Tarun Grover (UC Berkeley) - Stony Brook...

Many-Body Quantum Entanglement and

Topological Phases of Matter

Tarun Grover (UC Berkeley)

Collaborators:

Frank

ZhangAri

Turner

Masaki

Oshikawa

Ashvin

Vishwanath

Condensed Matter Physics:

Collective behavior of a very large number of

interacting particles

How to characterize “collective behavior”?

(Children)

Solids are rigid, liquids are flowy...

(Undergrads)

Thermodynamics: Pressure, Temperature,

Entropy,...

(Grad Students)

Thermodynamics, Correlation functions, phases of

matter ⇒ Solids are rigid, liquids are flowy .

Solids are rigid because they break translational

symmetry. Liquids don’t.

They are different phases of matter . Distinguishable

by an order parameter .

Symmetry breaking and Order parameters

Order parameter: collective degree of freedom



Ferromagnet

Cool

Tc

>T

Tc

>T

Order parameter = space-time averaged value of spins.



Superfluid

Order parameter = e iθ, ~∇θ ∝ superfluid velocity.



Solid

Order parameter = amplitude of the density wave.



Most known materials break some symmetry as tem-perature T → 0.

“Condensed” Matter Physics.

Can all phases have an order parameter description?

As T → 0, thermal fluctuations decrease but...

Quantum fluctuations increase.

Can quantum fluctuations prohibit all order?


Yes!

Most well-known: Quantum Hall liquids.

Others: frustrated quantum magnets, quantum dimer

models, ...

Positive characterization in the absence of any

order?

“Topological” (= non-local) order.


Yes!



models, ...


order?



Yes!



models, ...


order?


Topological Order

• Gapped phases characterized by non-local correlations.

Some features:

• Number of ground states depend on the topology.

• In two dimensions ⇒ “anyonic” excitations (i.e. neitherbosons, nor fermions).

• Potential application: topological quantum computing.

• Quantum entanglement essential to characterize topologicalorder.

Topological Order


Some features:





Topological Order


Some features:





Topological Order


Some features:





Plan of the talk

Part 1: Can quantum entanglement constrain or

predict new topological phases?

Part 2: How much information about a phase of

matter can be extracted from the ground state

wavefunction alone?

Part 3a: Is quantum entanglement useful for

detecting topological order in numerics?

Part 3b: Can quantum entanglement be measured in

experiments?

Part 1

Can quantum entanglement constrain or

predict new topological phases?

...but first, a brief introduction to topological order

and many-body quantum entanglement.

Topological Order: A Simple Example

Prerequisite:


H = g ~E 2 +(

~∇× ~A)2

with ~∇.~E = 0

• Simplify ⇒ Restrict ~E/~A to the numbers 0, 1 and set g = 0.

“Z2 electrodynamics”

• Easy to describe on a lattice.


H = g ~E 2 +(

~∇× ~A)2

with ~∇.~E = 0





H = g ~E 2 +(

~∇× ~A)2

with ~∇.~E = 0




Topological Order: A Simple ExampleSpin-1/2 spins on the links of a square lattice (Kitaev 1997):

H = −∏

�

σz with

∏

+

σx = 1

Analogy with the usual electrodynamics:

E ≡ σx

A ≡ σz


H = −∏

�

σz with

∏

+

σx = 1

Minimize energy ⇒ closed loops of ~A


H = −∏

�

σz with

∏

+

σx = 1

Minimize energy ⇒ closed loops of ~A


H = −∏

�

σz with

∏

+

σx = 1

Exactly solvable. Ground state = equal weight superposition of allclosed loop configurations:


Correlations of all local operators short ranged yet ground statenot a direct product state in real space. Non-local correlations.

Four degenerate ground states on torus:


Correlations of all local operators short ranged yet ground statenot a direct product state in real space. Non-local correlations.

Four degenerate ground states on torus:

Basic Intuition to detect Topological Order

Divide the total system into A and the rest (= A).

Allowed configurations intersect an even number of times at theboundary ∂A of region A.

⇒ Non-local entanglement.(Hamma, Ionicioiu, Zanardi 2005; Levin,Wen 2006; Kitaev,Preskill 2006).

What is “Entanglement Entropy”?

• Reduced density matrix for A:

ρA = traceA(|ψ〉〈ψ|)

A

• The von-Neumann entanglement entropySvN = −Trace(ρA log ρA).

• Sn (“n’th Renyi Entropy” ):

Sn = − 1

n− 1log(Trace(ρA)

n)






Sn = − 1


n)






Sn = − 1


n)






Sn = − 1


n)

Entanglement Entropy: Example # 1

Consider a spin singlet formed of two spins A and B :

|ψ〉 = 1√2(| ↑〉A | ↓〉B − | ↓〉A | ↑〉B)

Reduced density matrix ρA:

ρA = TraceB (|ψ〉〈ψ|)

=1

2(| ↑〉A A〈↑ |+ | ↓〉A A〈↓ |)

Entanglement entropy of spin A:

SA = −TraceA (ρA log ρA) = log 2

Entanglement Entropy: Example # 2

If |∂A| = L ⇒ Entanglement entropy SvN = L log(2)

“Area Law” of Entanglement Entropy

Sn(A) = Sn(A) ∀n

A

⇒ Entanglement entropy is a property of the boundary ∂A withS ∝ |∂A| for gapped phases. Connections with black-hole thermal

entropy (Bekenstein, Hawking).


Sn(A) = Sn(A) ∀n




Sn(A) = Sn(A) ∀n

A

AA




Sn(A) = Sn(A) ∀n

A

AA



Detecting Topological Order using Entanglement Entropy

• Subleading constant in entanglement entropy detectstopological order! (Kitaev-Preskill 2006, Levin-Wen 2006):

SvN = αL− γ + O(1/L)

γ is a topological invariant “Topological EntanglementEntropy”.

• γ 6= 0 if and only if the phase is topologically ordered.

• γ = log 2 for Z2 Electrodynamics.













Two Puzzles about “Topological Entanglement Entropy”

In two dimensions: S = αL− γ + O(1/L)

• Is “topological entanglement entropy” γ really non-local?

• YES, follows just from the rules of quantum mechanics!

• Are all γ’s made equal? Three dimensional topologicalphases?

• NO, different kinds of γ characterize different kinds oftopological orders, e.g., string Vs membrane correlations. ⇒Discovery of new topological phases using quantumentanglement!

(Grover, Turner, Vishwanath 2011)






















Solution to Puzzle # 1: Is γ really non-local?

• Recall: Entanglement entropy = property of boundary ∂A.

⇒ local contribution to entanglement expressible as localproperty of the boundary.

• Locally, boundary ∂A characterized by local curvature κ of ∂A.

• “Entanglement Entropy Functional” F (κ)

Slocal =

∫

∂A

F (κ, ∂iκ)






Slocal =

∫

∂A

F (κ, ∂iκ)






Slocal =

∫

∂A

F (κ, ∂iκ)






Slocal =

∫

∂A

F (κ, ∂iκ)


• S(A) = S(A).

• Under A←→ A, the curvature κ←→ −κ ⇒ “Entanglement

Entropy Functional” F (κ) is even in κ.



• S(A) = S(A).

• Under A←→ A, the curvature κ←→ −κ ⇒ “Entanglement

Entropy Functional” F (κ) is even in κ.



Two dimensions:

F (κ) = a0 + a2 κ2 + a4 κ

4...

⇒ Slocal =

∫

∂A

F (κ)

∼ A1L+ A−1/L+ A−3/L3 + ...

• Constant term not allowed in Slocal !

• Topological Entanglement Entropy γ must be non-local!.



Two dimensions:

F (κ) = a0 + a2 κ2 + a4 κ

4...

⇒ Slocal =

∫

∂A

F (κ)

∼ A1L+ A−1/L+ A−3/L3 + ...

• Constant term not allowed in Slocal !

• Topological Entanglement Entropy γ must be non-local!.


Structure of Many-body entanglement in All Dimensions

In general dimensions, S = Slocal + Stopological

Slocal = Ad−1Ld−1 + Ad−3L

d−3 + Ad−5Ld−5 + ...

• Not all terms allowed due to S(A) = S(A)

• A non-topological constant allowed in three dimensions.

Intriguing similarity with entanglement of strongly coupledconformal field theories (Takayanagi-Ryu’s “holographicentanglement entropy”).





d−3 + Ad−5Ld−5 + ...








d−3 + Ad−5Ld−5 + ...








d−3 + Ad−5Ld−5 + ...





Solution to Puzzle # 2: New Topological Phases Using

Quantum Entanglement?S = Slocal + Stopological

Stopological can detect topology of the boundary ∂A!



For Z2 Electrodynamics, Stopological = − log(2) when ∂A has onlyone connected component.

log(2)=

Stopological

A



In general, Stopological = −b0log(2) where b0 = number of con-nected components of ∂A.

b0 is called “zeroth Betti Number”.

-2 log(2)=

Stopological

A

-log(2)=

Stopological

A

b06 b = 2

06A


Quantum Entanglement?

Betti Numbers describe partial topology of boundary ∂A.

b0 = number of connected components.

b1 = number of two-dimensional or “circular” holes.

b2 = number of three-dimensional holes or “voids” and so on...



Betti Numbers describe partial topology of boundary ∂A.

b0 = number of connected components.

b1 = number of two-dimensional or “circular” holes.

b2 = number of three-dimensional holes or “voids” and so on...

Torus


Quantum Entanglement?Insight from Betti number dependence ...

• Generalize Z2 Electrodynamics e.g. |Ψ〉 =

| = p-dimensional membrane

p = 1 for Z Electrodynamics2

• Stopological = (bp−1 − bp−2 + bp−3 + ...+ (−)p−1b0) log(2)where bp is the p′th Betti number of the boundary ∂A.

• For a theory of membranes, (p = 2),Stopological = (b1 − b0) log(2).




















Ans:

In d = 2n and 2n + 1, at least n distinct topological ordered

phases with topological entanglement entropy proportional to n

distinct Betti numbers.


Question: Does Many-body Quantum Entanglement

prohibits certain types of topological phases?

Ans: Yes

Topological entanglement entropy always linear in Betti number b0and b1 in two and three dimensions!


⇒ No gapped phase with topological entanglement entropy ∝ b20

(say).

Summary of Part 1

• Topological ordered systems are intriguing: topological

degeneracy and non-local entanglement.

• Entanglement Entropy can detect topological order given just

the ground state wavefunction.

• General structure of entanglement entropy for gapped phases

is universal, independent of the nature of interactions.

• Quantum entanglement allows one to construct as well as

constrain new topologically ordered systems.

Summary of Part 1









Summary of Part 1









Summary of Part 1









Part 2

How much information about a phase can beextracted from the ground state wave-function alone?

Context: Quantum Spin-liquids. Realistic examples of topological

order.

Then generalize.

Part 2



order.

Then generalize.

Part 2



order.

Then generalize.

What are Quantum Spin-liquids?

Quantum magnets that

• Do not break any symmetry spontaneously even at zero

Kelvin.

• SU(2) spin-rotation symmetric with an odd number of

spin-1/2 per unit cell.

• Gapped quantum spin-liquids are topologically ordered

(Oshikawa-Hastings theorem).

Experimental candidates: triangular lattice organic Mott insulators

(Yamashita et al, Kanoda et al,...), NiGa2S4, Cs2CuCl4,...




Kelvin.










Kelvin.







Recent Numerical Sightings:

Honeycomb Hubbard Model (Meng et al 2010)

kagome Heisenberg (Yan et al 2011)

Square J1-J2 (Hongchen et al 2011)

Spin-liquid in pictures

Morally: Superposition of a large number of valence bond states(“Resonating Valence Bond”).




Quantum Spin-liquid: A Simple Example

• Consider a wave-function of superconducting electrons |BCS〉.

• Project |BCS〉 down to one-particle per site:

|Ψ〉projected =∏

i

(1− ni↑ni↓)|BCS〉

• This is a putative gapped spin-liquid!





i







i







i


• For dxy + idx2−y2 pairing, this is a putative “Chiral SpinLiquid”, the lattice analog of Laughlin ν = 1/2 state(Kalemeyer-Laughlin 1989, Wen, Wilczek, Zee 1989).

Applied Entanglement Entropy: Proof that Projected BCS

is topological ordered

Recall: Gapped spin-liquids have universal signatures inentanglement entropy:

S = Snon−universal − γ + O(ξ/L)

Kitaev, Preskill; Levin, Wen (2006)

Prediction from field theory: γ = 1

2log 2 for Chiral Spin-Liquid.



• γ calculable by a Monte Carlo technique.

• System size upto 18× 18. Sufficient to extract γ.

(Zhang, Grover, Vishwanath 2011)



• γ calculable by a Monte Carlo technique.

• System size upto 18× 18. Sufficient to extract γ.


Extracting Topological Entropy on a Lattice

γ = SA + SB + SC − SAB − SBC − SCA + SABC

Kitaev,Preskill 2006

Quantum Monte Carlo implementation: Zhang, Grover, Vishwanath 2011

Results

State Expected γ γcalculated/γexpectedChiral Spin Liquid log

√2 0.99 ± 0.12

Lattice ν = 1/3 Laughlin State log√3 1.07± 0.05

Z2 Spin Liquid log 2 0.85 ± 0.13


Can one do more?

Hallmark of topological order: excitations that are neither bosons,

nor fermions (“anyonic”).

Two mutually anyonic particles pick up a non-zero phase when

they encircle each other.

Can one do more?

Hallmark of topological order: excitations that are neither bosons,

nor fermions (“anyonic”).

Two mutually anyonic particles pick up a non-zero phase when

they encircle each other.

Can we extract the mutual fractional statistics of

quasiparticles from the ground state wavefunction

alone?

Yes!

Hint:

Degenerate ground states on a torus.

Different ground states related by threading a quasiparticlethrough the non-contractible loop.

# of ground states = # of quasiparticle types.

Useful to look at degenerate ground states to extract mutualstatistics of quasiparticles.

Hint:





Hint:





Quantum Entanglement and Mutual Statistics

Wavefunction for a quasiparticle along x̂ (say) =

Wavefunction that minimizes entanglement entropy for a cut

perpendicular to x̂ .

(Minimum Entropy ⇒ Maximum Knowledge about thequasiparticle.)

Quantum Entanglement and Mutual Statistics

Mutual statistics Sij : phase acquired by i ’th quasiparticle when itencircles j .

Consider following wavefunctions:

• |Σ〉i ,x ≡ threading of i ’th particle along x-direction.

• |Σ〉j ,y ≡ threading of j ’th particle along y -direction.

Sij ∝ i ,x〈Σ|Σ〉j ,y

Application: Mutual Statistics in Chiral Spin Liquid

• Chiral spin-liquid: two degenerate ground states |1〉 and |2〉.• Superpose: |Φ〉 = cos(φ)|1〉 + sin(φ)|2〉• Minimize entanglement entropy S(φ) numerically using

quantum Monte Carlo.








Overlap of minimum entropy states yields mutual statistics Sij :

Snumerical ≈

I semion

I 0.77 0.63semion 0.63 −0.77

• The negative sign on the diagonal ⇒ semionic self-statistics!

• Consistent with the Chern-Simons effective field theory.

(Zhang, Grover, Turner, Oshikawa, Vishwanath 2011)


Overlap of minimum entropy states yields mutual statistics Sij :

Snumerical ≈

I semion

I 0.77 0.63semion 0.63 −0.77

• The negative sign on the diagonal ⇒ semionic self-statistics!

• Consistent with the Chern-Simons effective field theory.

(Zhang, Grover, Turner, Oshikawa, Vishwanath 2011)

Summary of Part 2

Ground state wavefunctions “know” more than they naively ought

to, e.g. knowledge of excitations above the ground state.

Entanglement Entropy can detect fractional statistics of

quasiparticles ⇒ general result, applicable to realistic spin-liquids

and quantum Hall systems.

Summary of Part 2

Ground state wavefunctions “know” more than they naively ought

to, e.g. knowledge of excitations above the ground state.

Entanglement Entropy can detect fractional statistics of

quasiparticles ⇒ general result, applicable to realistic spin-liquids

and quantum Hall systems.

Part 3a

Is quantum entanglement useful for detecting

topological order in numerics?

(Jiang, Yao, Balents 2011)

Spin-liquid in Square lattice J1-J2 Model

“Topological Oscillations” in Entanglement Entropy

For topologically degenerate ground states, a linear combination

shows oscillations in entanglement entropy with universal amplitude

that is sensitive to the topology of entanglement boundary.

(T. Grover arxiv: arXiv:1112.2215)

|Ψ(φ)〉 = cos(φ)|1〉 + sin(φ)|2〉

“Topological Oscillations” in Entanglement EntropyNumerical confirmation in a spin-liquid (T. Grover arxiv:arXiv:1112.2215):

Part 3b

Can quantum entanglement be measured in

experiments?

Non-Local Correlations in a Cold Atoms Experiments

An Idea to Measure Quantum Entanglement

(work in progress)

A A

H = H + H + HA A AA


(work in progress)

A A

H = H + H + HA A AA

System no longer in the ground state

Thermal State Finite Entropy

Thermal Entropy related to Entanglement Entropy?

>


(work in progress)

A A

Li-Haldane (2008): log( ) = Hedge

ρA

Entanglement Entropy = Thermodynamic Entropy?

H = H + H + HA A AA

Directions for the Future

• Entanglement structure of interacting gapless phases? How to

detect “quantum order”?

• Topological order in D > 2:

• Topologically ordered phases beyond discrete gauge theories?

• Braiding and statistics of extended objects (string, membranes,

...) using entanglement?

• Measurement of entanglement entropy in experiments?

















Other Interests

• Intriguing phenomenology of spin-liquids in triangular latticeorganic Mott insulators. Proposed a gapless Z2 spin-liquid(with N. Trivedi, T. Senthil and P.A. Lee 2010).

• Deconfined Quantum criticality, collaboration with T. Senthil:

• Quantum spin-nematic to dimer transition (2007).• Quantum spin-hall insulator to a superconductor transition

(2008).• Frustrated Kondo lattice system beyond Hertz-Millis paradigm

(2009).

• A non-abelian quantum spin-liquid in frustrated quantummagnet (with T. Senthil, 2010).

• Universal features in entanglement entropy of symmetrybroken phases (with Max Metlitski,2011).

• Strongly interacting topological insulators (unpublished).

Other Interests





(2009).




Other Interests





(2009).




Other Interests





(2009).




Other Interests





(2009).




Thank you!

Renyi entropy S2 in Monte Carlo

• A simple and useful result:

e−S2 = trρ2A

=〈Φ|SwapA|Φ〉〈Φ|Φ〉

where

• |Φ〉 = |φ〉 ⊗ |φ〉• SwapA|a, b〉|a′, b′〉 = |a′, b〉|a, b′〉



e−S2 = trρ2A


where




e−S2 = trρ2A


where


Swap operator

β

α1

α2

1

β2

A B

Swap α1

α2A =


• 〈Φ|SwapA|Φ〉 can be expressed as a Monte Carlo sum akin toVariational Monte Carlo.

e−S2 = 〈SwapA〉 =∑

α1α2

ρα1ρα2

f (α1, α2)

Hastings et al 2010, Zhang et al 2010.

Example: Mutual Statistics in Z2 Electrodynamics

Construct states that carry definite electric and magnetic field per-pendicular to the entanglement cut.


Label them by a definite quasiparticle type:


This leads mutual statistics for Z2 electrodynamics:

S =1

2

I e m em

I 1 1 1 1e 1 1 −1 −1m 1 −1 1 −1em 1 −1 −1 1

e and m particles mutual semions: −1 sign upon encircling eachother.

What does Ground State Wavefunction “Know”?

X Topologically ordered states.

Similar question in totally different context...

Quantum Entanglement of Gapless Spin-liquids and

detection of emergent fermions.











Critical spin-liquids: An Example

• Recall: Chiral Spin-liquid = BCS superconductor with oneparticle per site.

• What if |BCS〉 ⇒ |Filled Fermi Sea〉 maintaining theconstraint one particle per site?

• Fermi Surface of neutral spin-1/2 particles(!)?









An insulator with metallic thermal transport!

• Material: EtMe3Sb[Pd(dmit)2]2 Yamashita et al 2010.

• κ/T (T → 0) extrapolates to non-zero value as T → 0.

An insulator with metallic thermal transport!

• Material: EtMe3Sb[Pd(dmit)2]2 Yamashita et al 2010.

• κ/T (T → 0) extrapolates to non-zero value as T → 0.

Physics of projected wavefunctions

Does the projected Fermi sea wave-function

has the correct entanglement properties to

describe a Fermi sea of spinons?

Violation of Area Law for Fermions with Fermi Surface

• For most phases (gapped and gapless): S2 ∼ LA whereLA = |∂A| (“Area Law”).ExceptFree as well as weakly interacting fermions with Fermi surface:S2 ∼ LA log(LA).

• ⇒ diagnostic of emergent Fermi surface of spinons.

Violation of Area Law for Fermions with Fermi Surface

• For most phases (gapped and gapless): S2 ∼ LA whereLA = |∂A| (“Area Law”).ExceptFree as well as weakly interacting fermions with Fermi surface:S2 ∼ LA log(LA).

• ⇒ diagnostic of emergent Fermi surface of spinons.

Triangular lattice Entanglement calculation

Unprojected

Fermi Surface

LA

kx

ky

Entanglement for Projected Fermi sea on triangular lattice

• Ltotal = 18, LA ≤ 7.

• Area-law violation in a bosonic wavefunction ⇒ fits LA log LAscaling. Signature of emergent fermions.


Entanglement for Projected Fermi sea on triangular lattice

• Ltotal = 18, LA ≤ 7.

• Area-law violation in a bosonic wavefunction ⇒ fits LA log LAscaling. Signature of emergent fermions.


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