SPIN LIQUIDS AND FRUSTRATED MAGNETISM

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1 SPIN LIQUIDS AND FRUSTRATED MAGNETISM Classical correlations, emergent gauge fields and fractionalised excitations John Chalker Physics Department, Oxford University For written notes see:

2 Outline Introduction Frustration and classical degeneracy Alternatives to symmetry-breaking Classical models and emergent gauge fields Triangular lattice Ising antiferromagnet Spin ice Classical dimer models Spin liquids Quantum dimer models Z 2 andu(1) spin liquids

3 Coulomb phases in classical dimer models General setting: bipartite lattice with oriented links & coord # z Dimer covering: n dimers touching every site Flux B l on linkl z n if dimer present n if no dimer

4 Coarse-grained description of Coulomb phases In three dimensions B = A P( B) e H with H = κ 2 B 2 d 3 r In two dimensions A = ẑh(x,y) equivalent to H = κ 2 h 2 d 2 r+h 1

5 U(1) vsz 2 dimer model phases What happens on non-bipartite lattices? Recall bipartite case A B Flux crossing A B not changed by local dimer moves

6 U(1) vsz 2 dimer model phases What happens on non-bipartite lattices? Swap locally conserved fluxes for parities

7 U(1) vsz 2 dimer model phases What happens on non-bipartite lattices? Swap locally conserved fluxes for parities Parity of dimer#crossing unchanged under local rearrangements

8 U(1) vsz 2 dimer model phases What happens on non-bipartite lattices? Swap locally conserved fluxes for parities Parity of dimer#crossing unchanged under local rearrangements

9 Z 2 dimer model phases Dimar correlations generically short range Entropic monomer-monomer potential short-range in 2D & 3D provided dimers not ordered

10 RVB picture Describe wavefn for spin liquid in basis of short range singlets 0 = Many questions: non-orthogonality, completeness, Néel state in this basis...

11 Quantum dimer models Quantum mechanics on Hilbert space with correlns of classical frustrated magnets built in Take set of dimer covering as basis states ψ = C A C C Pick quantum Hamiltonian to induce dynamics Simplest choice: plaquette resonances or

12 Quantum dimer models Quantum mechanics on Hilbert space with correlns of classical frustrated magnets built in Take set of dimer covering as basis states ψ = C A C C Pick quantum Hamiltonian to induce dynamics Simplest choice: plaquette resonances Hence Rokhsar-Kivelson model H = { t[ = + = ]+v[ + = = ]}

13 Rokhsar-Kivelson point At t = v > 0 Hamiltonian is a sum of projectors becomes H = { t[ = + = ]+v[ + = = ]} H RK = t ( = )( = ) Ground state wavefunction G = C C is equal-amplitude superposn of all dimer configs in given sector

14 Rokhsar-Kivelson point Ground state wavefunction G = C C is equal-amplitude superposn of all dimer configs in given sector Consequences Evaluate (diagonal) quantum correlators from classical avge

15 Rokhsar-Kivelson point Ground state wavefunction G = C C is equal-amplitude superposn of all dimer configs in given sector Consequences Evaluate (diagonal) quantum correlators from classical avge Dimer correlators Coulomb phase correlns on bipartite lattices Exponentially decaying correlns on non-bipartite lattices Static monomers zero-energy state for all monomer separations deconfined

16 QDM phase diagrams Is behaviour at RK point representative? For v > t H = H RK +(v t)[ + = = ] Staggered ground states avoid potential term or maximum flux states

17 QDM phase diagrams Is behaviour at RK point representative? For v H = { t[ = + = ]+v[ + = = ]} Maximise# flippable plaquettes columnar ground states

18 QDM phase diagrams Is behaviour at RK point representative? For v H = { t[ = + = ]+v[ + = = ]} Maximise# flippable plaquettes columnar ground states Intermediate v other possibilities e.g. plaquette states

19 QDM phase diagrams From Monte Carlo simulations - no sign problem Square lattice RK pt columnar plaquette staggered RK point is exceptional monomers confined for t v 1 v/t

20 QDM phase diagrams From Monte Carlo simulations - no sign problem Square lattice columnar RK pt plaquette staggered 1 v/t RK point is exceptional monomers confined for t v Triangular lattice RK pt columnar Z RVB DeconfinedZ 2 phase for v t 2 staggered v/t 1

21 QDM phase diagrams From Monte Carlo simulations - no sign problem Square lattice columnar RK pt plaquette staggered 1 v/t RK point is exceptional monomers confined for t v Diamond lattice columnar U(1) RVB In 3D: deconfinedu(1) phase for v t 1 staggered v/t

22 QDM phase diagrams Summary Bipartite lattices Confining in 2D except at RK point Deconfined U(1) phase neighbouring RK point in 3D Non-bipartite lattices DeconfinedZ 2 phases neighbouring RK point in 2D and 3D Consistent with general properties of gauge theories Z 2 has confined and deconfined phases in both (2+1)D and (3+1)D Compact U(1) always confined in (2+1)D has confined and deconfined phases in (3+1)D

23 Excited states in QDMs Outside the dimer model Hilbert space Monomer pairs spinon pairs deconfined in RVB phase energy not fixed by QDM natural to take as gapped

24 Excited states in QDMs Outside the dimer model Hilbert space Monomer pairs spinon pairs deconfined in RVB phase energy not fixed by QDM natural to take as gapped Within the dimer model Hilbert space ψ = C A C C must have nodes for orthogonality with ground state Two types of excitation Vison gapped Emergent photon inu(1) phase gapless

25 Vison trial wavefunction ψ vison = C ( 1) n C C n C =#of dimers crossing reference line Differs from ground state (only) near core expect finite energy cost With pbc appear in only pairs

26 Emergent photons Dimer counting operator σˆτ (r) for bond direction ˆτ at r σˆτ (q) = r σˆτ (r)e iq r Trail wavefunction ψ photon = σˆτ (q) G wave in dimer orientation emergent flux

27 Emergent photons Dimer counting operator σˆτ (r) for bond direction ˆτ at r σˆτ (q) = r σˆτ (r)e iq r Trail wavefunction ψ photon = σˆτ (q) G wave in dimer orientation emergent flux Estimate of energy E(q) G σˆτ( q)[h E 0 ]σˆτ (q) G G σˆτ ( q)σˆτ (q) G f(q) 2S(q) Express f(q) as f(q) = G [σˆτ ( q),[h,σˆτ (q)]] G

28 Gapless modes? Estimate of energy with and E(q) f(q) 2S(q) f(q) = G [σˆτ ( q),[h,σˆτ (q)]] G S(q) = G σˆτ ( q)σˆτ (q) G Can [H,σˆτ (q)] = 0? Require same contribution toσˆτ (q) from both configs in resonance Occurs for q = (π,π,π)+k (or 2D equivalent) as k 0

29 Gapless modes? Can [H,σˆτ (q)] = 0? Require same contribution toσˆτ (q) from both configs in resonance Occurs for q = (π,π)+k or q = (π,π,π)+k as k 0 Some details In fact [H,σˆτ (q)] = 0 for any k ˆτ But S(q) = [k 2 (k ˆτ) 2 ]/k 2 also has lines of zeros So E(q) k 2 for q = (π,π,π)+k (or 2D equivalent) 3D: linear dispersion away from RK point

30 Continuum action for (2+1)D? S = {( τ h) 2 +κ 2 ( 2 h) 2} d 2 rdτ Leads to h(k,ω) 2 1 ω 2 +κ 2 k 4 So that equal-time correlations are dω h(k,ω) 2 1 κk 2 as expected at RK point from height model

31 Continuum action for (2+1)D? S = {( τ h) 2 +κ 2 ( 2 h) 2} d 2 rdτ This is fine-tuned more general form is S = {( τ h) 2 +ρ 2 ( h) 2 +ρ 4 ( 2 h) 2 λcos2πh } d 2 rdτ Anticipate ρ 2 1 v/t Thenv > t favours max h staggered phase and forv < t cosine term is relevant pinned state so nou(1) phase for (2+1)D

32 Continuum action for (3+1)D? Equivalent argument in (3+1)D gives S = { ( τ A) 2 +ρ 2 ( A) 2 +ρ 4 ( ( A)) 2 } d 2 rdτ Again anticipate ρ 2 1 v/t For v < t in Coulomb gauge, omitting RG-irrelevant terms S = { E 2 +ρ 2 B 2} d 2 rdτ with E τ A and B A

33 Quantum dimer models Summary Quantum mechanics on Hilbert space with constraints inspired by classical frustrated magnets Classical topological sectors quantum topological order DeconfinedZ 2 phase in (2+1)D and (3+1)D Deconfined U(1) phase stable only in (3+1)D