2 If at time t = 0, λ is suddenly changed to b and λ = b is kept for t > 0, the system will evolve under sudden quench. The eigenvectors and eigenvalues of the final Hamiltonian H ( b ) are denoted as | φ$_{n}$ ( b ) 〉 and ω$_{n}$ respec- tively. The time-evolving wave function of the system straightforwardly, where ℜ ( z ) means the real part of a complex number z . One can expect that ∂$_{a}$ 〈 A ( a ) 〉 will behave singularly at a = λ$_{c}$ because the energy gap van- ishes in TL. at t > 0 is | Ψ( t > 0) 〉 = $^{∑}$$_{n}$〈 φ$_{n}$ ( b ) | ψ$_{0}$ ( a ) 〉 e − iω$_{n}$$^{t}$| φ$_{n}$ ( b ) 〉 , where we omit the relative phase. The time-averaged expectation value is So we have ∂$_{a}$ 〈 A ( a, b ) 〉 = 2 ∑ m =0 ∑ n 〈 φ$_{n}$ ( b ) | A | φ$_{n}$ ( b ) 〉 E$_{0}$ ( a ) − E$_{m}$ ( a ) × ℜ [( H$_{I}$ )$_{0}$$_{m}$ 〈 ψ$_{m}$ ( a ) | φ$_{n}$ ( b ) 〉〈 φ$_{n}$ ( b ) | ψ$_{0}$ ( a ) 〉 ] . (2) 〈 A ( a, b ) 〉 = lim t →∞ 1 t ∫ t 0 〈 A ( a, b, τ ) 〉 dτ = ∑ n 〈 ψ$_{0}$ ( a ) | φ$_{n}$ ( b ) 〉〈 φ$_{n}$ ( b ) | ψ$_{0}$ ( a ) 〉〈 φ$_{n}$ ( b ) | A | φ$_{n}$ ( b ) 〉 . There are two kinds of phase transition in anisotropic XY spin chain model. One belongs to the universal- ity class of the Ising phase transition and the other is anisotropic transition [10]. In this paper we focus on the Ising phase transition. The equilibrium Ising phase criti- cal points are | h | = 1. In J = 1 > 0 case, the system has anti-ferromagnetic interaction with the staggered magne- tization in the x direction 〈 σ $^{x}$〉$_{stag}$ = 1 N ∑ N − 1 i =0 ( − 1) $^{i}$〈 σ x i 〉 as the order parameter. For | h | < 1, the system is in an ordered phase with |〈 σ $^{x}$〉$_{stag}$ | = 1, while for | h | > 1, the system is in a disordered phase with 〈 σ $^{x}$〉$_{stag}$ = 0. Now we turn to the problem of dynamics under sudden quench. Using successive Jordan-Wigner, Fourier, and Bogoliubov transformations, the exact solutions of the evolved state of XY model under a sudden quench can be obtained [19], by which the quench dynamical proper- ties of many observables such as magnetization and two- spin correlation can be studied. In our work we focus on the quench two-spin correlation, which is defined as G α $_{n}$( a, b ) = lim$_{t}$$_{→∞}$ 1 t ∫ t $_{0}$〈 σ α $_{0}$( a, b, τ ) σ α $_{n}$( a, b, τ ) 〉 dτ , where α = x, y, z , n is the distance between two spins and a ( b ) is the initial (final) magnetic field. Because Eqs.(1) and (2) share the same denominator, we conclude that there must be similarity between the critical singular behaviors of ∂$_{a}$ 〈 A ( a ) 〉 and ∂$_{a}$ 〈 A ( a, b ) 〉 . The singularity and scaling behaviors come from the van- ishing of the energy gap of the initial Hamiltonian in TL. If there is a critical phenomenon of the initial Hamil- tonian with a vanishing energy gap which can be indi- cated by the singular behavior of ∂$_{a}$ 〈 A ( a ) 〉 at a = λ$_{c}$ , ∂$_{a}$ 〈 A ( a, b ) 〉 will also have a similar singular behavior at a = λ$_{c}$ if b = λ$_{c}$ . When b = λ$_{c}$ , the numerator in the above Eq.(2) is not zero. Therefore ∂$_{a}$ 〈 A ( a, b ) 〉 will show singularity at a = λ$_{c}$ because of the vanishing of the energy gap in TL. However if b = λ$_{c}$ = a , we have ψ = φ , so in Eq.(2) 〈 ψ$_{m}$ ( a ) | φ$_{n}$ ( b ) 〉〈 φ$_{n}$ ( b ) | ψ$_{0}$ ( a ) 〉 = δ$_{m,n}$δ$_{n,}$$_{0}$ . Consid- ering m = 0 in the sum, we can see both the numerator and denominator in Eq.(2) are zero at a = b = λ$_{c}$ , mean- ing that ∂$_{a}$ 〈 A ( a, λ$_{c}$ ) 〉 maybe has no singular behavior at a = λ$_{c}$ . III. XY SPIN CHAIN MODEL IV. QUENCH TWO-SPIN CORRELATION As the beginning we want to clarify our conclusion in Sec.II by considering ∂$_{a}$G z ( x ) 1 ( a, b ), namely the nearest- neighbor two-spin correlation. Due to the difficulty in the analytical calculation of them, we choose to numerically solve them. We let γ = 1 for convenience in this part and one should note that all conclusions below are also valid for γ = 1. In Fig.1(a), ∂$_{a}$G $_{1}$( a, b = 1) diverges at a = 1 for both the static and dynamical case. Moreover, we calculate that ∂$_{a}$G z $_{1}$( a, b ) = const × | a − 1 | − µ with µ ≈ 0 . 25, a universal exponent irrelevant with b (Fig.1(b)). While as predicted in Sec.II, the sudden quench case with b = 1 is very special. We can see from Fig.1(a) that ∂$_{a}$G z $_{1}$( a, 1) is not divergent at a = 1. Similar to G z $_{1}$( a, b ), where J is the coupling constant, σ x j , σ y $_{j}$, σ z j are the Pauli operators at the j th lattice site and J (1 ± γ ) / 2 is the measure of the interaction strength in x ( y ) component between two nearest-neighbor spins. γ is the anisotropy parameter which can change from 0 (XX model) to 1 (Ising model). h ( t ) is the time-dependent magnetic field. In our paper, we set J = 1 > 0 for convenience and the periodic boundary condition σ α N = σ α 0 , α = x, y, z is used. The phase diagram of this model is easy to be obtained. H = N − 1 ∑ j =0 { J 2$^{[}$ (1 + γ ) σ x j σ x j +1 + (1 − γ ) σ y $_{j}$σ y j +1 ] − h ( t ) σ z j $^{}}$, In this paper, we use the anisotropic XY model in a transverse field under a sudden quench at zero temper- ature to check our conclusion in the above section. The Hamiltonian of the spin-1/2 anisotropic XY spin chain is