Can you hear the shape of Schrödinger’s cat?eduardo.physics.illinois.edu/homepage/urbana... · Quantum Phase Transitions and Quantum Criticality Phase transitions in a macroscopic

Can you hear the shape of Schrödinger’s cat?Colloquium at the Department of Physics of the University of

Illinois,December 4, 2008

Eduardo Fradkin

Department of PhysicsUniversity of Illinois at Urbana Champaign

December 4, 2008

Outline

Outline

◮ Topological phases in condensed matter and quantum phasetransitions

Outline


◮ Quantum entanglement

Outline



◮ Scaling of the entanglement entropy and quantum criticality

Outline




◮ Entanglement entropy scaling in 1D and in 2D quantum criticalsystems

Outline





◮ Entanglement entropy in topological phases

Outline





◮ Entanglement entropy in topological phases

◮ Topological quantum computing?

◮ Conclusions

Quantum Phase Transitions and Quantum Criticality


◮ Phase transitions in a macroscopic quantum system at T = 0 as aparameter (coupling constant) is varied.



◮ Typical example: Ising spin chain in a transverse field,

H = −∑

n

σz (n)σz (n + 1) − g∑

n

σx (n)

{σz(n)}, {σx(n)} are 2 × 2 Pauli matrices defined at each site.




H = −∑

n

σz (n)σz (n + 1) − g∑

n

σx (n)


M

g0 gc

order disorder

M 6= 0 M = 0




H = −∑

n

σz (n)σz (n + 1) − g∑

n

σx (n)


M

g0 gc

order disorder

M 6= 0 M = 0

◮ Order parameter: M = 〈σz〉⇒ spontaneous symmetry

breaking




H = −∑

n

σz (n)σz (n + 1) − g∑

n

σx (n)


M

g0 gc

order disorder

M 6= 0 M = 0

◮ Order parameter: M = 〈σz〉⇒ spontaneous symmetry

breaking

◮ Quantum Critical Point gc :scale invariance, correlationlength: ξ ∼ |g − gc |−ν ,energy gap: m ∼ |g − gc |zν

z = 1 for the Ising model

Quantum Criticality

Quantum Criticality

◮ This description of a quantum critical point is very similar to aclassical critical point at a thermal phase transition

Quantum Criticality


◮ In both problems it describes the behavior of local fluctuating

operators: their correlations and scaling

Quantum Criticality




◮ This is not an accident. For z = 1, d quantum criticality ⇔ d + 1classical criticality

Quantum Criticality





◮ Quantum Mechanics dictates the dynamics: the dynamic criticalexponent z

Quantum Criticality






◮ The scaling behavior at and near a quantum critical point isdescribed by (and defines) an effective local Quantum Field Theory

Quantum Criticality







◮ Question: is there a more intrinsic quantum mechanical signature ofa quantum critical point?

Quantum Criticality







◮ Question: is there a more intrinsic quantum mechanical signature ofa quantum critical point?

◮ Natural candidate: Quantum Entanglement, measured by the vonNeumann Entanglement Entropy (to be defined shortly...)

Topological Phases of Matter


◮ Liquid phases of electron fluids and spin systems without long rangeorder, with or without time reversal symmetry breaking.



◮ Quasiparticles: vortices with fractional charge and fractionalstatistics (Abelian and non-Abelian).




◮ Hidden Topological Order and Topological Vacuum Degeneracy.





◮ Finite-dimensional quasiparticle Hilbert spaces ⇒ universaltopological quantum computer.





◮ Finite-dimensional quasiparticle Hilbert spaces ⇒ universaltopological quantum computer.

◮ Effective field theory description: Topological Field Theory, e.g.,Chern-Simons gauge theory, discrete gauge theory.

Experimentally “Known” Topological Quantum Liquids


Eisenstein and Störmer, 1990

◮ 2DEG Fractional QuantumHall Liquids.




◮ Laughlin and Jain FQHstates: fractional charge(noise experiments) andAbelian fractional statistics.





◮ ν = 5/2 a Moore-ReadFQH stateq = e/4 vortex:





◮ ν = 5/2 a Moore-ReadFQH stateq = e/4 vortex:Shote noise @ pointcontact (Heiblum)





◮ ν = 5/2 a Moore-ReadFQH stateq = e/4 vortex:Shote noise @ pointcontact (Heiblum)DC transport @ pointcontact (Marcus)






◮ Is the plateau at ν = 12/5a parafermion state?







◮ Rapidly rotating Bose gases: possible non-Abelian (Pfaffian) FQHstate of bosons at ν = 1 (still hard!)







◮ Rapidly rotating Bose gases: possible non-Abelian (Pfaffian) FQHstate of bosons at ν = 1 (still hard!)

◮ Time-Reversal Breaking Superconductors: Sr2RuO4 is a px + ipy

superconductor (strong evidence, not uncontroversial)

Fractional Quantum Hall states


◮ Laughlin States: 2DEG in a large magnetic field with filling factorν = 1/m for N electrons in Nφ = mN flux quanta are



Ψ(z1, . . . , zN) =∏

i<j

(zi − zj )m e−

P

i |zi |2/4ℓ2

, zj = xj + iyj



Ψ(z1, . . . , zN) =∏

i<j

(zi − zj )m e−

P

i |zi |2/4ℓ2

, zj = xj + iyj

◮ Non-Abelian States: Moore-Read Pfaffian States:



Ψ(z1, . . . , zN) =∏

i<j

(zi − zj )m e−

P

i |zi |2/4ℓ2

, zj = xj + iyj

◮ Non-Abelian States: Moore-Read Pfaffian States:

ΨMR(z1, . . . , zN) = Pf

(

1

zi − zj

)

∏

i<j

(zi − zj )m e−

P

i |zi |2/4ℓ2

Hydrodynamic picture: Abelian States


◮ The FQH state is an incompressible charged fluid in a magnetic fieldwith a conserved 3-current density jµ = (ρ,~j ):



∂ρ

∂t+ ~∇ ·~j = ∂µjµ = 0 ⇒ jµ =

1

2πǫµνλ∂

νAλ



∂ρ

∂t+ ~∇ ·~j = ∂µjµ = 0 ⇒ jµ =

1

2πǫµνλ∂

νAλ

◮ Hydrodynamic gauge invariance: Aµ → Aµ + ∂µΛ ⇒ jµ → jµ

◮ Effective field theory: Abelian Chern-Simons gauge theoryEffective action of the hydrodynamic gauge field Aµ



∂ρ

∂t+ ~∇ ·~j = ∂µjµ = 0 ⇒ jµ =

1

2πǫµνλ∂

νAλ



S(A) =m

4π

∫

S2×S1

d3x ǫµνλAµ∂νAλ



∂ρ

∂t+ ~∇ ·~j = ∂µjµ = 0 ⇒ jµ =

1

2πǫµνλ∂

νAλ



S(A) =m

4π

∫

S2×S1

d3x ǫµνλAµ∂νAλ

◮ The excitations are vortices with fractional charge q = e/m andfractional (braid) statistics θ = π/m.

Hydrodynamic picture: Non-Abelian States


◮ For the non-Abelian FQH states, e.g Moore-Read andgeneralizations, it is (essentially) an SU(2)2 Chern-Simons gaugetheory



◮ ◮ Quasiparticles◮ Half-vortices, σ-particles, with charge q = e/4 and non-Abelian

fractional (braid) statistics.




fractional (braid) statistics.◮ Charge neutral Majorana fermions ψ




fractional (braid) statistics.◮ Charge neutral Majorana fermions ψ◮ Laughlin vortices with charge e/m and abelian fractional statisticsπ/m

Statistics and Quantum Mechanics


In Quantum Mechanics the wave-function depends on the positions andquantum numbers of the particles:



Ψ(x1, x2, . . .)

In three dimensions the only allowed symmetry of the wave functionunder exchange is to be symmetric (bosons) or antisymmetric (fermions)



Ψ(x1, x2, . . .)

In three dimensions the only allowed symmetry of the wave functionunder exchange is to be symmetric (bosons) or antisymmetric (fermions)

Ψa(x1, x2, . . .) = ±Ψa(x2, x1, . . .)

Statistics and Adiabatic Evolution


In 2 + 1 dimensions there are more possibilities. We will regard theidentical particles as having a hard core and we will consider an adiabatictime evolution which corresponds to an exchange process:



time



time

◮ 3 + 1 dimensions: this path is topologically trivial



time


◮ 2 + 1 dimensions: this path is topologically non- trivial ⇒ Braids!



time



For Laughlin (and Jain) states



time




Ψa(x1, x2, . . .) = e iθΨa(x2, x1, . . .), θ =π

m



time




Ψa(x1, x2, . . .) = e iθΨa(x2, x1, . . .), θ =π

m

Anyons with Abelian (braid) fractional statistics!

Non-Abelian Braiding Statistics of Quasiholes


◮

Multivalued wave functionsnot uniquely determined bythe particle coordinates:Two linearly independentstates at fixed positions of theparticles

~r1~r1

~r2~r2 ~r3~r3

~r4 ~r4

|A〉 |B〉

Upon braiding particles 1 and 2|A〉 → a|A〉 + b|B〉


◮


~r1~r1

~r2~r2 ~r3~r3

~r4 ~r4

|A〉 |B〉


◮ Four non-Abelian anyons can be regarded as a topological qubit


◮


~r1~r1

~r2~r2 ~r3~r3

~r4 ~r4

|A〉 |B〉



◮ There are 2n−1 linearly independent states of 2n quasiholes at fixedpositions ⇒ non-Abelian statistics


◮


~r1~r1

~r2~r2 ~r3~r3

~r4 ~r4

|A〉 |B〉




◮ The braiding of two quasiholes induces to a unitary transformationin the two-dimensional Hilbert space of four-quasihole states (Nayakand Wilczek)


◮


~r1~r1

~r2~r2 ~r3~r3

~r4 ~r4

|A〉 |B〉




◮ The braiding of two quasiholes induces to a unitary transformationin the two-dimensional Hilbert space of four-quasihole states (Nayakand Wilczek)

Braiding matrix ⇒ e i π

4

√2

(

1 1−1 1

)

FQH Interferometers and Fractional Statistics


FQH fluidFQH fluidI1 I2Φ

Nq

t1 t2

edge states

A

B



Nq

t1 t2

edge states

A

B

Chamon, Freed, Kivelson, Sondhi and Wen (1997)

◮ Internal tunneling only!



Nq

t1 t2

edge states

A

B



◮ If we hold the electron number (and therefore the quasihole number)in the central region fixed, then the conductance will oscillate as afunction of Φ with period e

e∗Φ0, where e∗ is the quasihole charge.



Nq

t1 t2

edge states

A

B



◮ If we hold the electron number (and therefore the quasihole number)in the central region fixed, then the conductance will oscillate as afunction of Φ with period e

e∗Φ0, where e∗ is the quasihole charge.

◮ If, on the other hand, we vary Nq, we can probe the statistics.

Non-Abelian Interferometers


◮ A quasihole is injected at point A on the bottom edge and tunnelsat the 1st. point-contact arrives at point B in state |ψ〉.



◮ A quasihole tunnels at the 2nd. point contact is in the statee iα BNq

|ψ〉, where BNqis the braiding operator for the quasihole to

encircle the quasiholes in the central region






◮ The current measured at B is proportional to

1

2

(

|t1|2 + |t2|2)

+ Re{

t∗1 t2 e iα 〈ψ|BNq|ψ〉

}







1

2

(

|t1|2 + |t2|2)

+ Re{


}

◮ 〈ψ|BNq|ψ〉 is given by the expectation value of the Wilson lines

representing the world-lines of the quasiholes in the effectiveChern-Simons field theory







1

2

(

|t1|2 + |t2|2)

+ Re{


}



◮ In the non-Abelian case, it measures a topological invariant, theJones polynomial VNq

(e iπ/4) of the Wilson loops (Witten, 1989)!(Fradkin, Nayak, Tsvelik and Wilczek, (1998).)







1

2

(

|t1|2 + |t2|2)

+ Re{


}



◮ In the non-Abelian case, it measures a topological invariant, theJones polynomial VNq

(e iπ/4) of the Wilson loops (Witten, 1989)!(Fradkin, Nayak, Tsvelik and Wilczek, (1998).)

◮ Basis of current proposals (Das Sarma, Freedman and Nayak) toconstruct a “topological qubit”

Simple Model Systems with Topological Phases


◮ Models with time-reversal invariant topological phases



◮ The prototype: The Rokhsar-Kivelson Quantum Dimer Model




◮ Simple local models describing strongly frustrated and ring exchangequantum spin systems with a large spin gap and no long range spinorder ⇒ spin singlets ⇔ valence bonds





◮ Typical ground states: spin gap phases with valence bond (VB) order






◮ Solvable case: the Rokhsar-Kivelson (RK) point, exact ground statewave function has the short range RVB form







|ΨRVB〉 =∑

{C}

|C 〉, {C} = all dimer coverings of the lattice







|ΨRVB〉 =∑

{C}

|C 〉, {C} = all dimer coverings of the lattice

◮ square lattice: quantum critical point; triangular lattice: TopologicalZ2 deconfined phase (Moessner and Sondhi, 1998)

Effective field theory: the Quantum Lifshitz Model

Moessner, Sondhi and Fradkin; Ardonne, Fendley and Fradkin



◮ QDM on a square lattice ⇔ 2D height model




◮ Continuum limit: coarse grained height configurations ϕ(x)





◮ Quantum Lifshitz Model Hamiltonian:






H =

∫

d2x

[

1

2Π2 +

κ2

2

(

∇2ϕ)2

]

, Π = ϕ̇






H =

∫

d2x

[

1

2Π2 +

κ2

2

(

∇2ϕ)2

]

, Π = ϕ̇

◮ Energy ∝ (Momentum)2 ⇒ z = 2

◮ Action in imaginary time τ ⇔ 3D liquid crystal at thesmectic-nematic Lifshitz transition.






H =

∫

d2x

[

1

2Π2 +

κ2

2

(

∇2ϕ)2

]

, Π = ϕ̇

◮ Energy ∝ (Momentum)2 ⇒ z = 2

◮ Action in imaginary time τ ⇔ 3D liquid crystal at thesmectic-nematic Lifshitz transition.

S =

∫

d2x

∫

dτ

[

1

2(∂τϕ)

2+κ2

2

(

∇2ϕ)2

]

Scale Invariant Ground State Wave Functions


◮ The wave function of the Quantum Lifshitz Model is scale invariant



Ψ0[ϕ] ∝ e−κ

2

∫

d2x (∇ϕ(x))2



Ψ0[ϕ] ∝ e−κ

2

∫

d2x (∇ϕ(x))2

◮ The norm of the 2D wave function is the partition function of the2D Gaussian Model, a classical critical conformally invariant system!



Ψ0[ϕ] ∝ e−κ

2

∫

d2x (∇ϕ(x))2


‖Ψ0‖2 =

∫

Dϕ e−κ

∫

d2x (∇ϕ(x))2



Ψ0[ϕ] ∝ e−κ

2

∫

d2x (∇ϕ(x))2


‖Ψ0‖2 =

∫

Dϕ e−κ

∫

d2x (∇ϕ(x))2

◮ Matching the correlation functions of the Quantum Dimer andLifshitz models, one finds κ = 1

4π .

Entanglement in Quantum Mechanics


◮ Consider a system of two spins, A and B, each with S = 1/2



◮ Each spin has two possible states: | ↑〉 and | ↓〉



◮ Each spin has two possible states: | ↑〉 and | ↓〉◮ Consider two states:

◮ Singlet: |0, 0〉 = 1√

2(| ↑〉 ⊗ | ↓〉 − | ↓〉 ⊗ | ↑〉) ⇒ entangled state




◮ Singlet: |0, 0〉 = 1√

2(| ↑〉 ⊗ | ↓〉 − | ↓〉 ⊗ | ↑〉) ⇒ entangled state

◮ Fully Polarized: |1, 1〉 = | ↑〉 ⊗ | ↑〉 ⇒ product state




◮ Singlet: |0, 0〉 = 1√

2(| ↑〉 ⊗ | ↓〉 − | ↓〉 ⊗ | ↑〉) ⇒ entangled state


◮ In the entangled (singlet) state |0, 0〉




◮ Singlet: |0, 0〉 = 1√

2(| ↑〉 ⊗ | ↓〉 − | ↓〉 ⊗ | ↑〉) ⇒ entangled state


◮ In the entangled (singlet) state |0, 0〉◮ the state of each spin is not separately well defined




◮ Singlet: |0, 0〉 = 1√

2(| ↑〉 ⊗ | ↓〉 − | ↓〉 ⊗ | ↑〉) ⇒ entangled state


◮ In the entangled (singlet) state |0, 0〉◮ the state of each spin is not separately well defined◮ we only know the probability to measure a particular value of spin A




◮ Singlet: |0, 0〉 = 1√

2(| ↑〉 ⊗ | ↓〉 − | ↓〉 ⊗ | ↑〉) ⇒ entangled state


◮ In the entangled (singlet) state |0, 0〉◮ the state of each spin is not separately well defined◮ we only know the probability to measure a particular value of spin A◮ If A is measured to have spin ↑, we know that B must have spin ↓




◮ Singlet: |0, 0〉 = 1√

2(| ↑〉 ⊗ | ↓〉 − | ↓〉 ⊗ | ↑〉) ⇒ entangled state



“Spooky action at a distance” ⇔ non-locality (Einstein)




◮ Singlet: |0, 0〉 = 1√

2(| ↑〉 ⊗ | ↓〉 − | ↓〉 ⊗ | ↑〉) ⇒ entangled state




◮ In the (polarized) product state |1, 1〉




◮ Singlet: |0, 0〉 = 1√

2(| ↑〉 ⊗ | ↓〉 − | ↓〉 ⊗ | ↑〉) ⇒ entangled state




◮ In the (polarized) product state |1, 1〉◮ the value of each spin is separately well defined




◮ Singlet: |0, 0〉 = 1√

2(| ↑〉 ⊗ | ↓〉 − | ↓〉 ⊗ | ↑〉) ⇒ entangled state




◮ In the (polarized) product state |1, 1〉◮ the value of each spin is separately well defined◮ we know that A has spin ↑ with probability 1

Density Matrix and Entanglement Entropy


A ϕA

ℓAB

ϕB

ℓB


A ϕA

ℓAB

ϕB

ℓB

◮ Pure state in A ∪ B: Ψ[ϕA, ϕB ]


A ϕA

ℓAB

ϕB

ℓB


◮ Density Matrix:

〈ϕA, ϕB |ρA∪B |ϕ′A, ϕ

′B〉 = Ψ[ϕA, ϕB ] Ψ∗[ϕ′

A, ϕ′B ]


A ϕA

ℓAB

ϕB

ℓB


◮ Density Matrix:


′B〉 = Ψ[ϕA, ϕB ] Ψ∗[ϕ′

A, ϕ′B ]

◮ Observing only A ⇒ Mixed Statereduced density matrix ρA:


A ϕA

ℓAB

ϕB

ℓB


◮ Density Matrix:


′B〉 = Ψ[ϕA, ϕB ] Ψ∗[ϕ′

A, ϕ′B ]


〈ϕA|ρA|ϕ′A〉 = trB ρA∪B


A ϕA

ℓAB

ϕB

ℓB


◮ Density Matrix:


′B〉 = Ψ[ϕA, ϕB ] Ψ∗[ϕ′

A, ϕ′B ]



von Neumann Entanglement Entropy:


A ϕA

ℓAB

ϕB

ℓB


◮ Density Matrix:


′B〉 = Ψ[ϕA, ϕB ] Ψ∗[ϕ′

A, ϕ′B ]




SA = −tr(ρA ln ρA) = SB


A ϕA

ℓAB

ϕB

ℓB


◮ Density Matrix:


′B〉 = Ψ[ϕA, ϕB ] Ψ∗[ϕ′

A, ϕ′B ]




SA = −tr(ρA ln ρA) = SB

Symmetric for a pure state in A ∪ B

Back in the system of two spins...


◮ density matrix for spin A:

〈σA|ρA|σ′A〉 =

(

12

00 1

2

)

, σA, σ′A =↑, ↓




(

12

00 1

2

)

, σA, σ′A =↑, ↓

◮

SA = −tr (ρA ln ρA) = −∑

λ=↑,↓

pλ ln pλ = −2 × 1

2ln

(

1

2

)

= ln 2




(

12

00 1

2

)

, σA, σ′A =↑, ↓

◮

SA = −tr (ρA ln ρA) = −∑

λ=↑,↓

pλ ln pλ = −2 × 1

2ln

(

1

2

)

= ln 2

⇒ SA = SB = ln 2

Quantum Entanglement in Condensed Matter


The application of the concept of entanglement entropy in CondensedMatter raises some questions:



◮ It measures some very non-local correlations: the globalentanglement of regions A and B




◮ We would like to determine its behavior, e.g. universality,dependence on the geometry of the regions, etc.





◮ in ordered and disordered phases of matter





◮ in ordered and disordered phases of matter◮ at a quantum critical point





◮ in ordered and disordered phases of matter◮ at a quantum critical point◮ in a topological phase

Entanglement scaling


◮ ℓB ≫ ℓA ≫ ξ ≫ a, where ξ is the correlation length (if it is finite)and a is a short distance scale



◮ Massive phase, with a finite correlation length ξ = m−1, in d + 1dimensions: Area Law scaling,




SA ∼ µ ℓd−1A + . . . Sredinicki (1993)





◮ The Area Law is reminiscent of the Hawking-Beckenstein formula forthe entropy of extremal black holes






◮ However, the prefactor µ ∝ a−(d−1) is non-universal. It depends onthe short distance scale a and it is unrelated to Newton’s constant G







◮ For d > 1 the Area law is also the leading behavior at quantumcriticality, where ξ → ∞ (and the mass gap m → 0)








◮ Are there universal subleading terms?








◮ Are there universal subleading terms?

◮ For Fermi liquid in d dimensions (gapless but not critical),S ∼ ℓd−1 ln(ℓ/a) (Wolff, Klich)

Universal Scaling in D = 1 + 1 space-time dimensions


◮ In D = 1 + 1, quantum criticality ⇔ conformal invariance



◮ Quantum critical points in d = 1 space dimension are described by(unitary) Conformal Field Theory in D = 1 + 1 dimensions




◮ The CFT yields the universal data: the central charge c , thespectrum of critical exponents (scaling dimensions), etc.





◮ In D = 1 + 1 at criticality:






S =c

3ln

(

ℓ

a

)

+ O(ℓ0)






S =c

3ln

(

ℓ

a

)

+ O(ℓ0)

Callan and Wilczek (1993); Calabrese and Cardy (2004); Vidal,Latorre, Rico and Kitaev (2003)






S =c

3ln

(

ℓ

a

)

+ O(ℓ0)


◮ Logarithmic scaling also holds for random spin chains at infinitedisorder fixed points (Refael and Moore, 2004)






S =c

3ln

(

ℓ

a

)

+ O(ℓ0)


◮ Logarithmic scaling also holds for random spin chains at infinitedisorder fixed points (Refael and Moore, 2004)

◮ Away from criticality

S =c

3ln

(

ξ

a

)

+ finite terms

Entanglement Entropy of Scale Invariant Wave Functions:

Can you hear the shape of Schrödinger’s cat?

with Joel Moore



with Joel Moore◮ Scale invariant wave

functions, e.g.

Ψ0[ϕ] ∝ e−κ

2

∫

d2x (∇ϕ(x))2

◮ Two disjoint regions A andB, with a commonboundary Γ.

A ϕA

ℓAB

ϕB

ℓB




functions, e.g.

Ψ0[ϕ] ∝ e−κ

2

∫

d2x (∇ϕ(x))2


A ϕA

ℓAB

ϕB

ℓB◮

S = − ln

(

ZADZB

D

ZA∪B

)

≡ FA + FB − FA∪B




functions, e.g.

Ψ0[ϕ] ∝ e−κ

2

∫

d2x (∇ϕ(x))2


A ϕA

ℓAB

ϕB

ℓB◮

S = − ln

(

ZADZB

D

ZA∪B

)

≡ FA + FB − FA∪B

◮ ZAD = ||Ψ||2A, ZB

D = ||Ψ||2B and ZA∪B = ||Ψ||A∪B2 with Dirichlet

(fixed) boundary conditions

Universal Contributions to the Entanglement Entropy


◮ For a large bounded region of linear size ℓ and smooth boundary, F

obeys the ‘Mark Kac law’ (‘Can you hear the shape of a drum?’)




F = αℓ2 + βℓ− c

6χ ln

(

ℓ

a

)

+ O(1), (Cardy and Peschel)

α and β are non-universal constants, c is the central charge of theCFT, and χ is the Euler characteristic of the region:

χ = 2 − 2h − b, h = # handles, b = # boundaries




F = αℓ2 + βℓ− c

6χ ln

(

ℓ

a

)

+ O(1), (Cardy and Peschel)

α and β are non-universal constants, c is the central charge of theCFT, and χ is the Euler characteristic of the region:

χ = 2 − 2h − b, h = # handles, b = # boundaries

∆S = −c

6(χA + χB − χA∪B) log

(

ℓ

a

)

+ O(1)



◮ For regions A ⊆ B the coefficient of the log(ℓ/a) term vanishes since



χA + χB = χA∪B ⇒ ∆S = 0



χA + χB = χA∪B ⇒ ∆S = 0

◮ A and B are physically separate, without common intersection,χA + χB − χA∪B 6= 0.



χA + χB = χA∪B ⇒ ∆S = 0

◮ A and B are physically separate, without common intersection,χA + χB − χA∪B 6= 0.The system physically splits in two disjoint parts ⇒ log ℓ term in theentanglement entropy



χA + χB = χA∪B ⇒ ∆S = 0


◮ A 6⊆ B and share a common boundary ⇒ log ℓ term whosecoefficient is determined by the angles at the intersections



χA + χB = χA∪B ⇒ ∆S = 0


◮ A 6⊆ B and share a common boundary ⇒ log ℓ term whosecoefficient is determined by the angles at the intersections

◮ If the ln ℓ term cancels ⇒ the O(1) term is universal and determinedby the CFT

S = µ ℓ+ γQCP

(There are also geometry-dependent scale invariant terms)Hsu, Mulligan, Fradkin and Kim (2008)

Entanglement Entropy of 2D Topological States


◮ Universal topological entanglement entropy γ



S = µℓ−γtopo+O(1/ℓ) Kitaev and Preskill, Levin and Wen (2006)




µ: non-universal coefficient, γ = lnD is a universal finite topologicalinvariant, D =

√

∑

i d2i , di : quantum dimensions of the excitations,

i.e. the rate of growth of the topological degeneracy.





√

∑



◮ The topological entropy γ plays a crucial role in single pointcontacts in non-Abelian FQH states (Fendley, Fisher and Nayak).





√

∑



◮ The topological entropy γ plays a crucial role in single pointcontacts in non-Abelian FQH states (Fendley, Fisher and Nayak).

Entanglement in FQH fluids: Chern-Simons theory


◮ The FQH wave functions represent topological fluids with a finitecorrelation length ξ ∝ ℓ (ℓ is the magnetic length).



◮ The entanglement entropy of FQH states has be computednumerically (K. Schoutens and coworkers, 2007).




◮ One can compute the entanglement entropy directly from theeffective field theory, Chern-Simons gauge theory (Dong, Nowling,Leigh and Fradkin).





◮ This result can be applied directly to all known FQH states.





◮ This result can be applied directly to all known FQH states.

◮ It computes only γtopo, the topological invariant piece of theentanglement entropy.

Conclusions

Conclusions

◮ We discussed many of the current ideas in quantum criticality andtopological phases of matter

Conclusions


◮ This is an exciting area of research at the crossroads of condensedmatter physics, quantum information theory, quantum field theory,topology, and even string theory, as well as being at the cutting edgeof experimental condensed matter physics

Conclusions



◮ We discussed some current efforts to detect fractional (abelian andnon-abelian) statistics and to use these interferometers to maketopological qubits

Conclusions




◮ The universal scaling behavior of the entanglement entropy is a wayto characterize phases of quantum matter and their critical behavior.

Conclusions





◮ In a generic phase the entanglement entropy shows an area lawscaling

Conclusions





◮ In a generic phase the entanglement entropy shows an area lawscaling

◮ The universal behavior at a generic quantum critical point is not yetunderstood

Conclusions

Conclusions

◮ It is well understood only in 1D quantum critical systems where itobeys a logarithmic scaling law

Conclusions


◮ For 2D quantum critical points, with scale invariant wave functions,it has universal sub-leading terms which are either logarithmic orfinite, and determined by the scaling behavior of the wave function

Conclusions



◮ In a topological phase it has a subleading finite term which is auniversal property of the entire phase, and is determined by theeffective topological field theory

Conclusions




◮ In the case of the FQH states this was computed directly fromChern-Simons gauge theory

Conclusions




◮ In the case of the FQH states this was computed directly fromChern-Simons gauge theory

◮ It may be possible to determine the structure of the topological fieldtheory by means of entanglement entropy measurements

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