The Landscape of the Hubbard model Subir Sachdev Highly Frustrated Magnetism 2010 Johns Hopkins University August 1 Talk online: sachdev.physics.harvard.edu HARVARD The Hubbard Model 1 1 H = tijci†αcjα + U ni ni µ ci†αciα − ↑ − 2 ↓ − 2 − i<j i i t “hopping”. U local repulsion, µ chemical potential ij → → → Spin index α = , ↑ ↓ niα = ci†αciα ci†αcjβ + cjβci†α = δijδαβ ciαcjβ + cjβciα =0 Will study the honeycomb and square lattices. Outline 1. Honeycomb lattice: semi-metal and antiferromagnetism Dirac fermions and the Gross-Neveu model 2. Spin liquids Unified formulation as a SO(4) gauge theory 3. Instabilities of spin liquids Geometric phases, valence bond solids, and other competing orders Cenke Xu Harvard UCSB Liang Fu Harvard Physical Review Letters 105, 057201 (2010) and to appear Outline 1. Honeycomb lattice: semi-metal and antiferromagnetism Dirac fermions and the Gross-Neveu model 2. Spin liquids Unified formulation as a SO(4) gauge theory 3. Instabilities of spin liquids Geometric phases, valence bond solids, and other competing orders The Hubbard Model 1 1 H = tijci†αcjα + U ni ni µ ci†αciα − ↑ − 2 ↓ − 2 − i,j i i In the limit of large U, and at a density of one particle per site, this maps onto the Heisenberg antiferromagnet a a HAF = JijSi Sj i<j where a = x, y, z, a 1 a a S = c †σ c , i 2 iα αβ iβ with σa the Pauli matrices and 4t2 J = ij ij U Graphene t e2 e1 e3 Graphene t Q1 Semi-metal with massless Dirac fermions Q Brilluoin zone − 1 Graphene t t Insulating Dirac antiferromagnet semi-metal with Neel order U/t t εk H = t c† c ε c† c 0 − ij iα iα ≡ k kα kα i<j k k Begin with free electrons. • t εk H = t c† c ε c† c 0 − ij iα iα ≡ k kα kα i<j k k Begin with free electrons. • Add local antiferromagnetism with order pa- • rameter ϕ i H = ϕ (r )( 1) c† σ c sdw − i − iα αβ iβ i t εk H = t c† c ε c† c 0 − ij iα iα ≡ k kα kα i<j k k Begin with free electrons. • Add local antiferromagnetism with order pa- • rameter ϕ i H = ϕ (r )( 1) c† σ c sdw − i − iα αβ iβ i The phase with ϕ = 0 is an insulator with a • gap between conduction and valence bands. Honeycomb lattice at half filling. We define the unit length vectors e =(1, 0) , e =( 1/2, √3/2) , e =( 1/2, √3/2). 1 2 − 3 − − (1) Note that e e = 1/2fori = j,ande + e + e = 0. i · j − 1 2 3 We take the origin of co-ordinates of the honeycomb lattice at the center of an empty hexagon. The A sublattice sites closest to the origin are at e1, e2,ande3, while the B sublattice sites closest to the origin are at e , e ,and e . − 1 − 2 − 3 The reciprocal lattice is generated by the wavevectors 4π 4π 4π G = e , G = e , G = e (2) 1 3 1 2 3 2 3 3 3 The first Brillouin zone is a hexagon whose vertices are given by 1 1 1 Q = (G G ) , Q = (G G ) , Q = (G G ), 1 3 2 − 3 2 3 3 − 1 3 3 1 − 2 (3) and Q , Q ,and Q . − 1 − 2 − 3 We define the Fourier transform of the fermions by ik r cA(k)= cA(r)e− · (4) r and similarly for cB . The hopping Hamiltonian is H = t c† c + c† c (5) 0 − Aiα Bjα Bjα Aiα ij where α is a spin index. If we introduce Pauli matrices τ a in sublattice space (a = x, y, z), this Hamiltonian can be written as 2 d k x H = c†(k) t cos(k e ) + cos(k e ) + cos(k e ) τ 0 4π2 − · 1 · 2 · 3 + t sin(k e )+sin(k e )+sin(k e ) τ y c(k) (6) · 1 · 2 · 3 The low energy excitations of this Hamiltonian are near k Q . ≈ ± 1 t e2 e1 e3 Q1 Q Brilluoin zone − 1 In terms of the fields near Q and Q ,wedefine 1 − 1 ΨA1α(k)=cAα(Q + k) Ψ (k)=c ( Q + k) A2α Aα − ΨB1α(k)=cBα(Q + k) Ψ (k)=c ( Q + k) (7) B2α Bα − We consider Ψ to be a 8 component vector, and introduce Pauli matrices ρa which act in the 1, 2 valley space. Then the Hamiltonian is 2 d k y x z H = Ψ†(k) vτ k + vτ ρ k Ψ(k), (8) 0 4π2 x y z z where v =3t/2; below we set v = 1. Now define Ψ = Ψ†ρ τ . Then we can write the imaginary time Lagrangian as = iΨ (ωγ + k γ + k γ ) Ψ (9) L0 − 0 x 1 y 2 where γ = ρz τ z γ = ρz τ x γ = τ y (10) 0 − 1 2 − Antiferromagnetism We use the operator equation (valid on each site i): 1 1 2U U U n n = Sa2 + (11) ↑ − 2 ↓ − 2 − 3 4 Then we decouple the interaction via 2U 3U exp dτSa2 = Ja(τ)exp dτ Ja2 JaSa 3 i D i − 8 i − i i i i (12) We now integrate out the fermions, and look for the saddle point a of the resulting effective action for Ji . At the saddle-point we find that the lowest energy is achieved when the vector has opposite orientations on the A and B sublattices. Anticipating this, we look for a continuum limit in terms of a field ϕa where Ja = ϕa , Ja = ϕa (13) A B − The coupling between the field ϕa and the Ψ fermions is given by a a a a a J c† σ c = ϕ c† σ c c† σ c i iα αβ iβ Aα αβ Aβ − Bα αβ Bβ i a z a a z a = ϕ Ψ†τ σ Ψ = ϕ Ψρ σ Ψ (14) − From this we motivate the low energy theory 1 u 2 = Ψγ ∂ Ψ + (∂ ϕa)2 + sϕa2 + ϕa2 λϕaΨρz σaΨ L µ µ 2 µ 24 − (15) z a z a Note that the matrix ρ σ commutes with all the γµ;henceρ σ is a matrix in “flavor” space. This is the Gross-Neveu model, and it describes the quantum phase transition from the Dirac semi-metal to an insulating N´eelstate. In mean-field theory, the Dirac semi-metal is obtained for s > 0 with ϕa = 0. The N´eel a state obtains for s < 0, and we have ϕ = N0δaz (say), and so the dispersion of the electrons is ω = k2 + λ2N2 (16) k ± 0 near the points Q . These form the conduction and valence ± 1 bands of the insulator. Outline 1. Honeycomb lattice: semi-metal and antiferromagnetism Dirac fermions and the Gross-Neveu model 2. Spin liquids Unified formulation as a SO(4) gauge theory 3. Instabilities of spin liquids Geometric phases, valence bond solids, and other competing orders Outline 1. Honeycomb lattice: semi-metal and antiferromagnetism Dirac fermions and the Gross-Neveu model 2. Spin liquids Unified formulation as a SO(4) gauge theory 3. Instabilities of spin liquids Geometric phases, valence bond solids, and other competing orders Z. Y. Meng et al., Nature 464, 847 (2010). (x, τ) N´eel Rz Perform SU(2) rotation z on filled band of electrons: R c z z∗ ψ+ ↑ = ↑ − ↓ c z z∗ ψ ↓ ↓ ↑ − εk ψ states empty ± ψ states filled ± k Quantum “disordering” magnetic order c z z∗ ψ+ ↑ = ↑ − ↓ c z z∗ ψ ↓ ↓ ↑ − Quantum “disordering” magnetic order c z z∗ ψ+ ↑ = ↑ − ↓ c z z∗ ψ ↓ ↓ ↑ − SU(2)spin Quantum “disordering” magnetic order c z z∗ ψ+ ↑ = ↑ − ↓ c z z∗ ψ ↓ ↓ ↑ − U(1)charge Quantum “disordering” magnetic order c z z∗ ψ+ ↑ = ↑ − ↓ c z z∗ ψ ↓ ↓ ↑ − 1 U U − × SU(2)s;gauge S. Sachdev, M. A. Metlitski, Y. Qi, and S. Sachdev Phys. Rev. B 80, 155129 (2009) Quantum “disordering” magnetic order c z z∗ ψ+ ↑ = ↑ − ↓ c z z∗ ψ ↓ ↓ ↑ − The Hubbard model can be written exactly as a lattice gauge theory with a SU(2) SU(2) U(1) s;g× spin× charge invariance. The SU(2)s;g is a gauge invariance, while SU(2) U(1) is a global symmetry spin× charge Global symmetry operations: Spin rotations, SU(2) • spin Combine electromagnetic charge (electron number) • U(1)charge with particle-hole transformations to ob- tain SU(2)pseudospin. Global symmetry operations: Spin rotations, SU(2) • spin Combine electromagnetic charge (electron number) • U(1)charge with particle-hole transformations to ob- tain SU(2)pseudospin. Quantum “disordering” magnetic order c z z∗ ψ+ ↑ = ↑ − ↓ c z z∗ ψ ↓ ↓ ↑ − Global symmetry operations: Spin rotations, SU(2) • spin Combine electromagnetic charge (electron number) • U(1)charge with particle-hole transformations to ob- tain SU(2)pseudospin. Projected fermion wavefunctions (Fisher, Wen, Lee, Kim,Senthil) c b∗ b∗ f1 ↑ = 1 2 c† b2 b1 f2† ↓ − Global symmetry operations: Spin rotations, SU(2) • spin Combine electromagnetic charge (electron number) • U(1)charge with particle-hole transformations to ob- tain SU(2)pseudospin. Projected fermion wavefunctions (Fisher, Wen, Lee, Kim,Senthil) c b∗ b∗ f1 ↑ = 1 2 c† b2 b1 f2† ↓ − neutral fermionic spinons Global symmetry operations: Spin rotations, SU(2) • spin Combine electromagnetic charge (electron number) • U(1)charge with particle-hole transformations to ob- tain SU(2)pseudospin. Projected fermion wavefunctions (Fisher, Wen, Lee, Kim,Senthil) c b∗ b∗ f1 ↑ = 1 2 c† b2 b1 f2† ↓ − charged slave boson/rotor Global symmetry operations: Spin rotations, SU(2) • spin Combine electromagnetic charge (electron number) • U(1)charge with particle-hole transformations to ob- tain SU(2)pseudospin.