Singularities of dynamic response functions in the massless regime of the XXZ chain

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1 Singularities of dynamic response functions in the massless regime of the XXZ chain K K Kozlowski CNRS, Laboratoire de Physique, ENS de Lyon 10 th of September 2018 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" J Math Phys "Ludwig Faddeev memorial volume" (2018) K K Kozlowski " On singularities of dynamic response functions in the massless regime of the XXZ chain", to appear K K Kozlowski "Large-distance and long-time asymptotics of two-point dynamical correlation functions in the massless regime of the XXZ chain", to appear Recent Advances in Quantum Integrable Systems, LAPTh, Annecy-Le-Vieux K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

2 Outline Introduction 1 Introduction The dynamic response functions in the XXZ chain Universality and response functions The form factor expansion of dynamic response functions 2 The edge singular behaviour of dynamic response functions Singe excitations thresholds Equal velocity excitations The long-time and large-distance asymptotic behaviour 3 K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

3 The XXZ chain Introduction The dynamic response functions in the XXZ chain Universality and response functions The form factor expansion of dynamic response functions The XXZ spin-1/2 chain on h XXZ = L n=1 C2, σ α Pauli matrices L { H = σ x n σ x n+1 + σy nσ y n+1 + σz nσ z n+1 n} hσz n=1, σ α n+l σα n Local operators : σ α n = id id σ } {{ } α id id } {{ } n 1 times L n times Model solvable by Bethe Ansatz 31 (Bethe), 58 (Orbach): H Υ = Ê Υ Υ " Ground state, Completeness, Norms, Scalar products, Matrix elements of local operators Yang, Yang, Gaudin, McCoy, Wu, Korepin, Slavnov, Kitanine, Maillet, Terras, Mukhin, Tarasov, Varchenko " Spectrum above ground state lim L + {Ê Υ Ê Ω } Hulten, Yang, Yang, Des Cloiseau, Gaudin, Ishumira, Shiba, Faddeev, Takhtadjan, Dorlas, Samsonov, Gusev, K " Large understanding of the static correlators Izergin, Korepin, Jimbo, Miwa, Miki, Nakayashiki, Kitanine, Maillet, Slavnov, Terras K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

4 Dynamic response functions in the XXZ chain The dynamic response functions in the XXZ chain Universality and response functions The form factor expansion of dynamic response functions Time evolution of operators σ γ m (t) = eiht σ γ m e ith Connected dynamical two-point function at zero temperature ( γ σ 1 (t)) {( ( γ γ σ m+1 Ω, σ c = lim L + 1 (t)) σ γ m+1 Ω) ( Ω, σ γ 1 Ω) 2 } Experiments Dynamic Response Functions ( S (γ) γ (k, ω) = σ 1 (t)) γ σ m+1 c dt ei(ωt km) (2π) m Z 2 R Limited understanding of dynamical correlators at 0 95 Jimbo, Miwa, 04 Kitanine, Maillet, Slavnov, Terras : Series representation Not in a manageable form to extract physically pertinent information K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

5 The dynamic response functions in the XXZ chain Universality and response functions The form factor expansion of dynamic response functions Expermiental measurements on KCuF 3 Bragg spectroscopy KCuF 3 XXX chain at h = 0 05 ( Caux, Maillet, Hagemans) 95 ( Tennant, Cowley, Nagler, Tsvelik) 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 Singularities of dynamic response functions in the massless regime of the XXZ cha

6 State of the art for the response functions Non-linear luttinger liquid phenomenology The dynamic response functions in the XXZ chain Universality and response functions The form factor expansion of dynamic response functions Mahan, Noziére, De Dominicus, Glazman, Kamenev, Khodas, Pustilnik, Imambekov, Cheianov, Affleck, Pereira, White, Caux, Shashi S (z) (k, ω) A (k) Ξ (ω ε h (k)) [ω ε h (k)] ϑ Universal structure of the amplitudes A (k) = (2π) 2 Γ 1 (µ R + µ L ) [v(k) + v F ] µ R [vf v(k)] µ F (z) (k) 2 L, ϑ = µ R + µ L 1 Bethe Ansatz spectrum closed expressions for µ R,L ; F (z) (k) 2 volume-renormalised form factor; Integrability based investigation S (z) (k, ω) = n 0 S 2n (k, ω) for XXX h = 0 & (k, ω) behaviour from n = 1, 2 Jimbo, Miwa, Bougourzi, Couture, Kacir, Karbach, Müller, Mütter, Caux, Konno, Sorrel, Weston Numerics & Bethe Ansatz Biegel, Karbach, Müller, Sato, Shiroishi, Takahashi, Caux, Hagemans, Maillet, Formal saddle-point approach to form factor series in NLSM Kitanine, K, Maillet, Slavnov, Terras K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

7 The long-distance and large-time asymptotics The dynamic response functions in the XXZ chain Universality and response functions The form factor expansion of dynamic response functions Massless regime : 1 < 1, h < h c Universality asymptotics of ( γ σ 1 (t)) γ σ m+1 c Exact results: = 0: 82 (McCoy, Perk, Shrock), 84 (Schrock, Müller), 85 (McCoy, Tang) Predictions: 02 (Lukyanov, Terras) h = 0, 1 < < 1 09 (Affleck, Pereira, White) h c > h, 1 < < 1 12 (Imambekov, Schmidt, Glazman) h c > h, 1 < < 1 K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

8 The open problems Introduction The dynamic response functions in the XXZ chain Universality and response functions The form factor expansion of dynamic response functions Questions ( Can one carry out an ab inicio, exact, calculation of σ γ 1 (t)) γ σ m+1 c, S (γ) (k, ω)? 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 Singularities of dynamic response functions in the massless regime of the XXZ cha

9 The dynamic response functions in the XXZ chain Universality and response functions The form factor expansion of dynamic response functions Thermodynamic limit of the spectrum of the XXZ chain Long history of the L + analysis of the spectrum Hulten, Yang Yang, Des Cloiseau, Gaudin, Ishumira, Shiba, Faddeev, Takhtadjan, K Ê Υ\Ω = E(R Υ ) + O(L 1 ) Relative excitation momentum-energy: P(R) 1 E(R) = U (Y, v) + O(δ) v v = m t U (Y, v) = n r { ( (r) pr ν r N a=1 a ) 1 v ε r n ( )} (r) h { ν a pr (µ a ) 1 v ε r (µ a ) } + υl υ p F υ=± a=1 macroscopic variables of the massive excitations {{ } nh Y = µa ; {{ ν (r) } nr 1 a a=1 }r N ; { }} l υ with N = N st {1} K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

10 The dynamic response functions in the XXZ chain Universality and response functions The form factor expansion of dynamic response functions The form factor expansion of dynamic correlation functions Form factor expansion ( ( γ Ω, σ 1 (t)) } γ σ m+1 Ω) = exp {im P Υ\Ω itê Υ\Ω ( γ Υ, σ 1 Ω) 2 Υ [ 17 K ] Thermodynamic limit of the form factor expansion in massless regime ( γ σ 1 (t)) γ σ m+1 = ( 1) msγ d nr ν (r) n n S r N r!(2π) nr d n h µ n h!(2π) n F (γ) (Y) e imu (Y,v) h [ im υ ] ϑ2 υ (Y) C nr r,δ C n h h,δ Left right moving combinations m υ = υm v F t, s γ operator spin ( remainder r(y, m υ, δ) = O δ ln δ + { δ 2 m υ + δ ln m υ + e mυ δ} ) ; υ=± { S = n = ( ) } n h, {n r } r N, l υ : nh = l υ + rn r υ {±} First complete expression for a form factor expansion in a massless model υ=± ( ) 1 + r(y, m υ, δ) Non-perturbative finite-l resummation of low-energy modes constructive IR renormalisation r N K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

11 The two-particle-hole excitation thresholds The edge singular behaviour of dynamic response functions The long-time and large-distance asymptotic behaviour Form factor series representation for DRF ω S (γ) (k, ω) = S (γ) (k, ω) n S n Cp-2 h C 1 p-h C 2 p-h C p 3 C p 4 C p 1 C p 2 C h 1 C h 2 Pmin π - pf π pf + π Pmax 2 π pf 2 (π - pf ) k K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

12 The vicinity of a hole threshold The edge singular behaviour of dynamic response functions The long-time and large-distance asymptotic behaviour (k, ω) configuration close to the hole excitation line (P 0, E 0 ) = (p F t 0, e 1 (t 0 )) with t 0 ] p F ; p F [ The hole threshold ( ) S (z) (2π) 2 Ξ (δω) [δω] (h) P h 0, E 0 + δω = [v(t 0 ) + v F ] δ(h) + [v F v(t 0 )] δ(h) F (z) h (t 0 ) 2 Γ(δ (h) + + δ(h) ) (1 + O ( [δω] 1 0+ )) + S (zz) h;reg (δω) v(t 0 ) = e 1 (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 F (z) (t h 0 ) ( 2 L = lim L + 2π ) (h) δ + +δ(h) ( ) +1 Ω, σ z 1 Υ 2 t 0 Ω 2 Υt0 2 p F t 0 p F ground state Ω { E0 = e excitation Υ 1 (t 0 ) t0 P 0 = p F t 0 K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

13 Multi particle-hole thresholds The edge singular behaviour of dynamic response functions The long-time and large-distance asymptotic behaviour Particle and holes may have equal velocities v(k) = v(t(k)) t : ] K m ; K M [ ] p F ; p F [ (k, ω) configuration close to an equal-velocity particle-hole excitations ( ) (P 0, E 0 ) = n p k 0 n h t(k 0 ) + (n h n p )p F, n p e 1 (k 0 ) n h e 1 (t(k 0 )) with k 0 ] K m ; K M [ The particle threshold ( ) S (zz) F (z) p (k 0 ) 2 P ph 0, E 0 + δω = np n h t (k 0 ) (2π) np +nh 1 2 G(1 + n p )G(1 + n h ) [δω] (ph) Γ( (p) ) ( v ) n 2 p 1( v ) n 2 h (k 0 ) (t(k 0 )) [v(k 0 ) + v F ] δ(ph) + [v F v(k 0 )] δ(ph) { Ξ(δω) [ sin π ( )] δ (ph) π + + δ(ph) + Ξ( δω) [ } π sin π 2 (n2 p + n 2 h 3)] (1 + O ( )) [δω] S (zz) ph;reg (δω) Edge exponent: (ph) = δ (ph) + + δ (ph) ( n 2 p + n 2 h 3) K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

14 The edge singular behaviour of dynamic response functions The long-time and large-distance asymptotic behaviour The not-so-long time regime: v > v Regime m, t ±, m t = v ( σ γ 1 (t)) σ γ m+1 c = ( 1)msγ l Z F (γ) l e 2imlp F [ im υ ] ϑ2 υ;l υ=± ( 1 + O ln m ) υ + e cm m υ=± υ m υ = υm v F t; Critical exponents: ϑ υ;l = lz υsγ 2Z ; Amplitudes F (γ) : properly normalised form factor squared GS to l-umklapp excitation; l String excitations (bound states): produce exponentially small contributions in m; are necessary to carry out the steepest descent contour deformation Not related to space vs time-like regimes (v F > v ( ) ); K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

15 The edge singular behaviour of dynamic response functions The long-time and large-distance asymptotic behaviour The long time regime: v < v ( γ σ 1 (t)) ( γ σ m+1 c = ( 1)msγ C n F (γ) e iυlυmp F n n Sv υ=± [ im υ ] ϑ2 eiφn(m,t) υ;n m ϑsp;n 1 + O ln m ) υ + 1 m υ=± υ m Critical exponents: ϑ υ;n = κ v n 0 ϕ 1 (υq, ω 0 ) s γz υl υ n r ϕ 1 (υq, ω r ) l υ ϕ 1 (υq, υ q) r N ( ) 1 ϑ sp;n = n n r r N υ =± Fermi endpoints Saddle points Oscillating phases: [ φ n (m, t) = κ v n 0 mp1 (ω 0 ) tε 1 (ω 0 ) ] [ + n r mpr (ω r ) tε r (ω r ) ] F (γ) n : normalised FF squared GS Umklapp & multi-particle saddle point excitation; Universal pre-factor ( C n = ( κ v ) n 0 sgn[p r (ω r )] ) n r { ( ) 1 G(1 + nr ) 2 n 2 r } i r N r N {0} (2π) 1 2 nr p r (ωr ) 1 v ε r (ωr ) ; r N κ v = 1 if v F < v < v and κ v = 1 if v < v F ; { S v = n = ( ) {n r } r N {0} ; {l υ } υ=± : 0 = κv n 0 + l υ + } rn r υ {±} r N K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

16 and perspectives Review of the results " Rigorous analysis of auxiliary integrals; " Confirms predictions of non-linear Luttinger-liquid structure (hole, particle, strings); " Exhibits universal structure driven by equal velocity excitations; " Phenomenological form of correlators for the Luttinger liquid universality class; " Long-distance and large-time asymptotics; " universality singularity structure of form factors & classical saddle-point calculation; Further developments Solve the dynamical correlation functions at finite T program; K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

17 The form factor series ( γ σ 1 (t)) γ σ m+1 c = ( 1)msγ ( F (γ) (Y) 1 + r(y, m υ, δ) u r (λ; v) = p r (λ) 1 v ε r (λ) 1 nr dν (r) a n n S r N r! 2π a=1 C ) r,δ (r) (ν eimur Saddle-points ω r : u r (ω r ; v) away from integration contours n a ;v) 1 h dµ a n h! 2π e imu 1 (µa ;v) { } e iυlυmp F a=1 υ=± [ im C υ ] ϑ2 υ (Y) h,δ K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

18 The integration curves Particle and hole contours ω 1 Im[u 1 ] > 0 q O(δ) Im[u 1 ] < 0 λ ; λ ; C 1,δ ω 0 C h,δ λ +; q λ +; R + i π 2 Im[u 1 ] > 0 R r-string contour: ( 1) σr Im[u r ] > 0 ω r ( 1) σr Im[u r ] < 0 C r;δ = σ r R + i π 2 s r K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

19 The integration curves Deform contours C r,δ region where Im[u r (λ; v)] > 0 C h,δ region where Im[u (λ; v)] < 0 1 BUT Contours go to and u r ( ; v) = 0 Form factors have jump discontinuities in ν (1) ] q ; [ ± iζ, ν (r) ] q ; [ ± iζ r±1 2, r N st ( Res F (γ) (Y ph ) Form factors posses kinematical poles ν (r) a ν (p) = iζk, b k S (rp) n r r dν (r) a,k, r N st a=1 k=2 ) ν(r) a,k = ν(r) a,k 1 iζ, k = 2,, r = Rapidities : {{ } nh {{ Y ph = µa ; } (1) n1 ν 1 a {{{ ν (r) } r } nr } }, { } a=1 a;k ; { }} l k=1 a=1 υ r Nst r Nst, Y = r N st { ( i) (r 1)n r } F (γ) (Y) {{ } nh µa ; {{ ν (r) } nr 1 a a=1 }r N ; { }} l υ K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

20 The deformed contours Particle and hole contours ω 1 C 1,sd q ω 0 -A q -A Im[u 1 ] < 0 C h,sd R + i π 2 Im[u 1 ] > 0 R K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

21 The deformation of contours for the two-particle sector: 0 < ζ < π/2 Total contribution: I (2) tot = I ph + I 2 + I 0,1 I 2 = 1 2 C 2 1,δ dν 1 dν 2 (2π) 2 J(ν 1, ν 2 ), I 0,1 = I 2 = 1 2 C 1,δ dν 2 2π C 1,sd R dν (2) 2π J 0,1(ν (2) ) I ph = dν 1 2π J(ν 1, ν 2 ) C 1,δ dν 2 2π Γ ext C 1,δ dν 2π C h,δ dν 1 2π J(ν 1, ν 2 ) dµ 2π J ph(ν, µ) 1st integral C 1,δ C 1,sd without taking poles 2nd integral Γ ext residues poles: ν 1 = ν 2 + iζ, ν 2 J A and ν 1 = ν 2 iζ, ν 2 J A + i π 2 J A = R \ [ A ; A ] I 2 = 1 2 C 1,sd C 1,sd dν 1 dν 2 (2π) J(ν 1, ν 2 2 ) 1 2 J A +i 1 2 ζ dν (2) 2π J 0,1(ν (2) ) 1 2 J A +i 1 2 (π ζ) dν (2) 2π J 0,1(ν (2) ) K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

22 The deformation of contours for the two-particle sector: 0 < ζ < π/2 Final form A+i I (2) tot = 1 dν 1 dν 2 2 (2π) J(ν dν dµ 1, ν 2 2 ) + 2π 2π J ph(ν, µ) σ dν (2) 2 2π J 0,1(ν (2) ) C 1,sd C 1,sd C 1,sd C σ {ζ,π ζ} h,sd A+i 1 2 σ Direct to evaluate the integrals in m + Similar mechanism for 3-particle sector CONJECTURE Cancellation mechanism for contributions from extends to all particle sectors K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

23 The model integral Introduction I(x) = l { nr dp (r) r=1 I nr r a<b ( p (r) a p (r) ) } 2 b G ( p, z + (p; x), z (p; x) ) {Ξ ( z υ (p; x) ) [z υ (p; x) ] } υ(p) 1 υ=± G ( p, x, y) = G(p) + O ( x α + y α) { l } G = 0 on I nr r (R + ) 2 r=1 Edge function z υ (p; x) = E 0 + x l n r r=1 a=1 u r > 0 on I r ±v u r { u r (p (r) a ) + υv P 0 l ( Int(Ir ) ) n r r=1 a=1 } p (r) a I r = I (in) r I (out) r t r : I 1 I (in) r with ( u (out) r Int(I r ) ) u 1( Int(I1 ) ) = u 1 (k) = ( u r tr (k) ) Macroscopic momentum and energy on I 1, P(k) = l n r t r (k) and E(k) = l ( n r u r tr (k) ) P (k) 0 on Int(I 1 ) r=1 r=1 K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha

24 Asymptotic behaviour model integral Theorem 18 K a) The regular case ( ) {( ) } If P 0, E 0 P(k), E(k) : k I1 and min E 0 E(b) + υv(p 0 P(b)) > 0 b I 1 υ=± x I(x) is smooth at x = 0 b) The singular case If ( P 0, E 0 ) = ( P(k0 ), E(k 0 ) ), k 0 Int(I 1 ) and ϑ N, { I(x) = C x ϑ Ξ(x) sin[πν +] + Ξ( x) sin[πν } ] + r(x) + O ( x ϑ+α) π π C = G ( t(k 0 ) ) Γ(δ +)Γ(δ )Γ( ϑ) (2v) δ + +δ 1 l P (k 0 ) v υu 1 (k 0) δυ r=1 υ=± ϑ = 1 2 l n 2 r δ l + + δ, ν ± = 1 n 2 2 r r=1 r=1 : εr = 1 nr δ r,1 G(2 + n r ) (2π) 2 u r (t r (k 0 )) 1 2 (n 2 r δ r,1 ) 1 s 4 + δ υ υ=± : ±[v υu 1 (k 0 )]>0 and with δ υ = υ (t(k 0 )) s = sgn [ P (k 0 ) ] u 1 (k 0 ) ε r = sgn [ u r (t r (k 0 )) ] t(k 0 ) = ( t 1 (k 0 ),, t l (k 0 ) ) R n l with t r (k 0 ) = ( t r (k 0 ),, t r (k 0 ) ) R nr K K Kozlowski Singularities of dynamic response functions in the massless regime of the XXZ cha