7 We consider a time dependent g ( t ). For t < 0 we have g ( t ) = g , and we assume the system to be in equilibrium. We then assume that for t > 0, g ( t ) jumps to the value g $^{′}$. The equal-time correlation function of σ z i can be cal- culated exactly by expressing it in terms of the operator n$_{i}$ , i.e. 〈 σ z $_{i}$( t ) σ z $_{j}$( t ) 〉 = 1 − 4 〈 n$_{i}$ ( t ) 〉 + 4 〈 n$_{i}$ ( t ) n$_{j}$ ( t ) 〉 . It can be shown that the average density fermionic density (cor- responding to the z -component of the magnetization) is given by 〈 n$_{i}$ ( t ) 〉 = 〈 n$_{i}$ (0) 〉 + 1 M ∑ k F$_{g,g}$ ′ ( k, t ) (25) F$_{g,g}$ ′ ( k, t ) = (cos(2 ϵ$_{k,g}$ ′ t/ ℏ ) / 2 − 1 / 2)( g ′ − g ) sin $^{2}$k √ g 2 − 2 g cos k + 1( g ′ 2 − 2 g $^{′}$cos k + 1) (26) with and 〈 n$_{i}$ (0) 〉 = 1 2 − 1 2 M ∑ k g − cos k √ g 2 2 g cos k + 1 . (27) the density-density correlation function In turn the 〈 n$_{i}$ ( t ) n$_{j}$ ( t ) 〉 reads FIG. 7: The correlation function 〈 σ z 0 ( t ) σ z r ( t ) 〉 for a quench from g = 3 to g ′ = 1, in (a), and from g = 3 to g ′ = 0 . 5, in (b), as a function of time t , specifically of 2 Jt , and the spatial distance r . 〈 n$_{i}$ ( t ) n$_{j}$ ( t ) 〉 = 1 M 2 ∑ k 1 ,k 2 ( exp( − i ( k$_{1}$ − k$_{2}$ )( r$_{i}$ − r$_{j}$ )) ( ( v 2 k 1 ,g + F$_{g,g}$ ′ ( k$_{1}$, t ))( u 2 k 2 ,g − F$_{g,g}$ ′ ( k$_{2}$, t )) + ( u$_{k}$$_{1}$$_{,g}$v$_{k}$$_{1}$$_{,g}$ + G$_{g,g}$ ′ ( k$_{1}$, t ))( u$_{k}$$_{2}$$_{,g}$v$_{k}$$_{2}$$_{,g}$ + G ∗ g,g ′ ( k$_{2}$, t )) ) + ( v 2 k 1 ,g + F$_{g,g}$ ′ ( k$_{1}$, t ))( v 2 k 2 ,g + F$_{g,g}$ ′ ( k$_{2}$, t )) ) , (28) G$_{g,g}$ ′ ( k, t ) = ( i sin(2 ϵ$_{k,g}$t/ ℏ ) + 1 2 (cos(2 ϵ$_{k,g}$ ′ t/ ℏ ) − 1)( g ′ − cos k ) √ g ′ 2 − 2 g $^{′}$cos k + 1 )( ( g ′ − g ) sin k √ ( g 2 − 2 g cos k + 1)( g ′ 2 − 2 g $^{′}$cos k + 1) ) . (29) where Using these expressions we can easily analyze the quench dynamics. In Fig. 7 we show two examples showing spin- spin (density-density) correlation function after a quench. The first example corresponds to the ramp from g = 3 to g ′ = 1, i.e. a quench to the quantum critical point. At this point the dispersion (24) becomes gapless and linear at small energies. Then the light cone dynamics is an- ticipated because there is a well defined “speed of light” characterizing the propagation of excitations, which is equal to 2 J . Indeed Fig. 7a) shows clear signature of such dynamics. There is a clearly visible “light cone”, which separates into an instantaneous part (connecting