Yu. Stroganov, The importance of being odd Kiev 2000 Typeset by - PowerPoint PPT Presentation

Yu. Stroganov, “The importance of being odd” Kiev 2000 – Typeset by Foil T EX – Integrability and combinatorics, 2014

N. Kitanine Combinatorics of the form factors Combinatorics of the form factors of critical integrable models N. Kitanine IMB, Universit´ e de Bourgogne Integrability and Combinatorics Conference in memory of Yu. Stroganov Presqu’ˆ ıle de Giens, 26 June 2014 In collaboration with : K. K. Kozlowski, J.M. Maillet, N. Slavnov, V. Terras – Typeset by Foil T EX – Integrability and combinatorics, 2014 1

N. Kitanine Combinatorics of the form factors References: Form factor approach to the asymptotic behavior of correlation functions in critical models , N. K., K. K. Kozlowski, J. M. Maillet, N. Slavnov and V. Terras, J. Stat. Mech. (2011) P12010. arXiv:1110.0803 Form factor approach to dynamical correlation functions in critical models , N. K., K. K. Kozlowski, J. M. Maillet, N. Slavnov and V. Terras, J. Stat. Mech. (2012) P09001, arXiv:1206.2630 Long-distance asymptotic behavior of multi-point correlation functions in massless quantum integrable models , N. K., K. K. Kozlowski, J. M. Maillet and V. Terras, J. Stat. Mech., (2014) P05011 arXiv:1312.5089 – Typeset by Foil T EX – Integrability and combinatorics, 2014 2

N. Kitanine Combinatorics of the form factors 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 – Integrability and combinatorics, 2014 3

N. Kitanine Combinatorics of the form factors 2. The non-linear Schr¨ odinger (Lieb-Lineger) 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 – Integrability and combinatorics, 2014 4

N. Kitanine Combinatorics of the form factors Multipoint correlation functions � � r O j ( x j ) e − H/kT � tr H � r � j =1 � O j ( x j ) = , T > 0 � e − H/kT � tr H j =1 T r r � � � O j ( x j ) � = � Ψ g | O j ( x j ) | Ψ g � , T = 0 j =1 j =1 where | Ψ g � is the (normalized) ground state. We consider only T = 0 case. O ( x ) - local operators. Here we consider only equal-time correlation functions • 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 ) – Typeset by Foil T EX – Integrability and combinatorics, 2014 5

N. Kitanine Combinatorics of the form factors Form factor approach Our goal is to study the behavior of correlation functions using their form factor expansion r � � � O j ( x j ) � = � Ψ g | O 1 ( x 1 ) | Ψ 1 � � Ψ 1 | O 2 ( x 2 ) | Ψ 2 � . . . � Ψ r − 1 | O r ( x r ) | Ψ g � j =1 | ψ 1 � ,..., | ψr − 1 � Main difficulty : form factors scale to zero in the limit L → ∞ for critical models. � Ψ g | O † ( x ) | Ψ 1 � � Ψ 1 | O (0) | Ψ g � = L − θ e ix P ex A (Ψ 1 , 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 – Integrability and combinatorics, 2014 6

N. Kitanine Combinatorics of the form factors 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 – Integrability and combinatorics, 2014 7

N. Kitanine Combinatorics of the form factors The ground state 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 With bare momentum : p 0 ( λ ) = i log sinh( iζ/ 2 + λ ) for XXZ , p 0 ( λ ) = λ for NLS sinh( iζ/ 2 − λ ) and bare scattering phase ϑ ( λ ) = i log sinh( iζ + λ ) ϑ ( λ ) = i log ic − λ for XXZ , for NLS. sinh( iζ − λ ) ic + λ – Typeset by Foil T EX – Integrability and combinatorics, 2014 8

N. Kitanine Combinatorics of the form factors The particle-hole spectrum 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 New set of integers can be presented in terms of particle and hole quantum numbers 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 – Integrability and combinatorics, 2014 9

N. Kitanine Combinatorics of the form factors 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 – Integrability and combinatorics, 2014 10

N. Kitanine Combinatorics of the form factors Dressed quantities Lieb equation for the density of Bethe roots q ρ ( λ ) − 1 K ( λ − µ ) ρ ( µ ) dµ = 1 � 2 πp ′ K ( λ ) = ϑ ′ ( λ ) . 0 ( λ ) , with 2 π − q Similar integral equations can be written for the dressed charge , dressed momentum , dressed energy and dressed phase :  Z ( λ )   Z ( µ )   1  q 1 � p ( λ ) p ( µ ) p 0 ( λ )        − K ( λ − µ )  dµ =       ε ( λ ) ε ( µ ) ε 0 ( λ ) 2 π     − q φ ( λ, ν ) φ ( µ, ν ) ϑ ( λ − ν ) Boundary value of the dressed charge Z = Z ( ± q ) is related to the Luttinger liquid parameter K Ll = Z 2 – Typeset by Foil T EX – Integrability and combinatorics, 2014 11

N. Kitanine Combinatorics of the form factors Computation of form factors F ψ 1 ψ 2 ( x ) = |� ψ 1 |O ( x ) | ψ 2 �| 2 � ψ 1 � 2 · � ψ 2 � 2 1. Determinant representation ( quantum inverse problem, scalar product ) 2. Cauchy determinant extraction: 1999 Izergin, N.K., Maillet, Terras 1 F ψ 1 ψ 2 ( x ) = det × Smooth part λ a − µ b N 3. Thermodynamic limit L, N → ∞ (global density D = N/L fixed, related to the magnetic field h ) F ψ 1 ψ 2 ( x ) · F ψ 1 ψ 2 (0) ∼ L − θ e ix P ex S D , ¯ – Typeset by Foil T EX – Integrability and combinatorics, 2014 12