tarun grover (uc berkeley) - stony brook...
Post on 04-Sep-2020
2 Views
Preview:
TRANSCRIPT
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
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.
Symmetry breaking and Order parameters
Order parameter: collective degree of freedom
Superfluid
Order parameter = e iθ, ~∇θ ∝ superfluid velocity.
Symmetry breaking and Order parameters
Order parameter: collective degree of freedom
Solid
Order parameter = amplitude of the density wave.
Symmetry breaking and Order parameters
Order parameter: collective degree of freedom
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?
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.
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.
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.
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
• 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
• 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
• 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.
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:
Topological Order: A Simple Example
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.
Topological Order: A Simple Example
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.
Topological Order: A Simple Example
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.
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
Topological Order: A Simple ExampleSpin-1/2 spins on the links of a square lattice (Kitaev 1997):
H = −∏
�
σz with
∏
+
σx = 1
Minimize energy ⇒ closed loops of ~A
Topological Order: A Simple ExampleSpin-1/2 spins on the links of a square lattice (Kitaev 1997):
H = −∏
�
σz with
∏
+
σx = 1
Minimize energy ⇒ closed loops of ~A
Topological Order: A Simple ExampleSpin-1/2 spins on the links of a square lattice (Kitaev 1997):
H = −∏
�
σz with
∏
+
σx = 1
Exactly solvable. Ground state = equal weight superposition of allclosed loop configurations:
Topological Order: A Simple Example
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:
Topological Order: A Simple Example
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)
What is “Entanglement Entropy”?
• Reduced density matrix for A:
ρA = traceA(|ψ〉〈ψ|)
• The von-Neumann entanglement entropySvN = −Trace(ρA log ρA).
• Sn (“n’th Renyi Entropy” ):
Sn = − 1
n− 1log(Trace(ρA)
n)
What is “Entanglement Entropy”?
• Reduced density matrix for A:
ρA = traceA(|ψ〉〈ψ|)
• The von-Neumann entanglement entropySvN = −Trace(ρA log ρA).
• Sn (“n’th Renyi Entropy” ):
Sn = − 1
n− 1log(Trace(ρA)
n)
What is “Entanglement Entropy”?
• Reduced density matrix for A:
ρA = traceA(|ψ〉〈ψ|)
• The von-Neumann entanglement entropySvN = −Trace(ρA log ρA).
• Sn (“n’th Renyi Entropy” ):
Sn = − 1
n− 1log(Trace(ρA)
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).
“Area Law” of Entanglement Entropy
Sn(A) = Sn(A) ∀n
⇒ Entanglement entropy is a property of the boundary ∂A withS ∝ |∂A| for gapped phases. Connections with black-hole thermal
entropy (Bekenstein, Hawking).
“Area Law” of Entanglement Entropy
Sn(A) = Sn(A) ∀n
A
AA
⇒ Entanglement entropy is a property of the boundary ∂A withS ∝ |∂A| for gapped phases. Connections with black-hole thermal
entropy (Bekenstein, Hawking).
“Area Law” of Entanglement Entropy
Sn(A) = Sn(A) ∀n
A
AA
⇒ Entanglement entropy is a property of the boundary ∂A withS ∝ |∂A| for gapped phases. Connections with black-hole thermal
entropy (Bekenstein, Hawking).
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.
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.
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)
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)
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)
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)
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)
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)
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)
Solution to Puzzle # 1: Is γ really non-local?
• S(A) = S(A).
• Under A←→ A, the curvature κ←→ −κ ⇒ “Entanglement
Entropy Functional” F (κ) is even in κ.
(Grover, Turner, Vishwanath 2011)
Solution to Puzzle # 1: Is γ really non-local?
• S(A) = S(A).
• Under A←→ A, the curvature κ←→ −κ ⇒ “Entanglement
Entropy Functional” F (κ) is even in κ.
(Grover, Turner, Vishwanath 2011)
Solution to Puzzle # 1: Is γ really 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!.
(Grover, Turner, Vishwanath 2011)
Solution to Puzzle # 1: Is γ really 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!.
(Grover, Turner, Vishwanath 2011)
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”).
(Grover, Turner, Vishwanath 2011)
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”).
(Grover, Turner, Vishwanath 2011)
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”).
(Grover, Turner, Vishwanath 2011)
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”).
(Grover, Turner, Vishwanath 2011)
Solution to Puzzle # 2: New Topological Phases Using
Quantum Entanglement?S = Slocal + Stopological
Stopological can detect topology of the boundary ∂A!
Solution to Puzzle # 2: New Topological Phases Using
Quantum Entanglement?S = Slocal + Stopological
For Z2 Electrodynamics, Stopological = − log(2) when ∂A has onlyone connected component.
log(2)=
Stopological
A
Solution to Puzzle # 2: New Topological Phases Using
Quantum Entanglement?S = Slocal + Stopological
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
Solution to Puzzle # 2: New Topological Phases Using
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...
Solution to Puzzle # 2: New Topological Phases Using
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...
Torus
Solution to Puzzle # 2: New Topological Phases Using
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).
(Grover, Turner, Vishwanath 2011)
Solution to Puzzle # 2: New Topological Phases Using
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).
(Grover, Turner, Vishwanath 2011)
Solution to Puzzle # 2: New Topological Phases Using
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).
(Grover, Turner, Vishwanath 2011)
Solution to Puzzle # 2: New Topological Phases Using
Quantum Entanglement?
Ans:
In d = 2n and 2n + 1, at least n distinct topological ordered
phases with topological entanglement entropy proportional to n
distinct Betti numbers.
(Grover, Turner, Vishwanath 2011)
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!
(Grover, Turner, Vishwanath 2011)
⇒ 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
• 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
• 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
• 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.
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
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
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.
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,...
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,...
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,...
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”).
Spin-liquid in pictures
Spin-liquid in pictures
Spin-liquid in pictures
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!
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!
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!
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〉
• 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.
Applied Entanglement Entropy: Proof that Projected BCS
is topological ordered
• γ calculable by a Monte Carlo technique.
• System size upto 18× 18. Sufficient to extract γ.
(Zhang, Grover, Vishwanath 2011)
Applied Entanglement Entropy: Proof that Projected BCS
is topological ordered
• γ calculable by a Monte Carlo technique.
• System size upto 18× 18. Sufficient to extract γ.
(Zhang, Grover, Vishwanath 2011)
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
(Zhang, Grover, Vishwanath 2011)
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:
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:
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.
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.
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.
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.
Application: Mutual Statistics in Chiral Spin Liquid
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)
Application: Mutual Statistics in Chiral Spin Liquid
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
An Idea to Measure Quantum Entanglement
(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?
>
An Idea to Measure Quantum Entanglement
(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?
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?
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
• 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
• 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
• 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
• 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).
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′〉
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′〉
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′〉
Swap operator
β
α1
α2
1
β2
A B
Swap α1
α2A =
Renyi entropy S2 in Monte Carlo
• 〈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.
Example: Mutual Statistics in Z2 Electrodynamics
Label them by a definite quasiparticle type:
Example: Mutual Statistics in Z2 Electrodynamics
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.
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.
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(!)?
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(!)?
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.
(Zhang, Grover, Vishwanath 2011)
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.
(Zhang, Grover, Vishwanath 2011)
top related