Condensation properties of Bethe roots in the XXZ chain

Transcription

1 Condensation properties of Bethe roots in the XXZ chain K K Kozlowski CNRS, aboratoire de Physique, ENS de yon 25 th of August 206 K K Kozlowski "On condensation properties of Bethe roots associated with the XXZ chain" Math-ph: Recent Advances in Quantum Integrable Systems 206, Genève K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

2 Outline The particle-hole roots The condensation property 2 3 K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

3 The XXZ spin-/2 chain The particle-hole roots The condensation property The XXZ spin-/2 chain on h XXZ = a= C2 H XXZ = J { σ x n σ x n+ + σy nσ y n+ + cos(ζ)( σnσ z z n+ id)}, σ n+ σ n n= σ α Pauli matrices, σ α n = id id σ α id id : length of circle, cos(ζ) anisotropy parameter [H XXZ, S z ] = 0 with S z = a= σz a } h XXZ = N=0 h(n) with h {v (N) XXZ XXZ = h XXZ : S z v = ( 2N) v XXX ( 3 Bethe), XXZ ( 58 Orbach) quantum integrable by Bethe Ansatz Eigenvectors in h (N) XXZ : v(λ,, λ N ) ( ) sinh(iζ/2 λa ) sinh(iζ/2 + λ a ) N sinh(λ a λ b + iζ) sinh(λ b λ a + iζ) = ( )N+, a =,, N b= Eigenvalues H XXZ v({λ a } N ) = E({λ a} N )v({λ a} N ) E({λ a } N ) = N a= e(λ a ) with e(λ) = 2J sin 2 (ζ) sinh(λ iζ/2) sinh(λ + iζ/2), K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

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

5 Existence of particle-hole solutions The particle-hole roots The condensation property Proposition ( 5 K ) The og BAE with l a admit a real valued solution {λ a } N if for any J [[ ; N ]]: r J < a J ( la N + 2 ) ( π ζ < r J with r m = m 2π N(π 2ζ) ) + m 2 π 2ζ 2π 2π If < cos(ζ) < 0, the condition is necessary and the solution is unique Particle-hole solutions exist for any h < < h n and p < < p n such that π ζ π ( 2 N ) > p n N, p > π ζ π ( 2 N ) and π ζ ( π 2 N ) n Existence follows by showing that the Yang-Yang action blows up at infinity Necessariess and uniqueness follow from strict convexity emma ( 5 K ) If 0 N/ /2 ϵ, the ground state roots {λ a } N are bounded λ a Λ K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

6 The thermodynamic limit for the ground state The particle-hole roots The condensation property Thermodynamic limit of observables in fixed magnetisation sector N/ D [ 0 ; /2 ]: N Ground state per site energy e(λ a ) a= One assumes that the Bethe roots condense on [ q ; q ] with some density ρ( q): λ a+ λ a ρ(λ a q) N e(λ a ) a= q q e(s)ρ(s q) ds + Easy to characterise (ρ(µ q), q) if one assumes that the roots densify ( 38 Húlten, 64 Griffiths, 66 Yang,Yang ) ρ(λ Q) + Q Q (ρ(µ q), q) is the unique solution ( 66 Yang,Yang ) q ϑ (λ µ ζ)ρ(µ Q)dµ = ϑ ζ (λ 2 ) and D = ρ(λ q)dλ Densification used in thermodynamic limit of correlation functions, / corrections to GS and low-lying excitations, q K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

7 ( 09 Dorlas, Samsonov) proof of condensation of ground state roots for < cos(ζ) 0 Use of convex analysis on spaces of probability measures Theorem ( 5 K ) et {λ a } be any n particle-hole solution, n C, D [ 0 ; /2 ] For any bounded-ipschitz f it holds N f(λ a ) a= N, N/ D q q f(s)ρ(s q) ds There exists 0, such that for any such choice of l a, the og BAE solution is unique when 0 K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

8 The counting function Counting function for ground state roots {λ a } N ζ ξ(ω) = ϑ(ω 2 ) N ϑ(ω λ a ζ) + N + 2 a= so that ξ(λ a ) = a Characterise ξ by a non-linear integral equation AE for ξ control on roots ( 85 De Vega, Woynarovich, 90 Batchelor, Klümper, 9 Batchelor, Klümper, Pearce, 9 Destri, De Vega ) Main working assumption roots are bounded in : Λ λ a Λ ξ > c on [ 2Λ ; 2Λ ]; or ξ > c on [ 2Λ ; 2Λ ]; a priori control on growth of roots with and local behaviour of ξ at these roots The form taken by the NIE depends on these assumptions K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

9 The convergence to first order N/ D < /2; ξ { is a sequence in of holomorphic functions on R(z) 2Λ I(z) } ζ/4 ; ξ(ω) { } B for ω R(z) 2Λ I(z) ζ/4 ; Montel theorem: ξ e ξ e holomorphic on { R(z) 2Λ I(z) ζ/4 } ; ω Show that ξ e = ξ 0 for any extracted sequence, ξ 0 (ω) = ρ(s q) ds + D 2 ; ξ e changes sign on [ Λ ; Λ ] a finite number of times NIE and AE from NIE; 0 K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

10 ξ e > 0 on [ Λ ; Λ ]: the contour Γ+ N + /2 2 + iα + iα 2 N + /2 2 iα N + /2 iα Γ K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

11 ξ e > 0 on [ Λ ; Λ ] ξ e (λ) = ϑ ( λ 2 ζ) + N + 2 Ĉ ϑ(λ µ ζ) ξ e(µ) e 2iπ ξ e(µ) ( dµ with Ĉ = ξ e Γ + Γ ) ( q R = ξ e 2 (N + )) ( ) and q = ξ e 2 ; ξ sym = ξ e (N + )/(2) ξ sym (λ) + ϵ=± Γ ϵ q R ( θ (λ µ) ξ sym (µ) dµ = ϑ λ 2 ζ) N [ θ(λ qr ) + θ(λ q ) ] 2 q θ (λ ξ e (s)) ξ e( ξ e (s) ) ln [ e 2iπϵs ] ds 2iπ + ξ sym ξ sym = ξ e D/2 and ξ e ξe (ξ(o)) C ξ e ξ e (O ) ξ sym (λ) + q R θ (λ µ)ξ sym (µ) dµ = ϑ ( λ 2 ζ) D q 2 ξ sym (q R ) = ξ sym (q ) = D/2 The problem has a unique solution ξ e = ξ 0 and q R = q = q [ θ(λ qr ) + θ(λ q ) ] K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

12 ξ e changes sign on [ Λ ; Λ ]: the definitions ξ e(z (k) ) = 0, κ (k) = sgn ( ξ ]z (k) ;z (k+) [) ; J (±) = k : κ (k) =± ]z (k) ; z (k+) [; e 2iπ ξ e( q (k) R/ ) =, q (k) { } X = x [ Λ; Λ] : e 2iπ ξ e(x) =; R z(k+) maximal and z (k) + q (k) minimal;, X (in) = X { r [ q (k) k=0 Y = {λ a } N, Y (in) = Y X (in), Y (out) = Y \ Y (in) ; ; q(k) R ]}, X (out) = X \ X (in) ; 2δ z (0) z () q (k ) z (k) q (k) q (k) z (k+) z (r) z (r+) R R Λ κ (k) ξ Λ e > 0 δ J (κ(k) ) K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

13 ξ e changes sign on [ Λ ; Λ ]: the analysis ξ e(λ) + { J (+) J ( ) } θ (λ µ) ξ e(µ) dµ = ϑ ( λ 2 ζ) ϕ out (λ) + ϕ in (λ) + r r (k) [ ξ e ](λ) Driving terms : ϕ in (λ) = θ (λ x) 0 and ϕ out (λ) = θ (λ y) = O(δ) x X (in) \Y (in) y Y (out) Remainder z(k) r (k) [ ξ](λ) = κ (k){ q (k) + q R (k) z (k+) } θ (λ µ) ξ e(µ) dµ + ϵ Invert operators id + θ and then id [R J (+) J (+) ] J ( ) ξ e(λ) = ( id + [R J (+) ] J ( ))[ P ψ J (+) out + ψ in + r r (k) [ ξ e ] ] (λ) ( positivity of id + [ ] ) R J (+) [ ψ J ( ) in ] and then + and δ 0 + ϵ=± k=0 Γ (k) ϵ ξ e(λ) P (λ) > 0 J (+) k=0 θ ( λ ξ e (z)) e 2iπϵz dz = O( δ + ) K K Kozlowski Condensation properties of Bethe roots in the XXZ chain

14 Review of the results " Condensation of particle-hole Bethe roots irrespectively of anisotropy; " proof of existence and uniqueness of solutions; " method works also for complex solutions (proof of existence of strings); " closes the proof of numerous results relative to the thermodynamics K K Kozlowski Condensation properties of Bethe roots in the XXZ chain