Systematic strong coupling expansion for out-of-equilibrium dynamics in the Lieb-Liniger model
SSciPost Physics Submission
Systematic strong coupling expansion for out-of-equilibriumdynamics in the Lieb-Liniger model
Etienne Granet (cid:63) and Fabian H. L. Essler , The Rudolf Peierls Centre for Theoretical Physics, Oxford University, Oxford OX1 3PU,UK (cid:63) [email protected]
Abstract
We consider the time evolution of local observables after an interaction quench inthe repulsive Lieb-Liniger model. The system is initialized in the ground state forvanishing interaction and then time-evolved with the Lieb-Liniger Hamiltonianfor large, finite interacting strength c . We employ the Quench Action approach toexpress the full time evolution of local observables in terms of sums over energyeigenstates and then derive the leading terms of a /c expansion for several oneand two-point functions as a function of time t > after the quantum quench. Weobserve delicate cancellations of contributions to the spectral sums that depend onthe details of the choice of representative state in the Quench Action approach andour final results are independent of this choice. Our results provide a highly non-trivial confirmation of the typicality assumptions underlying the Quench Actionapproach. Contents (cid:104) σ (0) (cid:105) t (cid:104) σ ( x ) σ (0) (cid:105) t (cid:104) σ ( x, τ ) σ (0 , (cid:105) ∞ ,c in the stationary state 9 /c expansion 10 /c expansion 12 (cid:104) σ (0) (cid:105) t /c expansion and particle-hole excitations 134.3 Two particle-hole excitations 151 a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b ciPost Physics Submission5 Calculation of the two-point function (cid:104) σ ( x ) σ (0) (cid:105) t ( x, t ) 215.3.2 Second sum Σ ( x (cid:48) , t ) 225.3.3 Third sum Σ ( x (cid:48) , t ) 225.3.4 Result 225.4 Contributions arising from type II configurations 235.4.1 First term Σ (cid:48) ( x (cid:48) , t ) 255.4.2 Second term Σ (cid:48) ( x (cid:48) , t ) 265.4.3 Third term Σ (cid:48) ( x (cid:48) , t ) 265.4.4 Fourth term Σ (cid:48) ( x (cid:48) , t ) 275.4.5 Result for all contributions arising from type II configurations 275.5 Cancellation of the representative state dependence 28 (cid:104) σ ( x, τ ) σ (0 , (cid:105) ∞ in the steady state 28 /c expansion and particle-hole excitations 296.2 One particle-hole excitations 296.3 Two particle-hole excitations 29 A.1 Double principal values 31A.2 Proof of equation (163) 32
B Proof of Lemma 1 (101)
32C Proof of Lemma 2 (103)
34D Further results on (cid:104) σ ( x ) σ (0) (cid:105) D.1 Alternative expression for (22) 34D.2 Consistency check I: t → x → x, t → E Typicality and Quench Action method 36References 38 ciPost Physics Submission The non-equilibrium dynamics in isolated many-particle quantum systems has attracted agreat deal of attention over the last decade [1–6]. These developments were driven by theability to realize almost isolated many-particle quantum systems using trapped, ultra-coldatoms and investigate their time evolution when driven out of equilibrium in exquisite detail,see e.g. Refs [7–17]. It was realized early on that conservation laws play a crucial role in thelate time relaxational behaviour of isolated systems [8, 18]. This implies in particular thatin the thermodynamic limit integrable systems with extensive numbers of conservation lawswill typically relax to non-thermal stationary states [19–39]. The full time evolution of localobservables in integrable models is equally interesting, but significantly harder to determine.Early work focused on rational conformal field theories [5, 40, 41] and non-interacting models[19,42–44]. The low density regime after weak quantum quenches has be analyzed by means oflinked-cluster expansions [42,45–48] and semiclassical methods [49–51]. Arguably the methodof choice for studying the time evolution of local operators in interacting integrable models isthe so-called Quench-Action approach [23,52]. To date it mostly has been applied to determineand characterize the stationary state [26–29,32,34–37]. Exceptions are Refs [46], [53] and [54],which address respectively the asymptotic late-time regimes after quenches to the sine-Gordon,Lieb-Liniger and transverse field Ising models respectively. According to the Quench-Actionapproach the expectation values of local operators after a quantum quench from an initialstate | Ψ (cid:105) are given bylim L →∞ (cid:104)O ( t ) (cid:105) = lim L →∞ (cid:18) (cid:104) Ψ |O ( t ) | Φ s (cid:105) (cid:104) Ψ | Φ s (cid:105) + (cid:104) Φ s |O ( t ) | Ψ (cid:105) (cid:104) Φ s | Ψ (cid:105) (cid:19) . (1)Here L denotes the system size, the so-called representative state | Φ s (cid:105) is a simultaneouseigenstate of the Hamiltonian and of the (quasi)local [33] conservation laws I ( n ) of the theoryunder consideration, such that it correctly reproduces the extensive parts of the expectationvalues of the I ( n ) in the initial statelim L →∞ (cid:104) Ψ | I ( n ) | Ψ (cid:105) L = lim L →∞ (cid:104) Φ s | I ( n ) | Φ s (cid:105) L . (2)The structure of (1) is similar to that of response functions in equilibrium and provides aspectral representation in terms of (normalized) energy eigenstates | n (cid:105) by writing (cid:104) Ψ |O ( t ) | Φ s (cid:105) = (cid:88) n (cid:104) Ψ | n (cid:105)(cid:104) n |O (0) | Φ s (cid:105) e it ( E n − E s ) . (3)In practice the Quench Action approach faces two challenges: • It requires knowledge of the overlaps (cid:104) Ψ | n (cid:105) between the initial state and energy eigen-states. This is known as the “initial state problem”. To date such overlaps are known fora number of specific examples only [55–62], but many of these are physically interesting. • Determining the time evolution requires carrying out spectral sums like (3). Given thatthese generally involve an exponentially (in system size) large number of terms this is aformidable challenge. 3 ciPost Physics Submission
In this work we focus on the second of these problems, namely how to extract the timedependence of local observables after a quantum quench from the spectral representation.We consider the case of a quantum quench to the repulsive Lieb-Liniger model, and bring tobear strong-coupling expansion methods we recently developed in the context of equilibriumresponse functions [63].
We consider the Lieb-Liniger model [64–66] H = (cid:90) L d x (cid:104) ψ † ( x ) (cid:16) − (cid:126) m d dx (cid:17) ψ ( x ) + cψ † ( x ) ψ † ( x ) ψ ( x ) ψ ( x ) (cid:105) , (4)where ψ ( x ) is a canonical Bose field satisfying equal-time commutation relations[ ψ ( x ) , ψ † ( y )] = δ ( x − y ) . (5)In the following we set (cid:126) = 2 m = 1, impose periodic boundary conditions and restrict ourselvesto the repulsive case c >
0. For later convenience we define the local operators of interest,namely the density operator at position x and the interaction potential σ ( x ) = ψ † ( x ) ψ ( x ) ,σ ( x ) = (cid:0) ψ † ( x ) (cid:1) (cid:0) ψ ( x ) (cid:1) . (6)The Lieb-Liniger model is solvable by the Bethe ansatz [64–66]. Its eigenfunctions can beparametrized by N rapidity variables λ , ..., λ N that on a ring of radius L satisfy a set ofquantization conditions known as “Bethe equations” λ k π = I k L − L N (cid:88) j =1 π arctan λ k − λ j c , k = 1 , . . . , N. (7)Here I k are integer if N is odd and half-odd integer if N is even. The corresponding eigenstate | λλλ (cid:105) can be written as | λλλ (cid:105) = B ( λ ) ...B ( λ N ) | (cid:105) , (8)where B ( λ ) is a creation operator acting on a particular reference state | (cid:105) . The eigenvaluesof the Hamiltonian and other conserved quantities are expressed in terms of the rapidities aswell. For example the energy E ( λλλ ) and momentum P ( λλλ ) read E ( λλλ ) = N (cid:88) i =1 λ i , P ( λλλ ) = N (cid:88) i =1 λ i . (9)For c > λ i to the Bethe equations are real [66]. Following [28] we consider the following quantum quench protocol. We assume that the systemis prepared in the Bose-Einstein condensate (BEC) ground state for N particles in the absenceof interactions | Ψ BEC (cid:105) = 1 √ N ! L N (cid:90) L d x ... (cid:90) L d x N ψ † ( x ) ...ψ † ( x ) | (cid:105) . (10)4 ciPost Physics Submission At t = 0 we then suddenly turn on the interactions, so that for t > | Ψ( t ) (cid:105) is governed by the Hamiltonian (4) | Ψ( t ) (cid:105) = e − itH | Ψ BEC (cid:105) . (11)Our aim is to determine the full time evolution of a number of different observables afterthe quench in the framework of the systematic 1 /c -expansion developed in [63]. We haveconsidered the following one and two-point functions: • One-point function of the interaction potential (cid:104) σ (0) (cid:105) t ≡ (cid:104) Ψ( t ) | σ (0) | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | Ψ( t ) (cid:105) . (12) • Density-density correlation function (cid:104) σ ( x ) σ (0) (cid:105) t ≡ (cid:104) Ψ( t ) | σ ( x ) σ (0) | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | Ψ( t ) (cid:105) . (13) • Steady-state expectation value of the two-point function of the interaction potential (cid:104) σ ( x, τ ) σ (0 , (cid:105) ∞ ≡ lim t →∞ (cid:104) Ψ( t ) | σ ( x, τ ) σ (0 , | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | Ψ( t ) (cid:105) . (14)Here we have defined σ ( x, τ ) = e iHτ σ ( x ) e − iHτ . We note that we use a differentnotation for the time difference τ to avoid confusion with the time t according to whichthe system evolves after the quench. The analogous two-point function for the densityoperator was derived in [63] up to order 1 /c . As the derivations of our results are quite technical we start by presenting our final answersand discuss their physical implications. All correlators are expressed in terms of distributionfunctions of particles ρ ( λ ) and holes ρ h ( λ ) defined as follows [28] ρ ( λ ) = a ( λ/c ) ρ h ( λ ) = τ π (cid:0) a ( λ/c ) (cid:1) d a ( λ/c )d τ , (15)where τ = D c and a ( x ) = 2 πτx sinh(2 πx ) I − ix (4 √ τ ) I ix (4 √ τ ) , (16)with I the modified Bessel function. The particle density D is related to ρ ( λ ) by D = (cid:90) ∞−∞ ρ ( x )d x . (17)5 ciPost Physics Submission (cid:104) σ (0) (cid:105) t Our final result for the time evolution of the interaction potential σ (0) after the quench, validat all finite times t > /c up to and including order O ( c − ) is (cid:104) σ (0) (cid:105) t − (cid:104) σ (0) (cid:105) ∞ =lim (cid:15) → c (1 + 2 D /c ) (cid:90) ∞ (cid:90) ∞ λ (1 − µ c ) cos(2 t ( λ − µ )) ρ ( λ ) ρ h ( µ ) e − (cid:15)µ d λ d µ + O ( c − ) . (18)The steady-state value (cid:104) σ (0) (cid:105) ∞ has been previously calculated in [28]. From (18) the late-time asymptotics can be straightforwardly extracted with a saddle point approximation (cid:104) σ (0) (cid:105) t − (cid:104) σ (0) (cid:105) ∞ = 1 t π D /c ) c (cid:20) ρ (0) ρ (cid:48)(cid:48) h (0) − ρ (cid:48)(cid:48) (0) ρ h (0) − ρ (0) ρ h (0) c (cid:21) + O ( t − ) + O ( c − ) . (19)Here ρ (cid:48)(cid:48) (0) denotes the second derivative of ρ ( λ ) evaluated at λ = 0. The asymptotic t − dependence is in agreement with a previous conjecture [53]. However, our results show thatthis regime is reached only at rather late times when the expectation value is already negligiblysmall. This is shown in Figure 1, where we plot g ( t ) = (cid:104) σ (0) (cid:105) t D . (20)Our 1 /c -expansion provides us with the first few terms of an expression of the form g ( t ) = . . . . t g ( t ) − g ( ∞ ) − − − − log t l og ( g ( t ) − g ( ∞ )) Figure 1: Left: g ( t ) − g ( ∞ ) as a function of t (blue thick line), for c = 3 and D = 0 . ∝ t − . The inset shows the samequantities on a logarithmic scale. (cid:80) ∞ n =2 γ − n a n ( t ), where γ = c D and the functions a n ( t ) incorporate non-perturbative summa-tions of certain terms at all orders in 1 /c . In order to assess the parameter range in whichthe series may be convergent we consider the ratios r n ( t ) = (cid:12)(cid:12)(cid:12) a n ( t ) a ( t ) (cid:12)(cid:12)(cid:12) n − , n = 3 , . (21)6 ciPost Physics Submission In Figure 2 we plots these ratios as functions of t for c = 3 and D = 0 .
16. We see that both t r , ( t ) Figure 2: r (resp. r ) as a function of t , in light (resp. dark) green. These ratios give anestimate of the smallest value of γ for which the series is convergent.ratios grow at short times, indicating that the series is not likely to be uniformly convergentnear t = 0. Moreover, it follows from the fact that g (0) = 1 while g ( t ) = O ( c − ) for all t > t →
0. For t (cid:39) r , ( t ) suggest that the series could be convergent for γ (cid:39)
4. As a comparison, we recall thatthe series in γ for the ground state energy density is convergent for γ > .
527 [67]. (cid:104) σ ( x ) σ (0) (cid:105) t We find that the leading contributions in the 1 /c -expansion of the density-density correlationfunction can be cast in the form (cid:104) σ ( x ) σ (0) (cid:105) t = (cid:104) σ ( x ) σ (0) (cid:105) ∞ +(1 + 2 D /c ) (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) λµ cos( x (cid:48) ( µ − λ )) cos(2 t ( λ − µ )) − c sgn ( x ) (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) λ ( λ − µ ) µ sin( x (cid:48) ( λ − µ )) cos(2 t ( λ − µ ))+ 4 c (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) − (cid:90) d νρ ( ν ) F ( λ, µ, ν ; x (cid:48) ) cos(2 t ( λ − µ ))+ 4 c (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d νρ ( ν ) − (cid:90) d µρ h ( µ ) F ( λ, µ, ν ; x (cid:48) ) cos(2 t ( λ − µ )) + O ( c − ) . (22)where x (cid:48) = x (cid:18) D c (cid:19) , (23)7 ciPost Physics Submission and F ( λ, µ, ν ; x ) = (cid:20) λ ( ν − λ ) ν ( µ − ν ) + ν − λλ + µ (cid:21) cos( x ( ν − λ )) + (cid:20) ν ( ν − µ ) µ ( λ − ν ) + ν − µλ + µ (cid:21) cos( x ( ν − µ )) ,F ( λ, µ, ν ; x ) = (cid:20) λ ( µ − λ ) µ ( ν − µ ) + λ ( λ − µ ) µ ( ν − λ ) (cid:21) cos( x ( µ − λ )) . (24)Here − (cid:82) denotes a principal value integral defined as − (cid:90) f ( λ ) µ − λ d λ ≡ lim (cid:15) → (cid:90) | λ − µ | >(cid:15) f ( λ ) µ − λ d λ . (25)The limit c → ∞ of (22) was previously computed in [28]. The density-density correlator(22) is shown in Figs 3 and 4.Figure 3: Density plot of (cid:104) σ ( x ) σ (0) (cid:105) t (22) as a function of x, t for D = 1, c = ∞ (left) and c = 10 (right). The color coding is the same for both plots.We see that for the chosen parameters D = 1 and c = 10 the effects of the O ( c − ) termare clearly visible and significantly modify the c = ∞ result. In particular the oscillatorybehaviour as a function of distance for short times becomes more pronounced for smaller valuesof c . Perhaps the most striking feature of Fig. 3 is the apparent absence of any light coneeffect [41]. This can be understood by noting that (i) our initial state has an infinite correlationlength and any light cone like feature would therefore be weak; (ii) the local Hilbert space isinfinite dimensional and the dispersion relation of elementary excitations unbounded. Hencethe Lieb-Robinson bound [68] does not apply and “superluminal” effects [69] are allowed.An alternative representation of (22) more suitable for numerical evaluations and an anal-ysis of the x → t → x and t asymptotics of (22) at fixed ratio α = x t can be determined by astationary phase approximation, which results in (cid:104) σ ( x ) σ (0) (cid:105) t = (1 + D c ) π | t | ρ ( α (cid:48) ) ρ h ( α (cid:48) ) + o ( t − ) , (26)with α (cid:48) = x (cid:48) t . 8 ciPost Physics Submission . . . (cid:104) σ ( x ) σ ( ) (cid:105) t . . . x (cid:104) σ ( x ) σ ( ) (cid:105) t x x Figure 4: Two-point function (cid:104) σ ( x ) σ (0) (cid:105) t (22) as a function of x , for D = 1, c = ∞ (red) and c = 10 (blue), for different values of t = 0 . , . , . , . , . , ∞ (in reading direction). (cid:104) σ ( x, τ ) σ (0 , (cid:105) ∞ ,c in the stationary state We discussed how to determine the non-equal time density-density correlation function in anarbitrary energy eigenstate described by a root density ρ ( λ ) in our previous work [63]. Theresults in this Section are thus valid for a generic root density ρ , the steady state one (15)being a particular case. Applying the same method to the connected dynamical two-pointfunction of σ ( x ) gives the following result (cid:104) σ ( x, τ ) σ (0 , (cid:105) c = lim (cid:15) → c (cid:90) ∞−∞ (cid:90) ∞−∞ ρ ( λ ) ρ h ( µ ) G ( λ, µ ) e iτ ( λ − µ )+ ix ( µ − λ ) − (cid:15)µ d λ d µ + lim (cid:15) → c (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:90) ∞−∞ ρ ( λ ) ρ h ( µ ) ρ ( u ) ρ h ( v )( λ − u ) ( µ − v ) × e iτ ( λ − µ )+ ix ( µ − λ )+ iτ ( u − v )+ ix ( v − u ) − (cid:15)µ − (cid:15)v d λ d µ d u d v + O ( c − ) , (27)where we have defined G ( λ, µ ) = (cid:104) E + D λ − λ P + ( µ − λ )( D λ − P ) (cid:105) , P = (cid:90) ∞−∞ λρ ( λ )d λ , E = (cid:90) ∞−∞ λ ρ ( λ )d λ . (28)The result for the connected two-point function in the stationary state reached at late timesafter the quench is obtained by substituting the particle and hole densities (15) into (27) and(28). The leading asymptotic behaviour for large x and τ with α = x τ kept fixed can be9 ciPost Physics Submission obtained by a stationary phase approximation (cid:104) σ ( x, τ ) σ (0 , (cid:105) c = 16 πc | τ | ρ ( α ) ρ h ( α )( D α − P α + E ) + O ( τ − ) + O ( c − ) . (29)These results can be compared with predictions of Generalized Hydrodynamics [70]. Accord-ing to these the leading large time and distance asymptotics of connected correlations betweentwo local observables O and O is (cid:104)O ( x, τ ) O (0 , (cid:105) c = (cid:90) ∞−∞ δ ( x − v eff ( λ ) τ ) ρ ( λ ) ρ h ( λ ) ρ ( λ ) + ρ h ( λ ) V O ( λ ) V O ( λ )d λ + o ( τ − ) , (30)where V O ( λ ) is the so-called hydrodynamic projection of the operator O , and v eff ( λ ) theeffective velocity associated with the macro-state defined by the particle and hole densities ρ ( λ ) and ρ h ( λ ). The hydrodynamic projection V σ ( λ ) of σ has been determined in [70] V σ ( λ ) = 2 π (cid:90) ∞−∞ ρ ( µ ) ρ ( µ ) + ρ h ( µ ) g dr µ ( λ ) h dr1 ( λ ) ( h dr2 ( µ ) h dr1 ( λ ) − h dr1 ( µ ) h dr2 ( λ ))d µ . (31)Here h n ( λ ) = λ n − , g µ ( λ ) = µ − λ ( λ − µ ) + c , and the dressing operation h dr is defined by h dr ( λ ) = h ( λ ) + 12 π (cid:90) ∞−∞ cc + ( λ − µ ) ρ ( µ ) ρ ( µ ) + ρ h ( µ ) h dr ( µ )d µ . (32)We find that the asymptotics (29) agrees with this GHD prediction at leading order in 1 /c .The dynamical two-point function of σ ( x ) is related to the Drude weight D and theOnsager coefficient L by12 (cid:90) ∞−∞ x [ (cid:104) σ ( x, τ ) σ (0 , (cid:105) + (cid:104) σ ( x, − τ ) σ (0 , (cid:105) ] d x = Dτ + L | τ | + o ( τ ) . (33)In contrast to the density-density correlator the two-point function of σ ( x ) is expectedto exhibit diffusive behaviour, i.e. have a non-vanishing Onsager coefficient L (cid:54) = 0. Ourexpression for the two point function translates into the following results for D and L D = 128 πc (cid:90) ∞−∞ λ ρ ( λ ) ρ h ( λ )[ E + D λ − λ P ] d λ + O ( c − ) , L = O ( c − ) . (34)This shows that higher orders in the 1 /c -expansion are required to determine the Onsagercoefficient. We note that this specific result holds only for root densities ρ that decay suffi-ciently fast at infinity. In the case of the steady state root density (15), because of the slowdecay of the density, the next terms in the 1 /c expansion should be re-summed to yield aconvergent integral. /c expansion In this Section we discuss the implementation of a 1 /c expansion of the Quench Actionapproach [23], which we will then apply to several observables of interest in the remainder ofthe paper. 10 ciPost Physics Submission The time evolution of the expectation value of any operator O can always be expressed as adouble sum over a basis of energy eigenstates (cid:104)O(cid:105) t = (cid:88) λλλ,µµµ (cid:104) Ψ BEC | λλλ (cid:105)(cid:104) λλλ |O| µµµ (cid:105)(cid:104) µµµ | Ψ BEC (cid:105)(cid:104) λλλ | λλλ (cid:105)(cid:104) µµµ | µµµ (cid:105) e it ( E ( λλλ ) − E ( µµµ )) . (35)Here we have assumed that (cid:104) Ψ BEC | Ψ BEC (cid:105) = 1. The Quench Action approach [23] positsthat one of the two sums in (35) is completely dominated by states around a saddle pointcharacterized by a certain root distribution ρ s that is fixed by the overlaps. This allows oneto rewrite (35) in the formlim L →∞ (cid:104)O(cid:105) t = lim L →∞ | S L | (cid:88) λλλ ∈ S L Re (cid:20) (cid:88) µµµ (cid:104) Ψ BEC | µµµ (cid:105)(cid:104) λλλ |O| µµµ (cid:105)(cid:104) Ψ BEC | λλλ (cid:105)(cid:104) µµµ | µµµ (cid:105) e it ( E ( λλλ ) − E ( µµµ )) (cid:21) , (36)i.e. a generalized micro-canonical average [23, 71] over a set S L of microstates correspondingto the root density ρ s . Employing typicality ideas the micro-canonical average is then replacedby the expectation value with respect to a single “representative state” | λλλ (cid:105) [23]lim L →∞ (cid:104)O(cid:105) t = lim L →∞ Re (cid:20) (cid:88) µµµ (cid:104) Ψ BEC | µµµ (cid:105)(cid:104) λλλ |O| µµµ (cid:105)(cid:104) Ψ BEC | λλλ (cid:105)(cid:104) µµµ | µµµ (cid:105) e it ( E ( λλλ ) − E ( µµµ )) (cid:21) . (37)We note that this last step implies that in the thermodynamic limit (37) depends on therepresentative state | λλλ (cid:105) only through its root density ρ ( λ ). To be of practical use the representation (37) requires closed-form expressions for the overlaps (cid:104) Ψ BEC | λλλ (cid:105) . For our quench protocol an efficient representation for the overlaps was derivedin [27, 28]. Importantly, the overlaps are non-zero only for “pair” states, i.e. states whoserapidities are of the form − λ N/ , ..., − λ , λ , ..., λ N/ with 0 < λ j ∀ j . We will denote a set ofpositive λ j by λλλ >
0. We will use the notation ¯ λ ¯ λ ¯ λ = ( − λλλ ) ∪ λλλ for such sets of rapidities. Theoverlaps are then given by (cid:104) Ψ BEC | ¯ λ ¯ λ ¯ λ (cid:105) (cid:112) (cid:104) ¯ λ ¯ λ ¯ λ | ¯ λ ¯ λ ¯ λ (cid:105) = ( − N/ (cid:114) N ! L N (cid:115) det G + ( λλλ )det G − ( λλλ ) 1 (cid:81) N/ j =1 λ j (cid:113) λ j c + , (38)where G ± ( λλλ ) are ( N/ × ( N/
2) matrices of the form G ± ij ( λλλ ) = δ ij (cid:18) L N/ (cid:88) k =1 cc + ( λ i − λ k ) + 2 cc + ( λ i + λ k ) (cid:19) − (cid:18) L cc + ( λ i − λ j ) ± L cc + ( λ i + λ j ) (cid:19) . (39)For our quench protocol the saddle point root distribution was determined in Ref. [28] and isgiven in (15). 11 ciPost Physics Submission /c expansion Our objective is to combine the Quench Action approach to non-equilibrium dynamics (36)with a strong coupling expansion around c = ∞ . A detailed exposition of the 1 /c expansiontechnique for dynamical correlation functions in equilibrium has been given in [63]. In thefollowing we recall the key steps of the method and then extend it to the out-of-equilibriumcase.In order to facilitate the 1 /c -expansion of the form factors and Bethe equations we firstfix an arbitrary, large Λ > ∞ at the end of the calculation. We thenselect an arbitrary averaging state λλλ by fixing its Bethe numbers III , impose the constraintthat ∀ i, | λ i | < Λ, and define the following overlap-weighted spectral sum (cid:104)O(cid:105) [ λλλ ] , Λ t ≡ Re (cid:20) (cid:88) µµµ ∀ i, | µ i | < Λ (cid:104) Ψ BEC | µµµ (cid:105)(cid:104) λλλ |O| µµµ (cid:105)(cid:104) Ψ BEC | λλλ (cid:105)(cid:104) µµµ | µµµ (cid:105) e it ( E ( λλλ ) − E ( µµµ )) (cid:21) . (40)The overlap-weighted form factor can then be expanded in powers of 1 /c at fixed L, III (cid:104) Ψ BEC | µµµ (cid:105)(cid:104) λλλ |O| µµµ (cid:105)(cid:104) Ψ BEC | λλλ (cid:105)(cid:104) µµµ | µµµ (cid:105) = ∞ (cid:88) n =0 F n ( III, JJJ ) c n , (41)where JJJ denotes the Bethe numbers of µµµ . We also expand the argument of the phase E ( λλλ ) − E ( µµµ ) = ∞ (cid:88) n =0 E n ( III ) − E n ( JJJ ) c n , (42)but do not expand the phase e it ( E ( λλλ ) − E ( µµµ )) itself in powers of 1 /c . The truncation of theresulting series at a given order O ( c − m ) defines the m -th term of our expansion. Once thistruncation has been done, the thermodynamic limit and (if necessary) the average in (36) canbe performed. By construction, the result depends only on the root density ρ of the fixedaveraging state λλλ .Finally one would like to take the limit Λ → ∞ . As we will see, the thermodynamic limitof the quantity (cid:104)O(cid:105) [ λλλ ] , Λ t at finite Λ > I n (Λ | t, x ) = (cid:90) Λ − Λ µ n e − itµ + ixµ d µ . (43)The limit Λ → ∞ of these integrals for n > x, t has a well-defined limit when Λ → ∞ . The resultinglimits are denoted by I n ( t, x ) and have been worked out in [63] for n = 0 , , I ( t, x ) = x t I ( x, t ) ,I ( t, x ) = (cid:18)(cid:16) x t (cid:17) + 12 it (cid:19) I ( x, t ) ,I ( t, x ) = (cid:90) ∞−∞ e − itµ + ixµ d µ . (44)An equivalent representation is I n ( t, x ) = lim (cid:15) → (cid:90) ∞−∞ µ n e − itµ + ixµ − (cid:15)µ d µ . (45)12 ciPost Physics Submission The process described above provides closed-form expressions at order O ( c − m ) for the quan-tities (cid:104)O(cid:105) [ ρ ] t ≡ lim Λ →∞ lim L →∞ | S L | (cid:88) λλλ ∈ S L (cid:104)O(cid:105) [ λλλ ] , Λ t . (46)Finally, in order to obtain the out-of-equilibrium time evolution (36) this result needs to beevaluated for the saddle point root density ρ describing the quench protocol of interest. (cid:104) σ (0) (cid:105) t In this Section we apply the Quench Action approach combined with a 1 /c expansion tocompute the one-point function (cid:104) σ (0) (cid:105) t . In order to evaluate the expression (36), one requires a closed-form expression for the formfactors of σ between energy eigenstates. In the case of interest, because of the structure ofthe non-vanishing overlaps with the initial state | Ψ BEC (cid:105) , the states entering (36) have a pairstructure and will be denoteed | ¯ λ ¯ λ ¯ λ (cid:105) and | ¯ µ ¯ µ ¯ µ (cid:105) . Hence they have same (vanishing) momentum.In this situation the normalized form factors have been calculated previously and read [78] (cid:104) µµµ | σ (0) | λλλ (cid:105) (cid:112) (cid:104) λλλ | λλλ (cid:105) (cid:104) µµµ | µµµ (cid:105) = ( − i ) N +1 ( − N ( N − / cL N (cid:112) det G ( λλλ ) det G ( µµµ ) ( E ( µµµ ) − E ( λλλ )) N (cid:89) j (cid:54) = p ( V + j − V − j ) × (cid:81) i Let f ( λ, µ, ν ) be a regular function that grows sufficiently slowly at infinity. Thenin the thermodynamic limit L (cid:88) λ a ∈ Λ (cid:88) n (cid:54) =0 (cid:88) λ i ∈ Λ f ( λ a , λ i + πnL , λ i )( λ i + πnL )( − πnL ) = 12 π (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d νρ ( ν ) − (cid:90) d µ f ( λ, µ, ν ) µ ( ν − µ )+ 12 π ρ (0) π (cid:90) ∞−∞ f ( λ, , ρ ( λ ) d λ − Ω( λλλ ) (cid:90) ∞−∞ ρ ( λ ) f ( λ, , d λ + O ( L − ) . (103)A proof of Lemma 1 is given in Appendix C. Σ ( x, t )Writing out the various constraints in the summations explicitly we haveΣ ( x, t ) = 1 L (cid:88) λ a ∈ Λ (cid:88) n (cid:54) =0 (cid:88) λ i ∈ Λ g ( λ a , λ i + πnL )( λ i + πnL )( − πnL ) − L (cid:88) λ a ∈ Λ (cid:88) λ j ∈ Λ (cid:88) λ i ∈ Λ i (cid:54) = j g ( λ a , λ j ) λ j ( λ i − λ j ) − L (cid:88) λ a ∈ Λ (cid:88) µ a µ a / ∈ Λ g ( λ a , µ a ) µ a ( λ a − µ a ) − L (cid:88) λ a ∈ Λ (cid:88) µ a µ a / ∈ Λ g ( λ a , µ a ) µ a ( − λ a − µ a ) . (104)We note that since the states have a pair structure, N is even and the Bethe numbers arehalf-odd integers, and so neither µ a or λ i + πnL can vanish in the denominators.The last two sums are two-dimensional sums with a prefactor 1 /L but only simple polesand hence vanish in the thermodynamic limit. The remaining two sums in (104) can be carriedout using Lemma 2 and Lemma 1 respectively. This givesΣ ( x (cid:48) , t ) = − (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) λµ ( µ − λ ) ˜ ρ ( µ ) e it ( λ − µ )+ ix (cid:48) ( µ − λ ) + (cid:104) λλλ ) − π ρ (0) (cid:16) π − ρ (0)2 (cid:17)(cid:105) (cid:90) ∞−∞ λ ρ ( λ ) e itλ − ix (cid:48) λ d λ , (105)where ˜ ρ ( λ ) denotes the Hilbert transform of ρ ( λ ) defined by˜ ρ ( λ ) = − (cid:90) ρ ( ν ) λ − ν d ν . (106)21 ciPost Physics Submission5.3.2 Second sum Σ ( x (cid:48) , t )Writing out the constraints on the various summations explicitly we haveΣ ( x (cid:48) , t ) = 1 L (cid:88) λ a ∈ Λ (cid:88) n (cid:88) λ i ∈ Λ λ i (cid:54) = λ a g ( λ a , π ( n +1 / L ) π ( n +1 / L ( λ i − λ a ) − L (cid:88) λ a ∈ Λ (cid:88) λ j ∈ Λ (cid:88) λ i ∈ Λ i (cid:54) = a g ( λ a , λ j ) λ j ( λ i − λ a )+ 12 L (cid:88) λ a ∈ Λ (cid:88) µ a µ a / ∈ Λ g ( λ a , µ a ) µ a λ a . (107)The third sum is a two-dimensional sum with a prefactor 1 /L and no double poles and hencevanishes in the thermodynamic limit. The first two sums can be turned into principal valueintegrals, which givesΣ ( x (cid:48) , t ) = − (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) λµ ( µ − λ ) ˜ ρ ( λ ) e it ( λ − µ )+ ix (cid:48) ( µ − λ ) . (108) Σ ( x (cid:48) , t )The third sum is a two-dimensional sum with a prefactor 1 /L , so can contribute in thethermodynamic limit only if there is a double pole. It follows thatΣ ( x (cid:48) , t ) = 1 L (cid:88) λ a ∈ Λ (cid:88) µ a / ∈ Λ λ a µ a e it ( λ a − µ a )+ ix (cid:48) ( µ a − λ a ) + O ( L − ) . (109)By writing out the constraint explicitly we haveΣ ( x (cid:48) , t ) = 1 L (cid:88) λ a ∈ Λ (cid:88) n λ a ( π ( n +1 / L ) e it ( λ a − ( 2 π ( n +1 / L ) )+ ix (cid:48) ( 2 π ( n +1 / L − λ a ) − L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ λ a λ i e it ( λ a − λ i )+ ix (cid:48) ( λ i − λ a ) . (110)We see that the sum over λ a can be turned into an integral, while the remaining sums can berespectively carried out explicitly and expressed in terms of Ω( λλλ ) (102). This givesΣ ( x (cid:48) , t ) = (cid:20) − Ω( λλλ ) (cid:21) (cid:90) ∞−∞ λ ρ ( λ ) e itλ − ix (cid:48) λ d λ + O ( L − ) . (111) Putting everything together, we obtain the following result for the contribution of two oneparticle-hole excitations to the spectral sum (85) C , λ ¯ λ ¯ λ ( x, t ) =(1 + 2 D /c ) (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) λµ (cid:18) − c ( µ − λ )( ˜ ρ ( µ ) − ˜ ρ ( λ )) (cid:19) e it ( λ − µ )+ ix (cid:48) ( µ − λ ) + 1 c (cid:20) − πρ (0) + 2 π ρ (0) + Ω( λλλ ) (cid:21) (cid:90) ∞−∞ λ ρ ( λ ) e itλ − ix (cid:48) λ d λ . (112)We stress that C , λ ¯ λ ¯ λ ( x, t ) depends on the representative state λλλ not only through the rootdensity ρ , but via the quantity Ω( λλλ ) (102) as well.22 ciPost Physics Submission Let us denote by C , λ ¯ λ ¯ λ ( x, t ) the sum of contributions of type-II configurations to (85), i.e.configurations where ννν corresponds to a one particle-hole excitation above ¯ λ ¯ λ ¯ λ (¯ µ ¯ µ ¯ µ ) and a twoparticle-hole excitation above ¯ µ ¯ µ ¯ µ (¯ λ ¯ λ ¯ λ ) respectively . There are altogether four cases:(i) The Bethe numbers of ννν are those of ¯ λ ¯ λ ¯ λ except for the replacement of I a or − I a by K a .Denoting the corresponding root by ν a we have the following restrictions: ∀ i, µ a (cid:54) = λ i ; ∀ i, ν a (cid:54) = λ i ; ν a (cid:54) = µ a , − µ a .(ii) The Bethe numbers of ννν are those of ¯ µ ¯ µ ¯ µ with only J a or − J a replaced by a K a . Denoting ν a the corresponding root, we have the restrictions ∀ i, µ a (cid:54) = λ i and ∀ i, ν a (cid:54) = λ i and ν a (cid:54) = µ a , − µ a . Cases (i) and (ii) are sketched in Figure 6. | ¯ λ ¯ λ ¯ λ (cid:105) . . . − I a . . . I a . . . | ννν (cid:105) . . . − I a . . . K a . . . | ¯ µ ¯ µ ¯ µ (cid:105) . . . − J a . . . J a . . .Figure 6: Cases (i) and (ii) of type II excitations.(iii) The Bethe numbers of ννν are those of ¯ λ ¯ λ ¯ λ with only I b or − I b ( b (cid:54) = a ) replaced by J a or − J a . The restrictions on the rapidities are λ b (cid:54) = λ a , − λ a ; ∀ i, µ a (cid:54) = λ i .(iv) The Bethe numbers of ννν are those of ¯ µ ¯ µ ¯ µ with only I b or − I b ( b (cid:54) = a ) replaced by I a or − I a . The restrictions on the rapidities are λ b (cid:54) = λ a , − λ a ; ∀ i, µ a (cid:54) = λ i . Cases (iii) and(iv) are sketched in Figure 7. | ¯ λ ¯ λ ¯ λ (cid:105) . . . − I a . . . I a . . I b . . | ννν (cid:105) . . . − I a . . . I a . . J a . . | ¯ µ ¯ µ ¯ µ (cid:105) . . . − J a . . . J a . . I b . .Figure 7: Cases (iii) and (iv) of type II excitations.Case (i) can be accounted for by always changing λ a for ν a , but allowing λ a to range between −∞ and ∞ . One can also allow µ a to range between −∞ and ∞ by introducing a combi-natorial factor . In case (ii) the same holds true with λ a and µ a interchanged. Case (iii)can be accounted for by always changing λ b for µ a , but allowing both λ b and µ a to rangebetween −∞ and ∞ . One can also allow λ a to range between −∞ and ∞ by introducing acombinatorial factor . In case (iv) the same holds true with λ a and µ a interchanged.In cases (i) and (ii) the product of all the signs appearing in (89) and (90) give a factor − sgn ( λ a µ a ). In cases (iii) and (iv) they give a factor sgn ( λ a µ a ). It follows that in these four23 ciPost Physics Submission cases we have (cid:104) Ψ BEC | ¯ µ ¯ µ ¯ µ (cid:105)(cid:104) ¯ λ ¯ λ ¯ λ | σ (0) | ννν (cid:105)(cid:104) ννν | σ (0) | ¯ µ ¯ µ ¯ µ (cid:105)(cid:104) Ψ BEC | ¯ λ ¯ λ ¯ λ (cid:105)(cid:104) ¯ µ ¯ µ ¯ µ | ¯ µ ¯ µ ¯ µ (cid:105)(cid:104) ννν | ννν (cid:105) e it ( E ( λλλ ) − E ( µµµ ))+ ix (cid:48) P ( ννν ) = − cL ( ν a − λ a ) λ a ( ν a + λ a )( ν a − µ a )( λ a − µ a ) e it ( λ a − µ a ) e ix (cid:48) ( ν a − λ a ) case (i) − cL ( ν a − µ a ) λ a ( ν a + µ a )( ν a − λ a )( µ a − λ a ) µ a e it ( λ a − µ a ) e ix (cid:48) ( ν a − µ a ) case (ii) cL ( λ b − µ a ) λ a ( µ a + λ b )( λ b − λ a )( µ a − λ a ) µ a e it ( λ a − µ a ) e ix (cid:48) ( µ a − λ b ) case (iii) cL ( λ b − λ a ) λ a ( λ b + λ a )( λ b − µ a )( λ a − µ a ) e it ( λ a − µ a ) e ix (cid:48) ( λ a − λ b ) case (iv) + O ( c − ) . (113)In order to proceed it is convenient to decompose the rational functions in (113) using( ν a − λ a ) λ a ( ν a + λ a )( ν a − µ a )( λ a − µ a ) = ν a − λ a λ a + µ a + λ a − ν a µ a − λ a + λ a ( ν a − λ a ) ν a ( µ a − ν a ) + λ a ( λ a − ν a ) ν a ( ν a + µ a ) , ( ν a − µ a ) λ a ( ν a + µ a )( ν a − λ a )( µ a − λ a ) µ a = ν a − µ a µ a + λ a + µ a − ν a λ a − µ a + ν a ( ν a − µ a ) µ a ( λ a − ν a ) + ν a ( µ a − ν a ) µ a ( λ a + ν a ) , ( λ b − µ a ) λ a ( µ a + λ b )( λ b − λ a )( µ a − λ a ) µ a = µ a − λ b λ a − µ a + λ b − µ a λ a + µ a + λ b ( λ b − µ a ) µ a ( λ a − λ b ) + λ b ( µ a − λ b ) µ a ( λ a + λ b ) , ( λ b − λ a ) λ a ( λ b + λ a )( λ b − µ a )( λ a − µ a ) = λ a − λ b µ a − λ a + λ b − λ a λ a + µ a + λ a ( λ b − λ a ) λ b ( µ a − λ b ) + λ a ( λ a − λ b ) λ b ( λ b + µ a ) . (114)Using (113) and (114) we can express the sum of all type-II contributions to (85) in the form C , λ ¯ λ ¯ λ ( x, t ) = 2 c (cid:104) − Σ (cid:48) ( x (cid:48) , t ) − Σ (cid:48) ( x (cid:48) , t ) + Σ (cid:48) ( x (cid:48) , t ) + Σ (cid:48) ( x (cid:48) , t ) (cid:105) , (115)whereΣ (cid:48) ( x (cid:48) , t ) = 1 L (cid:88) λ a ∈ Λ (cid:88) µ a µ a / ∈ Λ (cid:88) ν a ν a / ∈ Λ ν a (cid:54) = µ a , − µ a (cid:20) ν a − λ a λ a + µ a + λ a ( ν a − λ a ) ν a ( µ a − ν a ) (cid:21) e it ( λ a − µ a ) e ix (cid:48) ( ν a − λ a ) , Σ (cid:48) ( x (cid:48) , t ) = 1 L (cid:88) λ a ∈ Λ (cid:88) µ a µ a / ∈ Λ (cid:88) ν a ν a / ∈ Λ ν a (cid:54) = µ a , − µ a (cid:20) ν a − µ a µ a + λ a + ν a ( ν a − µ a ) µ a ( λ a − ν a ) (cid:21) e it ( λ a − µ a ) e ix (cid:48) ( ν a − µ a ) , Σ (cid:48) ( x (cid:48) , t ) = 1 L (cid:88) λ a ∈ Λ (cid:88) µ a µ a / ∈ Λ (cid:88) λ b ∈ Λ λ b (cid:54) = λ a , − λ a (cid:20) µ a − λ b λ a − µ a + λ b ( λ b − µ a ) µ a ( λ a − λ b ) (cid:21) e it ( λ a − µ a ) e ix (cid:48) ( µ a − λ b ) , Σ (cid:48) ( x (cid:48) , t ) = 1 L (cid:88) λ a ∈ Λ (cid:88) µ a µ a / ∈ Λ (cid:88) λ b ∈ Λ λ b (cid:54) = λ a , − λ a (cid:20) λ a − λ b µ a − λ a + λ a ( λ b − λ a ) λ b ( µ a − λ b ) (cid:21) e it ( λ a − µ a ) e ix (cid:48) ( λ a − λ b ) . (116)In all four contributions Σ (cid:48) j ( x (cid:48) , t ) the respective first term only involves simple poles andtherefore can be straightforwardly expressed in terms of principal value integrals in the ther-modynamic limit. The other terms involve two simple poles and require a more elaboratetreatment. 24 ciPost Physics Submission5.4.1 First term Σ (cid:48) ( x (cid:48) , t )The contribution to Σ (cid:48) ( x (cid:48) , t ) involving two simple poles is of the form S L [ f ] = 1 L (cid:88) λ a ∈ Λ (cid:88) µ a / ∈ Λ (cid:88) ν a / ∈ Λ ν a (cid:54) = ± µ a f ( λ a , µ a , ν a ) ν a ( µ a − ν a ) , (117)where f ( λ, µ, ν ) = λ ( ν − λ ) e it ( λ − µ )+ ix (cid:48) ( ν − λ ) . (118)Resolving all the constraints, we have at leading order in 1 /cS L [ f ] = 1 L (cid:88) λ a ∈ Λ (cid:88) m (cid:88) n (cid:54) =0 f ( λ a , λ a + πmL , λ a + π ( n + m ) L )( λ a + π ( n + m ) L )( − πnL ) − L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) m (cid:54) =0 f ( λ a , λ i + πmL , λ i ) λ i πmL − L (cid:88) λ a ∈ Λ (cid:88) λ j ∈ Λ (cid:88) n (cid:54) =0 f ( λ a , λ j , λ j + πnL )( λ j + πnL )( − πnL )+ 1 L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:54) = λ i f ( λ a , λ j , λ i ) λ i ( λ j − λ i )+ 12 L (cid:88) λ a ∈ Λ (cid:88) n (cid:54) =0 f ( λ a , π ( n +1 / L , − π ( n +1 / L )( π ( n +1 / L ) − L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ f ( λ a , λ i , − λ i ) λ i . (119)The first two contributions can be computed by first summing over m , and then summing over n and λ i respectively, which involves one-dimensional sums with only a single simple pole. Inthe thermodynamic limit they can be readily turned into principal value integrals. The fifthand sixth terms are double sums with a factor L − and hence are completely dominated bytheir respective double poles. They yield12 L (cid:88) λ a ∈ Λ (cid:88) n (cid:54) =0 f ( λ a , π ( n +1 / L , − π ( n +1 / L )( π ( n +1 / L ) − L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ f ( λ a , λ i , − λ i ) λ i = (cid:20) − Ω( λλλ )2 (cid:21) (cid:90) ∞−∞ f ( λ, , ρ ( λ )d λ + O ( L − ) . (120)The third term is of a very similar structure to Lemma 2 (103) and can be treated analogously.We write1 L (cid:88) λ a ∈ Λ (cid:88) λ j ∈ Λ (cid:88) n (cid:54) =0 f ( λ a , λ j , λ j + πnL )( λ j + πnL )( − πnL ) = 1 L (cid:88) λ a ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:88) n f ( λ a , λ j , λ j + πnL ) λ j + πnL − L (cid:88) λ a ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:88) n (cid:54) =0 f ( λ a , λ j , λ j + πnL ) πnL − L (cid:88) λ a ∈ Λ (cid:88) λ j ∈ Λ f ( λ a , λ j , λ j ) λ j . (121)In the thermodynamic limit this becomes1 L (cid:88) λ a ∈ Λ (cid:88) λ j ∈ Λ (cid:88) n (cid:54) =0 f ( λ a , λ j , λ j + πnL )( λ j + πnL )( − πnL ) = (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µ ρ ( µ ) µ − (cid:90) d ν f ( λ, µ, ν )2 πν − (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µ ρ ( µ ) µ − (cid:90) d ν f ( λ, µ, ν )2 π ( ν − µ ) − Ω( λλλ ) (cid:90) ∞−∞ f ( λ, , ρ ( λ )d λ . (122)25 ciPost Physics Submission The two principal values can be brought under a single principal value as in (163), cf. Appendix A.2. Finally the fourth term in (119) can be calculated using Lemma 1 (101).Putting everything together we then obtainΣ (cid:48) ( x (cid:48) , t ) = (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) − (cid:90) d νρ h ( ν ) ν − λλ + µ e it ( λ − µ )+ ix (cid:48) ( ν − λ ) + (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d νρ h ( ν ) − (cid:90) d µρ h ( µ ) λ ( ν − λ ) ν ( µ − ν ) e it ( λ − µ )+ ix (cid:48) ( ν − λ ) + 12 (cid:16) πρ (0) − π ρ (0) − (cid:17) (cid:90) ∞−∞ λ ρ ( λ ) e itλ − ix (cid:48) λ dλ . (123) Σ (cid:48) ( x (cid:48) , t )The contribution to Σ (cid:48) ( x (cid:48) , t ) involving two simple poles is of the form1 L (cid:88) λ a ∈ Λ (cid:88) µ a / ∈ Λ (cid:88) ν a / ∈ Λ ν a (cid:54) = ± µ a f ( λ a , µ a , ν a ) µ a ( λ a − ν a ) = 1 L (cid:88) λ a ∈ Λ (cid:88) m (cid:88) n (cid:54) =0 f ( λ a , π ( m +1 / L , λ a + πnL ) (cid:0) − π ( m +1 / L (cid:1) πnL − L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) n (cid:54) =0 f ( λ a , λ i , λ a + πnL )( − λ i ) πnL − L (cid:88) λ a ∈ Λ (cid:88) m (cid:88) λ j ∈ Λ λ j (cid:54) = λ a f ( λ a , π ( m +1 / L , λ j ) π ( m +1 / L ( λ a − λ j )+ 1 L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:54) = λ a f ( λ a , λ i , λ j ) λ i ( λ a − λ j ) − L (cid:88) λ a ∈ Λ (cid:88) µ a / ∈ Λ (cid:104) f ( λ a , µ a , µ a ) µ a ( λ a − µ a ) + f ( λ a , µ a , − µ a ) µ a ( λ a + µ a ) (cid:105) , (124)where f ( λ, µ, ν ) = ν ( ν − µ ) e it ( λ − µ )+ ix (cid:48) ( ν − µ ) . (125)The first terms on the right-hand side can all be computed by performing successive one-dimensional sums with only a single simple pole, which allows them to be turned into principalvalue integrals in the thermodynamic limit. The last term involves a two-dimensional sumwith a factor L − and a summand featuring only simple poles. Hence it vanishes in thethermodynamic limit. We conclude thatΣ (cid:48) ( x (cid:48) , t ) = (cid:90) ∞−∞ d λρ ( λ ) (cid:90) ∞−∞ d µρ h ( µ ) − (cid:90) d νρ h ( ν ) ν − µµ + λ e it ( λ − µ )+ ix (cid:48) ( ν − µ ) + (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) − (cid:90) d νρ h ( ν ) ν ( ν − µ ) µ ( λ − ν ) e it ( λ − µ )+ ix (cid:48) ( ν − µ ) + O ( L − ) . (126) Σ (cid:48) ( x (cid:48) , t )This contribution is straightforward to deal with. After writing the sum over µ a as thedifference of a sum over vacancies and holes the sums over λ a,b can be factorized and willinvolve only single simple poles. It then follows thatΣ (cid:48) ( x (cid:48) , t ) = (cid:90) ∞−∞ d λρ ( λ ) (cid:90) ∞−∞ d µρ h ( µ ) − (cid:90) d νρ ( ν ) µ − νλ − µ e it ( λ − µ )+ ix (cid:48) ( µ − ν ) + (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) − (cid:90) d νρ ( ν ) ν ( ν − µ ) µ ( λ − ν ) e it ( λ − µ )+ ix (cid:48) ( µ − ν ) + O ( L − ) . (127)26 ciPost Physics Submission5.4.4 Fourth term Σ (cid:48) ( x (cid:48) , t )The contribution to Σ (cid:48) ( x (cid:48) , t ) involving two simple poles is of the form1 L (cid:88) λ a ∈ Λ (cid:88) µ a / ∈ Λ (cid:88) λ b ∈ Λ f ( λ a , µ a , λ b ) λ b ( µ a − λ b ) = 1 L (cid:88) λ a ∈ Λ (cid:88) n (cid:54) =0 (cid:88) λ b ∈ Λ f ( λ a , λ b + πnL , λ b ) λ b πnL − L (cid:88) λ a ∈ Λ (cid:88) λ b ∈ Λ (cid:88) λ i ∈ Λ λ i (cid:54) = λ b f ( λ a , λ i , λ b ) λ b ( λ i − λ b ) , (128)where f ( λ, µ, ν ) = λ ( ν − λ ) e it ( λ − µ )+ ix (cid:48) ( λ − ν ) . (129)The first sum can be straightforwardly turned into a principal value integral and the secondsum can be carried out using Lemma 1 (101). This givesΣ (cid:48) ( x (cid:48) , t ) = (cid:90) ∞−∞ d λρ ( λ ) (cid:90) ∞−∞ d µρ h ( µ ) − (cid:90) d νρ ( ν ) λ − νµ − λ e it ( λ − µ )+ ix (cid:48) ( λ − ν ) + (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) − (cid:90) d νρ ( ν ) λ ( ν − λ ) ν ( µ − ν ) e it ( λ − µ )+ ix (cid:48) ( λ − ν ) + π ρ (0) − Ω( λλλ )2 (cid:90) ∞−∞ λ ρ ( λ ) e itλ + ix (cid:48) λ d λ . (130) The combined contribution of all Σ (cid:48) n ( x (cid:48) , t ) can be brought into a simpler form by using that(i) the root distribution is even; (ii) at leading order in 1 /c we can write ρ ( λ ) + ρ h ( λ ) = 12 π + O ( c − ) , (131)and (iii) for x (cid:54) = 0 we have in a distribution sense (cid:90) ∞−∞ e ixν d ν = 0 , (cid:90) ∞−∞ νe ixν d ν = 0 , (cid:90) ∞−∞ e ixν ν d ν = iπ sgn ( x ) . (132)This allows us to combine the contributions of the terms in Σ (cid:48) n ( x (cid:48) , t ) involving only a singlesimple pole into the following expression2 (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) − (cid:90) d νρ ( ν ) ν − λλ + µ cos( x (cid:48) ( ν − λ )) e it ( λ − µ ) +2 (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) − (cid:90) d νρ ( ν ) ν − µλ + µ cos( x (cid:48) ( ν − µ )) e it ( λ − µ ) − sgn ( x ) (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) λ ( λ − µ ) µ sin( x (cid:48) ( λ − µ )) e it ( λ − µ ) . (133)27 ciPost Physics Submission Our final result for the thermodynamic limit of all contributions to (85) arising from type IIconfigurations is then C , λ ¯ λ ¯ λ ( x (cid:48) , t ) = 4 c (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d νρ ( ν ) − (cid:90) d µρ h ( µ ) F ( λ, µ, ν ; x (cid:48) ) cos (cid:0) t ( λ − µ ) (cid:1) − x (cid:48) ) c (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µρ h ( µ ) λ ( λ − µ ) µ sin( x ( λ − µ )) e it ( λ − µ ) + 1 c (cid:18) − πρ (0) + 2 π ρ (0) − Ω( λλλ ) (cid:19) (cid:90) ∞−∞ λ ρ ( λ ) e itλ − ix (cid:48) λ d λ , (134)where F ( λ, µ, ν ; x (cid:48) ) is the function defined in (24).We stress that C , λ ¯ λ ¯ λ ( x, t ) depends on the representative state λλλ not only through the rootdensity ρ , but via the quantity Ω( λλλ ) (102) as well. Once both contributions (112) and (134) to the spectral sum are summed up, we observethat the dependence on the representative state through the quantity Ω( λλλ ) exactly vanishes!This non-trivial cancellation suggests that the typicality assumption underlying (37) is indeedcorrect, even though the partial contributions do carry an additional dependence on the chosenrepresentative state.To arrive at the expression (22) written in the introduction, we sum up (112) and (134)and use that at leading order in 1 /cρ (0) = 12 π + O ( c − ) . (135) (cid:104) σ ( x, τ ) σ (0 , (cid:105) ∞ in thesteady state We saw in (50) that the expectation value of an observable (cid:104)O(cid:105) t after the quench convergeswhen t → ∞ to (cid:104)O(cid:105) ∞ given in (49). This limit value is thus expressed as an equilibriumexpectation value of O in a representative state corresponding to the steady-state root density ρ that is fixed by the quench protocol. An interesting question is then how to characterizethe physical properties of this steady state through its response functions.The dynamical correlation function of an observable O in an energy eigenstate | λλλ (cid:105) has aspectral representation in a basis of (unnormalized) energy eigenstates | µµµ (cid:105) of the form (cid:104)O ( x, τ ) O (0 , (cid:105) = (cid:88) µµµ |(cid:104) λλλ |O (0) | µµµ (cid:105)| (cid:104) λλλ | λλλ (cid:105) (cid:104) µµµ | µµµ (cid:105) e iτ ( E ( λλλ ) − E ( µµµ ))+ ix ( P ( µµµ ) − P ( λλλ )) . (136)We have previously considered the case where O ( x ) = σ ( x ) in [63]. This case is quite special as σ ( x ) is the density of a conserved charge. In the following we consider the case O ( x ) = σ ( x ).An expression for the form factors of this operator between two states of equal momenta waspresented previously in (47). To determine the dynamical two-point function we require form28 ciPost Physics Submission factors between states with different momenta as well, which can be expressed in the form [78] (cid:104) µµµ | σ (0) | λλλ (cid:105) (cid:112) (cid:104) λλλ | λλλ (cid:105) (cid:104) µµµ | µµµ (cid:105) = i c J ( λλλ, µµµ )( P ( λλλ ) − P ( µµµ )) (cid:104) µµµ | σ (0) | λλλ (cid:105) (cid:112) (cid:104) λλλ | λλλ (cid:105) (cid:104) µµµ | µµµ (cid:105) , (137)where the density form factor given in (87) and J ( λλλ, µµµ ) = ( P ( λλλ ) − P ( µµµ )) − P ( λλλ ) − P ( µµµ ))( Q ( λλλ ) − Q ( µµµ )) + 3( E ( λλλ ) − E ( µµµ )) . (138) /c expansion and particle-hole excitations Let us again follow the same reasoning as in the previous sections, and investigate the leadingbehaviour of the form factor when c → ∞ , for generic λλλ, µµµ satisfying the Bethe equations. Thesimple relation (137) allows us to directly use the results of [63] for the density correlations.Denoting ν the number of Bethe numbers of µµµ that do not appear among those of λλλ , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) µµµ | σ (0) | λλλ (cid:105) (cid:112) (cid:104) λλλ | λλλ (cid:105) (cid:104) µµµ | µµµ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( c − ν ) . (139)Hence the 1 /c expansion is also an expansion in the number of particle-hole excitations. Byrestricting our analysis to O ( c − ), we can focus only on one and two-particle-hole excitations. We now consider a one-particle-hole excitation above λλλ , namely a state µµµ such that all itsBethe numbers are those of λλλ except for I a which is replaced by I a + n . This results inconstraints on the Bethe numbers n (cid:54) = 0 , ∀ i (cid:54) = a, I a + n (cid:54) = I i . (140)We then can use the results of [63] because of the simple relation (137), which always holdssince the momenta between the two states involved are necessarily different. We obtain |(cid:104) λλλ | σ (0) | µµµ (cid:105)| (cid:104) λλλ | λλλ (cid:105)(cid:104) µµµ | µµµ (cid:105) = 16 c L (cid:20) −E − D λ a + 2 λ a P + 2 πnL ( P − D λ a ) (cid:21) + O ( c − ) . (141)Interestingly, the a priori leading order O ( c − ) contribution vanishes. As a result the one andtwo particle-hole excitations contribute at the same order in 1 /c . Since (141) does not havepoles the corresponding contribution to the spectral sum (136) is straightforward to computeand gives the first line of (27). The other class of intermediate states contributing at order O ( c − ) are two particle-holeexcitations, i.e. states with rapidities µµµ such that the corresponding Bethe numbers are thoseof λλλ with the exception of I a and I b which are replaced by I a + n and I b respectively. The29 ciPost Physics Submission Bethe numbers are subject to the following constraints n, m (cid:54) = 0 , ∀ i (cid:54) = a, b, I a + n (cid:54) = I i , I b + m (cid:54) = I i ,I a + n (cid:54) = I b + m ,I a + n (cid:54) = I b ,I b + m (cid:54) = I a . (142)Assuming that the momenta of the two states are different, i.e. that n (cid:54) = − m , one can againuse (137) and [63] to obtain |(cid:104) λλλ | σ (0) | µµµ (cid:105)| (cid:104) λλλ | λλλ (cid:105)(cid:104) µµµ | µµµ (cid:105) = 16 c L ( λ a − λ b ) ( λ a − λ b + 2 π ( n − m ) L ) + O ( c − ) . (143)This expression has no singularities and the corresponding contribution to the spectral sum isstraightforwardly expressed as an integral in the thermodynamic limit. This gives the secondline of (27).When n = − m , i.e. when the momenta of the two states are identical, we obtain from(79) that |(cid:104) λλλ | σ (0) | µµµ (cid:105)| (cid:104) λλλ | λλλ (cid:105)(cid:104) µµµ | µµµ (cid:105) = O ( L − ) , (144)and that there are no singularities in n . As in this case there are only three sums we concludethat such contributions vanish in the thermodynamic limit. In this work we have combined the Quench Action approach with our recently developed1 /c -expansion method for form factor sums in the Lieb-Liniger model to analyze a number ofdifferent observables after a quantum quench starting in the ground state of a non-interactingBose gas. To the best of our knowledge our work is the first to obtain analytic results forquench dynamics in an interacting integrable theory beyond the asymptotic late-time regime.Our work also uncovered a novel aspect regarding the application of typicality ideas to theanalysis of quantum quenches in integrable models. We observed that carrying out partialsummations of the spectral sums in the Quench Action approach can lead to results thatviolate the underlying typicality assumption and depend on details of the particular represen-tative state selected. In the case at hand this dependence arises from the singular behaviourof overlaps at zero rapidity. But remarkably, we observe that this representative-state depen-dence cancels out between different types of particle-hole excitations at the order in 1 /c of ourcalculation, yielding a significant check of typicality in an out-of-equilibrium setting. However,we are able to construct ad hoc initial states in a free theory for which these cancellationsdo not occur. This results in a failure of typicality, but this failure is weak in the sense thatthe problematic representative states are rare and can be avoided through a regularizationprocedure. A brief discussion of these findings is given in Appendix E.Our work raises a number of interesting questions that should be investigated further.First, it is important to work out higher orders in the 1 /c -expansion both for dynamicalresponse functions and in the quench context. In particular, conjectured extensions of GHD30 ciPost Physics Submission predict that the two-point functions of σ ( x ) will exhibit diffusive behaviour [70]. This is notseen in the leading order of the 1 /c -expansion worked out here, but supposedly will appear atthe next order. Second, it should be explored how to define truncations of the spectral sumthat would be finite in the thermodynamic limit (not divergent and not exponentially small)for finite c . Indeed, the spectral sum truncation induced by the 1 /c expansion genericallyexhibits terms polynomial in the system size that cross-cancel between different numbers ofparticle-hole excitations. Third, it would be very interesting to apply our strong couplingexpansion method to dynamical correlations in other models like the Heisenberg XXZ chain[79–82]. These typically will involve bound states, and an important question is how to extendthe strong coupling expansion in order to take their contributions into account. Fourth, itwould be interesting to extend the analysis presented above to quantum quenches startingin inhomogeneous initial states [83, 84]. Finally, we think it is important to arrive at amore complete understanding of the scope and limitations for applying typicality ideas to thecalculation of dynamical correlations in and out of equilibrium. Acknowledgements We are grateful to Jacopo de Nardis and Karol Kozlowski for helpfuldiscussions and comments. This work was supported by the EPSRC under grant EP/S020527/1. A Principal value integrals In this appendix we present details on principal value integrals used in the main text and theproofs of Lemma 1 and 2. A.1 Double principal values Given a function F ( λ, µ, ν ), we define its integral with successive double principal value as − (cid:90) F ( λ, µ, ν )( λ − µ )( µ − ν ) d λ d µ d ν = (cid:90) d µ − (cid:90) d ν µ − ν − (cid:90) d λ F ( λ, µ, ν ) λ − µ , (145)where the − (cid:82) symbols appearing in the right-hand side of this expression denote single principalvalues defined in (25). As shown in [63], the following relations hold − (cid:90) F ( λ, µ, ν )( λ − µ )( µ − ν ) d λ d µ d ν = (cid:90) d ν − (cid:90) d µ µ − ν − (cid:90) d λ F ( λ, µ, ν ) λ − µ = (cid:90) d λ − (cid:90) d µ λ − µ − (cid:90) d ν F ( λ, µ, ν ) µ − ν = (cid:90) d µ − (cid:90) d λ λ − µ − (cid:90) d ν F ( λ, µ, ν ) µ − ν , (146)and − (cid:90) F ( λ, µ, ν )( λ − µ )( µ − ν ) d λ d µ d ν = lim (cid:15),(cid:15) (cid:48) → (cid:90) | λ − µ | >(cid:15) | µ − ν | >(cid:15) (cid:48) F ( λ, µ, ν )( λ − µ )( µ − ν ) d λ d µ d ν . (147)The integral with simultaneous double principal value is defined by= (cid:90) F ( λ, µ, ν )( λ − µ )( µ − ν ) d λ d µ d ν = lim (cid:15) → (cid:90) | λ − µ | >(cid:15) | µ − ν | >(cid:15) | λ − ν | >(cid:15) F ( λ, µ, ν )( λ − µ )( µ − ν ) d λ d µ d ν . (148)31 ciPost Physics Submission As shown in [63], it is related to the integral with successive double principal value throughthe Poincar´e-Bertrand-like formula= (cid:90) F ( λ, µ, ν )( λ − µ )( µ − ν ) d λ d µ d ν = − (cid:90) F ( λ, µ, ν )( λ − µ )( µ − ν ) d λ d µ d ν + π (cid:90) ∞−∞ F ( λ, λ, λ )d λ . (149) A.2 Proof of equation (163) Using the identity (149) we obtain (cid:90) ∞−∞ d λ − (cid:90) d µ µ − (cid:90) d ν F ( λ, µ, ν ) ν = −− (cid:90) F ( λ, µ − λ, ν − λ )( ν − λ )( λ − µ ) d λ d µ d ν = − = (cid:90) F ( λ, µ − λ, ν − λ )( ν − λ )( λ − µ ) d λ d µ d ν + π (cid:90) ∞−∞ F ( λ, , λ , (150)and (cid:90) ∞−∞ d λ − (cid:90) d µ µ − (cid:90) d ν F ( λ, µ, ν ) ν − µ = − (cid:90) F ( λ, µ − λ, ν − λ )( ν − µ )( µ − λ ) d λ d µ d ν = = (cid:90) F ( λ, µ − λ, ν − λ )( ν − µ )( µ − λ ) d λ d µ d ν − π (cid:90) ∞−∞ F ( λ, , λ . (151)In (163) the sum of these two quantities appears. The latter can be brought under a singlesimultaneous principal value because the excluded regions of the integral are identical (whichis not the case of the successive principal values). Hence (cid:90) ∞−∞ d λ − (cid:90) d µ µ − (cid:90) d ν F ( λ, µ, ν ) ν − (cid:90) ∞−∞ d λ − (cid:90) d µ µ − (cid:90) d ν F ( λ, µ, ν ) ν − µ = = (cid:90) F ( λ, µ − λ, ν − λ )( λ − ν )( ν − µ ) d λ d µ d ν + 2 π (cid:90) ∞−∞ F ( λ, , λ = − (cid:90) F ( λ, µ − λ, ν − λ )( λ − ν )( ν − µ ) d λ d µ d ν + π (cid:90) ∞−∞ F ( λ, , λ . (152)Using (146) we arrive at (163). B Proof of Lemma 1 (101) We start by adding the condition λ j (cid:54) = − λ i L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:54) = λ i f ( λ a , λ j , λ i ) λ i ( λ j − λ i ) = 1 L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:54) = λ i , − λ i f ( λ a , λ j , λ i ) λ i ( λ j − λ i ) − L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ f ( λ a , − λ i , λ i ) λ i . (153)32 ciPost Physics Submission The second sum is two-dimensional and comes with a factor L − . Hence it is dominated bythe double pole and its thermodynamic limit reads12 L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ f ( λ a , − λ i , λ i ) λ i = Ω( λλλ )2 (cid:90) ∞−∞ f ( λ, , ρ ( λ )d λ + O ( L − ) . (154)To compute the first term on the right-hand side in (153) we symmetrize in i, j and ± λ i,j using the pair structure of the state1 L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:54) = λ i , − λ i f ( λ a , λ j , λ i ) λ i ( λ j − λ i ) = 18 L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:54) = λ i , − λ i G ( λ a , λ j , λ i ) , (155)Here we have defined G ( λ a , λ j , λ i ) = g ( λ a , λ j , λ i ) − g ( λ a , − λ j , λ i ) − g ( λ a , λ j , − λ i ) + g ( λ a , − λ j , − λ i ) λ i λ j ,g ( λ a , λ j , λ i ) = λ j f ( λ a , λ j , λ i ) − λ i f ( λ a , λ i , λ j ) λ j − λ i . (156)The right-hand side in (155) is a Riemann sum of a regular function without singularities,hence converges to an integral in the thermodynamic limit1 L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:54) = ± λ i f ( λ a , λ j , λ i ) λ i ( λ j − λ i ) = 18 (cid:90) (cid:90) (cid:90) ∞−∞ G ( x, y, z ) ρ ( x ) ρ ( y ) ρ ( z )d x d y d z + O ( L − ) . (157)To proceed, we remove from the integration region the points where | y | < (cid:15) or | z | < (cid:15) . Thisincurs only an error O ( (cid:15) ) since the integrand is regular and allows us to split the integral intofour. We then replace y and z by y − x and z − x and use (147) to obtain1 L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:54) = ± λ i f ( λ a , λ j , λ i ) λ i ( λ j − λ i ) = − − (cid:90) g ( x, y − x, z − x ) ρ ( x ) ρ ( y − x ) ρ ( z − x )( y − x )( x − z ) d x d y d z . (158)Under the successive principal value we cannot use the definition of g in terms of f and splitthe integral into two since we do not necessarily have | z − y | > (cid:15) . However, we can use (149)to obtain an expression in terms of a simultaneous principal value integral1 L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:54) = ± λ i f ( λ a , λ j , λ i ) λ i ( λ j − λ i ) = − 12 = (cid:90) g ( x, y − x, z − x ) ρ ( x ) ρ ( y − x ) ρ ( z − x )( y − x )( x − z ) d x d y d z + π ρ (0) (cid:90) ∞−∞ f ( x, , ρ ( x )d x . (159)We now express g in terms of f , split the integral and swap the variables y, z in one of thetwo resulting integrals to obtain1 L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ (cid:88) λ j ∈ Λ λ j (cid:54) = ± λ i f ( λ a , λ j , λ i ) λ i ( λ j − λ i ) = = (cid:90) f ( x, y − x, z − x ) ρ ( x ) ρ ( y − x ) ρ ( z − x )( y − z )( z − x ) d x d y d z + π ρ (0) (cid:90) ∞−∞ f ( x, , ρ ( x )d x . (160)33 ciPost Physics Submission Finally we employ (149) to arrive at Eq (101). C Proof of Lemma 2 (103) We start by rewriting the multiple sum of interest as1 L (cid:88) λ a ∈ Λ (cid:88) n (cid:54) =0 (cid:88) λ i ∈ Λ f ( λ a , λ i + πnL , λ i )( λ i + πnL )( − πnL ) = − L (cid:88) λ a ∈ Λ (cid:88) n (cid:54) =0 (cid:88) λ i ∈ Λ f ( λ a , λ i + πnL , λ i ) λ i πnL + 1 L (cid:88) λ a ∈ Λ (cid:88) n (cid:88) λ i ∈ Λ f ( λ a , λ i + πnL , λ i )( λ i + πnL ) λ i − L (cid:88) λ a ∈ Λ (cid:88) λ i ∈ Λ f ( λ a , λ i , λ i ) λ i . (161)The first and second terms on the right-hand side can be turned into principal part integrals inthe thermodynamic limit by first summing over n and then over λ i . The third sum, althoughtwo-dimensional with a prefactor 1 /L , is not negligible in the thermodynamic limit since itinvolves a double pole in λ i . Its thermodynamic in fact depends on the representative state λλλ through the quantity Ω( λλλ ) defined in (102).1 L (cid:88) λ a ∈ Λ (cid:88) n (cid:54) =0 (cid:88) λ i ∈ Λ f ( λ a , λ i + πnL , λ i )( λ i + πnL )( − πnL ) = − π (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d ν ρ ( ν ) ν − (cid:90) d µ µ − ν f ( λ, µ, ν )+ 12 π (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d ν ρ ( ν ) ν − (cid:90) d µ µ f ( λ, µ, ν ) − Ω( λλλ ) (cid:90) ∞−∞ ρ ( λ ) f ( λ, , λ . (162)The two principal values can be brought under a single principal value according to thefollowing relation, proved in Appendix A.2 (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µ ρ ( µ ) µ − (cid:90) d ν f ( λ, µ, ν ) ν − (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d µ ρ ( µ ) µ − (cid:90) d ν f ( λ, µ, ν ) ν − µ = (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d ν − (cid:90) d µρ ( µ ) f ( λ, µ, ν ) ν ( µ − ν ) + π ρ (0) (cid:90) ∞−∞ f ( λ, , ρ ( λ )d λ . (163)This gives the desired result1 L (cid:88) λ a ∈ Λ (cid:88) n (cid:54) =0 (cid:88) λ i ∈ Λ f ( λ a , λ i + πnL , λ i )( λ i + πnL )( − πnL ) = 12 π (cid:90) ∞−∞ d λρ ( λ ) − (cid:90) d νρ ( ν ) − (cid:90) d µ f ( λ, µ, ν ) µ ( ν − µ )+ 12 π ρ (0) π (cid:90) ∞−∞ f ( λ, , ρ ( λ )d λ − Ω( λλλ ) (cid:90) ∞−∞ ρ ( λ ) f ( λ, , λ . (164) D Further results on (cid:104) σ ( x ) σ (0) (cid:105) In this appendix we collect a number of additional results on the two-point function after ourinteraction quench (22). D.1 Alternative expression for (22) In this section we present an alternative expression for the two-point function after the quench(22), that is particularly useful for numerical purposes. It is based on the observation that34 ciPost Physics Submission the first terms in the 1 /c -expansion of the steady state root density (15) take the simple form ρ s ( λ ) = 1 + D c π (cid:32) λ D (cid:104) 1+ 2 D c (cid:105) (cid:33) + 1 + O ( c − ) , (165)which allows one to carry out some of the integrals in (22). To that end we introduce I x,t [ f ( λ )] = (cid:90) ∞−∞ e − ixλ +2 itλ f ( λ )d λ . (166)We then find (cid:104) σ ( x ) σ (0) (cid:105) t −(cid:104) σ ( x ) σ (0) (cid:105) ∞ = (1+ D c ) (cid:18) D π (cid:19) F (¯ x, ¯ t )+ 4 πc (1+ D c ) (cid:18) D π (cid:19) Re F (¯ x, ¯ t ) , (167)where we have defined ¯ x = 2 D (1 + D c ) x , ¯ t = [2 D (1 + D c )] t , F ( x, t ) = (cid:12)(cid:12)(cid:12) I x,t [ λ λ ] (cid:12)(cid:12)(cid:12) ,F ( x, t ) =2 (cid:16) I x,t [ λ λ ] I x,t [ λ (1+ λ ) ] ∗ − I x,t [ λ λ ] I x,t [ λ (1+ λ ) ] ∗ (cid:17) + i sgn ( x ) (cid:16) I x,t [ λ λ ] I x,t [ λ λ ] ∗ − I x,t [ λ λ ] I x,t [ λ λ ] ∗ (cid:17) + 2 i sgn ( x ) (cid:16) I x,t [ λ λ ] I x,t [ λ (1+ λ ) ] ∗ − I x,t [ λ λ ] I x,t [ λ (1+ λ ) ] ∗ (cid:17) + 2 e −| x | (cid:16) I x,t [ λ λ ] I ,t [ λ ) ] ∗ − I x,t [ λ (1+ λ ) ] I ,t [ λ ] ∗ (cid:17) + 2 i sgn ( x ) e −| x | (cid:16) I x,t [ λ (1+ λ ) ] I ,t [ λ ] ∗ − I x,t [ λ λ ] I ,t [ λ ) ] ∗ (cid:17) − e −| x | I x,t [ λ (1+ λ ) H t ( λ )] + 2 i sgn ( x ) e −| x | I x,t [ λ (1+ λ ) H t ( λ )] , (168)and H t ( λ ) = − (cid:90) e − itµ λ − µ d µ . (169) D.2 Consistency check I: t → limit Since the expression (22) for the two-point function (cid:104) σ ( x ) σ (0) (cid:105) t holds for all t > t → O ( c − ) result for the correspondingcorrelation function within the BEC state. The latter are simple (cid:104) Ψ BEC | σ ( x ) σ (0) | Ψ BEC (cid:105) = D . (170)In order to investigate the t → O ( c − )for its infinite time limit (cid:104) σ ( x ) σ (0) (cid:105) ∞ . Using [63] we find (cid:104) σ ( x ) σ (0) (cid:105) ∞ = D − D e − D (1+ 2 D c ) | x | (cid:18) D c | x | (cid:19) + O ( c − ) . (171)35 ciPost Physics Submission At t = 0, all integrals appearing in (168) can be carried out explicitly by noting that (cid:90) ∞−∞ e ixλ (1 + λ ) d λ = π | x | ) e −| x | . (172)The integrals in (168) can be deduced by differentiating this with respect to x . A straight-forward calculation then shows that at t = 0 we indeed recover the two-point function in theBEC state at order O ( c − ) (cid:104) σ ( x ) σ (0) (cid:105) t =0 = D + O ( c − ) . (173) D.3 Consistency check II: x → limit Since for x (cid:54) = 0 we have σ ( x ) σ (0) = ψ † ( x ) ψ † (0) ψ ( x ) ψ (0) , (174)the correlation function (cid:104) σ ( x ) σ (0) (cid:105) t should approach (cid:104) σ (0) (cid:105) t = O ( c − ) in the x → x = 0 and simplifying (22) by exploiting that the root density ρ ( λ ) is even wefind after some calculations that indeedlim x → (cid:104) σ ( x ) σ (0) (cid:105) t = O ( c − ) . (175) D.4 Some remarks on the limit x, t → As we have noted in the main text the limit t → (cid:104) σ (0) (cid:105) t does not recoverthe correct result for the expectation value of σ in the BEC initial state, D . On the otherhand, we have just shown that the limit x → (cid:104) σ ( x ) σ (0) (cid:105) t =0 does reduce to D . On atechnical level it can be traced back to properties of the integral (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞−∞ λ λ e − ixλ +2 itλ d λ (cid:12)(cid:12)(cid:12)(cid:12) , (176)which vanishes if one first takes the limit x → t → 0, but gives a finite result ifone takes first t → x → E Typicality and Quench Action method In this Appendix we present an ad hoc initial state in a free theory for which the QuenchAction spectral sum for the out-of-equilibrium dynamics is representative state dependent.We consider a simple tight-binding Hamiltonian on a ring H = L (cid:88) j =1 a † j a j +1 + a † j +1 a j − a † j a j , (177)where a † j , a j are fermionic creation and annihilation operators satisfying canonical anticom-mutation relations { a j , a † k } = δ j,k . The Hamiltonian is straightforwardly diagonalized by acanonical transformation to Bogoliubov fermions in momentum space H = − L (cid:88) n =1 sin ( k n / b † k n b k n , (178)36 ciPost Physics Submission where k n = πnL and { b p , b † k } = δ p,k . We denote the Bogoliubov vacuum state by | (cid:105) . We nowconsider a quantum quench where the system is initialized in a Gaussian state parametrizedby a fixed arbitrary function K ( p ) | I (cid:105) = L/ − (cid:89) m =1 (cid:112) K ( k m ) exp i L/ − (cid:88) n =1 K ( k n ) b †− k n b † k n | (cid:105) . (179)For our purposes it is sufficient to focus on the Green’s function G ( n, t ) = (cid:104) I ( t ) | a n +1 a | I ( t ) (cid:105) . (180)Since the model is free G ( n, t ) can be straightforwardly calculated G ( n, t ) = 12 π (cid:90) π − π iK ( k )1 + K ( k ) e it sin ( k/ e ikn d k + O ( L − ) . (181)Let us now try to recover this with the Quench Action approach. The normalized overlaps ofthe initial state with an eigenstate | ¯ λ ¯ λ ¯ λ (cid:105) = (cid:81) k ∈ λλλ b †− k b † k | (cid:105) are (cid:104) ¯ λ ¯ λ ¯ λ | I (cid:105) = (cid:81) k ∈ λλλ iK ( k ) (cid:81) L/ − n =1 (cid:112) K ( k n ) , (182)from which one finds the root density characterizing the non-equilibrium steady state reachedat late times after the quench ρ ( k ) = 12 π K ( k )1 + K ( k ) . (183)The form factor of the operator of interest between two pair states ¯ λ ¯ λ ¯ λ, ¯ µ ¯ µ ¯ µ is (cid:104) ¯ λ ¯ λ ¯ λ | a n +1 a | ¯ µ ¯ µ ¯ µ (cid:105) = (cid:40) e iπkn L if µµµ = λλλ ∪ { k } and k / ∈ λλλ , . (184)Let us now choose a representative pair state λλλ of the root density ρ , and write the QuenchAction spectral sum (cid:104) ¯ λ ¯ λ ¯ λ | ( a n +1 a )( t ) | I (cid:105)(cid:104) ¯ λ ¯ λ ¯ λ | I (cid:105) = (cid:88) µµµ (cid:104) ¯ λ ¯ λ ¯ λ | a n +1 a | ¯ µ ¯ µ ¯ µ (cid:105) (cid:104) ¯ µ ¯ µ ¯ µ | I (cid:105)(cid:104) ¯ λ ¯ λ ¯ λ | I (cid:105) e it ( E ( λλλ ) − E ( µµµ )) = 1 L (cid:88) k / ∈ λλλ iK ( k ) e it sin ( k/ e ikn . (185)If K ( k ) is a regular function of k this sum can be turned into an integral over the density ofholes ρ h ( k ) = 12 π 11 + K ( k ) , (186)and the Quench Action approach precisely recovers the result (181)lim L →∞ (cid:104) ¯ λ ¯ λ ¯ λ | ( a n +1 a )( t ) | I (cid:105)(cid:104) ¯ λ ¯ λ ¯ λ | I (cid:105) = 12 π (cid:90) π − π iK ( k )1 + K ( k ) e it sin ( k/ e ikn d k . (187)37 ciPost Physics Submission So far we have closely followed the discussion in [23]. However, let us now consider thefollowing singular behaviour K ( k ) = 1 k m , (188)with m ≥ | λ (cid:48) λ (cid:48) λ (cid:48) (cid:105) by replacing k ∈ λλλ by k (cid:48) . Byconstruction | λ (cid:48) λ (cid:48) λ (cid:48) (cid:105) is a micro-state that for any choice of k , k (cid:48) corresponds to the macro-statewith particle density ρ in the thermodynamic limit, and in particular the extensive partsof all local conservation laws are the same for | λ (cid:48) λ (cid:48) λ (cid:48) (cid:105) and | λλλ (cid:105) . Let us choose k (cid:48) finite in thethermodynamic limit, and k = O ( L − ). We observe that (cid:104) ¯ λ ¯ λ ¯ λ | ( a n +1 a )( t ) | I (cid:105)(cid:104) ¯ λ ¯ λ ¯ λ | I (cid:105) = (cid:104) ¯ λ (cid:48) ¯ λ (cid:48) ¯ λ (cid:48) | ( a n +1 a )( t ) | I (cid:105)(cid:104) ¯ λ (cid:48) ¯ λ (cid:48) ¯ λ (cid:48) | I (cid:105) + iL (cid:104) K ( k (cid:48) ) e it sin ( k (cid:48) / e ik (cid:48) n − K ( k ) e it sin ( k / e ik n (cid:105) . (189)This shows that the two choices of representative state lead to different results in the ther-modynamic limit, which generally does not even exist as K ( k ) ∝ L m . This shows that forthis particular initial state a naive application of typicality ideas fails.However, a few comments are in order. First, since ρ h ( k ) ∼ k m at small k , the smallesthole in a representative state λλλ is typically of order L − / (2 m +1) , and in this case the additionalterms are negligible indeed. Hence for such ”typical” states, typicality ideas can be applied.This fact is confirmed numerically by observing that when one averages (185) over representa-tive states, one indeed recovers (181). Second, in the problem at hand one can slightly changethe initial state by imposing for example K ( k ) = K ( δ ) for k < δ for a fixed small δ . With this”regularisation” one obtains (187), which is now well-behaved and allows for the limit δ → References [1] A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore, Colloquium: Nonequilibriumdynamics of closed interacting quantum systems , Rev. Mod. Phys. , 863 (2011).[2] F. H. L. Essler and M. Fagotti, Quench dynamics and relaxation in isolated integrablequantum spin chains , J. Stat. Mech. 064002 (2016).[3] L. D’Alessio, Y. Kafri, A. Polkovnikov and M. Rigol, From quantum chaos and eigen-state thermalization to statistical mechanics and thermodynamics , Adv. Phys. (3), 239(2016).[4] C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statisticalmechanics in closed quantum systems , Rep. Prog. Phys. (5), 056001 (2016).[5] P. Calabrese and J. Cardy, Quantum quenches in dimensional conformal fieldtheories , J. Stat. Mech. 064003 (2016).[6] T. Langen, T. Gasenzer and J. Schmiedmayer, Prethermalization and universal dynamicsin near-integrable quantum systems , J. Stat. Mech. 064009,(2016).[7] M. Greiner, O. Mandel, T.W. H¨ansch, and I. Bloch, Collapse and revival of the matterwave field of a Bose-Einstein condensate , Nature , 51-54 (2002).[8] T. Kinoshita, T. Wenger, D. S. Weiss, A quantum Newton’s cradle , Nature , 900(2006). 38 ciPost Physics Submission [9] S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Non-equilibrium coherence dynamics in one-dimensional Bose gases , Nature , 324-327(2007).[10] S. Trotzky Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollw¨ock, J. Eisert, and I. Bloch, Probing the relaxation towards equilibrium in an isolated strongly correlated 1D Bose gas ,Nature Phys. , 325 (2012).[11] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A.Smith, E. Demler, and J. Schmiedmayer, Relaxation Dynamics and Pre-thermalizationin an Isolated Quantum System , Science , 1318 (2012).[12] U. Schneider, L. Hackerm¨uller, J. P. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch, E.Demler, S. Mandt, D. Rasch, and A. Rosch, Fermionic transport and out-of-equilibriumdynamics in a homogeneous Hubbard model with ultracold atoms , Nature Phys. , 213(2012).[13] M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauss, T. Fukuhara, C. Gross,I. Bloch, C. Kollath, and S. Kuhr, Light-cone-like spreading of correlations in a quantummany-body system , Nature , 484 (2012).[14] F. Meinert, M.J. Mark, E. Kirilov, K. Lauber, P. Weinmann, A.J. Daley, and H.-C.N¨agerl, Quantum Quench in an Atomic One-Dimensional Ising Chain , Phys. Rev. Lett. , 053003 (2013).[15] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau, P. Schauß, S. Hild, D. Bellem, U.Schollw¨ock, T. Giamarchi, C. Gross, I. Bloch, and S. Kuhr, Quantum dynamics of amobile spin impurity , Nature Physics , 235 (2013).[16] N. Navon, A.L. Gaunt, R.P. Smith and Z. Hadzibabic, Critical Dynamics of SpontaneousSymmetry Breaking in a Homogeneous Bose gas , Science , 167 (2015).[17] M. Schemmer, I. Bouchoule, B. Doyon and J. Dubail, Generalized HydroDynamics on anAtom Chip , Phys. Rev. Lett. , 090601 (2019).[18] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Relaxation in a Completely IntegrableMany-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly ExcitedStates of 1D Lattice Hard-Core Bosons , Phys. Rev. Lett. , 50405 (2007).[19] P. Calabrese, F.H.L. Essler, and M. Fagotti, Quantum Quench in the Transverse-FieldIsing Chain , Phys. Rev. Lett. , 227203 (2011).[20] P. Calabrese, F.H.L. Essler, and M. Fagotti, Quantum Quench in the Transverse FieldIsing Chain II: Stationary State Properties , J. Stat. Mech. (2012) P07022.[21] B. Pozsgay, The generalized Gibbs ensemble for Heisenberg spin chains , J. Stat. Mech.(2013) P07003.[22] M. Fagotti and F.H.L. Essler, Stationary behaviour of observables after a quantum quenchin the spin-1/2 Heisenberg XXZ chain , J. Stat. Mech. (2013) P07012.[23] J.-S. Caux and F.H.L. Essler, Time Evolution of Local Observables After Quenching toan Integrable Model , Phys. Rev. Lett. , 257203 (2013).[24] M. Kormos, A. Shashi, Y.-Z. Chou, J.-S. Caux and A. Imambekov, Interaction quenchesin the one-dimensional Bose gas , Phys. Rev. B , 205131 (2013).[25] M, Fagotti, M, Collura, F.H.L. Essler, and P. Calabrese, Relaxation after quantumquenches in the spin-1/2 Heisenberg XXZ chain , Phys. Rev. B , 125101 (2014).[26] B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, M. Rigol, and J.-S. Caux, Quench-ing the Anisotropic Heisenberg Chain: Exact Solution and Generalized Gibbs EnsemblePredictions , Phys. Rev. Lett. , 117202 (2014).39 ciPost Physics Submission [27] M. Brockmann, B. Wouters, D. Fioretto, J. De Nardis, R. Vlijm and J.-S. Caux, Quenchaction approach for releasing the N´eel state into the spin-1/2 XXZ chain , Stat. Mech.(2014) P12009.[28] J. De Nardis, B. Wouters, M. Brockmann, and J.-S. Caux, Solution for an interactionquench in the Lieb-Liniger Bose gas , Phys. Rev. A , 033601 (2014).[29] B. Pozsgay, M. Mesty´an, M.A. Werner, M. Kormos, G. Zar´and, and G. Tak´acs, Correla-tions after Quantum Quenches in the XXZ Spin Chain: Failure of the Generalized GibbsEnsemble , Phys. Rev. Lett. , 117203 (2014).[30] B. Pozsgay, Quantum quenches and generalized Gibbs ensemble in a Bethe Ansatz solvablelattice model of interacting bosons , J. Stat. Mech. (2014) P10045.[31] G. Goldstein and N. Andrei, Failure of the GGE hypothesis for integrable models withbound states , Phys. Rev. A , 043625 (2014).[32] M. Mesty´an, B. Pozsgay, G. Tak´acs, and M.A. Werner, Quenching the XXZ spin chain:quench action approach versus generalized Gibbs ensemble , J. Stat. Mech. (2015) P04001.[33] E. Ilievski, M. Medenjak, T. Prosen and L. Zadnik, Quasilocal charges in integrable latticesystems , J. Stat. Mech. 064008, (2016).[34] E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F.H.L. Essler, T. Prosen, CompleteGeneralized Gibbs Ensemble in an interacting Theory , Phys. Rev. Lett. , 157201(2015).[35] E. Ilievski, E. Quinn, J. De Nardis and M. Brockmann, String-charge duality in integrablelattice models , J. Stat. Mech. 063101 (2016).[36] L. Piroli, P. Calabrese, and F.H.L. Essler, Multiparticle Bound-State Formation followinga Quantum Quench to the One-Dimensional Bose Gas with Attractive Interactions , Phys.Rev. Lett. , 070408 (2016).[37] L. Piroli, P. Calabrese, and F.H.L. Essler, Quantum quenches to the attractive one-dimensional Bose gas: exact results SciPost Phys. , 001 (2016).[38] S. Sotiriadis, Memory-preserving equilibration after a quantum quench in a 1d criticalmodel , Phys. Rev. A , 031605 (2016).[39] J. Cardy, Quantum quenches to a critical point in one dimension: some further results ,J. Stat. Mech. (2016) 023103.[40] P. Calabrese and J. Cardy, Time Dependence of Correlation Functions Following a Quan-tum Quench , Phys. Rev. Lett. , 136801 (2006).[41] P. Calabrese and J. Cardy, Quantum quenches in extended systems , J. Stat. Mech. (2007)P06008.[42] P. Calabrese, F.H.L. Essler, and M. Fagotti, Quantum quench in the transverse field Isingchain: I. Time evolution of order parameter correlators . J. Stat. Mech. (2012) P07016.[43] J. De Nardis and J.-S. Caux, Analytical expression for a post-quench time evolution ofthe one-body density matrix of one-dimensional hard-core bosons , J. Stat. Mech. (2014)P12012.[44] M. Kormos, M. Collura and P. Calabrese, Analytic results for a quantum quench fromfree to hard-core one-dimensional bosons , Phys. Rev. A , 013609 (2014).[45] D. Schuricht and F.H.L. Essler, Dynamics in the Ising field theory after a quantumquench , J. Stat. Mech. P04017 (2012).[46] B. Bertini, D. Schuricht, and F.H.L. Essler, Quantum quench in the sine-Gordon model ,J. Stat. Mech. (2014) P10035.[47] A. Cort´es Cubero and D. Schuricht, Quantum quench in the attractive regime of thesine-Gordon model , J. Stat. Mech. 103106 (2017).40 ciPost Physics Submission [48] D. X. Horvath, M. Kormos and G. Takacs, Overlap singularity and time evolution inintegrable quantum field theory , JHEP 08 170 (2018).[49] H. Rieger and F. Igl´oi, Semiclassical theory for quantum quenches in finite transverseIsing chains , Phys. Rev. B , 165117 (2011).[50] S. Evangelisti, Semi-classical theory for quantum quenches in the O(3) non-linear sigmamodel , J. Stat. Mech. (2013) P04003.[51] M. Kormos and G. Zar´and, Quantum quenches in the sine-Gordon model: a semiclassicalapproach , Phys. Rev. E , 062101 (2016).[52] J.-S. Caux, The Quench Action , J. Stat. Mech. , 064006 (2016).[53] J. De Nardis, L. Piroli and J.-S. Caux, Relaxation dynamics of local observables in inte-grable systems , Phys. A: Math. Theor. Finite temperature and quench dynamics in theTransverse Field Ising Model from form factor expansions , SciPost Phys. , 033 (2020).[55] M. Brockmann, J. De Nardis, B. Wouters, and J.-S. Caux, A Gaudin-like determinantfor overlaps of N´eel and XXZ Bethe states , J. Phys. A47 , 145003 (2014).[56] B. Pozsgay, Overlaps between eigenstates of the XXZ spin-1/2 chain and a class of simpleproduct states , J. Stat. Mech. , P06011 (2014).[57] L. Piroli, B. Pozsgay, and E. Vernier, What is an integrable quench? , Nucl. Phys. B 925 ,362 (2017).[58] M. de Leeuw, C. Kristjansen, and S. Mori, AdS/dCFT one-point functions of the SU(3)sector , Phys. Lett. B 763 , 197 (2016).[59] M. Mestyan, B. Bertini, L. Piroli and P. Calabrese Exact solution for the quench dynamicsof a nested integrable system , J. Stat. Mech. (2017).[60] B. Pozsgay, Overlaps with arbitrary two-site states in the XXZ spin chain , J. Stat. Mech.053103 (2018).[61] B. Pozsgay, L. Piroli, and E. Vernier, “Integrable Matrix Product States from boundaryintegrability,” SciPost Phys. , 062 (2019).[62] Y. Jiang and B. Pozsgay, On exact overlaps in integrable spin chains , JHEP 22 (2020).[63] E. Granet and F. H. L. Essler, A systematic /c -expansion of form factor sums fordynamical correlations in the Lieb-Liniger model , SciPost Phys. , 082 (2020).[64] E. H. Lieb and W. Liniger, Exact Analysis of an Interacting Bose Gas. I. The GeneralSolution and the Ground State , Phys. Rev. , 1605 (1963).[65] F.A. Br´ezin, G.P. Pohil and V.M. Finkelberg, The Schr¨odinger equation for a system ofone-dimensional particles with point interactions. , Viest. Mosk. Uni. , 21-28 (1964).[66] V. E. Korepin, N. M. Bogoliubov and A. G. Izergin, Quantum inverse scattering methodand correlation functions , Cambridge University Press (1993).[67] G. Lang, Conjectures about the structure of strong- and weak-coupling expansions of afew ground-state observables in the Lieb-Liniger and Yang-Gaudin models , SciPost Phys. , 055 (2019).[68] E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems ,Commun. Math. Phys. , 251 (1972).[69] B. Bertini, Approximate light cone effects in a non-relativistic quantum field theory aftera local quench , Phys. Rev. B , 075153 (2017).[70] B. Doyon, Exact large-scale correlations in integrable systems out of equilibrium , SciPostPhys. , 054 (2018).[71] A. C. Cassidy, C. W. Clark, and M. Rigol, Generalized Thermalization in an IntegrableLattice System , Phys. Rev. Lett. , 140405 (2011).41 ciPost Physics Submission [72] V. E. Korepin, Calculation of norms of Bethe wave functions , Comm. Math. Phys. ,391 (1982).[73] N.A. Slavnov, Calculation of scalar products of wave functions and form factors in theframework of the algebraic Bethe ansatz , Theor. Math. Phys. , 502 (1989).[74] N.A. Slavnov, Nonequal-time current correlation function in a one-dimensional Bose gas ,Theor. Math. Phys. , 273 (1990).[75] V.E. Korepin and N.A. Slavnov, The form factors in the finite volume. , Int. J. Mod.Phys. B , 2933 (1999).[76] T. Oota, Quantum projectors and local operators in lattice integrable models. , J. Phys.A: Math. Gen. , 441 (2004).[77] K.K. Kozlowski, On form factors of the conjugated field in the non-linear Schr¨odingermodel. , J. Math. Phys. , 083302 (2011).[78] L. Piroli and P. Calabrese, Exact formulas for the form factors of local operators in theLieb-Liniger model , J. Phys. A: Math. Theor. , 454002 (2015).[79] A.J.A. James, W.D. Goetze and F.H.L. Essler, Finite Temperature Dynamical StructureFactor of the Heisenberg-Ising Chain , Phys. Rev. B , 214408 (2009).[80] M. Collura and F.H.L. Essler, How order melts after quantum quenches , Phys. Rev. B , 041110(R) (2020).[81] L. Zadnik and M. Fagotti, The Folded Spin-1/2 XXZ Model: I. Diagonalisation, Jam-ming, and Ground State Properties , arXiv:2009.04995.[82] L. Zadnik, K. Bidzhiev and M. Fagotti, The Folded Spin-1/2 XXZ Model: II. Thermo-dynamics and Hydrodynamics with a Minimal Set of Charges , arXiv:2011.01159.[83] A. de Luca, G. Martelloni and J. Viti,