content_image/434050.jpg content_image/434042.jpg content_image/434049.jpg
<img src='content_image/434050.jpg'>

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- 

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. 

## V. A FURTHER DISCUSSION: FIXED INITIAL MAGNETIC FIELD 

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 

<img src='content_image/434042.jpg'>

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 

<img src='content_image/434049.jpg'>

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. z 

In the following we use ∂$_{b}$G $_{1}$( a, b ) as an example to analyze (see Fig.6). As we expect above, ∂$_{b}$G z $_{1}$( a, b ) is 

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. 

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 

