Condensation properties of Bethe roots in the XXZ chain K. K. - PowerPoint PPT Presentation

The setting of the problem The main result Conclusion Condensation properties of Bethe roots in the XXZ chain K. K. Kozlowski CNRS, Laboratoire de Physique, ENS de Lyon. 25 th of August 2016 K. K. Kozlowski "On condensation properties of Bethe roots associated with the XXZ chain." Math-ph:1508.05741 Recent Advances in Quantum Integrable Systems 2016, Genève . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski Condensation properties of Bethe roots in the XXZ chain

The setting of the problem The main result Conclusion Outline The setting of the problem 1 The particle-hole roots The condensation property The main result 2 Main steps of the proof for the ground state Conclusion 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski Condensation properties of Bethe roots in the XXZ chain

The setting of the problem The particle-hole roots The main result The condensation property Conclusion The XXZ spin-1 / 2 chain ⊛ The XXZ spin-1 / 2 chain on h XXZ = ⊗ L a = 1 C 2 { ( )} ∑ L n + 1 + σ y n σ y σ x n σ x σ z n σ z H XXZ = J n + 1 + cos ( ζ ) n + 1 − id , σ n + L ≡ σ n n = 1 σ α Pauli matrices, σ α n = id ⊗ · · · ⊗ id ⊗ σ α ⊗ id ⊗ · · · ⊗ id . L : length of circle, cos ( ζ ) anisotropy parameter. [ H XXZ , S z ] = 0 with S z = ∑ L a = 1 σ z a . { } v ∈ h XXZ : S z · v = ( L − 2 N ) · v N = 0 h ( N ) h ( N ) h XXZ = ⊕ L with XXZ = , XXZ XXX ( ’31 Bethe), XXZ ( ’58 Orbach) quantum integrable by Bethe Ansatz Eigenvectors in h ( N ) XXZ : v ( λ 1 , . . . , λ N ) ⇝   ( sinh ( i ζ/ 2 − λ a ) ) L ∏ N      sinh ( λ a − λ b + i ζ )   = ( − 1 ) N + 1 , · a = 1 , . . . , N .      sinh ( i ζ/ 2 + λ a ) sinh ( λ b − λ a + i ζ ) b = 1 H XXZ · v ( { λ a } N 1 ) = E ( { λ a } N 1 ) v ( { λ a } N 1 ) Eigenvalues − 2 J sin 2 ( ζ ) N ∑ E ( { λ a } N 1 ) = e ( λ a ) e ( λ ) = with sinh ( λ − i ζ/ 2 ) sinh ( λ + i ζ/ 2 ) a = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski Condensation properties of Bethe roots in the XXZ chain

The setting of the problem The particle-hole roots The main result The condensation property Conclusion The particle-hole excited states and the ground state ⊛ Distinguish solutions by taking logarithm ℓ a ∈ Z , λ a ∈ R , ( sinh ( i η + λ ) ) ∑ N ϑ ( λ a − λ b | ζ ) + N + 1 2 ζ ) − 1 = ℓ a i ϑ ( λ | 1 and ϑ ( λ | η ) = 2 π ln sinh ( i η − λ ) L 2 L L a = 1 ⊛ ( ’38 Húlten) Ground state in h ( N ) XXZ ℓ a = a λ a ∈ R and ⊛ ( ’64 Griffiths , ’66 Yang,Yang ) Existence for all cos ( ζ ) , uniqueness when − 1 < cos ( ζ ) ≤ 0. ⊛ Real-valued particle-hole excitation, λ a ∈ R ℓ a = a for a ∈ [ [ 1 ; N ] ] \ { h 1 , . . . , h n } ℓ h a = p a for a = 1 , . . . n and ⊛ ( ’64 Griffiths , ’83 Gaudin ) Existence for cos ( ζ ) ≥ 1, for some subsets of ℓ a ’s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski Condensation properties of Bethe roots in the XXZ chain

The setting of the problem The particle-hole roots The main result The condensation property Conclusion Existence of particle-hole solutions Proposition ( ’15 K ) The Log BAE with ℓ a admit a real valued solution { λ a } N 1 if for any J ⊂ [ [ 1 ; N ] ] : ( π − ζ ) ∑ ( ) − r |J| < 1 ℓ a − N + 1 − N ( π − 2 ζ ) + m 2 π − 2 ζ r m = m < r |J| with L 2 2 π 2 π L 2 π L a ∈J If − 1 < cos ( ζ ) < 0, the condition is necessary and the solution is unique. ⊛ Particle-hole solutions exist for any h 1 < · · · < h n and p 1 < · · · < p n such that ( 1 ) ( 1 ) ( 1 ) π − ζ 2 − N − 1 > p n − N p 1 − 1 > − π − ζ 2 − N − 1 π − ζ 2 − N ≥ n , and π L L L π L π L L ♦ Existence follows by showing that the Yang-Yang action blows up at infinity. ♦ Necessariess and uniqueness follow from strict convexity. Lemma ( ’15 K ) If 0 ≤ N / L ≤ 1 / 2 − ϵ , the ground state roots { λ a } N 1 are bounded | λ a | ≤ Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski Condensation properties of Bethe roots in the XXZ chain

The setting of the problem The particle-hole roots The main result The condensation property Conclusion The thermodynamic limit for the ground state ⊛ Thermodynamic limit of observables in fixed magnetisation sector N / L → D ∈ [ 0 ; 1 / 2 ] : ∑ N ♦ Ground state per site energy 1 e ( λ a ) L a = 1 ⊛ One assumes that the Bethe roots condense on [ − q ; q ] with some density ρ ( ∗ | q ) : ∫ q 1 1 ∑ N λ a + 1 − λ a ≃ e ( λ a ) ≃ e ( s ) ρ ( s | q ) · d s + · · · L ρ ( λ a | q ) L a = 1 − q ⊛ Easy to characterise ( ρ ( µ | q ) , q ) if one assumes that the roots densify. ( ’38 Húlten , ’64 Griffiths , ’66 Yang,Yang ) ∫ Q ∫ q ϑ ′ ( λ − µ | ζ ) ρ ( µ | Q ) d µ = ϑ ′ ( λ | ζ ρ ( λ | Q ) + 2 ) D = ρ ( λ | q ) d λ and − Q − q ⊛ ( ρ ( µ | q ) , q ) is the unique solution ( ’66 Yang,Yang ). ♦ Densification used in thermodynamic limit of correlation functions, 1 / L corrections to GS and low-lying excitations, ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski Condensation properties of Bethe roots in the XXZ chain

The setting of the problem The main result Main steps of the proof for the ground state Conclusion The main result ⊛ ( ’09 Dorlas, Samsonov) proof of condensation of ground state roots for − 1 < cos ( ζ ) ≤ 0. ♦ Use of convex analysis on spaces of probability measures. Theorem ( ’15 K ) Let { λ a } be any n particle-hole solution, n ≤ C , D ∈ [ 0 ; 1 / 2 ] . ⊛ For any bounded-Lipschitz f it holds ∫ q ∑ N 1 f ( λ a ) −→ f ( s ) ρ ( s | q ) · d s L N , L →∞ a = 1 N / L → D − q There exists L 0 , such that for any such choice of ℓ a , the Log BAE solution is unique when L ≥ L 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski Condensation properties of Bethe roots in the XXZ chain

The setting of the problem The main result Main steps of the proof for the ground state Conclusion The counting function ⊛ Counting function for ground state roots { λ a } N 1 ∑ N 2 ) − 1 ϑ ( ω − λ a | ζ ) + N + 1 ξ ( λ a ) = a ξ ( ω ) = ϑ ( ω | ζ � � so that L 2 L L a = 1 ♦ Characterise � AE for � ξ by a non-linear integral equation control on roots ⇝ ξ ⇝ ( ’85 De Vega, Woynarovich , ’90 Batchelor, Klümper , ’91 Batchelor, Klümper, Pearce , ’91 Destri, De Vega ) ⊛ Main working assumption roots are bounded in L : − Λ ≤ λ a ≤ Λ ξ ′ > c on [ − 2 Λ ; 2 Λ ] ; � or ξ ′ > c on [ − 2 Λ ; 2 Λ ] ; � ξ ′ at these roots. a priori control on growth of roots with L and local behaviour of � The form taken by the NLIE depends on these assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski Condensation properties of Bethe roots in the XXZ chain

The setting of the problem The main result Main steps of the proof for the ground state Conclusion The convergence to first order N / L → D < 1 / 2; { } � |ℜ ( z ) | ≤ 2 Λ |ℑ ( z ) | ≤ ζ/ 4 ξ is a sequence in L of holomorphic functions on ; { } � � � � �� � ≤ B for ω ∈ ξ ( ω ) |ℜ ( z ) | ≤ 2 Λ |ℑ ( z ) | ≤ ζ/ 4 ; { } Montel theorem: � ξ e → ξ e holomorphic on |ℜ ( z ) | ≤ 2 Λ |ℑ ( z ) | ≤ ζ/ 4 ; ∫ ω ρ ( s | q ) · d s + D Show that ξ e = ξ 0 for any extracted sequence, ξ 0 ( ω ) = 2 ; 0 ξ ′ e changes sign on [ − Λ ; Λ ] a finite number of times ⇝ NLIE and AE from NLIE; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski Condensation properties of Bethe roots in the XXZ chain

b b The setting of the problem The main result Main steps of the proof for the ground state Conclusion ξ ′ e > 0 on [ − Λ ; Λ ] : the contour � Γ + N + 1 / 2 1 + i α 2 L + i α L 1 N + 1 / 2 2 L L 1 2 L − i α N + 1 / 2 − i α � Γ − L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. K. Kozlowski Condensation properties of Bethe roots in the XXZ chain