Criticality in topologically ordered systems: a case study

Transcription

1 Criticality in topologically ordered systems: a case study Fiona Burnell Schulz & FJB 16 FJB 17?

2 Phases and phase transitions ~ 194 s: Landau theory (Liquids vs crystals; magnets; etc.) Local order parameter ~198 s: Topological order (RVB; FQHE; spin liquids) No local order parameter; TQFT; topology-dependent ground state degeneracy; anyons

3 Transitions between different topological orders Condense a boson : New topological order Mutual statistics with some other excitations: confinement (i.e. magnetic flux quantization in a superconductor) e e i)e i / Challenging for field theoretic approaches need alternative methods (Duality)

4 Today s examples gauge theory (Toric code) e (Charge 1, modulo ) : e e =1 m (flux, modulo : m m =1 Isi (non-chiral bilayer) (fermion with charge 1, modulo ) (flux, modulo, with bound Majorana zero mode) : =1 : =1+

5 Bosons to condense: gauge theory (Toric code) : e, m Isi (non-chiral bilayer) Similar to Ising spins Unlike any spin due to nonabelian statistics!

6 Bosons to condense: gauge theory (Toric code) : e, m Isi (non-chiral bilayer) similar to Ising spins Unlike any spin due to nonabelian statistics! 1 1

7 The plan Duality mapping to Ising transitions Extending this to the non-abelian case?

8 Duality on the lattice: example Wegner 71 No e excitations: edges with = domain walls z = 1 Represent by spins on edges of (dual) honeycomb lattice i i j σ z (i) S z (i)s z (j) Constraint: down spins form closed loops

9 Duality on the lattice: example Wegner 71 i σ z (i) i j S z (i)s z (j) ) i 3 Y x (i) S x (i)

10 Duality on the lattice: example Wegner 71 H = h X i S x i J X hiji S z i S z j

11 Duality on the lattice: example Wegner 71 H = h X i S x i J X hiji S z i S z j H = h X P Y xi J X i=edges z i z =1 z = 1

12 Duality on the lattice: Toric code H = X V Y zi h X P Y xi J X i=edges z i Kitaev 3 Dual Hamiltonian is almost the Toric code Hamiltonian, a wellknown model for topological order (plus the J term )

13 Duality on the lattice: Toric code Excitations: H = X Y zi h X Y xi V Vertex constraint P Kinetic term Kitaev 3 charges (e) fluxes (m)

14 What can we learn? Confining transition: same as ordering in transverse field Ising model Fradkin & Shenker 79 Consequences for dynamics Chandran, FJB, Khemani, & Sondhi 13 Fradkin & Shenker 79

15 What about Isi topological order? Can also study a lattice model, and look for best approximation to spin dual. Lattice Hilbert space: Toric code ( ): z =1 z = 1 Isi : 1

16 What about Isi topological order? Toric code ( ): z =1 z = 1 Isi : 1 Vertex constraints: Closed red loops

17 The example: Isi topological order Hamiltonian Levin & Wen 6 H = X B V X (BP + BP 1 ) +edge terms V P Energetically imposes vertex Flips Flips constraint 1 1

18 Edge terms: operators that create bosons i )ŝ i ŝ i =( 1) n,i creates pairs of 8 >< 1 if i is a 1 edge ŝ i = 1 if i is a edge >: if i is a edge bosons creates pairs of bosons

19 Vertex constraints and domain walls Even number of green loops ending on red loops: domain walls in Ashkin-Teller model ( ) Not true if there are excitations!

20 Counting: same or different, depending on BC s No non-contractible loops: Non-contractible loop: Isi : 1 4N P 1 + N P 1 4 N P 1 :4 N P 1 (See also Gils 9)

21 Operators: Operator algebra (products, commutators) is essentially the same for and Isi except for commutators related to non-abelian statistics, which are different

22 What about Isi topological order? Isi (Spin dual: Ashkin Teller) (NO spin dual) Hilbert spaces different, but scale the same way Operators to create/ measure plaquette fluxes have equivalent fusion rules Different braiding: difference in some operator commutation relations

23 Phase diagram H = X P B P X J ŝ e + J ŝ e Schulz & FJB 16 e tan 1 J e /J p col 1 trivial Z tan 1 (J e /Jp) AF- Ising col plaq - -

24 Phase diagram H = X P B P X J ŝ e + J ŝ e Schulz & FJB 16 e tan 1 J e /J p col 1 Z trivial tan 1 (J e /Jp) AF- Ising Isi region: models not equivalent; differences apparent if there are fluxes col plaq - -

25 Phase diagram H = X P B P X J ŝ e + J ŝ e e tan 1 J e /J p col 1 Z trivial tan 1 (J e /Jp) AF- Ising Toric code topological order (possibly with translation symmetry breaking) col plaq - Schulz & FJB 16 -

26 Phase diagram H = X P B P X J ŝ e + J ŝ e e tan 1 J e /J p tan 1 (J e /Jp) col 1 AF- Z Ising trivial Various phases that are topologically trivial (but may break translation; different combinations of ferromagnetic & antiferromagnetic order). col plaq - Schulz & FJB 16 -

27 Phase diagram H = X P B P X J ŝ e + J ŝ e e col 1 Z tan 1 J e /J p trivial Inequivalent: Symmetry-broken in tan 1 (J e /Jp) AF- Ising Topologically ordered in Isi col plaq - Schulz & FJB 16 -

28 Lessons from duality H = X P B P X J ŝ e + J ŝ e tan 1 J e /J p e col 1 Z trivial No fluxes: spin dual tan 1 (J e /Jp) AF- col Ising plaq Low energy theories equivalent everywhere along this line! - -

29 Lessons from duality H = X P B P X J ŝ e + J ŝ e tan 1 J e /J p e col 1 Z trivial tan 1 (J e /Jp) AF- Ising - col 3D XY transition plaq - 3D Ising transition

30 Lessons from duality H = X P B P X J ŝ e + J ŝ e tan 1 (J e /Jp) col 1 AF- Z tan 1 Ising e J e /J p trivial No lines: exact spin dual! 3D Ising transition col plaq 3D XY transition - -

31 New transitions? tan 1 (J e /Jp) col 1 AF- col Z tan 1 Ising J e /J p trivial plaq Direct transition from to trivial topological order: condense Same as in Ashkin- Teller, if braiding is irrelevant Isi - -

32 Morals Phase transitions between topological orders: better tools needed! Exact duality describes some, but not all Approximate duality (differences arising only because of nonabelian statistics) is suggestive, but not sufficient to determine transition