Form factor approach to the correlation functions of quantum integrable models N. Kitanine IMB, Universit´ e de Bourgogne Quantum Integrable Systems and Geometry Olh˜ ao, September 2012 In collaboration with : K. K. Kozlowski, J.M. Maillet, N. A. Slavnov, V. Terras – Typeset by Foil T EX – Olh˜ ao, September 2012
N. Kitanine Form factor approach Critical Integrable models 1. The XXZ spin- 1 2 Heisenberg chain in a magnetic field Defined on a one-dimensional lattice with M sites, with Hamiltonian, H = H (0) − hS z , M H (0) = σ x m σ x m +1 + σ y m σ y m +1 + ∆( σ z m σ z � � � m +1 − 1) , m =1 M S z = 1 σ z [ H (0) , S z ] = 0 . � m , 2 m =1 σ x,y,z are the local spin operators (in the spin- 1 2 representation) associated with each site m m of the chain and ∆ = cos( ζ ) , ζ real, is the anisotropy parameter. h - external magnetic field; h > 0 . We impose the periodic (or quasi-periodic) boundary conditions – Typeset by Foil T EX – Olh˜ ao, September 2012 1
N. Kitanine Form factor approach 2. The non-linear Schr¨ odinger model • NLSE ≡ 1D limit of 3D Bose gas. • Simplest possible interacting massless integrable model. L � � � ∂ y Ψ † ( y ) ∂ y Ψ( y ) + c Ψ † ( y )Ψ † ( y )Ψ( y )Ψ( y ) − h Ψ † ( y )Ψ( y ) H = dy, 0 Ψ( x ) , Ψ † ( x ) quantum Bose fields � � Ψ( x ) , Ψ † ( y ) = δ ( x − y ) , L : length, c > 0 coupling constant ( repulsive regime ), h > 0 chemical potential. Both models: solution by algebraic Bethe ansatz , ground state finite Fermi zone – Typeset by Foil T EX – Olh˜ ao, September 2012 2
N. Kitanine Form factor approach Time-dependent correlation functions � O † ( x, t ) O (0 , 0) e − H/kT � tr H � � O † ( x, t ) O (0 , 0) T = , T > 0 � e − H/kT � tr H � O † ( x, t ) O (0 , 0) � = � ψ g | O † ( x, t ) O (0 , 0) | ψ g � , T = 0 where | ψ g � is the (normalized) ground state. We consider only T = 0 case O ( x ) - local operators • For the XXZ spin chain local spin operators σ ± m , σ z m , odinger Model (NLSM) local fields Ψ( x ) , Ψ † ( x ) and local • For the Non-linear Schr¨ densities j ( x ) = Ψ † ( x )Ψ( x ) O ( x, t ) ≡ e itH O ( x ) e − itH – Typeset by Foil T EX – Olh˜ ao, September 2012 3
N. Kitanine Form factor approach Dynamical structure factors ∞ ∞ � � dt e i ( ωt − kx ) � O † ( x, t ) O (0 , 0) � S ( k, ω ) = dx −∞ −∞ DSF gives the response functions under external perturbations of the system and can be experimentally measured. Numerical computation from the algebraic Bethe ansatz results for the form factors XXZ - Caux, Hagemans, Maillet (2005) NLSM - Calabrese, Caux (2006) – Typeset by Foil T EX – Olh˜ ao, September 2012 4
N. Kitanine Form factor approach Asymptotic problems We consider the correlation functions in the thermodynamic limit L → ∞ • Equal time correlation functions t = 0 : Large distance asymptotics x → ∞ • Time-dependent correlation functions : Long time large distance asymptotics x x → ∞ , t = const • Dynamical structure factors : Edge exponents (near the excitation dispersion curves ω = ε h ( k ) ) S ( k, ω ) ≃ A ( k ) H ( ω − ε h ) [ ω − ε h ] θ , θ > 0 Edge exponent near “ the hole excitation threshold ”. – Typeset by Foil T EX – Olh˜ ao, September 2012 5
N. Kitanine Form factor approach Predictions and previous results • Equal time correlation functions 1. Luttinger liquid: 1975 Luther, Peschel, 1981 Haldane 2. Conformal field theory : 1984 Cardy 3. Algebraic Bethe ansatz and Riemann-Hilbert analisys 2009 N.K., Kozlowski, Maillet, Slavnov, Terras • Time-dependent correlation functions 1. Conformal field theory: Only for x >> t . 2. Algebraic Bethe ansatz and Riemann-Hilbert analisys: 2011 Kozlowski, Terras • Dynamical structure factors, edge exponents: 1. Non-linear Luttinger liquid approach 2008 Imambekov, Glazman – Typeset by Foil T EX – Olh˜ ao, September 2012 6
N. Kitanine Form factor approach Form factor approach Our goal is to study the behavior of correlation functions in critical models using their form factor expansion �O † ( x, t ) O (0 , 0) � = � ¯ F ψg ψ ′ ( x, t ) F ψ ′ ψg (0 , 0) | ψ ′� Main difficulty : form factors scale to zero in the infinite size limit ( L → ∞ ) for critical models. F ψg ψ ′ ( x, t ) F ψ ′ ψg (0 , 0) = L − θ e ix P ex − it E ex A ( ψ ′ , ψ g ) ¯ Analyze the form factor series for large (but finite) system size. Hence we need to describe states that will contribute to the leading behavior of the series in the limits x → ∞ and L → ∞ with x << L , and also to compute the corresponding form factors and their behavior in these limits – Typeset by Foil T EX – Olh˜ ao, September 2012 7
N. Kitanine Form factor approach Algebraic Bethe ansatz and form factors 1. Diagonalise the Hamiltonian using ABA 1979 Faddeev, Sklyanin, Takhtajan → key point : Yang-Baxter algebra A ( λ ) , B ( λ ) , C ( λ ) , D ( λ ) → | ψ � = B ( λ 1 ) . . . B ( λ N ) | 0 � with { λ } satisfying Bethe equations 2. Describe the ground state and excited states → Bethe equations. 3. Act with local operators on eigenstates → solve the quantum inverse problem 1999 N.K., Maillet, Terras O ( x ) = f ( A, B, C, D ) → use Yang-Baxter commutation relations 4. Compute the resulting scalar products (determinant representation) 1989 Slavnov → determinant representation for the form factors in finite volume 1999 N.K., Maillet, Terras 5. Analysis of the form factors in the thermodynamic limit. 2010 N.K., Kozlowski, Maillet, Slavnov, Terras – Typeset by Foil T EX – Olh˜ ao, September 2012 8
N. Kitanine Form factor approach Algebraic Bethe ansatz L.D. Faddeev, E.K. Sklyanin, L.A. Takhtajan (1979): Algebraic Bethe ansatz • Yang-Baxter equation : R 12 ( λ 1 − λ 2 ) R 13 ( λ 1 − λ 3 ) R 23 ( λ 2 − λ 3 ) = = R 23 ( λ 2 − λ 3 ) R 13 ( λ 1 − λ 3 ) R 12 ( λ 1 − λ 2 ) . • Monodromy matrix . � A ( λ ) � B ( λ ) T a ( λ ) = C ( λ ) D ( λ ) [ a ] → Yang-Baxter algebra : ◦ generators A , B , C , D ֒ ◦ commutation relations given by the R-matrix of the model R ab ( λ, µ ) T a ( λ ) T b ( µ ) = T b ( µ ) T a ( λ ) R ab ( λ, µ ) – Typeset by Foil T EX – Olh˜ ao, September 2012 9
N. Kitanine Form factor approach • Transfer matrix : T ( λ ) = tr a T a ( λ ) = A ( λ ) + D ( λ ) Commuting charges: [ T ( λ ) , T ( µ )] = 0 • Hamiltonian can be reconstructed from the transfer matrix using the trace identities Conserved quantities: [ H, T ( λ )] = 0 • Eigenstates. Construction of the space of states by action of B on a reference state A ( λ ) | 0 � = a ( λ ) | 0 � , D ( λ ) | 0 � = d ( λ ) | 0 � , C ( λ ) | 0 � = 0 . States | ψ � = � B ( λ k ) | 0 � with { λ k } solution of the Bethe equations are eigenstates k of the transfer matrix: T ( µ ) | ψ � = τ ( µ, { λ } ) | ψ � , � � N And hence of the Hamiltonian: H � � � B ( λ k ) | 0 � = ε 0 ( λ j ) B ( λ k ) | 0 � . k j =1 k – Typeset by Foil T EX – Olh˜ ao, September 2012 10
N. Kitanine Form factor approach The particle-hole spectrum Ground state solution of the Bethe equations can be described in terms of real rapidities λ j densely filling (with a density ρ ( λ ) ) the Fermi zone [ − q, q ] : N j − N + 1 � � � Lp 0 ( λ j ) − ϑ ( λ j − λ k ) = 2 π , j = 1 , . . . , N. 2 k =1 Excited states parametrized by numbers { µ ℓa } N ′ with N ′ = N + k , involving other 1 choices of integers ℓ 1 < · · · < ℓ N ′ in the rhs : N ′ ℓ j − N ′ + 1 � � j = 1 , . . . , N ′ . � Lp 0 ( µ ℓj ) − ϑ ( µ ℓj − µ ℓk ) = 2 π , 2 k =1 ℓ a = a , a ∈ { 1 , . . . , N ′ } \ { h 1 , . . . , h n } ℓ ha = p a , p a ∈ Z \ { 1 , . . . , N ′ } . To every choice of integers { p a } and { h a } there is an associated configuration of rapidities for the particles { µ pa } and for the holes { µ ha } . We don’t consider complex solutions for XXZ (open problem, can contribute for the dynamical correlation functions). – Typeset by Foil T EX – Olh˜ ao, September 2012 11
N. Kitanine Form factor approach Thermodynamics of the excited states • ”holes” in continuous distribution of rapidities at µ h 1 , . . . , µ hn • new ”particle” rapidities at µ p 1 , . . . , µ pn 1 F Lρ Lρ ✛ ✲ ✲ ✛ Ground State × × • • • • • • • • • • × × × × • ◦ ◦ • Exited × • • • • • • • • × × × ⇒ Excited state’s roots µ j shifted infinitesimally in respect to the ground state roots λ j . 1 � µ p 1 , . . . , µ pn � � � + O( L − 2 ) µ j − λ j = Lρ ( λ j ) · F λ j � µ h 1 , . . . , µ hn ⇒ Additive excitation spectrum. n n � � P ex − P G.S. = p ( µ pa ) − p ( µ ha ) and E ex − E G.S. = ε ( µ pa ) − ε ( µ ha ) a =1 a =1 – Typeset by Foil T EX – Olh˜ ao, September 2012 12
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