Topological phases of matter at finite
temperature
Claudio Castelnovo
University of Oxford
Claudio Chamon
Boston University
University of Hertfordshire, June 23, 2008
Outline
Brief introduction to quantum topological order
A toy model: the toric code in two dimensions
Finite temperature behaviour
Thermal fragility in spite of gap “protection”
Classical origin of quantum topological information?
Topological order at finite temperature in three dimensions?
Conclusions
Outline
Brief introduction to quantum topological order
A toy model: the toric code in two dimensions
Finite temperature behaviour
Thermal fragility in spite of gap “protection”
Classical origin of quantum topological information?
Topological order at finite temperature 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.”
Characteristic properties
I
ground state degeneracy that depends on the topology
{|ψα i}
I
α = 1, . . . , #g
∀ α, β and for all local O
no local operators can ‘connect’ different ground states
hψα |O|ψβ i = 0
I
(on a surface of genus g )
no local observables can distinguish the different ground states
hψα |O|ψα i = hψβ |O|ψβ i
I
(in the limit T , ξ → 0)
∀ α, β and for all local O
(anyonic low-energy excitations, gapless edge modes, ...)
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
Fundamental questions:
Question # 1:
I How can we study and characterise a topologically ordered
phase at finite T ?
Question # 2:
I What properties and behaviours arise as we leave T = 0?
Let us cosider a ‘simple’ example...
Outline
Brief introduction to quantum topological order
A toy model: the toric code in two dimensions
Finite temperature behaviour
Thermal fragility in spite of gap “protection”
Classical origin of quantum topological information?
Topological order at finite temperature in three dimensions?
Conclusions
The toric code (I)
H = −λA
X
As − λB
s
Bp =
Y
j∈p
σ̂jz
Kitaev 1997
X
Bp
p
As =
Y
σ̂jx
j∈s
where λA , λB are real, positive parameters
The toric code (I)
H = −λA
X
Kitaev 1997
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
z
Thor
=
Y
i∈ξhor
σ̂iz
z
Tvert
=
Y
i∈ξvert
σ̂iz
(p.b.c.)
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
z
Thor
=
Y
σ̂iz
i∈ξhor
x
Thor
=
Y
i∈τhor
z
Tvert
=
Y
σ̂iz
i∈ξvert
σ̂ix
x
Tvert
=
Y
i∈τvert
σ̂ix
(p.b.c.)
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
z
Thor
=
Y
σ̂iz
i∈ξhor
x
Thor
=
Y
i∈τhor
z
Tvert
=
Y
σ̂iz
i∈ξvert
σ̂ix
x
Tvert
=
Y
i∈τvert
σ̂ix
(p.b.c.)
Outline
Brief introduction to quantum topological order
A toy model: the toric code in two dimensions
Finite temperature behaviour
Thermal fragility in spite of gap “protection”
Classical origin of quantum topological information?
Topological order at finite temperature in three dimensions?
Conclusions
Outline
Brief introduction to quantum topological order
A toy model: the toric code in two dimensions
Finite temperature behaviour
Thermal fragility in spite of gap “protection”
Classical origin of quantum topological information?
Topological order at finite temperature in three dimensions?
Conclusions
Compare with a ferromagnetically ordered phase
at zero temperature T = 0:
M0 =
hσ̂iz i0
independent of i
→
at finite temperature T :
1 X z
M(T ) =
hσ̂i 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
Intuitive approach to a topologically ordered phase
order parameter to distinguish topological sectors:
at zero temperature T = 0:
*
+
Y
z
z
T (0) =
σ̂i
= ±1
i∈ξ
independent of ξ
0
→
at finite temperature T :
*
+
X Y
1
T z (T ) =
σ̂iz
Nξ
{ξ}
i∈ξ
T
Intuitive approach to a topologically ordered phase
order parameter to distinguish topological sectors:
at zero temperature T = 0:
*
+
Y
z
z
T (0) =
σ̂i
= ±1
i∈ξ
0
independent of ξ
low energy defects:
I
plaquettes with Bp = −1
I
they appear in pairs
→
at finite temperature T :
*
+
X Y
1
T z (T ) =
σ̂iz
Nξ
{ξ}
i∈ξ
T
Intuitive approach to a topologically ordered phase
order parameter to distinguish topological sectors:
at zero temperature T = 0:
*
+
Y
z
z
T (0) =
σ̂i
= ±1
i∈ξ
0
independent of ξ
low energy defects:
I
plaquettes with Bp = −1
I
they appear in pairs
→
at finite temperature T :
*
+
X Y
1
T z (T ) =
σ̂iz
Nξ
{ξ}
i∈ξ
T
Intuitive approach to a topologically ordered phase
order parameter to distinguish topological sectors:
at zero temperature T = 0:
*
+
Y
z
z
T (0) =
σ̂i
= ±1
i∈ξ
0
independent of ξ
low energy defects:
I
plaquettes with Bp = −1
I
they appear in pairs
→
at finite temperature T :
*
+
X Y
1
T z (T ) =
σ̂iz
Nξ
{ξ}
i∈ξ
T
Intuitive approach to a topologically ordered phase
order parameter to distinguish topological sectors:
at zero temperature T = 0:
*
+
Y
z
z
T (0) =
σ̂i
= ±1
i∈ξ
0
independent of ξ
Two (deconfined) defects immediately spoil the order parameter: T z (T ) ' 0
⇓
protection only if
NO defects at all!
→
at finite temperature T :
+
*
X Y
1
σ̂iz
T z (T ) =
Nξ
{ξ}
i∈ξ
T
Intuitive approach to a topologically ordered phase
order parameter to distinguish topological sectors:
at zero temperature T = 0:
*
+
Y
z
z
T (0) =
σ̂i
= ±1
i∈ξ
0
independent of ξ
Confined defects preserve the
‘fabric’ of topological order
even at finite temperature!
→
at finite temperature T :
+
*
X Y
1
σ̂iz
T z (T ) =
Nξ
{ξ}
i∈ξ
T
These non-local operators vanish trivially at finite T !
I
the number of loops that do not pass through a defect is
vanishingly small with respect to all others
Prob. ∼ (1 − ρdefect )L
I
L→∞
−→
0
loop operators change sign if the cross a defect ⇒ their
average expectation value at any finite temperature is zero
We need to look for a better topological order parameter...
Outline
Brief introduction to quantum topological order
A toy model: the toric code in two dimensions
Finite temperature behaviour
Thermal fragility in spite of gap “protection”
Classical origin of quantum topological information?
Topological order at finite temperature in three dimensions?
Conclusions
The topological entropy (I)
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 (II)
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
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)
H = −λA
XY
s
j∈s
σ̂jx − λB
XY
p
σ̂jz
j∈p
(For finite N and assuming for simplicity
λA < λB )
The vanishing occurs
when the average
number
of
defects in the system
approaches
one:
−λA/B /T
Ne
∼1
Further considerations
I
in the limit of λB → ∞, λA → 0 (λB → 0, λA → ∞), half of
the topological entropy is preserved at finite T , and the
system becomes a classical hard-constrained model ⇒
CC + Chamon 2006
classical topological order
Further considerations
I
in the limit of λB → ∞, λA → 0 (λB → 0, λA → ∞), 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 )
The non-vanishing quantum topological entropy arises from the
plaquette and star terms in the Hamiltonian as two equal and
independent (i.e., classical) contributions
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
Outline
Brief introduction to quantum topological order
A toy model: the toric code in two dimensions
Finite temperature behaviour
Thermal fragility in spite of gap “protection”
Classical origin of quantum topological information?
Topological order at finite temperature in three dimensions?
Conclusions
What about three dimensions?
I
CC + Chamon (in preparation)
on a simple cubic lattice, the plaquette operators remain
planar (4-spin) terms, but the star operators become
three-dimensional (6-spin) terms
H = −λA
XY
s
i∈s
σ̂ix − λB
XY
p
i∈p
σ̂iz
Membranes vs Loops (I)
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
Membranes vs Loops (II)
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
Membranes vs Loops (III)
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
ROBUST!
fragile
The topological entropy (I)
I
an exact expression for the von Neumann entropy can be
derived at finite temperature
I
The two contributions (λA and λB ) are again additive:
(P)
(S)
SA (T ) = SA (T /λB ) + SA (T /λA )
Topological entropy (II)
I
it is able to distinguish
between the fragile and
robust behaviour
I
the low temperature
phase has non vanishing
topological entropy
I
the classical origin of
topological information is
confirmed (no need for
hard constraints in 3D)
Topological order survives at finite temperature!
I
prepare the system in a superposition of topological sectors
|ψi i = a1 |GS1 i + a2 |GS2 i
I
heat up to T < Tc
I
cool back to T = 0 (in thermodynamic equilibrium)
|ψf i = |a1 | e iφ1 |GS1 i + |a2 | e iφ2 |GS2 i
A local thermal bath does not allow amplitude excange between
different loop sectors ⇒ same final and initial probability to be in
each sector: topological protection at finite T
However, phase coherence is lost (the membrane sector is fragile):
the original qubit got demoted to a pbit
Outline
Brief introduction to quantum topological order
A toy model: the toric code in two dimensions
Finite temperature behaviour
Thermal fragility in spite of gap “protection”
Classical origin of quantum topological information?
Topological order at finite temperature in three dimensions?
Conclusions
Conclusions
I
we investigated the behaviour of quantum topological order at
finite temperature, from a condensed matter perspective
I
it is related to the confined vs. deconfined nature of the
thermal excitations (fragile vs. robust)
I
(average) expectation values of loop operators are intrinsically
flawed at finite temperature
I
topological entropy appears to be a valuable tool to investigate
both zero and finite temperature systems
Conclusions
I
I
we investigated the behaviour of quantum topological order at
finite temperature, from a condensed matter perspective
I
it is related to the confined vs. deconfined nature of the
thermal excitations (fragile vs. robust)
I
(average) expectation values of loop operators are intrinsically
flawed at finite temperature
I
topological entropy appears to be a valuable tool to investigate
both zero and finite temperature systems
the topological nature of these quantum systems can be
interpreted as the result of classical topological structures
the quantum nature originates from the ability to create
coherent superpositions
Two final thoughts...
◦ Can we use our intuition on classical systems to engineer /
investigate new classes of quantum topologically ordered
systems?
(e.g., superpose two dimer models instead of two loop models
– P. Fendley arXiv:0804.0625v2)
◦ What about quantum Hall states and other topologically
ordered systems in the continuum?
Is there a “classical description” and what does it correspond
to?