On singularities of dynamic response functions in the massless - PowerPoint PPT Presentation

Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion On singularities of dynamic response functions in the massless regime of the XXZ chain K. K. Kozlowski CNRS, Laboratoire de Physique, ENS de Lyon. 20 th of July 2017 K. K. Kozlowski "Form factors of bound states in the XXZ chain." J. Phys. A: Math. & Theor., 50 , 184002, (2017). K. K. Kozlowski "On the thermodynamic limit of form factor expansions of dynamical correlation functions in the massless regime of the XXZ spin-1/2 chain." math.-ph. 1706.09459 K. K. Kozlowski " On singularities of dynamic response functions in the massless regime of the XXZ chain." to appear. Integrability in Gauge and String Theory 2017 ENS, Paris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski On singularities of dynamic response functions in the massless regime of the XXZ

Introduction The edge singular behaviour of dynamic response functions The massless form factor expansion Conclusion Outline Introduction 1 The dynamic response functions in the XXZ chain Universality and response functions The edge singular behaviour of dynamic response functions 2 Singe excitations thresholds Beyond the Luttinger liquid: Equal velocity excitations The massless form factor expansion 3 The main ingredients The overall strategy Conclusion 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski On singularities of dynamic response functions in the massless regime of the XXZ

Introduction The edge singular behaviour of dynamic response functions The dynamic response functions in the XXZ chain The massless form factor expansion Universality and response functions Conclusion Dynamic response functions in the XXZ chain n = 1 C 2 , σ α Pauli matrices ⊛ The XXZ spin-1 / 2 chain on h XXZ = ⊗ L { } ∑ L n + 1 + σ y n σ y σ x n σ x n + 1 + ∆ σ z n σ z n + 1 − h σ z H = , σ n + L ≡ σ n n n = 1 ⊛ Massless regime − 1 < ∆ < 1, 0 < h < h c ⇝ Ground state Ω . ⊛ Space-time evolution of operators σ γ m + 1 ( t ) = e i m P + i H t · σ γ 1 · e − i t H − i m P ⊛ Connected dynamical two-point function at zero temperature {( � 2 } ⟨( ) † σ γ ⟩ ( ) † σ γ ) ( )� � � � σ γ σ γ Ω , σ γ � 1 ( t ) c = lim Ω , 1 ( t ) m + 1 Ω − 1 Ω m + 1 L → + ∞ ♦ Experiments ⇝ Dynamic Response Functions ∫ ∑ ⟨( ) † σ γ ⟩ d t σ γ S ( γ ) ( k , ω ) = c · e i ( ω t − km ) 1 ( t ) m + 1 ( 2 π ) 2 m ∈ Z R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski On singularities of dynamic response functions in the massless regime of the XXZ

Introduction The edge singular behaviour of dynamic response functions The dynamic response functions in the XXZ chain The massless form factor expansion Universality and response functions Conclusion Expermiental measurements on KCuF 3 ⊛ Bragg spectroscopy. XXX chain at h = 0. ⊛ KCuF 3 ⇝ Observations for S ( z ) ( k , ω ) Dominant intensity delimited by viking helmet curves Large intensity concentrated on lower curves Decrease of intensity when approaching top curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski On singularities of dynamic response functions in the massless regime of the XXZ

Introduction The edge singular behaviour of dynamic response functions The dynamic response functions in the XXZ chain The massless form factor expansion Universality and response functions Conclusion Universal features of response functions ⊛ ’67 (Mahan), ’67 (Noziére, De Dominicus) Arguments for a power-law behaviour near edges. ⊛ ’80’s Luttinger liquid, CFT methods ⇝ predictions of a local behaviour at ( k , ω ) = ( 0 , 0 ) . ⊛ ’08 (Glazman, Kamenev, Khodas, Pustilnik) X-Ray edge singularities S ( z ) ( k , ω ) ≃ A ( k ) · Ξ ( ω − ε h ( k )) · [ ω − ε h ( k )] ϑ ⊛ ’08 (Glazman, Imambekov) Non-linear Luttinger liquid ( 2 π ) 2 Γ − 1 ( µ R + µ L ) � � � � � 2 � F ( z ) ( k ) A ( k ) = [ v ( k ) + v F ] µ R [ v F − v ( k )] µ L · , ϑ = µ R + µ L − 1 ⊛ ’08 (Glazman, Imambekov) ’08 (Cheianov, Pustilnik) ’09 (Affleck, Pereira, White) Bethe Ansatz spectrum closed expressions for µ R , L . ⇝ � � � � 2 ⇝ volume-renormalised form factor; � � F ( z ) ( k ) ⊛ ’10 (Caux, Glazman, Imambekov, Shashi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski On singularities of dynamic response functions in the massless regime of the XXZ

Introduction The edge singular behaviour of dynamic response functions The dynamic response functions in the XXZ chain The massless form factor expansion Universality and response functions Conclusion Exact results ⊛ ’79 (Beck,Bonner,Müller) Closed formula for S ( z ) ( k , ω ) for XX ( ∆ = 0) ∀ h . ⊛ ’95 (Jimbo,Miwa) Closed formula for S ( γ ) ( k , ω ) for XXX, h=0. ⊛ ’98-’00 (Bougourzi, Couture,Kacir, Fledderjohann,Karbach,Müller,Mütter,.... ) Analysis of 2 and 4 spinon contributions ⊛ ’00-’07 (Biegel,Karback,Müller,Sato,Shiroishi,Takahashi,Caux, Hagemans,Maillet,... ) Numerics & Bethe Ansatz ⊛ ’12 (Caux,Konno,Sorrel,Weston) 2 spinon for XXZ and h = 0 ⊛ ’12 (Kitanine,K.,Maillet,Slavnov,Terras) Formal saddle-point approach to form factor series in NLSM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski On singularities of dynamic response functions in the massless regime of the XXZ

Introduction The edge singular behaviour of dynamic response functions The dynamic response functions in the XXZ chain The massless form factor expansion Universality and response functions Conclusion The open problems Questions Can one carry out an ab inicio , exact, calculation of the DRF? Can one access to the predicted universal features for an integrable model? Is the non-linear Luttinger picture complete? Can one bring the description of this universality to a singularity theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski On singularities of dynamic response functions in the massless regime of the XXZ

ω π Introduction The edge singular behaviour of dynamic response functions Singe excitations thresholds The massless form factor expansion Beyond the Luttinger liquid: Equal velocity excitations Conclusion The two-particle-hole excitation thresholds ⊛ Form factor series representation for DRF ∑ S ( γ ) S ( γ ) ( k , ω ) = ( k , ω ) n n ∈ S C p - 2 h 1 2 C p - h C p - h 3 4 C p C p 1 2 C p C p C h 1 C h 2 k P min π - p F p F + π P max 2 π 0 2 p F 2 ( π - p F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski On singularities of dynamic response functions in the massless regime of the XXZ

Introduction The edge singular behaviour of dynamic response functions Singe excitations thresholds The massless form factor expansion Beyond the Luttinger liquid: Equal velocity excitations Conclusion The vicinity of a hole threshold ( k , ω ) configuration close to the hole excitation line t 0 ∈ ] − p F ; p F [ . ( P 0 , E 0 ) = ( p F − t 0 , − e ( t 0 )) with ⊛ The hole threshold � � � � ( ) Ξ ( δω ) · [ δω ] ∆( h ) � 2 ( [ δω ] 1 − 0 + )) � F ( z ) ( ( t 0 ) S ( z ) + S ( zz ) h P 0 , E 0 + δω = 1 + O h ; reg ( δω ) . · h [ v ( t 0 ) + v F ] δ ( h ) Γ( δ ( h ) + + δ ( h ) + [ v F − v ( t 0 )] δ ( h ) − ) − ⊛ v ( t 0 ) = e ′ ( t 0 ) : velocity of the hole at t 0 v F : velocity excitations on Fremi boundary. ⊛ Edge exponent: ∆ ( h ) = δ ( h ) + δ ( h ) − 1 − + ⊛ δ ( h ) ± : microscopic shift of right(+)/left(-) Fermi boundary due to excitation. � )� ( � �  2  ( L ) δ ( h ) + + δ ( h ) � � Ω , σ z  − + 1 1 Υ t 0  � �  � �  � 2 =   � � � F ( z ) ( t 0 ) lim  � � � � � � � �     � � � � 2 · � � � � �  h � 2 2 π L → + ∞ � � Ω � � � Υ t 0 � ⋆ ground state Ω ❝ ❫ { E 0 rrrrrrrrrrrrrr rrrrs = − e ( t 0 ) t 0 − p F p F ⋆ excitation Υ t 0 P 0 = p F − t 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski On singularities of dynamic response functions in the massless regime of the XXZ