Kane-Mele 05; Bernevig-Zhang 06
and 3D Moore-Balents 07; Fu-Kane-Mele 07
1D 2D 3DXiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Interacting bosonic SPT phase: A group-cohomologytheory
Chen-Liu-Wen 11, Chen-Gu-Liu-Wen 11
For any symmetry group Gand in any dimensions d
Two key observations:• Short-range-entangled states
have a simple canonical form:A rep.
A rep.
Nota rep.
after we treat each block as an effective site.
- Each effective site has several independent degrees of freedomsentangled with its neighbors.
- The combined degrees of freedoms on a site form a rep. of G
• Each degree of freedoms on the effective site may not form arepresentation of G , but the whole state is still inv. under GXiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Non-trivial short-range entangled states w/ symmetry
• Haldane phase w/ SO(3) symm.: spin-1/2 is not a rep. of SO(3)
+spin−1
spin−1/2 spin−1/2
spin−1/2 x spin−1/2spin−0
one siteone site one site
• 2D SPT phase w/ Z2 symm.:Chen-Liu-Wen 2011
CZ 121 2
34
One
Site
(spin−1/2)4
a b
cd
e
f
gh
i
j
k l
- Physical states on each site:(spin- 1
2 )4 = |α〉 ⊗ |β〉 ⊗ |γ〉 ⊗ |λ〉- The ground state wave function:|ΨCZX 〉 = ⊗all squares(| ↑↑↑↑〉+ | ↓↓↓↓〉)
- The on-site Z2 symmetry: (acting on each site |α〉 ⊗ |β〉 ⊗ |γ〉 ⊗ |λ〉):UCZX = UCZUX , UX = X1X2X3X4, UCZ = CZ12CZ23CZ34CZ41
CZ : | ↑↑〉 → | ↑↑〉, | ↑↓〉 → | ↑↓〉, | ↓↑〉 → | ↓↑〉, | ↓↓〉 → −| ↓↓〉
- Z2 symm. Hamiltonian H =∑
�Hp, Hp = −XabcdPef PghPijPkl ,Xabcd = | ↑↑↑↑〉〈↓↓↓↓ |+ | ↓↓↓↓〉〈↑↑↑↑ |, P = | ↑↑〉〈↑↑ |+ | ↓↓〉〈↓↓ |.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Non-trivial short-range entangled states w/ symmetry
• Haldane phase w/ SO(3) symm.: spin-1/2 is not a rep. of SO(3)
+spin−1
spin−1/2 spin−1/2
spin−1/2 x spin−1/2spin−0
one siteone site one site
• 2D SPT phase w/ Z2 symm.:Chen-Liu-Wen 2011
CZ 121 2
34
One
Site
(spin−1/2)4
a b
cd
e
f
gh
i
j
k l
- Physical states on each site:(spin- 1
2 )4 = |α〉 ⊗ |β〉 ⊗ |γ〉 ⊗ |λ〉- The ground state wave function:|ΨCZX 〉 = ⊗all squares(| ↑↑↑↑〉+ | ↓↓↓↓〉)
- The on-site Z2 symmetry: (acting on each site |α〉 ⊗ |β〉 ⊗ |γ〉 ⊗ |λ〉):UCZX = UCZUX , UX = X1X2X3X4, UCZ = CZ12CZ23CZ34CZ41
CZ : | ↑↑〉 → | ↑↑〉, | ↑↓〉 → | ↑↓〉, | ↓↑〉 → | ↓↑〉, | ↓↓〉 → −| ↓↓〉
- Z2 symm. Hamiltonian H =∑
�Hp, Hp = −XabcdPef PghPijPkl ,Xabcd = | ↑↑↑↑〉〈↓↓↓↓ |+ | ↓↓↓↓〉〈↑↑↑↑ |, P = | ↑↑〉〈↑↑ |+ | ↓↓〉〈↓↓ |.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Non-trivial short-range entangled states w/ symmetry
• Haldane phase w/ SO(3) symm.: spin-1/2 is not a rep. of SO(3)
+spin−1
spin−1/2 spin−1/2
spin−1/2 x spin−1/2spin−0
one siteone site one site
• 2D SPT phase w/ Z2 symm.:Chen-Liu-Wen 2011
CZ 121 2
34
One
Site
(spin−1/2)4
a b
cd
e
f
gh
i
j
k l
- Physical states on each site:(spin- 1
2 )4 = |α〉 ⊗ |β〉 ⊗ |γ〉 ⊗ |λ〉- The ground state wave function:|ΨCZX 〉 = ⊗all squares(| ↑↑↑↑〉+ | ↓↓↓↓〉)
- The on-site Z2 symmetry: (acting on each site |α〉 ⊗ |β〉 ⊗ |γ〉 ⊗ |λ〉):UCZX = UCZUX , UX = X1X2X3X4, UCZ = CZ12CZ23CZ34CZ41
CZ : | ↑↑〉 → | ↑↑〉, | ↑↓〉 → | ↑↓〉, | ↓↑〉 → | ↓↑〉, | ↓↓〉 → −| ↓↓〉
- Z2 symm. Hamiltonian H =∑
�Hp, Hp = −XabcdPef PghPijPkl ,Xabcd = | ↑↑↑↑〉〈↓↓↓↓ |+ | ↓↓↓↓〉〈↑↑↑↑ |, P = | ↑↑〉〈↑↑ |+ | ↓↓〉〈↓↓ |.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Non-trivial short-range entangled states w/ symmetry
• Haldane phase w/ SO(3) symm.: spin-1/2 is not a rep. of SO(3)
+spin−1
spin−1/2 spin−1/2
spin−1/2 x spin−1/2spin−0
one siteone site one site
• 2D SPT phase w/ Z2 symm.:Chen-Liu-Wen 2011
CZ 121 2
34
One
Site
(spin−1/2)4
a b
cd
e
f
gh
i
j
k l
- Physical states on each site:(spin- 1
2 )4 = |α〉 ⊗ |β〉 ⊗ |γ〉 ⊗ |λ〉- The ground state wave function:|ΨCZX 〉 = ⊗all squares(| ↑↑↑↑〉+ | ↓↓↓↓〉)
- The on-site Z2 symmetry: (acting on each site |α〉 ⊗ |β〉 ⊗ |γ〉 ⊗ |λ〉):UCZX = UCZUX , UX = X1X2X3X4, UCZ = CZ12CZ23CZ34CZ41
CZ : | ↑↑〉 → | ↑↑〉, | ↑↓〉 → | ↑↓〉, | ↓↑〉 → | ↓↑〉, | ↓↓〉 → −| ↓↓〉
- Z2 symm. Hamiltonian H =∑
�Hp, Hp = −XabcdPef PghPijPkl ,Xabcd = | ↑↑↑↑〉〈↓↓↓↓ |+ | ↓↓↓↓〉〈↑↑↑↑ |, P = | ↑↑〉〈↑↑ |+ | ↓↓〉〈↓↓ |.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Edge excitations for the 2D Z2 SPT state
• Bulk Hamiltonian H =∑
�Hp, Hp = −XabcdPef PghPij ,Pkl ,Xabcd = | ↑↑↑↑〉〈↓↓↓↓ |+ | ↓↓↓↓〉〈↑↑↑↑ |, P = | ↑↑〉〈↑↑ |+ | ↓↓〉〈↓↓ |.
• Edge excitations: gapless or break the Z2
symmetry, robust against any perturba-tions that do not break the Z2 symmetry.
• Edge effective spin |↑〉 and |↓〉.• Edge effective Z2 symmetry : exp
(∑i
14 (Zi Zi+1 − 1)
)∏i Xi
which cannot be written as UZ2 =∏
i Oi , such as UZ2 =∏
Xi .Not an on-site symmetry!
• Edge effective Hamiltonian (c = 1 gapless if the Z2 is not broken)Hedge = −J
∑Zi Zi+1 + Bx
∑[Xi + Zi−1Xi Zi+1]
+By
∑[Yi − Zi−1Yi Zi+1]
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
SPT states for any symmetry in any dimensions
• Generic SPT state: (| ↑↑↑↑〉+ | ↓↓↓↓〉 →∑
g∈G |gggg〉)|SPT state〉 = ⊗all squares
∑g∈G|gggg〉
CZ 121 2
34
One
Site
(spin−1/2)4
a b
cd
e
f
gh
i
j
k l
• Generic twisted symmetry transformations
|g1, g2, g3, g4〉 → η(g1, g2, g3, g4)|gg1, gg2, gg3, gg4〉
where the twisting phase factor η(g1, g2, g3, g4) correspond tococycles in Hd+1[G ,UT (1)].
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Group cohomology Hd+1[G ,UT (1)]→ bosonic SPT phasesSymmetry G d = 0 d = 1 d = 2 d = 3
U(1) o ZT2 (top. ins.) Z Z2 (0) Z2 (Z2) Z2
2 (Z2)U(1) o ZT
2 × trans Z Z× Z2 Z× Z32 Z× Z8
2
U(1)× ZT2 (spin sys.) 0 Z2
2 0 Z32
U(1)× ZT2 × trans 0 Z2
2 Z42 Z9
2
ZT2 (top. SC) 0 Z2 (Z) 0 (0) Z2 (0)ZT
2 × trans 0 Z2 Z22 Z4
2
U(1) Z 0 Z 0U(1)× trans Z Z Z2 Z4
Zn Zn 0 Zn 0Zn × trans Zn Zn Z2
n Z4n
D2h = Z2 × Z2 × ZT2 Z2
2 Z42 Z6
2 Z92
SO(3) 0 Z2 Z 0SO(3)× ZT
2 0 Z22 Z2 Z3
2
Table of Hd+1[G ,UT (1)]“ZT
2 ”: time reversal,“trans”: translation,others: on-site symm.0 → only trivial phase.(Z2)→ free fermion result
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)
SPT phases
(tensor category
(group cohomology
theory)
LRE 1 LRE 2
SET orders
w/ symmetry)intrinsic topo. order
topological order
(tensor category)
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
A classification of gauge anomalies
• SPT states in d + 1 space-time dimensions are classified byHd+1[G ,U(1)].
• The non-on-site symmetries in d space-time dimensions areclassified by Hd+1[G ,U(1)]
• The non-on-site symmetries = the gauge anomalies
• The gauge anomalies in d space-time dimensions are classified byHd+1[G ,U(1)]
The free part of Hd+1[G ,U(1)]→ Adler-Bell-Jackiw chiral-anomalies
The torsion part of Hd+1[G ,U(1)]→ new global anomalies (?)
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Many (non-exact) examples of SPT states
• U(1) bosonic SPT state in 2+1D: Lu Vishwanath 12, Senthil Levin 12, Chen Wen 12
L =1
4πKIJa
Iµ∂µa
Jλε
µνλ +1
2πqIAµ∂µa
Iλε
µνλ, K =
(0 11 2k
), q =
(11
)Even-integer quantize Hall conductance (σxy = (2k − 2)/2π).
• U f (1) fermionic SPT state in 2+1D: Lu Vishwanath 12, Wen 12
K =
(1 00 −1
), q =
(2k1 + 12k2 + 1
)Hall conductance quantized as 8 times integers.
• SU(2) bosonic SPT state in 2+1D (λ→∞):
S =
∫M
1
λ|∂g |2 − i
θ
24π2
∫MTr(g−1dg)3, θ = 2πk , g ∈ SU(2)
SU(2) symmetry acts as g(x)→ hg(x), h ∈ SU(2)Quantized spin-Hall conductance.
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Entanglementis everything
Xiao-Gang Wen, Perimeter Institute/MIT, Feb, 2013 The world of long-range entanglement – from new states of quantum matter to a unification of gauge inter. and Fermi statistics
Top Related