Applications of algebraic Bethe ansatz matrix elements to spin chains

Transcription

1 1/22 Applications of algebraic Bethe ansatz matrix elements to spin chains Rogier Vlim Montreal, July 17, 215

2 2/22 Integrable quantum spin chains Outline H XXZ = J N =1 ( [S x S x+1 + S y Sy+1 + S z S+1 z 1 )] 4 Dynamical phenomena in XXZ (and higher spin) Momentum/energy resolved experiments Real space-time tracking Algebraic Bethe ansatz / QISM Bethe states {λ} M = M B(λ ) =1 Determinant formula for matrix elements {µ} M S a {λ} M

3 3/22 Scattering of KdV-solitons Korteweg-de Vries equation: t φ + 3 xφ + 6φ x φ = Classical solitons in the KdV equation Localised particles Stabilised by interplay of non-linear and dispersive effects Interactions - unaltered shape, phase shift only

4 Scattering of KdV-solitons Classical Inverse Scattering Method Quantum Inverse Scattering Method Goal: provide quantum analogue in XXZ chain using ABA N.J. Zabusky and M.D.Kruskal, PRL /22

5 5/22 Bethe ansatz Bethe wave functions {λ} H. Bethe, Zur Theorie der Metalle (1931) {λ} = M A Q ({λ}) e iap(λ Qa ) S a Q a=1 Magnon Bethe equations θ 2s (λ ) 1 N M k Bound state θ 2 (λ λ k ) = 2π N J

6 6/22 Bethe ansatz Bound states - String Hypothesis ( ) Two-strings λ (2),± = λ (2) ± i δ(2) Deviated n-strings λ (n),a = λ(n) + iζ 2 (n + 1 2a) + iπ 4 (1 ν ) + iδ (n),a ζ = acos ( < 1), ζ = acosh ( > 1) Bethe-Gaudin-Takahashi equations (δ (n),a =, s = 1 2 ) θ n (λ (n) ) 1 N M n m k Θ nm (λ (n) λ (m) k ) = 2π N I(n)

7 Quasi-solitons Wave packets of Bethe states Non-decaying excitation (Translationally-invariant) Gaussian wave packets of Bethe states Ψ() = p e ipx 1 4 α(p p)2 {λ (2) } p tj 4 8 Time evolution e iht = δ 2 Evaporation rate 2 E p 2 p=p Two-string Three-string /22

8 8/22 Algebraic Bethe ansatz / QISM Matrix elements Overlap formula Determinant representation - Slavnov (1989) {µ} {λ} = M M C(µ k ) B(λ ) k=1 =1 Norm - Gaudin, McCoy, Wu (1981); Korepin (1982) Matrix elements N ({λ}) = (φ 2 ()) M a b {λ} S z {µ} = ϕ 1 ({µ}) φ 1 (µ ) ϕ 1 ({λ}) φ 1 (λ ) φ 2 (λ a λ b ) det Φ({λ}) φ (λ a λ b ) det(h 2P ) < φ (µ i µ )φ (λ λ i ) N. Kitanine, J. M. Maillet, and V. Terras, NPB 1999 O.A. Castro-Alvaredo and J.M. Maillet, JPA 27

9 9/22 Quasi-soliton scattering ABA time evolution Ψ(t = ) = n c n ψ n ({λ}) S z (t) = Ψ(t) S z Ψ(t) = n,m e i(en Em)t c n c m ψ m S z ψ n }{{} Matrix elements.5 tj S z (t) Unaltered traectories, up to a displacement

10 1/22 Quasi-soliton scattering General single stable particle dp ψ(x, t) = 2π ψ(p) eipx e ite(p) Prescattering (far apart) ψ(x 1, x 2, t) ψ 1 (x 1, t) ψ 2 (x 2, t) Postscattering (far apart) ψ(x 1, x 2, t) = dp1 2π dp 2 2π ψ it (E(p1)+E(p2)) 1(p 1 )ψ 2 (p 2 )e [S(p 1, p 2 ) e i(p1x1+p2x2)] Scattering matrix S(p 1, p 2 ) = e i θ BA(p 1,p 2 ) e iθ(p 1,p 2 )+i p i θ(p 1,p 2 )(p i p i )+...

11 Quasi-soliton scattering Momentum derivative of string phase shift { x (t) = ϑ (n,m) x + v t x + v t ϑ (p 1, p 2 ) before, after. (p, p k ) = Θ nm(λ (n) λ (m) k ) p p =p tj Displ. () Magnons Two-magnons ϑ (1,1) ϑ (2,2) /22

12 12/22 Quasi-soliton scattering Quasi-soliton scattering tj tj 2 4 p 2 = π p 2 = π Displ.() Displ.() ϑ ϑ1 ϑ 2 ϑ ϑ ϑ1 ϑ 2 ϑ ( θ s λ (n) λ (m)) lim = 1 p 1 lim ϑ(n,m) = 2 min(n, m) δ nm ABA time evolution of scattering quasi-solitons matches scattering theory predictions In collaboration with M. Ganahl, D. Fioretto, M. Brockmann, M. Hague, H.G. Evertz, J.-S. Caux

13 13/22 Beyond integrability Inspired by quantum dots - R. van den Berg et al PRB 214 Integrable model + perturbation H = H int + V (t) 2 nd order Suzuki-Trotter decomposition U(t, t + dt) U (2) dt i = e 2 H int e idtv (t+dt) dt i e 2 H int Time evolution Ψ(t + dt) i ψ i ψ i U (2) Ψ(t)

14 14/22 Beyond integrability Particle production H = H XXZ + BS z one-str two-str one+inf tj 2.4 S z (t).3 n cn(t) Two-string wave packet t Local magnetic field B = 4, (J = 1) at = 24

15 Dynamical correlations S aā (k, ω) = 1 N e iq( ) dt e iωt S a (t)sā N () c, = 2π e βωω Ω Sq a α 2 δ(ω ω α + ω Ω ) Z Ω,α = 2π Ω T Sq a α 2 δ(ω ω α + ω ΩT ) α Zero T XXZ spin- 1 2 GS + Spinons Inelastic Neutron Scattering Finite T XXZ spin- 1 2 TBA Ω T LiebLin: Panfil, Caux PRA 214 Higher spin integrable 15/22

16 16/22 Spinons Ψ() = 2S + GS Animation hs z (t)i hs z (t)i hs z (t)i.4.2 hs z (t)i t = hs z (t)i hs z (t)i t =1/J t =2/J

17 17/22 Babuan-Takhtaan spin-1 chain Dynamical structure factor Fusion of R-matrices - polynomial in nearest neigbor int. Babuan-Takhtaan model (spin-1 chain) H BT = J 4 N =1 [Ŝ Ŝ+1 (Ŝ Ŝ+1) 2] GS Sea of deviated two-strings

18 18/22 Babuan-Takhtaan spin-1 chain Ground state GS Deviated two-strings: λ (2),± = λ (2) ± i Solve parameterized BE in λ (2) and δ (2) ( ) δ(2) Parametrisation iterations results Pure two-strings Two-string solutions groundstate N=5.5 Im λ Re λ α Spinons: break one or more two-strings to one strings, three-strings + (higher)

19 Babuan-Takhtaan spin-1 chain Dynamical structure factor S aā (q, ω) = 2π α GS S a q α 2 δ(ω ω α ) ME higher spin: O.A. Castro-Alvaredo and J.M. Maillet, JPA 27 RV and J.-S. Caux, JSTAT 214 H = J 4 N=1 [Ŝ Ŝ+1 (Ŝ Ŝ+1) 2] 19/22

20 2/22 Babuan-Takhtaan spin-1 chain Real space-time correlations versus CFT Correlations in real space-time S a (t)sā () = 1 GS Sq a α 2 e iqα iωαt N SU(2) level-2s WZW-model Antiferromagnetic correlations S a Sā ( 1) 2 = h + h, h = h = n2 1 2n(n+k), α k = 2s

21 21/22 Babuan-Takhtaan spin-1 chain Arbitrary precision sumrule saturation t + dω 1 2π N q S + (q, ω) = 4 3 Perfect up to arbitrary number of digits. Not possible without string deviations. N = 6 t + 1 one-str 2 two-str two-str 1 three-str one-str 1 three-str one-str 1 four-str five-str

22 22/22 Applications of ABA matrix elements Conclusions & Outlook Quasi-soliton scattering & particle production (int breaking) tj Displ. () Magnons Two-magnons ϑ (1,1) ϑ (2,2) tj S z (t) n cn(t) one-str two-str one+inf t Spinons XXZ S z (t).1 S z (t) t = 2/J DSF spin-1 Outlook Integrability breaking Driving Thermal correlations