© Simon Trebst
Topology driven quantum phase transitions
Alexei KitaevAndreas Ludwig
Matthias TroyerZhenghan Wang
Charlotte Gils
Simon TrebstMicrosoft Station QUC Santa Barbara
Dresden July 2009
© Simon Trebst
Spontaneous symmetry breaking• ground state has less symmetry than high-T phase
• Landau-Ginzburg-Wilson theory
• local order parameter
Topological quantum liquids
Topological order• ground state has more symmetry than high-T phase
• degenerate ground states
• non-local order parameter
• quasiparticles have fractional statistics = anyons
cosmic microwave background
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Topological order
quantum Hall liquids magnetic materials
2a
time reversal symmetry
broken invariant
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Quantum phase transitions
topological order!
!c
some other (ordered) state
no local order parameter ↓
no Landau-Ginzburg-Wilson theory
How can we describe these phase transitions?
complex field theoretical framework ↓
doubled (non-Abelian) Chern-Simons theories
Liquids on surfaces can provide a “topological framework”.
time-reversal invariant systems
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Time-reversal invariant liquids
chiral excitations
✗
✗ “flux” excitationsnon-chiral
anyonic liquid L
anyonic liquid L̄
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Flux excitations
anyonic liquid
flux through “hole”
flux through “tube”
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Flux excitations
anyonic liquid
flux through “hole”
flux through “tube”
skeletonlattice
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Abelian liquids → loop gas / toric code
111
Semion fusion rulesloop gas
s! s = 1
1s s
Abelian liquids → loop gas / toric code
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Extreme limits
The “two sheets” ground state exhibits topological order. In the toric code model this is the flux-free, loop gas ground state
no flux through “holes”↓
close holes↓
“two sheets”
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Extreme limits
The “decoupled spheres” ground state exhibits no topological order. In the toric code model this is the fully polarized, classically ordered state.
no flux through “tubes”↓
pinch off tubes↓
“decoupled spheres”
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Semions (Abelian): Toric code in a magnetic field / loop gas with loop tension .Je/Jp
Connecting the limits: a microscopic model
H = !Jp
!
plaquettes p
!!(p),1 ! Je
!
edges e
!"(e),1
Varying the couplings we can connect the two extreme limits.Je/Jp
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The quantum phase transition
vary loop tension Je/Jp
“two sheets”topological order
“decoupled spheres”no topological order
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The quantum phase transition
“two sheets”topological order
“decoupled spheres”no topological order
topology fluctuationson all length scales
“quantum foam”
continuous transition
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Universality classes (semions)
“gapless” toric code
Ardonne, Fendley, Fradkin (2004)Castelnovo & Chamon (2008)
Fendley (2008)
z = 2 2D Isinguniversality
wavefunctiondeformations |!!
toric codeclassical
(polarized)state
Fradkin & Shenker (1979)ST et al. (2007)
Tupitsyn et al. (2008)Vidal, Dusuel, Schmidt (2009)
z = 1
3D Isinguniversality
RG flow
HHamiltoniandeformations
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The non-Abelian case
! !!
! !1 1
11
Fibonacci anyon fusion rules ! ! ! = 1 + !
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Non-abelian liquids → string net
! !!
! !1 1
11
Fibonacci anyon fusion rules ! ! ! = 1 + !string net
Non-abelian liquids → string net
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The quantum phase transition
vary string tension Je/Jp
“two sheets”topological order
“decoupled spheres”no topological order
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One-dimensional analog
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A continuous transition
• 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.
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Phase diagram
c = 14/15exactly solvable
c = 7/5exactly solvable
non-abeliantopological phase
non-abeliantopological phase
“quantum foam”c = 7/5
Jp
Jr
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Gapless theory & exact solution
(1, 1)
(1, !)
(!, 1)(!, !)1 !
The gapless theory is a CFT with .c = 14/15
The Hamiltonian maps onto the Dynkin diagram D6
The operators in the Hamiltonian form aTemperley-Lieb algebra
(Xi)2 = d · Xi XiXi±1Xi = Xi [Xi, Xj ] = 0|i! j| " 2for
d =!
2 + !
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c = 14/15exactly solvable
c = 7/5exactly solvable
non-abeliantopological phase
non-abeliantopological phase
“quantum foam”c = 7/5
Jp
Jr
0 2 4 6 8 10 12momentum kx [2π/L]
0 0
0.5 0.5
1 1
1.5 1.5
2 2
resc
aled
ene
rgy
E(k
x , k y )
L = 12, ky = 0L = 12, ky = πprimary fieldsdescendant fields
02/452/154/15
2/3
14/15
56/45
8/5
1+2/451+2/151+4/15
1+2/3
1+14/15
τ-flux
τ-fluxno-flux
(τ+1)-flux
(τ+1)-fluxτ-flux(τ+1)-flux
no-flux
τ-fluxτ-flux
τ-flux
τ-fluxno-flux
τ-flux
Gapless theory & exact solution
© Simon Trebst
c = 14/15exactly solvable
c = 7/5exactly solvable
non-abeliantopological phase
non-abeliantopological phase
“quantum foam”c = 7/5
Jp
Jr
0 2 4 6 8momentum kx [2π/L]
0 0
0.5 0.5
1 1
1.5 1.5
2 2
resc
aled
ene
rgy
E( k
x, ky )
L = 8, ky = 0L = 8, ky = πprimary fields
03/20 1/5
τ-flux
(τ+1)-flux
(τ+1)-flux
(τ+1)-flux
τ-flux
2/5
19/20
6/5
7/4
7/5
τ-flux
τ-fluxno-flux
Gapless theory & exact solution
© Simon Trebst
Dressed flux excitations
0 π/2 π 3π/2 2πmomentum Kx
0 0
0.5 0.5
1 1
1.5 1.5
2 2en
ergy
E(K
x)
θ = 0.3π
θ = 0.4π
θ = 0.6π
θ = 0.7π
θ = 0.8π
θ = 0.6π
c = 14/15exactly solvable
c = 7/5exactly solvable
non-abelian
topological phase
non-abelian
topological phase
“quantum foam”c = 7/5
Jp
Jr
© Simon Trebst
Back to two dimensions
wavefunctiondeformations |!!
z = 2c = 14/15Fendley & Fradkin (2008)
string net classicalstate
z = 1
continuoustransition?
HHamiltoniandeformations
string netz = 1
c = 14/15classical
state1+1 D
2+1 D
2+0 D
© Simon Trebst
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. WangarXiv:0906.1579 → Nature Physics (accepted)