insight on quantum topological order from finite temperature ...insight on quantum topological order...
TRANSCRIPT
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Insight on quantum topological order fromfinite temperature considerations
Claudio CastelnovoUniversity of Oxford
Claudio ChamonBoston University
Conference on Classical and Quantum Information Theory,Santa Fe NM, March 24, 2008
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Outline
General context
A toy model: the toric code in two dimensions
Finite temperature behaviourthermal fragility in spite of gap “protection”classical origin of quantum topological information?
From qubits to pbits in three dimensions
Conclusions
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Outline
General context
A toy model: the toric code in two dimensions
Finite temperature behaviourthermal fragility in spite of gap “protection”classical origin of quantum topological information?
From qubits to pbits in three dimensions
Conclusions
![Page 4: Insight on quantum topological order from finite temperature ...Insight on quantum topological order from finite temperature considerations Claudio Castelnovo University of Oxford](https://reader034.vdocument.in/reader034/viewer/2022051605/600ab044cc9e286e3b5e9290/html5/thumbnails/4.jpg)
Topological phases of matter Das Sarma, Freedman, Nayak, Simon, Stern 2007
“A system is in a topological phase if, at low temperatures,energies, and wavelengths*, all observable properties (e.g.correlation functions) are invariant under smooth deformations(diffeomorphisms) of the spacetime manifold in which the systemlives.”
(i.e., all observable properties are independent of the choice ofspacetime coordinates).
[*] “By ‘at low temperatures, energies, and wavelengths’, we meanthat diffeomorphism invariance is only violated by terms whichvanish as ∼ max{e−∆/T , e−|x |/ξ} for some non-zero energy gap ∆and finite correlation length ξ.”
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How robust is really the topological protection?
Non locality ⇒ strong pro-tection from zero-temperatureperturbations (non-local tun-neling events are suppressedexponentially in system size)
What about thermal fluctuations?
I above the energy barrier: notopological protection
I there is no local orderparameter: ‘ordinary’protection arguments (e.g.,magnets) do not apply
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How robust is really the topological protection?
Non locality ⇒ strong pro-tection from zero-temperatureperturbations (non-local tun-neling events are suppressedexponentially in system size)
What about thermal fluctuations?
I above the energy barrier: notopological protection
I there is no local orderparameter: ‘ordinary’protection arguments (e.g.,magnets) do not apply
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The topological entropy (I)
Given a system described by the ground state density matrixρ = |Ψ0〉〈Ψ0|
ρA = TrB ρ
SA = −Tr [ρA log ρA]
= αL− γ + . . .
where γ > 0 is a universal constant of global origin, dubbed thetopological entropy (Kitaev, Preskill 2006, Levin, Wen 2006)
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The topological entropy (II) Levin, Wen 2006
The topological contribution γ can be obtained directly bychoosing the following four bipartitions
and taking an appropriate linear combination
Stopo ≡ limr ,R→∞
(−S1A + S2A + S3A − S4A) = 2γ
where each term is given by the Von Neumann (entanglement)entropy of the corresponding bipartition
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Outline
General context
A toy model: the toric code in two dimensions
Finite temperature behaviourthermal fragility in spite of gap “protection”classical origin of quantum topological information?
From qubits to pbits in three dimensions
Conclusions
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The toric code (I) Kitaev 1997
H = −λA
∑s
As − λB
∑p
Bp
Bp =∏j∈p
σ̂zj As =
∏j∈s
σ̂xj
where λA, λB are real, positive pa-rameters
[Bp,Bp′ ] = 0 [As ,As′ ] = 0 [As ,Bp] = 0
One can diagonalise the Hamiltonian at the same time as theN − 1 independent Bp operators, and N − 1 independent As
operators (N being the number of sites).
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The toric code (I) Kitaev 1997
H = −λA
∑s
As − λB
∑p
Bp
Bp =∏j∈p
σ̂zj As =
∏j∈s
σ̂xj
where λA, λB are real, positive pa-rameters
[Bp,Bp′ ] = 0 [As ,As′ ] = 0 [As ,Bp] = 0
One can diagonalise the Hamiltonian at the same time as theN − 1 independent Bp operators, and N − 1 independent As
operators (N being the number of sites).
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The toric code (II)
The ground state is 4-fold degenerate 22N−2(N−1) = 4,
⇒ four topological sectors identifiedby the eigenvalues of the non-localoperators
Γhor =∏
i∈γhor
σ̂zi Γvert =
∏i∈γvert
σ̂zi
Thor =∏
i∈τhor
σ̂xi Tvert =
∏i∈τvert
σ̂xi
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The toric code (II)
The ground state is 4-fold degenerate 22N−2(N−1) = 4,
⇒ four topological sectors identifiedby the eigenvalues of the non-localoperators
Γhor =∏
i∈γhor
σ̂zi Γvert =
∏i∈γvert
σ̂zi
Thor =∏
i∈τhor
σ̂xi Tvert =
∏i∈τvert
σ̂xi
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The toric code (II)
The ground state is 4-fold degenerate 22N−2(N−1) = 4,
⇒ four topological sectors identifiedby the eigenvalues of the non-localoperators
Γhor =∏
i∈γhor
σ̂zi Γvert =
∏i∈γvert
σ̂zi
Thor =∏
i∈τhor
σ̂xi Tvert =
∏i∈τvert
σ̂xi
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Outline
General context
A toy model: the toric code in two dimensions
Finite temperature behaviourthermal fragility in spite of gap “protection”classical origin of quantum topological information?
From qubits to pbits in three dimensions
Conclusions
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Finite T behaviour of the topological entropy (I)
ρ(0) = |Ψ0〉〈Ψ0| −→ ρ(T ) = e−βH/Z , yet one can obtain anexact expression for both SA and Stopo CC + Chamon 2007
I at finite system size N and temperature T , Stopo is a functionof N ln[tanh(λA/B/T )] ∼ NT , and the N →∞ and T → 0limits do not commute:
I if the thermodynamic limit is taken first, the topologicalentropy vanishes identically
I if the zero temperature limit is taken first, we recover theknown result Stopo(T = 0) = 2 ln 2
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Finite T behaviour of the topological entropy (II)
H = −λA
∑s
∏j∈s
σ̂xj − λB
∑p
∏j∈p
σ̂zj
(For finite N and as-suming for simplicityλA < λB)
The vanishing occurswhen the averagenumber of de-fects in the systemapproaches one:
N e−λ
A/B/T ∼ 1
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Intuitive picture
order parameter to distinguish topological sectors:
at zero temperature T = 0:
Γ0 =
⟨∏i∈γ
σ̂zi
⟩0
= ±1
independent of γ
→at finite temperature T :
Γ(T ) =1
Nγ
∑{γ}
⟨∏i∈γ
σ̂zi
⟩T
low energy defects:
plaquettes with Bp = −1
I they appear in pairs
I they are deconfined
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Intuitive picture
order parameter to distinguish topological sectors:
at zero temperature T = 0:
Γ0 =
⟨∏i∈γ
σ̂zi
⟩0
= ±1
independent of γ
→at finite temperature T :
Γ(T ) =1
Nγ
∑{γ}
⟨∏i∈γ
σ̂zi
⟩T
low energy defects:
plaquettes with Bp = −1
I they appear in pairs
I they are deconfined
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Intuitive picture
order parameter to distinguish topological sectors:
at zero temperature T = 0:
Γ0 =
⟨∏i∈γ
σ̂zi
⟩0
= ±1
independent of γ
→at finite temperature T :
Γ(T ) =1
Nγ
∑{γ}
⟨∏i∈γ
σ̂zi
⟩T
low energy defects:
plaquettes with Bp = −1
I they appear in pairs
I they are deconfined
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Intuitive picture
order parameter to distinguish topological sectors:
at zero temperature T = 0:
Γ0 =
⟨∏i∈γ
σ̂zi
⟩0
= ±1
independent of γ
→at finite temperature T :
Γ(T ) =1
Nγ
∑{γ}
⟨∏i∈γ
σ̂zi
⟩T
low energy defects:
plaquettes with Bp = −1
I they appear in pairs
I they are deconfined
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Intuitive picture
order parameter to distinguish topological sectors:
at zero temperature T = 0:
Γ0 =
⟨∏i∈γ
σ̂zi
⟩0
= ±1
independent of γ
→at finite temperature T :
Γ(T ) =1
Nγ
∑{γ}
⟨∏i∈γ
σ̂zi
⟩T
Two (deconfined) defects im-mediately spoil the order pa-rameter: Γ(T ) ' 0
⇓protection only ifNO defects at all!
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Compare with a ferromagnetically ordered phase
at zero temperature T = 0:
M0 = 〈σ̂zi 〉0
independent of i
→at finite temperature T :
M(T ) =1
N
∑i
〈σ̂zi 〉T
Thermal fluctuations act on individual spins, andone needs a finite density of them to affect theorder parameter, which is therefore exponentiallyprotected: δM ∼ e−∆/T
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Further considerations
I in the limit of λB →∞, λA → 0 (λB → 0, λA →∞), half ofthe topological entropy is preserved at finite T , and thesystem becomes a classical hard-constrained model ⇒classical topological order CC + Chamon 2006
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Further considerations
I in the limit of λB →∞, λA → 0 (λB → 0, λA →∞), half ofthe topological entropy is preserved at finite T , and thesystem becomes a classical hard-constrained model ⇒classical topological order CC + Chamon 2006
I the two contributions in the Hamiltonian (λA and λB) are infact additive:
SA(T ) = S(P)A (T/λB) + S
(S)A (T/λA)
Stopo = S(P)topo(T/λB) + S
(S)topo(T/λA)
The non-vanishing quantum topological entropy arises from theplaquette and star terms in the Hamiltonian as two equal and
independent (i.e., classical) contributions
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Conclusions from the 2D toric code
I topological order is fragile ascompared to Landau-Ginzburgordered systems with a gap
I in experiments, larger systemsgive higher topological quantumprotection, but requireT ∼ 1/ lnN to be safe fromthermal fluctuations
I the topological nature of this model has a classical origin
I quantum mechanics arises from the ability to create coherentsuperpositions
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Conclusions from the 2D toric code
I topological order is fragile ascompared to Landau-Ginzburgordered systems with a gap
I in experiments, larger systemsgive higher topological quantumprotection, but requireT ∼ 1/ lnN to be safe fromthermal fluctuations
I the topological nature of this model has a classical origin
I quantum mechanics arises from the ability to create coherentsuperpositions
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Outline
General context
A toy model: the toric code in two dimensions
Finite temperature behaviourthermal fragility in spite of gap “protection”classical origin of quantum topological information?
From qubits to pbits in three dimensions
Conclusions
![Page 29: Insight on quantum topological order from finite temperature ...Insight on quantum topological order from finite temperature considerations Claudio Castelnovo University of Oxford](https://reader034.vdocument.in/reader034/viewer/2022051605/600ab044cc9e286e3b5e9290/html5/thumbnails/29.jpg)
What about three dimensions? CC + Chamon (in preparation)
I on a simple cubic lattice, the plaquette operators remainplanar (4-spin) terms, but the star operators becomethree-dimensional (6-spin) terms
H = −λA
∑s
∏i∈s
σ̂xi − λB
∑p
∏i∈p
σ̂zi
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Membranes vs Loops (I)
I on a simple cubic lattice, the plaquette operators remainplanar (4-spin) terms, but the star operators becomethree-dimensional (6-spin) terms
I the dual loop structure is replaced by closed membranes onthe dual lattice
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Membranes vs Loops (II)
I on a simple cubic lattice, the plaquette operators remainplanar (4-spin) terms, but the star operators becomethree-dimensional (6-spin) terms
I the dual loop structure is replaced by closed membranes onthe dual lattice
I non-local operators distinguishing the different sectors arewinding loops and winding membranes
fragile ROBUST!
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Finite temperature behavior
The two contributions (λA and λB) are again additive:
SA(T ) = S(P)A (T/λB) + S
(S)A (T/λA)
⇓Classical origin of topological information is confirmed (no need for
hard constraints in 3D)
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From qubits to pbits
The quantum topological information becomes probabilistic(classical) topological information at finite temperature:
I the loop sectors are still well defined
I loss of membrane contribution ⇒ loss of coherence withineach sector
|Ψ(T = 0)〉 = ψ1
∑α∈1
|α〉+ ψ2
∑α′∈2
|α′〉+ ...
T 6= 0 :
P1(T ) = |ψ1|2
P2(T ) = |ψ2|2
...
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Outline
General context
A toy model: the toric code in two dimensions
Finite temperature behaviourthermal fragility in spite of gap “protection”classical origin of quantum topological information?
From qubits to pbits in three dimensions
Conclusions
![Page 35: Insight on quantum topological order from finite temperature ...Insight on quantum topological order from finite temperature considerations Claudio Castelnovo University of Oxford](https://reader034.vdocument.in/reader034/viewer/2022051605/600ab044cc9e286e3b5e9290/html5/thumbnails/35.jpg)
Conclusions
I quantum topological order is proteced (e−∆/T ) from thermalfluctuations by the presence of a finite gap...
I ... but the nature of the protection is radically different fromlocally ordered systems
I in particular, the protection gets weaker for larger systems!
I the topological nature of these quantum systems can beinterpreted as the result of classical topological structures
I the quantum nature originates from the ability to createcoherent superpositions
Can we use our intuition on classical systems to engineer /investigate new classes of quantum topologically ordered systems?
What about quantum Hall states and other topologically orderedsystems in the continuum?
![Page 36: Insight on quantum topological order from finite temperature ...Insight on quantum topological order from finite temperature considerations Claudio Castelnovo University of Oxford](https://reader034.vdocument.in/reader034/viewer/2022051605/600ab044cc9e286e3b5e9290/html5/thumbnails/36.jpg)
Conclusions
I quantum topological order is proteced (e−∆/T ) from thermalfluctuations by the presence of a finite gap...
I ... but the nature of the protection is radically different fromlocally ordered systems
I in particular, the protection gets weaker for larger systems!
I the topological nature of these quantum systems can beinterpreted as the result of classical topological structures
I the quantum nature originates from the ability to createcoherent superpositions
Can we use our intuition on classical systems to engineer /investigate new classes of quantum topologically ordered systems?
What about quantum Hall states and other topologically orderedsystems in the continuum?
![Page 37: Insight on quantum topological order from finite temperature ...Insight on quantum topological order from finite temperature considerations Claudio Castelnovo University of Oxford](https://reader034.vdocument.in/reader034/viewer/2022051605/600ab044cc9e286e3b5e9290/html5/thumbnails/37.jpg)
Conclusions
I quantum topological order is proteced (e−∆/T ) from thermalfluctuations by the presence of a finite gap...
I ... but the nature of the protection is radically different fromlocally ordered systems
I in particular, the protection gets weaker for larger systems!
I the topological nature of these quantum systems can beinterpreted as the result of classical topological structures
I the quantum nature originates from the ability to createcoherent superpositions
Can we use our intuition on classical systems to engineer /investigate new classes of quantum topologically ordered systems?
What about quantum Hall states and other topologically orderedsystems in the continuum?