3 FIG. 1. γ = 1. (a) ∂$_{a}$G z $_{1}$( a, b ) for static case and dynamical cases b = 0 , 0 . 5 , 1 , 1 . 5. (b) ln( ∂$_{a}$G z $_{1}$( a, b )) as a function of ln | a − 1 | . One can see the slopes of the lines for different b are all the same, namely a universal constant -0.25. FIG. 3. γ = 0 . 6. G z $_{n}$( a, b ) for (a) The static case b = a , (b) b = 0, (c) b = 1 and (d) b = 1 . 5. FIG. 4. γ = 1. G x $_{n}$( a, b ) in (a) static system and dynamical systems with (b) b = 0 . 5, (c) b = 1 and (d) b = 2. ∂$_{a}$G x $_{1}$( a, b = 1) diverges at a = 1 for static and sud- den quench case. After numerical analysis, we can find ∂$_{a}$G x $_{1}$( a, b ) = 1 π ln | a − 1 | + f ( b ), with f ( b ) a constant which we do not concern with (Fig.2(b)). We also can see from Fig.2(a) that ∂$_{a}$G x $_{1}$( a, 1) is not divergent at a = 1. Through the comparison between static and sudden quench cases of XY model, we find that in general sudden quench case ( b = 1), the quench system and static system share similar critical scaling form of ∂$_{a}$G z ( x ) 1 ( a, b ). This is consistent with our conclusion in Sec.II and reflects the memory of the quench system to the initial Hamiltonian. Now we shift our focus from the nearest-neighbor two- spin correlation to the correlation between two spins with arbitrary distance n . We show G z $_{n}$( a, b ) as a function of a and n for both static and sudden quench case in Fig.3, where γ = 0 . 6 is used. From this figure, we know as n increases, G z $_{n}$( a, b ) reaches a constant quickly after a little increase of n . There is no qualitative difference between static and general dynamical case. For G x $_{n}$( a, b ), there will be a big difference. We let γ = 1 for convenience here. In a static system ( a = b ), from Fig.4(a) we can see when a > 1, G x $_{n}$( a ) tends to zero very quickly and is nonzero only if n is very small, demonstrating no long-range correlation in x direction. While in another phase a < 1, the ground state is degen- erate and possesses long-range correlation in x direction. However, in a sudden quench system, the case will be obviously different. If the final magnetic field b < 1, af- ter a long time evolution the long-range correlation in x direction only exists at the point a = b < 1 (that is just the static case), meaning that when b < 1 a change of magnetic field at t = 0 will eliminate the long-range cor- relation in x direction. In Fig.4(b), we plot the G x $_{n}$( a, b ) up to n = 150 with b = 0 . 5. Through numerical fit of the two typical data G x $_{n}$(0 . 4 , 0 . 5) and G x $_{n}$(0 . 45 , 0 . 5) (Fig.5), we find G x $_{n}$( a, b ) ∼ n − $^{c$_{1}$}$e − c$_{2}$$^{n}$, decaying to zero with n even when a and b is slightly different. So we think the G x $_{n}$( a, b ) is destroyed after an arbitrarily small quench. While b > 1, no matter what a is, even for a < 1, there is no long-range correlation in x direction. This means that there is a possibility that a system possessing a long- range correlation in x direction at the beginning will lose (a) (b) 2 (a) (b) FIG. 2. γ = 1. (a) ∂$_{a}$G x $_{1}$( a, b ) for static case and dynamical cases b = 0 , 0 . 5 , 1 , 1 . 5. (b) ∂$_{a}$G x $_{1}$( a, b ) as a function of ln | a − 1 | . One can see the slopes of the lines for different b are all the same, namely a universal constant 1 /π . (b) b=0.5 (c) b=1 (d) b=2.0 Static (a) n n n n