Quantum Field Theory of Two-Dimensional Spin Liquids Flavio S. - PowerPoint PPT Presentation

Quantum Field Theory of Two-Dimensional Spin Liquids Flavio S. Nogueira Theoretische Physik III, Ruhr-Universit¨ at Bochum Hamburg, Oct. 2013

Outline: Mott insulators and the Hubbard model spin liquids and U (1) gauge theories Chiral spin liquids Quantum dimer model Lecture notes with full details of the derivations will be provided. Some additional material not included here will be also discussed there. First (incomplete) version will become available at the homepage of the student seminar next week (possibly on Monday).

Band theory of solids Gap Metal Insulator = Fermi energy = wavevector Response to small external perturbation

Landau Fermi liquid theory Effective quantum field theory of interacting Fermi systems Quasi-particle concept: one-to-one correspondence between particles in a free electron gas and elementary excitations in the interacting Fermi systems Non-interacting Fermi system at T=0 1 Lev Landau 1 Interacting Fermi system at T=0 }

Mott insulators Band structure calculations predict metallic behavior in some materials, which is contradicted by experiments Failure of one-particle theory = ⇒ Many-body theory of interacting systems is necessary Landau Fermi liquid theory describes well the metallic behavior of interacting systems, but is insufficient to deal with Mott insulators Paradigmatic model for Mott insulators: Hubbard model � � f † � � H = − t iσ f jσ − µ n iσ + U n i ↑ n i ↓ σ i,σ i � i,j � n iσ ≡ f † iσ f iσ U > 0

Mott insulators Schematic picture of half-filled band (or, more generally, one electron per unit cell): Band theory does not forbid double occupancy, provided electrons have opposite spin: = ⇒ predicts metallic behavior Double occupancy is actually forbidden for U ≫ t

Mott-Hubbard metal-insulator transition Typically a Mott insulator is magnetically ordered AF Metal n ( ) ε 1 } Z k F ε ε F U U> 0 c U c ∼ t

Mott insulators Fermionic Hubbard model at half-filling = ⇒ Metal-Mott insulator transition Bosonic Hubbard model at integer filling = ⇒ Superfluid-Mott insulator transition Insulating phase: Interaction driven gapped excitations, unbroken U (1) symmetry Superfluid phase: Interaction driven gapless excitations, broken U (1) symmetry Metallic phase: Fermi surface, unbroken U (1) symmetry Spin liquid: Mott insulator without broken symmetries and with fractionalized excitations

Symmetries of the Hubbard model Particle-hole symmetry at half-filling, i.e., 1 � j,σ � n jσ � = 1 , and L bipartite lattices: U (1) symmetry: f jσ → e iθ f jσ , f † jσ → e iθ f † jσ

Symmetries of the Hubbard model 2 ψ † j S j , where S j = 1 SU (2) spin symmetry: S = � j � σψ j , with ψ j = [ f j ↑ f j ↓ ] T and � σ = ( σ x , σ y , σ z ) = ⇒ [ S , H ] = 0 SU (2) pseudo-spin symmetry (valid for bipartite lattices): 2 η † ση j , with η j = [ f j ↑ f † j e Q · R j J j , where J j = 1 j ↓ ] T J = � j � = ⇒ [ J , H ] = 0 Full symmetry of the Hubbard model is SO (4) = SU (2) × SU (2) The SO (4) symmetry allowed to complete the exact solution for the one-dimensional Hubbard model by obtaining the full excitation spectrum; see book by Essler, Frahm, G¨ ohmann, Kl¨ umper, and Korepin, The one-dimensional Hubbard model (Cambridge University Press, 2005) Strong-coupling ( U ≫ t ) limit of the Hubbard model: Heisenberg antiferromagnet ⇒ H = 2 t 2 � � i,j � S i · S j U

Symmetries of the Hubbard model Hubbard model in bipartite lattices at half-filling: µ = U/ 2 ( exact ) Proof: particle-hole transformation f iσ → e i Q · R i f † iσ , f † iσ → e i Q · R i f iσ , H ′ = U − 2 µ − t � � f † � � iσ f jσ + ( µ − U ) n iσ + U n i ↑ n i ↓ σ σ i � i,j � ⇒ H ′ = H . If F and F ′ are the free energy µ = U/ 2 = densities associated to Hamiltomians H and H ′ , 2 − n = − ∂ F ′ n = − ∂ F ∂µ , ∂µ µ = U/ 2 = ⇒ n = 2 − n = ⇒ n = 1

Mean-field theory for the Hubbard model At half-filling the Hubbard Hamiltonian can be rewritten as iσ f jσ − 2 U � � f † � S 2 H = − t i 3 σ i � i,j � Magnetic (mean-field) ground states: � S j � = m (FM) or � S j � = e i Q · R j m (AF) Due to nesting, AF instabilities arise at half-filling, so an AF ordered ground state is favored (lower energy) over a FM ground state. Away from half-filling a FM ground state is favored Spin liquid mean-field ground states (more later) arise in a square lattice with nearest neighbor hopping and half-filling only when generalizing SU (2) → SU ( N ) , with N sufficiently large.

Hubbard-Stratonovich transformation: m i · S i + 3 U � � f † � � m 2 H = − t iσ f jσ − U i 8 σ i i � i,j � Staggered magnetization: m i = e i Q · R i m . The SU (2) symmetry allows us to choose m = m ˆ z without loss of generality. A B A B A B A A B k ,σ ψ † k σ M k σ ψ k σ + 3 UL 8 m 2 Mean-field Hamiltonian: H MF = � � c k σ � − σUm � � ε k 2 ψ † � c † c † � ψ k σ = , k σ = , M k σ = ¯ k σ k σ σUm c k σ ¯ ε k 2

� k + ∆ 2 , where ∆ 2 ≡ U 2 m 2 / 4 Energy spectrum: E ± ε 2 k = ± � ′ Ground state energy density: E 0 = − 2 k E + 2 U ∆ 2 3 k + L � 0 d 2 k ∂E 0 3 1 1 � √ ∂m = 0 = ⇒ 2 U = k +∆ 2 = − 4 t dε √ (2 π ) 2 ε 2 ε 2 +∆ 2 Approximate form of the density of states in two dimensions: ρ ( ε ) ≈ ln( t/ε ) 4 π 2 t (see lecture notes) Solution of gap equation: � � � 12 t ∆ = 2 π 2 t exp − U At half-filling mean-field theory yields an AF ground state for all U > 0 = ⇒ no metal-insulator transition at finite U , i.e., U c = 0

Disordering Mott insulators Our mean-field theory fails to describe a metal-insulator transition with U c � = 0 . Quantum fluctuations around the mean-field solution are expected to destroy the AF order at weak-coupling. We will give an example with long-range Coulomb interaction in a honeycomb lattice (i.e., interacting graphene), where the gap has the form � � � t ∆ ∼ exp − const , such that the system becomes U − U c metallic for U < U c , when the gap vanishes. However, such a system features an excitonic condensate for U > U c rather than an AF phase (this is a consequence of the long-range Coulomb interaction). In this case it is not the SU (2) symmetry that is being spontaneously broken, but the so-called chiral symmetry. The chiral symmetry will also be important in our study of U (1) spin liquids. Important question: can a Mott insulator also be disordered in the strong-coupling regime?

Disordering Mott insulators Tight-binding in a honeycomb lattice: B B A A B A B A A B A B A B A A B A B B A B A B A A A B B A B A B

Disordering Mott insulators Tight-binding in a honeycomb lattice: B B A A B A B A A B A B A B A A B A B B A B A B A A A B B A B A B

Disordering Mott insulators Tight-binding in a honeycomb lattice: B B A A B A B A A B A B A B A A B A B B A B A B A A A B B A B A B

Disordering Mott insulators Tight-binding in a honeycomb lattice: B B A A B A B A A B A B A B A A B A B B A B A B A A A B B A B A B

Disordering Mott insulators Tight-binding in a honeycomb lattice: √ 3 x − 1 � c iσ , i ∈ A a 1 = ˆ y , a 2 = 2 ˆ 2 ˆ y f iσ = ¯ c iσ , i ∈ B √ 3 x − 1 k ,σ ψ † H 0 = � a 3 = − 2 ˆ 2 ˆ k σ M k ψ k σ y � � � � c k σ 0 − tϕ k � � ψ † c † c † ψ k σ = , k σ = , M k = ¯ k σ k σ − tϕ ∗ ¯ c k σ 0 k � √ � ϕ k = � 3 i =1 e i k · a i = e ik y + 2 cos 3 e − ik y / 2 2 k x Energy eigenvalues: � √ � √ � � � � � 3 � 3 3 � � 1 + 4 cos 2 E k = ± t 2 k x + 4 cos 2 k x cos 2 k y

4 π The spectrum E k has (independent) nodes at k 1 = 3 ˆ x and √ 3 2 π x − 2 π k 2 = 3 ˆ 3 ˆ y √ 3 Expanding the tight-binding Hamiltonian around the nodes: � � ψ † H 0 ≈ − t k + k i ,σ M k + k i ψ k + k i , σ k ,σ i =1 , 2 ϕ k + k 1 ≈ 3 k + k 1 ≈ 3 ϕ ∗ 2( − k x + ik y ) , 2( − k x − ik y ) , ϕ k + k 2 ≈ 3 e iπ/ 3 k + k 2 ≈ 3 e − iπ/ 3 ϕ ∗ ( − k x − ik y ) , ( − k x + ik y ) 2 2 c k + k 1 ,σ ] T and Define ψ 1 σ ( k ) = [ c k + k 1 ,σ ¯ ψ 2 σ ( k ) = [ e iπ/ 3 ¯ c k + k 2 ,σ c k + k 2 ,σ ] T � � 0 k x − ik y 3 t � � ψ † = ⇒ H 0 ≈ iσ ( k ) ψ iσ ( k ) 2 k x + ik y 0 k ,σ i =1 , 2 � � ψ † = v F iσ ( k ) k · � σψ iσ ( k ) k ,σ i =1 , 2 Here v F ≡ 3 t/ 2 and � σ = σ x ˆ x + σ y ˆ y

Four-component Dirac fermion representation: Ψ σ = [ ψ 1 σ ψ 2 σ ] T , Ψ σ = Ψ † γ 0 ¯ Dirac γ matrices: � σ z � iσ y � − iσ x 0 � 0 � 0 � γ 0 = γ 1 = γ 2 = , , 0 − σ z 0 − iσ y 0 iσ x Dirac Lagrangian: L 0 = ¯ Q = γ µ Q µ ); Ψ i / ∂ Ψ (Dirac “slash” notation: / ∂ = γ 0 ∂ t − v f � ∂ µ = ( ∂ t , v F ∇ ) and / γ · ∇ = ⇒ massless Dirac fermions Action including Coulomb interaction: H Coulomb = U � � 1 d 2 r ′ ¯ � d 2 r Ψ α ( r ) γ 0 Ψ α ( r ) Ψ β ( r ′ ) γ 0 Ψ β ( r ′ ) ¯ 2 | r − r ′ | α,β U ≡ e 2 /ǫ