Download Breakdown of a topological phase

Breakdown of a topological phase
Quantum phase transition(s) in a loop gas with tension
Simon Trebst
Microsoft Research
University of California, Santa Barbara
http://StationQ.ucsb.edu
Outline
•
•
•
Topological phases
•
Fault-tolerant quantum computing
•
A simple, soluble spin model - the toric code
Breakdown of topological order
•
Perturbing the toric code around soluble point
•
Numerical simulations
Outlook
Introduction
•
•
Spontaneous symmetry breaking
•
ground state has less symmetry than Hamiltonian
•
Landau-Ginzburg-Wilson theory
•
local order parameter
Topological order
•
ground state has more symmetry more than Hamiltonian
•
degenerate ground states
•
non-local order parameter
Topological quantum liquids
•
•
Gapped spectrum
No broken symmetry
•
Degenerate ground state on torus
•
Fractional statistics of excitations
•
Hilbert space split into topological sectors
No local matrix element mixes the sectors
∆
∆
iθ
e
0
H
1
Fault-tolerant quantum computing
A. Kitaev, Ann. Phys. 303, 2 (2003).
Idea: Use degenerate
groundstates as qubits
Topological order makes it robust
with respect to local perturbations
excitations
∆
|0! |1!
Energy gap ∆.
protects from
thermal excitations
|0!
|1!
States are locally indistinguishable → no phase errors
States do not couple → no bit flip errors
Liquid is required since environment
couples to broken symmetries
Example systems
•
Experimental realization of topological quantum liquid
•
Fractional quantum Hall effect (FQHE)
but gapless edge states are a problem
•
First implementation proposal in Josephson junction arrays
•
Other candidate systems
2a
•
Frustrated magnets
•
Quantum dimer models
•
Ultra-cold atoms
Stability of the topological phase
•
•
Topological phases have not been directly observed
in experiment (beyond FQHE), because
•
they are unstable?
•
exist only in small regions of phase space?
•
we miss the appropriate tools?
We will look at the simplest model of an
(abelian) topological phase: the quantum loop gas
Quantum loop gases
•
are the “Ising models” of topological phases
•
are hidden also in the quantum dimer model
reference
configuration
dimer
configuration
+
loop
confguration
=
The toric code: the simplest loop gas
A. Kitaev, Ann. Phys. 303, 2 (2003).
HTC = −A
!
v
"
j∈vertex(v)
σjz
−B
!
"
σjx
p j∈plaquette(p)
similar to ring exchange
introduces frustration
σi
Hamiltonian has only local terms.
All terms commute → exact solution!
The vertex term
A. Kitaev, Ann. Phys. 303, 2 (2003).
HTC = −A
!
v
"
j∈vertex(v)
σjz
−B
!
"
σjx
p j∈plaquette(p)
•
is minimized by an even number of down-spins around a vertex.
•
Replacing down-spins by loop segments maps ground state to closed loops.
•
Open ends are (charge) excitations costing energy 2A.
σz = ↓
A→∞
The plaquette term
A. Kitaev, Ann. Phys. 303, 2 (2003).
HTC = −A
!
v
"
j∈vertex(v)
σjz
−B
!
"
σjx
p j∈plaquette(p)
•
flips all spins on a plaquette.
•
favors equal amplitude superposition of all loop configurations.
•
Sign changes upon flip (vortices) cost energy 2B.
fugacity = 1
free loop creation
isotopy
free local distortions
surgery
reconnections
The toric code
A. Kitaev, Ann. Phys. 303, 2 (2003).
Ground-state manifold is a quantum loop gas.
Ground-state wavefunction is equal superposition of loop configurations.
The toric code
A. Kitaev, Ann. Phys. 303, 2 (2003).
Ground-state manifold is a quantum loop gas.
Py = 1
Topological sectors defined by winding number parity Py/x =
Py = −1
!
i∈cx/y
σiz
The toric code
A. Kitaev, Ann. Phys. 303, 2 (2003).
Ground-state manifold is a quantum loop gas.
Py = 1
Topological sectors defined by winding number parity Py/x =
Py = −1
!
i∈cx/y
σiz
Perturbing the toric code
H = HTC − h
!
i
σiz − J
magnetic field
!
σiz σjz
!ij"
Ising interaction
local perturbations
chemical potential
for loops
small loops
are favored
Magnetic field / Ising interaction introduce bare loop tension.
Mapping to plaquette variables
Express bond spins by product of two plaquette spins.
plaquette spin
µp plaquette spin µq
σi
σiz
1 z z
= ± µp µq
2
Choice of local signs is gauge-invariant
up to fixing the topological sector.
Mapping of interaction terms
plaquette spin
µp plaquette spin µq
σi
µk
σiz
σiz σjz
1 z z
= µp µq
2
=
=
µm
µl
!
1 z z
µk µl
2
1 z z
µk µm
4
Both magnetic field and Ising interaction give
Ising coupling between plaquette spins.
"!
1 z z
µl µm
2
"
Mapping to 3D Ising model
Hcl = −Kτ
!
τ,p
Sp (τ )Sp (τ + ∆τ ) − K
1
Kτ = − ln [tanh(∆τ · B)]
2
plaquette flips
!
Sp (τ )Sq (τ )
τ,!p,q"
1
K = ∆τ · h
2
magnetic field
Kc = 0.2216595(26)
imaginary time
A. M. Ferrenberg and D. P. Landau,
Phys. Rev. B 44, 5081 (1991).
B
Kτ
real space
= 1
= K
gives
∆τ
hc
=
=
0.761403
0.58224
→ numerical simulation
Lattice gauge theories
J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).
Without dynamical electric charges (A → ∞),
the toric code becomes equivalent to the “even” Ising gauge theory.
S = − βτ
!
{Pτ }
σz σz σz σz − β
imaginary time
!
σz σz σz σz
{Ps }
real space
In the context of QCD, the finite-temperature physics
of such gauge theories has been studied extensively.
We are interested in the zero-temperature physics of this gauge theory.
Magnetic phase transition
0.5
0.5
0.5
no equal
superposition
of loops
magnetization
0.4
0.4
0.3
0.3
0.2
0.2
susceptibility
1
L=8
L = 12
L = 16
L = 24
0.4
0.5
0
0.4
0.5
0.6 0.7 0.8
loop tension h
0.9
!
!
σiz "/2N
i
1
topological
topological
phase
phase
0.1
0.1
0.3
classically
classically
orderedphase
phase
ordered
0.2
0.1
hc = 0.58824
00
00
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
0
11
loop
loop tension
tension hh
Continuous quantum phase transition between topological phase
and classically ordered phase (3D Ising universality class).
Magnetic vortices
dual lattice
Magnetic vortices are plaquettes with
!
j
σjx = −1
Massive in topological phase.
Correlation function for two vortices is
given by product of bond spins along path
!
! σiz " = !µzp µzq "
i
which is equivalent to spin-spin correlation
function of plaquette spins.
Vortex condensation
j
0.8
gap Δ / 2B
!
1
σjx = −1
exp(−β∆)
0.8
0.6
0.6
0.4
0.4
topological phase
0.2
0.2
0
0
0.2
0.4
error probability
1
0.6
0.8
0
1
loop tension h
Gap estimated from imaginary time correlation function ∆ ∝ 1/ξτ .
Degeneracy splitting
odd odd
energy splitting δE(L)
20
20
topological phase
0
0
-20
-20
-40
-40
even even
-60
-60
0
0.2
0.4
0.6
0.8
loop tension h
1
1.2
Back to the Ising model
•
Where are the topological sectors in the Ising model?
plaquette spin
µp plaquette spin µq
σi
σiz
1 z z
= µp µq
2
spin flip changes
sign of Ising interaction
along cut
change of boundary condition
“Topological sectors” correspond to
variations of the boundary conditions.
Degeneracy splitting
odd odd
energy splitting δE(L)
20
20
topological phase
0
0
-20
-20
-40
√
δE ∝ L 2
-40
even even
-60
-60
0
0.2
0.4
0.6
0.8
1
1.2
loop tension h
Strong tension limit well understood.
δE ∝ L
Degeneracy splitting
4
4
topological
phase
energy splitting δE(L)
2
2
0
-2
0
δE ∝ exp(−∆L/v)
L=8
L = 12
L = 16
L = 24
L = 32
L = 40
L = 48
-4
-6
-8
0.54
-2
-4
-6
δE ∝ L
0.55
0.56
0.57
0.58
0.59
loop tension h
0.6
0.61
-8
0.62
Transition from exponential suppression to power-law growth.
Finite-size scaling
-2
10
classically
ordered phase
2
energy splitting
δE(L)
energy
splitting !E(L) / L
-2
10
4
4
-3
-3
10
10
h = 0.585
h = 0.584
h = 0.583
h = 0.58224
h = 0.581
h = 0.580
2
1
2
topological
phase
1
-4
-4
10
0.5
critical exponent
z = 1.42 ± 0.02
8
12
16
20
24
system size L
0.5
10
28 32 36 40 44 48
0.25
exponential accuracy
for larger systems
0.25
8
12
16
20
24
system size L
28 32 36 40 44 48
100,000 cpu hours
Transition from power-law to exponential scaling.
Charge confinement
Charge excitations are open loop ends.
Deconfined in topological phase
We can define a confinement length ξc :
distance between loop ends.
At soluble point:
1
!ξc2 " = 2
L
L/2
!
L/2
!
∆x=−L/2 ∆y=−L/2
"
∆x + ∆y
2
2
#
L2 + 2
=
6
What happens for finite loop tension?
Charge confinement
confinement length !c
1
confinementlength
length !ξc c· 6/(L2 + 2)
confinement
topological phase
crossing point
topological
topologicalphase
phase
1
11
1
0.95
0.5
L=8
L = 12
LL == 16
8
0.95
0.5
0.5
0.9
0.56
0
0
0
00
0.1 0
L = 12
L = 16
0.57
0.9
0.58
0.59
0.56
0.57
0.58
0.1
0.2
0.3
0.4
0.3
0.20.1 0.3
0.2
0.6
0.59
0.6
0.4
0.7
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
0.5
0.6
0.4
0.7
0.8
0.9
11
0.5
0.6
loop tension h
loop tension
h h
loop tension
Magnetic and confinement transitions occur simultaneously.
There is only one length scale ξ = ξc .
Dissipation
Coupling the environment to the classical state of the system
H = −2B
!
p
# !
!"
†
†
x
z
µp +
Ck (bp,k + bp,k )µp +
ωi,k bi,k bi,k
p,k
plaquette spin flips
(toric code)
i,k
heat bath
coupling to
environment
Integrate out bath degrees of freedom (Ohmic dissipation)
Hcl = −Kτ
!
τ,p
α ! " π #2 Sp (τ )Sp (τ ! )
Sp (τ )Sp (τ + ∆τ ) −
2
Nτ sin2 ( Nπ |τ − τ ! |)
!
τ <τ ,p
τ
Dissipation
Hcl = −Kτ
!
τ,p
α ! " π #2 Sp (τ )Sp (τ ! )
Sp (τ )Sp (τ + ∆τ ) −
2
Nτ sin2 ( Nπ |τ − τ ! |)
!
τ <τ ,p
τ
spins couple along imaginary time only
imaginary time
no real space coupling
system decouples into 1-dimensional chains
(with long-range interactions)
real space
is just the Caldeira-Legget model
Dissipation
1
linear
0.8
0.8
0.6
0.6
crossover α ≈ 0.1
0.4
0.4
topological
phase
0.2
exponential
0.2
0
0
0.1
0.2
0.3
0.4
dissipation strength α
0.5
error probability
gap Δ / 2B
1
0.6
0
0.7
Thouless-type
phase transition
Topological phase is stable for small dissipation strength α < αc ≈ 0.7 .
Summary and Outlook
•
The topological phase in the toric code exists for
an extended range around the soluble point.
•
Local perturbations can drive a continuous quantum
phase transition to a classically ordered phase.
•
No need to fine-tune system to have topological order.
•
Paucity of experimental observations not due to
intrinsic delicateness of such phases.
•
Does this picture hold for non-abelian phases?
S. Trebst, P. Werner, M. Troyer, K. Shtengel, and C. Nayak
cond-mat/0609048
Non-abelian topological quantum computing
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