Download Topological order at finite temperature?

Topological order at finite temperature?
Claudio Castelnovo
University of Oxford
Claudio Chamon
Boston University
University of Manchester, Manchester UK, February 20, 2008
Outline
General context → The central question
Topological entropy
A toy model: the toric code in 2D
Finite temperature fluctuations
gap “protection”
coherent picture of classical and quantum topological order
From qubits to pbits in three dimensions
Conclusions
Outline
General context → The central question
Topological entropy
A toy model: the toric code in 2D
Finite temperature fluctuations
gap “protection”
coherent picture of classical and quantum topological order
From qubits to pbits in three dimensions
Conclusions
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 system
lives.”
(i.e., all observable properties are independent of the choice of
spacetime coordinates).
[*] “By ‘at low temperatures, energies, and wavelengths’, we mean
that diffeomorphism invariance is only violated by terms which
vanish as ∼ max{e −∆/T , e −|x|/ξ } for some non-zero energy gap ∆
and finite correlation length ξ.”
Properties
I
energy gap separating the ground state(s) from the lowest
excited states
Properties
I
energy gap separating the ground state(s) from the lowest
excited states
I
the only local operator which has non-vanishing correlation
functions is the identity ⇒ any local perturbations, such as
the environment, cannot have non-trivial matrix elements
between different ground states
Properties
I
energy gap separating the ground state(s) from the lowest
excited states
I
the only local operator which has non-vanishing correlation
functions is the identity ⇒ any local perturbations, such as
the environment, cannot have non-trivial matrix elements
between different ground states
I
Topological invariance does not imply trivial low-energy
physics!
I
I
topological degeneracy ⇔ anyonic quasiparticles (in 2D)
chiral (i.e. parity, P and time reversal T -violating) topological
phase + boundary −→ gapless excitations at the boundary
Why do we care?
(fractional) Quantum Hall physics:
I
ν = 1/3 quantum Hall state −→ Abelian quasiparticles
Why do we care?
(fractional) Quantum Hall physics:
I
ν = 1/3 quantum Hall state −→ Abelian quasiparticles
I
ν = 5/2 quantum Hall state −→ non-Abelian quasiparticles
Why do we care?
Topological (fault tolerant) quantum computation:
I
ν = 12/5 quantum Hall state → non-Abelian quasiparticles +
universal gates (Fibonacci anyons)
Why do we care?
Topological (fault tolerant) quantum computation:
I
ν = 12/5 quantum Hall state → non-Abelian quasiparticles +
universal gates (Fibonacci anyons)
Other realisations:
I
p-wave superconductors
I
p-wave Feshbach resonance induced fermionic cold atom
superfluids
I
ultra-cold atoms in optical lattices
I
arrays of Josephson junctions
How robust is really the topological protection?
Non locality ⇒ strong protection from zero-temperature
perturbations (non-local tunneling events are suppressed
exponentially in system size)
How robust is really the topological protection?
Non locality ⇒ strong protection from zero-temperature
perturbations (non-local tunneling events are suppressed
exponentially in system size)
What about thermal fluctuations?
I
above the energy barrier: no
topological protection
I
there is no local order
parameter: ‘ordinary’
protection arguments (e.g.,
magnets) do not apply
Outline
General context → The central question
Topological entropy
A toy model: the toric code in 2D
Finite temperature fluctuations
gap “protection”
coherent picture of classical and quantum topological order
From qubits to pbits in three dimensions
Conclusions
How can we detect topological order?
“the necessary and sufficient conditions for the existence of
topological ground states are not known even in theoretical
models. [...] the topological symmetry of the ground state is an
emergent symmetry at low energy, which is not present in the
microscopic Hamiltonian of the system.”
How can we detect topological order?
“the necessary and sufficient conditions for the existence of
topological ground states are not known even in theoretical
models. [...] the topological symmetry of the ground state is an
emergent symmetry at low energy, which is not present in the
microscopic Hamiltonian of the system.”
How can we detect it?
no local order: we cannot use a Landau-Ginzburg approach
⇓
non-local order parameter: topological entropy
(Kitaev, Preskill 2006, Levin, Wen 2006)
The von Neumann entropy SA
Given a system described by the ground state density matrix
ρ = |Ψ0 ihΨ0 |
ρA = TrB ρ
SA = −Tr [ρA log ρA ]
= αL − γ + . . .
where γ > 0 is a universal constant of global origin, dubbed the
topological entropy (Kitaev, Preskill 2006, Levin, Wen 2006)
The topological entropy Stopo
Levin, Wen 2006
The topological contribution γ can be obtained directly by
choosing the following four bipartitions
and taking an appropriate linear combination
Stopo ≡ lim (−S1A + S2A + S3A − S4A ) = 2γ
r ,R→∞
where each term is given by the Von Neumann (entanglement)
entropy of the corresponding bipartition
Outline
General context → The central question
Topological entropy
A toy model: the toric code in 2D
Finite temperature fluctuations
gap “protection”
coherent picture of classical and quantum topological order
From qubits to pbits in three dimensions
Conclusions
The toric code (I)
H = −λA
X
As − λB
s
Bp =
Y
j∈p
σ̂jz
Kitaev 2003
X
Bp
p
As =
Y
σ̂jx
j∈s
where λA , λB are real, positive parameters
The toric code (I)
H = −λA
X
Kitaev 2003
As − λB
s
Bp =
Y
σ̂jz
j∈p
X
Bp
p
As =
Y
σ̂jx
j∈s
where λA , λB are real, positive parameters
[Bp , Bp0 ] = 0
[As , As 0 ] = 0
[As , Bp ] = 0
One can diagonalise the Hamiltonian at the same time as the
N − 1 independent Bp operators, and N − 1 independent As
operators (N being the number of sites).
The toric code (II)
The ground state is 4-fold degenerate 22N−2(N−1) = 4,
⇒ four topological sectors identified
by the eigenvalues of the non-local
operators
Γhor =
Y
σ̂iz
i∈γhor
Thor =
Y
i∈τhor
Γvert =
Y
σ̂iz
i∈γvert
σ̂ix
Tvert =
Y
i∈τvert
σ̂ix
The underlying (gauge) loop structures
Y
σz = 1
−→
closed loop
←−
Y
closed dual loop
σx = 1
The ground state wavefunction
Within each sector, the ground state wavefunction |Ψ0 i is given by
the equal amplitude superposition of all states
In the z-basis (x-basis)
I
choose any one state |0i in that sector
I
define the (Abelian) group G generated by the star operators
{As } (plaquette operators {Bp })
|Ψ0 i =
1 X
g |0i,
|G |1/2 g ∈G
|G | = 2N−1
Calculation of the von Neumann entropy (I) Hamma et al. 2005
SA = −Tr [ρA log ρA ] = log
|G |
dA dB
dA = |{g ∈ G | g = gA ⊗ 11B }|
dB = |{g ∈ G | g = 11A ⊗ gB }|
Calculation of the von Neumann entropy (I) Hamma et al. 2005
SA = −Tr [ρA log ρA ] = log
|G |
dA dB
dA = |{g ∈ G | g = gA ⊗ 11B }|
dB = |{g ∈ G | g = 11A ⊗ gB }|
When subsystem A is connected
dA = 2ΣA
dB = 2
ΣB
ΣA = number of star operators acting solely on A
ΣB = number of star operators acting solely on B
Calculation of the von Neumann entropy (I) Hamma et al. 2005
SA = −Tr [ρA log ρA ] = log
|G |
dA dB
dA = |{g ∈ G | g = gA ⊗ 11B }|
dB = |{g ∈ G | g = 11A ⊗ gB }|
When subsystem A is connected
dA = 2ΣA
dB = 2
SA = log
ΣB
2N−1
2ΣA 2ΣB
ΣA = number of star operators acting solely on A
ΣB = number of star operators acting solely on B
= N − 1 − ΣA − ΣB log 2 = ΣAB − 1 log 2
Calculation of the von Neumann entropy (II)
What happens if subsystems A and B have more than one
connected component? CC + Chamon
SA = log
dA = 2ΣA
dB = 2ΣB
|G |
dA dB
Calculation of the von Neumann entropy (II)
What happens if subsystems A and B have more than one
connected component? CC + Chamon
SA = log
|G |
dA dB
dA = 2ΣA +mB −1
dB = 2ΣB +mA −1
where
A = A 1 ∪ . . . ∪ Am
A
B = B1 ∪ . . . ∪ Bm
B
with A1 , . . . , Am and B1 , . . . , Bm disjoint and connected
A
B
Calculation of the von Neumann entropy (III)
Therefore,
SA = log
2N−1
2ΣA +mB −1 2ΣB +mA −1
=
N − 1 − ΣA − mB + 1 − ΣB − mA + 1 log 2
=
ΣAB − mA − mB + 1 log 2
The topological entropy
Stopo =
=
lim (−S1A + S2A + S3A − S4A )
(1)
(2)
(3)
(4)
lim −ΣAB + 2 + ΣAB − 1 + ΣAB − 1 − ΣAB + 2 log 2
r ,R→∞
r ,R→∞
= 2 log 2 = log D 2 ,
D=2
(1)
(4)
(2)
(3)
where we used the fact that ΣAB + ΣAB = ΣAB + ΣAB
Outline
General context → The central question
Topological entropy
A toy model: the toric code in 2D
Finite temperature fluctuations
gap “protection”
coherent picture of classical and quantum topological order
From qubits to pbits in three dimensions
Conclusions
Intuitive picture
order parameter to distinguish topological sectors:
at zero temperature T = 0:
*
+
Y
z
Γ0 =
σi
i∈γ
independent of γ
0
→
at finite temperature T :
*
+
1 X Y z
Γ(T ) =
σi
Nγ
{γ}
i∈γ
T
Intuitive picture
order parameter to distinguish topological sectors:
at zero temperature T = 0:
*
+
Y
z
Γ0 =
σi
i∈γ
0
independent of γ
low energy defects:
plaquettes with Bp = −1
I
they appear in pairs
I
they are deconfined
→
at finite temperature T :
*
+
1 X Y z
Γ(T ) =
σi
Nγ
{γ}
i∈γ
T
Intuitive picture
order parameter to distinguish topological sectors:
at zero temperature T = 0:
*
+
Y
z
Γ0 =
σi
i∈γ
0
independent of γ
Two (deconfined) defects immediately spoil the order parameter: Γ(T ) ' 0
⇓
protection only if
NO defects at all!
→
at finite temperature T :
*
+
1 X Y z
σi
Γ(T ) =
Nγ
{γ}
i∈γ
T
Compare with a ferromagnetically ordered phase
at zero temperature T = 0:
M0 =
hσiz i0
independent of i
→
at finite temperature T :
1 X z
M(T ) =
hσi iT
N
i
Thermal fluctuations act on individual spins, and
one needs a finite density of them to affect the
order parameter, which is therefore exponentially
protected: δM ∼ e −∆/T
Finite T behaviour of the topological entropy (I)
ρ(0) = |Ψ0 ihΨ0 | −→ ρ(T ) = e −βH /Z , yet one can obtain an
exact expression for both SA and Stopo
CC + Chamon 2007
I
at finite system size N and temperature T , Stopo is a function
of N ln[tanh(λA/B /T )] ∼ NT , and the N → ∞ and T → 0
limits do not commute:
I
if the thermodynamic limit is taken first, the topological
entropy vanishes identically
I
if the zero temperature limit is taken first, we recover the
known result Stopo (T = 0) = 2 ln 2
Finite T behaviour of the topological entropy (II)
For finite N and assuming for simplicity λA < λB
The vanishing occurs when the average number of defects in the
−λ
/T
system approaches one: N e A/B ∼ 1
Further considerations
I
in the limit of λA → ∞ (λB → ∞), half of the topological
entropy is preserved at finite T , and the system becomes a
classical hard-constrained model → classical topological order
CC + Chamon 2006
I
the two contributions in the Hamiltonian (λA and λB ) are in
fact additive:
(P)
(S)
SA (T ) = SA (T /λB ) + SA (T /λA )
(P)
(S)
Stopo = Stopo (T /λB ) + Stopo (T /λA )
We do not need to minimise simultaneously both the plaquette and
the star term in the Hamiltonian to obtain a non-vanishing
topological entropy: they contribute equally and independently
(i.e., classically)
Conclusions from the 2D toric code
I
topological order is fragile as
compared to Landau-Ginzburg
ordered systems with a gap
I
in experiments, larger systems
give higher topological quantum
protection, but require
T ∼ 1/ ln N to be safe from
thermal fluctuations
Conclusions from the 2D toric code
I
topological order is fragile as
compared to Landau-Ginzburg
ordered systems with a gap
I
in experiments, larger systems
give higher topological quantum
protection, but require
T ∼ 1/ ln N to be safe from
thermal fluctuations
I
the topological nature of this model has a classical origin
I
quantum mechanics arises from the ability to create coherent
superpositions (Highly non-trivial: e.g., a quantum spin-1/2 has a
much richer behaviour than a classical Ising spin)
Outline
General context → The central question
Topological entropy
A toy model: the toric code in 2D
Finite temperature fluctuations
gap “protection”
coherent picture of classical and quantum topological order
From qubits to pbits in three dimensions
Conclusions
What about three dimensions? (I)
CC + Chamon (in preparation)
I
on a simple cubic lattice, the plaquette operators remain
planar (4-spin) terms, but the star operators become
three-dimensional (6-spin) terms
I
the dual loop structure is replaced by closed membranes on
the dual lattice
I
non-local operators distinguishing the different sectors are
winding loops and winding membranes
fragile
ROBUST!
What about three dimensions? (II)
I
classical origin confirmed (classical topological order at finite
T without hard constraints in 3D)
I
the quantum topological information becomes probabilistic
classical information at finite temperature: loss of membrane
contribution equals loss of coherence, while the loop sectors
are still well defined
Outline
General context → The central question
Topological entropy
A toy model: the toric code in 2D
Finite temperature fluctuations
gap “protection”
coherent picture of classical and quantum topological order
From qubits to pbits in three dimensions
Conclusions
Conclusions
I
quantum topological order is proteced from thermal
fluctuations by the presence of a finite gap...
I
... but the nature of the protection is radically different from
locally ordered systems
I
in particular, the protection gets weaker for larger systems!
Conclusions
I
quantum topological order is proteced from thermal
fluctuations by the presence of a finite gap...
I
... but the nature of the protection is radically different from
locally ordered systems
I
in particular, the protection gets weaker for larger systems!
I
the topological nature of these quantum systems can be
interpreted as the result of classical topological structures
I
the quantum nature originates from the ability to create
coherent superpositions
Conclusions
I
quantum topological order is proteced from thermal
fluctuations by the presence of a finite gap...
I
... but the nature of the protection is radically different from
locally ordered systems
I
in particular, the protection gets weaker for larger systems!
I
the topological nature of these quantum systems can be
interpreted as the result of classical topological structures
I
the quantum nature originates from the ability to create
coherent 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 ordered
systems in the continuum?