Local Density of States in 1D Mott Insulators with a Boundary - PowerPoint PPT Presentation

Local Density of States in 1D Mott Insulators with a Boundary F.H.L. Essler (Oxford) D. Schuricht (Oxford), E. Fradkin and A. Jaefari (Urbana) D. Schuricht and F.H.L. Essler, JSTAT P11004 (2007) D. Schuricht, F.H.L. Essler, A. Jaefari and E. Fradkin, PRL 101, 086403 (2008). GGI, September 2008.

Outline • Statement of the problem. • Introduction and experimental motivation. • 1D Mott insulators. • Low-energy limit and bosonization. • Correlators is presence of a boundary. • Fermion autocorrelator. • Boundary state form factor approach to correlation functions. • Some results. • Summary.

Statement of the Problem Take the U(1) Thirring model on the half-line � � � 0 2 � � ¯ g α J α µ ( t, x ) J αµ ( t, x ) H = dx iv F Ψ a ( t, x ) γ 1 ∂ x Ψ a ( t, x ) − , g x = g y . −∞ a =1 α      R †  R ↑ ( t, x ) ↓ ( t, x ) µ = 1 ¯ J α Ψ a ( t, x ) γ µ σ α  , Ψ 2 =  . ab Ψ b ( t, x ) , Ψ 1 = 2 L † L ↑ ( t, x ) ↓ ( t, x ) Boundary Conditions: e.g. R σ (0) = − L σ (0) . Calculate the retarded fermion autocorrelation function � ∞ � 0 dte iωt dxe − iqx � 0 |{ R ( t, x ) , L † (0 , x ) }| 0 � 0 −∞ by combining form factor bootstrap and boundary state approaches.

Introduction � Reduced Dimensionality → large � Unusual Collective Many-Body Strong Interactions Physics at T = 0 • Numerous experimental realizations and measurements. • Integrable models allow calculation of measurable quantities: (effects of integrability breaking perturbations smaller than experimental error) – Neutron scattering − → 2-point function of spin operators � � δ αγ − Q α Q γ � S αγ ( ω, Q ) , I ( ω, Q ) ∝ Q · Q α,γ � ∞ 2 π e iωt 1 dt � l ( t ) S γ S αγ ( ω, Q ) e − i Q · ( R l − R l ′ ) � 0 | S α = l ′ | 0 � N −∞ l,l ′ – Photoemission − → 2-point function of electron operators – Scanning Tunneling Spectroscopy ?

Scanning Tunneling Spectroscopy (STS) Precise measurement of local single-particle density of states (LDOS) Energy STM tip V eV − e E F y Sample STM tip Sample x � eV Current : I ( V, x ) ∝ dE N sample ( E F + E, x ) N tip ( E F + E − eV ) 0 dI ( V, x ) ∝ N sample ( eV, x ) N tip ≈ const ⇒ dV

LDOS related to local single-particle Green’s function: � ∞ − 1 dt e iEt G ret ( t, x ) , N sample ( E, x ) = π Im 0 − iθ ( t ) � 0 | Tc ( t, x ) c † (0 , x ) | 0 � . G ret ( t, x ) = Translational invariance: no x -dependence. Idea of STS: impurities break translational invce − → x -dependence emerges. Measure N sample ( E, x ) for many x , use it to extract bulk dynamical properties “ Use impurities as measurement device”. In practice: determine � e − i k · x N ( E, x ) . N ( E, k ) = x Can this technique see spin-charge separation 1D Mott insulators?

Field Theory of 1D Mott Insulators “Standard Model” of Mott insulator: (extended) Hubbard model ( Lieb/Wu ’68 ) ( n j, ↑ − 1 2)( n j, ↓ − 1 � � c † H = − t j,σ c j +1 ,σ + h.c. + U 2) j j,σ = ↑ , ↓ � n j = n j, ↑ + n j, ↓ , n j,σ = c † + V ( n j − 1)( n j +1 − 1) , j,σ c j,σ . j • U = V = 0: metal. Gapless fermionic excitations. • U > V > 0: “Mott insulator” (dynamical mass generation). Single-electron excitations have gap, but gapless spin excitation. Low-energy continuum limit for U, V � t : linearize dispersion around ± k F = ± π 2 a 0 c j,σ → √ a 0 � R σ ( x ) e ik F x + L σ ( x ) e − ik F x � , x = ja 0 . H − → 2 Dirac fermions with 4-fermion interactions

Bosonization � � � � η σ − i − if σ 2 π e if σ π/ 4 exp L † σ ( τ, x ) = √ 2 ¯ ϕ c ( τ, x ) exp 2 ¯ ϕ s ( τ, x ) , � i � � if σ � η σ 2 π e if σ π/ 4 exp R † σ ( τ, x ) = √ 2 ϕ c ( τ, x ) exp 2 ϕ s ( τ, x ) . ϕ c,s chiral Bose fields, f ↑ = 1 = − f ↓ , η a Klein factors. Z H = dx [ H c ( x ) + H s ( x )] , v c g ( ∂ x Φ c ) 2 + ( ∂ x Θ c ) 2 ˜ ˆ H c = − (2 π ) 2 cos( β Φ c ) + irrelevant , 16 π v s ( ∂ x Φ s ) 2 + ( ∂ x Θ s ) 2 ˜ ˆ H s = + irrelevant . 16 π ϕ c , 1 Φ s = ϕ s + ¯ ϕ s , Θ s = ϕ s − ¯ ϕ s , β Φ c = ϕ c + ¯ β Θ c = ϕ c − ¯ ϕ c Free spin boson and sine-Gordon model in charge sector − → integrable.

Single Impurity Place strong potential impurity at x = 0 − → cuts line into two. Hard-wall boundary conditions: R σ (0) = − L σ (0) (have also considered additional phase shift) Z 0 h i H B c ( x ) + H B H = dx s ( x ) , −∞ v c g ( ∂ x Φ c ) 2 + ( ∂ x Θ c ) 2 ˜ H B ˆ = − (2 π ) 2 cos( β Φ c ) , c 16 π v s ( ∂ x Φ s ) 2 + ( ∂ x Θ s ) 2 ˜ H B ˆ = . s 16 π Boundary Conditions: Φ c,s (0) = 0 . Compatible with spin-charge separation. Ground state in presence of boundary | 0 B � = | 0 B,c � ⊗ | 0 B,s � .

Imaginary-time Fermion Autocorrelator −� 0 B | T τ c j,σ ( τ ) c † G ( τ, x, x ) = j,σ | 0 B � � � � 0 B | T τ R σ ( τ, x ) R † σ (0 , x ) | 0 B � + � 0 B | T τ L σ ( τ, x ) L † − → − a 0 σ (0 , x ) | 0 B � a 0 e 2 ik F x � 0 B | T τ R σ ( τ, x ) L † − σ (0 , x ) | 0 B � a 0 e − 2 ik F x � 0 B | T τ L σ ( τ, x ) R † − σ (0 , x ) | 0 B � Fourier transform � 0 dx e − iqx G ( τ, x, x ) G ( τ, q ) = −∞ Dominant contribution at q ≈ 2 k F � 0 dx e − ikx � 0 B | T τ R σ ( τ, x ) L † G ( τ, 2 k F + k ) ≈ a 0 σ (0 , x ) | 0 B � . � �� � −∞ G RL ( τ,x ) Focus on this from now on (other pieces work the same).

Bosonize: G RL ( τ, x ) = � 0 B,s | T τ e − i 2 ϕ s ( τ,x ) e − i 2 ¯ ϕ s (0 ,x ) | 0 B,s �� 0 B,c | T τ O 1 4 ( τ, x ) O − 1 4 (0 , x ) | 0 B,c � − β − β spin: n 4 β Θ , 4 = e − i β 4 Φ c − in O n 4 , topological charge: n − β Spin Sector Free boson on the halfline. Calculate the correlator by mode expansion 1 � 0 B,s | T τ e − i 2 ϕ s ( τ,x ) e − i 2 ¯ ϕ s (0 ,x ) | 0 B,s � = 2 π √ v s τ − 2 ix.

Charge Sector: sine-Gordon on the halfline Want to calculate O ( τ, x ) = e H B c τ O (0 , x ) e −H B G c ( τ, x ) = � 0 B,c | T τ O 1 4 ( τ, x ) O − 1 c τ 4 (0 , x ) | 0 B,c � , − β − β Now change “transfer direction” Ghoshal/Zamolodchikov ’94 τ τ B H c H c −x −x 0 0 4 ( τ, x ) O − 1 � 0 | T x O 1 4 (0 , x ) | B � − β − β , O ( τ, x ) = e −H c x e − iP τ O (0 , 0) e iP τ e H c x G c ( τ, x ) = � 0 | B � − → particular matrix element in bulk SGM!

Boundary State Ghoshal/Zamolodchikov ’94: | B � can be constructed from bulk sine-Gordon scattering states. | θ 1 , . . . , θ n � a 1 ,...,a n = A † a 1 ( θ 1 ) . . . A † a n ( θ n ) | 0 � Faddeev-Zamolodchikov algebra: S b 1 b 2 A a 1 ( θ 1 ) A a 2 ( θ 2 ) = a 1 a 2 ( θ 1 − θ 2 ) A b 2 ( θ 2 ) A b 1 ( θ 1 ) , a 1 a 2 ( θ 1 − θ 2 ) A † b 2 ( θ 2 ) A † S b 1 b 2 A † a 1 ( θ 1 ) A † a 2 ( θ 2 ) = b 1 ( θ 1 ) , 2 πδ ( θ 1 − θ 2 ) δ a 1 a 2 + S b 2 a 1 a 2 b 1 ( θ 1 − θ 2 ) A † A a 1 ( θ 1 ) A † a 2 ( θ 2 ) = b 2 ( θ 2 ) A b 1 ( θ 1 ) . � ∞ � 1 � dξ 2 π K ab ( ξ ) A † a ( − ξ ) A † | B � = exp b ( ξ ) | 0 � , 2 −∞

K ab ( ξ ) obtained from solution of reflection equations K ab ( ξ ) = R b a ( iπ/ 2 − ξ ) ¯ c 1 ( θ 1 ) S b 2 b 1 c 1 ( θ 1 ) S d 2 b 1 c 2 d 1 ( θ 1 + θ 2 ) R b 2 R c 2 a 2 ( θ 2 ) S c 1 d 2 a 1 c 2 ( θ 1 + θ 2 ) R d 1 d 2 d 1 ( θ 1 − θ 2 ) = S c 1 c 2 a 1 a 2 ( θ 1 − θ 2 ) R d 1 d 2 ( θ 2 ) “Boundary cross-unitarity” K ab ( ξ ) = S ab cd (2 ξ ) K dc ( − ξ ) . Boundary unitarity R c a ( θ ) R b c ( − θ ) = δ b a . √ Form of K ab ( ξ ) generally quite complicated; for β = 1 / 2 and Φ c ( τ, 0) = Φ 0 � � 4 ± i Φ 0 i π 2 + θ cosh 2 R + − ( θ ) = R − R ∓ � , + ( θ ) = − ± ( θ ) = 0 . � i π 4 ± i Φ 0 2 − θ cosh 2 Ameduri/Konik/LeClair ’95 For all cases we study: K aa ( ξ ) = 0, K + − ( ξ ) = K − + ( ξ ) ≡ K ( ξ )

Spectral Representation � dθ 1 . . . dθ n ∞ � � � 0 |O 1 4 ( τ, x ) O − 1 (2 π ) n n ! � 0 |O 1 4 (0 , x ) | B � = 4 ( τ, x ) | θ 1 , . . . , θ n � a 1 ...a n − β − β − β n =0 a j × a n ...a 1 � θ n , . . . , θ 1 |O − 1 4 (0 , x ) | B � − β Idea: • Expand boundary state � dξ a ( − ξ ) A † | B � = | 0 � + 1 2 π K ab ( ξ ) A † b ( ξ ) | 0 � + . . . 2 • Evaluate terms in spectral rep with lowest numbers of particles using Form Factor Bootstrap Approach • Observe that terms quadratic/cubic in K ab and/or large n are (very) small except at x ≈ 0. To get x � ξ = v c ∆ must sum full series. • But x ≈ has small contribution to Fourier transform in x . • Rapidly “converging” expansion for Fourier transform.

Form Factor Bootstrap Approach Karowski/Weisz ’78, Smirnov ’93, Lukyanov ’95, Delfino/Mussardo ’95... Basis of scattering states: n n ∆ X X | θ 1 , . . . , θ n � a 1 ...a n , E n = ∆ cosh θ n , P n = v sinh θ n . k =1 k =1 Matrix Elements (“Form Factors”): � 0 |O (0 , 0) | θ 1 , . . . , θ n � a 1 ...a n Idea: analytic properties of S-matrix − → analytic properties of form factors − → form factors.