4 range correlation in the z direction can survive after the quench but the long-range correlation in the x direction is destroyed by the quench. We note that the long range correlation is also re- searched by Sengupta et al . in Ref.[3]. They give some analytic results of the two point correlation function per- pendicular to the magnetic field direction in quantum Ising model. However, they only consider two limit cases, namely the initial magnetic field is fixed at a = 0 and a = ∞ . In both cases, G x $_{n}$( a, b ) tends to zero when n → ∞ but in different ways which depend on the value of the final magnetic field b . In Fig.4, one can see that G x $_{n}$(0 , 0 . 5) tends to zero much slower than G x $_{n}$(0 , 2) and there is a clear spatial oscillation of G x $_{n}$(2 , 0 . 5). These phenomena are consistent with Ref.[3]. Therefore, our numerical results confirm their analytic results about long range correlation and give an extension to the gen- eral situation beyond the two limit cases. Up to now we only consider the case in which the fi- nal magnetic field b is fixed and all quench quantities are regarded as functions of initial magnetic field a . For completeness, we discuss the case in which a is fixed and all quench quantities are regarded as functions of b . As one can expect, the quench quantities will have different behaviors. If we make a similar calculation with that in Sec.II using perturbation theory, we can obtain V. A FURTHER DISCUSSION: FIXED INITIAL MAGNETIC FIELD FIG. 5. Numerical fitting of G x $_{n}$( a, b ) at b = 0 . 5 up to n = 150. Through numerical fit of the two typical data G x $_{n}$(0 . 4 , 0 . 5) and G x $_{n}$(0 . 45 , 0 . 5), we find G x $_{n}$( a, b ) ∼ n − $^{c$_{1}$}$e − c$_{2}$$^{n}$, with G x $_{n}$(0 . 4 , 0 . 5) = n − 0 . $^{0000404236}$e( − 0 . 074916 n − 0 . 0031414) and G x $_{n}$(0 . 45 , 0 . 5) = n − 0 . $^{0000460715}$e( − 0 . 07259538 n − 0 . $^{00080686)}$. We extrapolate those two curves to n = 400. it after a long enough time evolution (Fig.4(d)). So, only when static case a = b < 1 , the long-range correlation in x direction is preserved. In other cases we can have when n → ∞ , 〈 σ x $_{0}$( a, b ) σ x $_{n}$( a, b ) 〉 → 0, leading to 〈 σ$_{x}$ ( a, b ) 〉 = 0 straightforwardly. A physical intuition tells us that be- cause the magnetic field is in the z direction, the long- ∂$_{b}$ 〈 A ( a, b ) 〉 = 2 ∑ n ∑ m = n 〈 φ$_{n}$ ( b ) | A | φ$_{n}$ ( b ) 〉 ω$_{n}$ ( b ) − ω$_{m}$ ( b ) × ℜ [( H$_{I}$ )$_{nm}$ 〈 ψ$_{0}$ ( a ) | φ$_{n}$ ( b ) 〉〈 φ$_{m}$ ( b ) | ψ$_{0}$ ( a ) 〉 ] + 2 ∑ n ∑ m = n 〈 ψ$_{0}$ ( a ) | φ$_{n}$ ( b ) 〉〈 φ$_{n}$ ( b ) | ψ$_{0}$ ( a ) 〉 ω$_{n}$ ( b ) − ω$_{m}$ ( b ) × ℜ [( H$_{I}$ )$_{nm}$ 〈 φ$_{m}$ ( b ) | A | φ$_{n}$ ( b ) 〉 ] . (3) divergent at b = 1 when a = 1. While a = 1, it is discontinuous at b = 1. How to explain the discontinuity is still an open question to us. It is difficult to study the behavior of Eq.(3) near b = λ$_{c}$ for a general a . However, the case in which a = λ$_{c}$ is easy to analyze. When a = λ$_{c}$ = b , φ = ψ . There- fore in the first term of the right hand side of Eq.(3), 〈 ψ$_{0}$ ( a ) | φ$_{n}$ ( b ) 〉〈 φ$_{m}$ ( b ) | ψ$_{0}$ ( a ) 〉 = δ$_{m,}$$_{0}$δ$_{n,}$$_{0}$ , meaning that this term is 0 because m = n in the sum. Similarly, we have 〈 ψ$_{0}$ ( a ) | φ$_{n}$ ( b ) 〉〈 φ$_{n}$ ( b ) | ψ$_{0}$ ( a ) 〉 = δ$_{n,}$$_{0}$ in the second term of Eq.(3). So we have In summary, starting from a general Hamiltonian which may undergo a QPT with the change of a con- trollable parameter, we obtain a general conclusion that in a sudden quench system, when the final Hamiltonian is fixed, the behavior of the time-averaged expectation of any observable has close relationship with the gapless excitation of the initial Hamiltonian. This general con- clusion, clarified by our example of XY model, to a large extent explains the similarity between the critical phe- VI. SUMMARY This equation is the same with Eq.(1) and means that ∂$_{b}$ 〈 A ( λ$_{c}$, b ) 〉 will diverge at b = λ$_{c}$ because the vanish of the energy gap. In the following we use ∂$_{b}$G z $_{1}$( a, b ) as an example to analyze (see Fig.6). As we expect above, ∂$_{b}$G z $_{1}$( a, b ) is ∂$_{b}$ 〈 A ( λ$_{c}$, b ) 〉|$_{b}$$_{=}$$_{λ}$$_{c}$ = 2 ∑ m =0 ℜ [( H$_{I}$ )$_{0}$$_{m}$ 〈 φ$_{m}$ ( λ$_{c}$ ) | A | φ$_{0}$ ( λ$_{c}$ ) 〉 ] ω$_{0}$ ( λ$_{c}$ ) − ω$_{m}$ ( λ$_{c}$ ) .