Download Topology driven quantum phase transitions

Topology driven
quantum phase transitions
Dresden
July 2009
Simon Trebst
Microsoft Station Q
UC Santa Barbara
Charlotte Gils
Alexei Kitaev
Andreas Ludwig
© Simon Trebst
Matthias Troyer
Zhenghan Wang
Topological quantum liquids
Spontaneous symmetry breaking
•
ground state has less symmetry than high-T phase
•
Landau-Ginzburg-Wilson theory
•
local order parameter
Topological order
© Simon Trebst
•
ground state has more symmetry than high-T phase
•
degenerate ground states
•
non-local order parameter
•
quasiparticles have fractional statistics = anyons
cosmic microwave
background
Topological order
time reversal symmetry
broken
quantum Hall liquids
invariant
magnetic materials
2a
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Quantum phase transitions
time-reversal invariant systems
topological order
some other (ordered) state
λ
λc
How can we describe these phase transitions?
no local order parameter
↓
no Landau-Ginzburg-Wilson theory
complex field theoretical framework
↓
doubled (non-Abelian) Chern-Simons theories
Liquids on surfaces can provide a “topological framework”.
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Time-reversal invariant liquids
✗
✗
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chiral excitations
anyonic liquid
L
“flux” excitations
non-chiral
anyonic liquid
L̄
Flux excitations
anyonic liquid
flux through
“hole”
flux through
“tube”
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Flux excitations
anyonic liquid
flux through
“hole”
flux through
“tube”
skeleton
lattice
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Abelian
Abelian liquids
liquids →
→ loop
loop gas
gas // toric
toric code
code
Semion fusion rules
s
s×s=1
s
1
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loop gas
1
1
1
Extreme limits
no flux through “holes”
↓
close holes
↓
“two sheets”
The “two sheets” ground state exhibits topological order.
In the toric code model this is the flux-free, loop gas ground state
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Extreme limits
no flux through “tubes”
↓
pinch off tubes
↓
“decoupled spheres”
The “decoupled spheres” ground state exhibits no topological order.
In the toric code model this is the fully polarized, classically ordered state.
© Simon Trebst
Connecting the limits: a microscopic model
H = −Jp
!
plaquettes p
δφ(p),1 − Je
!
δ"(e),1
edges e
Varying the couplings Je /Jp we can connect the two extreme limits.
Semions (Abelian):
Toric code in a magnetic field / loop gas with loop tension Je /Jp .
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The quantum phase transition
“two sheets”
topological order
“decoupled spheres”
no topological order
vary loop tension Je /Jp
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The quantum phase transition
“two sheets”
topological order
“decoupled spheres”
no topological order
“quantum foam”
topology fluctuations
on all length scales
continuous transition
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Universality classes (semions)
Hamiltonian
deformations
Fradkin & Shenker (1979)
ST et al. (2007)
Tupitsyn et al. (2008)
Vidal, Dusuel, Schmidt (2009)
H
toric code
3D Ising
universality
z=1
classical
(polarized)
state
RG flow
“gapless” toric code
z=2
wavefunction
deformations
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|ψ!
2D Ising
universality
Ardonne, Fendley, Fradkin (2004)
Castelnovo & Chamon (2008)
Fendley (2008)
The non-Abelian case
Fibonacci anyon fusion rules
τ
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τ
τ
τ
τ ×τ =1+τ
τ
1
1
1
1
Non-abelian liquids → string net
Fibonacci anyon fusion rules
τ
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τ
τ
τ
string net
τ ×τ =1+τ
τ
1
1
1
1
The quantum phase transition
“two sheets”
topological order
“decoupled spheres”
no topological order
vary string tension Je /Jp
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One-dimensional analog
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A continuous transition
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•
A continuous quantum phase transition connects
the two extremal topological states.
•
The transition is driven by topology fluctuations
on all length scales.
•
The gapless theory is exactly sovable.
Phase diagram
Jr
c = 14/15
exactly solvable
non-abelian
topological phase
Jp
“quantum foam”
c = 7/5
non-abelian
topological phase
c = 7/5
exactly solvable
© Simon Trebst
Gapless theory & exact solution
The operators in the Hamiltonian form a
Temperley-Lieb algebra
(Xi )2 = d · Xi
d=
!
[Xi , Xj ] = 0
Xi Xi±1 Xi = Xi
for |i − j| ≥ 2
2+φ
The Hamiltonian maps onto
the Dynkin diagram D6
(1, 1)
(1, τ )
1
(τ, τ )
τ
The gapless theory is a CFT with c = 14/15.
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(τ, 1)
Gapless theory & exact solution
2
2
1+14/15
Jr
τ-flux
rescaled energy E(kx , ky )
no-flux
8/5
1.5
1+2/3
(τ+1)-flux
14/15
1
L = 12, ky = 0
no-flux
non-abelian
primary
fields
topological
phase
L = 12, ky = π
0.5
0
exactly solvable
1+4/15 τ-flux
1+2/15 (τ+1)-flux
1+2/45 τ-flux
τ-flux
2/3
4/15
2/15
2/45
0
0.5
descendant fields
τ-flux
(τ+1)-flux
τ-flux
no-flux
0
c = 14/15
1.5
τ-flux
56/45
1
τ-flux
2
0
4
6
8
10
“quantum foam”
momentum kx [2π/L]
12
c = 7/5
non-abelian
topological phase
c = 7/5
exactly solvable
© Simon Trebst
Jp
Gapless theory & exact solution
rescaled energy E( kx, ky )
2
2
Jr
7/4
τ-flux
1.5
exactly solvable
6/5
(τ+1)-flux
1
c = 14/15
1.5
7/5
τ-flux
1
19/20
(τ+1)-flux
L = 8, ky = 0
non-abelian
topological
phase
primary
fields
L = 8, ky = π
0.5
2/5
τ-flux
0
1/5
(τ+1)-flux
3/20
τ-flux
0 no-flux
0
2
0.5
0
4
6
“quantum foam”
momentum kx [2π/L]
8
c = 7/5
non-abelian
topological phase
c = 7/5
exactly solvable
© Simon Trebst
Jp
Dressed flux excitations
Jr
c = 14/15
exactly solvable
non-abelian
topological phase
Jp
2
“quantum foam”
2
c = 7/5
c = 7/5
exactly solvable
energy E(Kx)
non-abelian
topological phase
θ = 0.6π
1.5
1.5
θ = 0.3π
θ = 0.4π
1
1
θ = 0.6π
0.5
0.5
θ = 0.7π
θ = 0.8π
0
0
© Simon Trebst
π/2
π
momentum Kx
0
3π/2
2π
Back to two dimensions
z=1
string net
c = 14/15
Hamiltonian
deformations
H
z=1
1+1 D
classical
state
2+1 D
continuous
transition?
classical
state
string net
2+0 D
z=2
wavefunction
deformations
© Simon Trebst
|ψ!
c = 14/15
Fendley & Fradkin (2008)
Summary
•
A “topological” framework for the description
of topological phases and their phase transitions.
•
Unifying description of loop gases and string nets.
•
Quantum phase transition is driven by fluctuations of topology.
•
Visualization of a more abstract mathematical description,
namely doubled non-Abelian Chern-Simons theories.
•
The 2D quantum phase transition out of
a non-Abelian phase is still an open issue:
A continuous transition in a novel universality class?
C. Gils, ST, A. Kitaev, A. Ludwig, M. Troyer, and Z. Wang
arXiv:0906.1579 → Nature Physics (accepted)
© Simon Trebst