content_image/408862.jpg content_image/408857.jpg content_image/408860.jpg content_image/408856.jpg content_image/408864.jpg content_image/408865.jpg
<img src='content_image/408862.jpg'>

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 

with 

<img src='content_image/408857.jpg'>

and 

<img src='content_image/408860.jpg'>

In 〈 n$_{i}$ ( t ) n$_{j}$ ( t ) 〉 reads 

<img src='content_image/408856.jpg'>

<img src='content_image/408864.jpg'>

where 

<img src='content_image/408865.jpg'>

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 

