Topology driven quantum phase transitions

Transcription

1 Topology driven quantum phase transitions Dresden July 2009 Simon Trebst Microsoft Station Q UC Santa Barbara Charlotte Gils Alexei Kitaev Andreas Ludwig Matthias Troyer Zhenghan Wang

2 Topological quantum liquids Spontaneous symmetry breaking ground state has less symmetry than high-t phase Landau-Ginzburg-Wilson theory local order parameter cosmic microwave background Topological order ground state has more symmetry than high-t phase degenerate ground states non-local order parameter quasiparticles have fractional statistics = anyons

3 Topological order time reversal symmetry broken invariant quantum Hall liquids magnetic materials

4 Quantum phase transitions time-reversal invariant systems topological order λ c some other (ordered) state λ 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.

5 Time-reversal invariant liquids chiral excitations anyonic liquid L flux excitations non-chiral anyonic liquid L

6 Flux excitations anyonic liquid flux through hole flux through tube

7 Flux excitations anyonic liquid flux through hole flux through tube skeleton lattice

8 Abelian liquids loop gas / toric code Semion fusion rules s s =1 loop gas s s

9 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

10 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.

11 Connecting the limits: a microscopic model H = J p δ φ(p),1 J e δ l(e),1 plaquettes p edges e Varying the couplings J e /J p we can connect the two extreme limits. Semions (Abelian): Toric code in a magnetic field / loop gas with loop tension. J e /J p

12 The quantum phase transition two sheets topological order decoupled spheres no topological order vary loop tension Je /Jp

13 The quantum phase transition two sheets topological order decoupled spheres no topological order quantum foam topology fluctuations on all length scales continuous transition

14 Universality classes (semions) Hamiltonian deformations H 3D Ising universality Fradkin & Shenker (1979) ST et al. (2007) Tupitsyn et al. (2008) Vidal, Dusuel, Schmidt (2009) z = 1 toric code gapless toric code RG flow classical (polarized) state z = 2 2D Ising universality wavefunction deformations ψ Ardonne, Fendley, Fradkin (2004) Castelnovo & Chamon (2008) Fendley (2008)

15 The non-abelian case Fibonacci anyon fusion rules τ τ τ τ τ τ = 1 + τ τ

16 Non-abelian liquids string net Fibonacci anyon fusion rules τ τ τ τ string net τ τ = 1 + τ τ

17 The quantum phase transition two sheets topological order decoupled spheres no topological order vary string tension Je /Jp

18 One-dimensional analog

19 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.

20 Phase diagram J r c = 14/15 exactly solvable non-abelian topological phase J p quantum foam c =7/5 non-abelian topological phase c = 7/5 exactly solvable

21 Gapless theory & exact solution The operators in the Hamiltonian form a Temperley-Lieb algebra (X i ) 2 = d X i X i X i±1 X i = X i [X i, X j ] = 0 d = 2+φ for i j 2 The Hamiltonian maps onto the Dynkin diagram D6 (1, 1) 1 (τ, τ) τ (1, τ) (τ, 1) The gapless theory is a CFT with c = 14/15.

22 Gapless theory & exact solution rescaled energy E(k x, k y ) /15 τ-flux no-flux τ-flux momentum k x [2π/L] quantum foam c =7/5 J r 1+2/3 8/5 1.5 τ-flux /45 1+4/15 τ-flux (τ+1)-flux 1+2/15 (τ+1)-flux 1+2/45 τ-flux /15 τ-flux 0.5 no-flux L = 12, k y = 0 2/3 L = 12, non-abelian k y = π primary topological fields phase descendant fields 4/15 τ-flux 0.5 2/15 (τ+1)-flux 0 2/45 0 τ-flux no-flux 0 non-abelian topological phase c = 14/15 exactly solvable J p c = 7/5 exactly solvable

23 Gapless theory & exact solution 2 2 rescaled energy E( k x, k y ) 7/4 τ-flux non-abelian topological phase quantum foam c =7/5 J r /5 τ-flux 6/5 1 (τ+1)-flux 19/20 1 (τ+1)-flux L = 8, k y = L = 8, k y = π 0.5 2/5 primary fields τ-flux 1/5 3/20 (τ+1)-flux τ-flux 0 0 no-flux momentum k x [2π/L] non-abelian topological phase c = 14/15 exactly solvable J p c = 7/5 exactly solvable

24 Dressed flux excitations J r c = 14/15 exactly solvable non-abelian topological phase quantum foam c =7/5 c = 7/5 exactly solvable non-abelian topological phase J p energy E(K x ) 2 2 θ = 0.6π θ = 0.3π θ = 0.4π 1 1 θ = 0.6π θ = 0.7π θ = 0.8π π/2 π 3π/2 2π momentum K x

25 Back to two dimensions string net z = 1 c = 14/ D classical state Hamiltonian deformations H z = D continuous transition? string net classical state wavefunction deformations ψ z = 2 c = 14/ D Fendley & Fradkin (2008)

26 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: Nature Physics (accepted)