To answer the above question, we plot the panoramic diagram of rate function Eq. (3) with respect to chiral phases ϕitalic-ϕ\phiitalic_ϕ and time t𝑡titalic_t in the polar coordinates [Fig. 4(a)]. The results exhibit that the distribution of the non-analytic points form a “honeycomb lattice” structure with the “lattice constant” of a0=2⁢3⁢π9subscript𝑎023𝜋9a_{0}=\frac{2\sqrt{3}\pi}{9}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = divide start_ARG 2 square-root start_ARG 3 end_ARG italic_π end_ARG start_ARG 9 end_ARG. Since the non-analytic points of the rate function all fall on the vertex of the “hexagonal lattice” structure, the emergence of DQPT in the quench dynamics is possible only when a suitable phase angle is selected.

To further reveal the conditions for the emerged DQPT, we analyze the return amplitude G⁢(t)𝐺𝑡G(t)italic_G ( italic_t ) below. From Eq. (6), one can get the Hamiltonian of the PM phase (f=1𝑓1f=1italic_f = 1), i.e.,

It is not difficult to find that the matrix is zero except for the elements on the diagonal. In other words, there is only on-site operator in the quench process. The fully polarized pure state of Eq. (5) is chosen as the initial state of quench dynamics. Under the condition, one can derive the analytical expression of the dynamical partition function with respect to ϕitalic-ϕ\phiitalic_ϕ and t𝑡titalic_t, i.e.,

We show the behaviors of dynamical partition functions through analytic expression (11) for different ϕitalic-ϕ\phiitalic_ϕ.

First, let’s consider the case of ϕ=0italic-ϕ0\phi=0italic_ϕ = 0. The corresponding dynamical partition function reads

It can be seen clearly that |G⁢(0,t)|≥13𝐺0𝑡13\left|G(0,t)\right|\geq\frac{1}{3}| italic_G ( 0 , italic_t ) | ≥ divide start_ARG 1 end_ARG start_ARG 3 end_ARG, which means, no matter what time t𝑡titalic_t evolves to, the dynamical partition function can never be equal to zero, so DQPT can never happen in this case [see blue solid line in Fig. 4(b)]. This once again proves that the conclusion of reference [24] is correct [24].

Next, let’s have a look at what will happen after the chiral phases have been introduced. Under the condition of ϕ=π6,π4,arctan⁡(127)italic-ϕ𝜋6𝜋4arctangent127\phi=\frac{\pi}{6},\frac{\pi}{4},\arctan(\frac{1}{\sqrt{27}})italic_ϕ = divide start_ARG italic_π end_ARG start_ARG 6 end_ARG , divide start_ARG italic_π end_ARG start_ARG 4 end_ARG , roman_arctan ( start_ARG divide start_ARG 1 end_ARG start_ARG square-root start_ARG 27 end_ARG end_ARG end_ARG ), one can obtain the corresponding expressions for dynamical partition function, respectively, i.e.,

We plot the results of  (13) (orange dashed line), (14) (green dotted line) and (15) (red dash-dotted line) in Fig. 4(b), respectively. The results show that under certain circumstances (such as ϕ=π/6italic-ϕ𝜋6\phi=\pi/6italic_ϕ = italic_π / 6 or arctan⁡(127)arctangent127\arctan{\frac{1}{\sqrt{27}}}roman_arctan ( start_ARG divide start_ARG 1 end_ARG start_ARG square-root start_ARG 27 end_ARG end_ARG end_ARG )), the G⁢(t)𝐺𝑡G(t)italic_G ( italic_t ) curve versus time will touch zero. Then, the corresponding rate function will have points with non-analytical behavior, so DQPT emerges. On the other hand, in most cases (such as ϕ=π/4italic-ϕ𝜋4\phi=\pi/4italic_ϕ = italic_π / 4), the dynamical partition function can never fall on zero, as a result, the DQPT will not occur.

Let’s further analyze why DQPT occurs by splitting up the G⁢(ϕ,t)𝐺italic-ϕ𝑡G(\phi,t)italic_G ( italic_ϕ , italic_t ). One can divide the dynamical partition function G⁢(ϕ,t)𝐺italic-ϕ𝑡G(\phi,t)italic_G ( italic_ϕ , italic_t ) into three parts, i.e.,

|ψm=1,2,3⟩=|1⟩ketsubscript𝜓𝑚123ket1|\psi_{m=1,2,3}\rangle=|1\rangle| italic_ψ start_POSTSUBSCRIPT italic_m = 1 , 2 , 3 end_POSTSUBSCRIPT ⟩ = | 1 ⟩, |ω⟩ket𝜔|\omega\rangle| italic_ω ⟩, |ω2⟩ketsuperscript𝜔2|\omega^{2}\rangle| italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ are the basis vectors in the spinor subspace and the corresponding energy Em=2⁢cos⁡[ϕ+(m−1)⁢2⁢π3]subscript𝐸𝑚2italic-ϕ𝑚12𝜋3E_{m}=2\cos\left[\phi+(m-1)\frac{2\pi}{3}\right]italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 2 roman_cos [ italic_ϕ + ( italic_m - 1 ) divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ] as depicted in Eq. (10).

Note that, the modulus of Gm=1,2,3subscript𝐺𝑚123G_{m=1,2,3}italic_G start_POSTSUBSCRIPT italic_m = 1 , 2 , 3 end_POSTSUBSCRIPT are the same, all equal to 1/3131/\sqrt{3}1 / square-root start_ARG 3 end_ARG. Gm=1,2,3subscript𝐺𝑚123G_{m=1,2,3}italic_G start_POSTSUBSCRIPT italic_m = 1 , 2 , 3 end_POSTSUBSCRIPT is a periodic function with period of 2⁢π/Em2𝜋subscript𝐸𝑚2\pi/E_{m}2 italic_π / italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT. The arguments of subspace dynamical partition function arg⁡[Gm]subscript𝐺𝑚\arg{[G_{m}]}roman_arg [ italic_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ] show periodic evolution over time [see Fig.5(a)-(d)]. Then, by changing the chiral phase ϕitalic-ϕ\phiitalic_ϕ, Emsubscript𝐸𝑚E_{m}italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT can be adjusted to affect the behavior of the Gm⁢(t)subscript𝐺𝑚𝑡G_{m}(t)italic_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) curve. For example, in the case of introducing different phase angles ϕitalic-ϕ\phiitalic_ϕ, the values of Gm=1,2,3subscript𝐺𝑚123G_{m=1,2,3}italic_G start_POSTSUBSCRIPT italic_m = 1 , 2 , 3 end_POSTSUBSCRIPT will rotate in the complex plane [see Fig.5(e)-(h)]. Since the modulus are equal, when the three phase angles form a certain relationship, the vectors’ sum of the three complex numbers will be zero, i.e., they satisfy the equation

In other words, after fixing some specific chiral phase angle ϕitalic-ϕ\phiitalic_ϕ, the time t satisfying the above expression will lead to the emergence of DQPT. These specific times are often called the critical time, denoted as t∗superscript𝑡t^{*}italic_t start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. For example, here we provide four typical chiral phase angles and calculate the corresponding arg⁡[Gm⁢(t)]subscript𝐺𝑚𝑡\arg[G_{m}(t)]roman_arg [ italic_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ]. The result reveals that for the case of ϕ=0italic-ϕ0\phi=0italic_ϕ = 0 and ϕ=π/4italic-ϕ𝜋4\phi=\pi/4italic_ϕ = italic_π / 4, the arguments cannot form a suitable composite pattern at any time to make the sum of G1,G2,G3subscript𝐺1subscript𝐺2subscript𝐺3G_{1},G_{2},G_{3}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT be zero in the complex plane. However, when ϕ=π/6italic-ϕ𝜋6\phi=\pi/6italic_ϕ = italic_π / 6 or ϕ=arctan⁡(1/27)italic-ϕarctangent127\phi=\arctan{1/\sqrt{27}}italic_ϕ = roman_arctan ( start_ARG 1 / square-root start_ARG 27 end_ARG end_ARG ), the summation of G1,G2,G3subscript𝐺1subscript𝐺2subscript𝐺3G_{1},G_{2},G_{3}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT can be zero at t∗superscript𝑡t^{*}italic_t start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT, resulting in the dynamical partition function G⁢(ϕ,t)=0𝐺italic-ϕ𝑡0G(\phi,t)=0italic_G ( italic_ϕ , italic_t ) = 0. Therefore, in these cases, the system will have DQPT. The analytical results here agree well with the results of Fisher Zero analysis, which once again prove that DQPT can be induced by introducing the chiral phase.

Now, we discuss the rules for the selection of chiral phases. Mathematically, by analyzing the equation G⁢(ϕ,t)=0𝐺italic-ϕ𝑡0G(\phi,t)=0italic_G ( italic_ϕ , italic_t ) = 0, we obtain that if DQPT emerges, ϕitalic-ϕ\phiitalic_ϕ and t𝑡titalic_t must satisfy the following two analytic expressions, i.e.,

and

where (p,q)∈ℤ𝑝𝑞ℤ(p,q)\in\mathbb{Z}( italic_p , italic_q ) ∈ blackboard_Z and p−qmod3≠0modulo𝑝𝑞30p-q\mod 3\neq 0italic_p - italic_q roman_mod 3 ≠ 0.

From the above expressions (18) and (19), one can obtain the position where zeros appear in the two dimensional polar coordinate plane. In other words, the ϕitalic-ϕ\phiitalic_ϕ and t𝑡titalic_t corresponding to any integer pair (p,q)𝑝𝑞(p,q)( italic_p , italic_q ) are the coordinates of the nonanalytic points of DQPT. However, for some phases ϕitalic-ϕ\phiitalic_ϕ that have no integer pair (p,q)𝑝𝑞(p,q)( italic_p , italic_q ) to correspond, DQPT can never be induced (e.g. ϕ=0italic-ϕ0\phi=0italic_ϕ = 0 or ϕ=π/4italic-ϕ𝜋4\phi=\pi/4italic_ϕ = italic_π / 4).

Specifically, when (p,q)=(1,0)𝑝𝑞10(p,q)=(1,0)( italic_p , italic_q ) = ( 1 , 0 ), one can obtain ϕ⁢(1,0)=π/6italic-ϕ10𝜋6\phi(1,0)=\pi/6italic_ϕ ( 1 , 0 ) = italic_π / 6, which makes the expression (18) valid. Besides, ϕ⁢(3,−1)=arctan⁡(1/27)italic-ϕ31arctangent127\phi(3,-1)=\arctan{1/\sqrt{27}}italic_ϕ ( 3 , - 1 ) = roman_arctan ( start_ARG 1 / square-root start_ARG 27 end_ARG end_ARG ) can also satisfy the above analytical expression (18). For ϕ=π/4italic-ϕ𝜋4\phi=\pi/4italic_ϕ = italic_π / 4 or ϕ=0italic-ϕ0\phi=0italic_ϕ = 0, no suitable integer pair (p,q)𝑝𝑞(p,q)( italic_p , italic_q ) can be found to meet the condition. That is to say, zero point of the dynamical partition function appears only when ϕitalic-ϕ\phiitalic_ϕ satisfies the expression (18), such as ϕ=π/6italic-ϕ𝜋6\phi=\pi/6italic_ϕ = italic_π / 6 or arctan⁡(1/27)arctangent127\arctan{1/\sqrt{27}}roman_arctan ( start_ARG 1 / square-root start_ARG 27 end_ARG end_ARG ).

Note that, the expression (18) and (19) are universal, therefore, one can find more chiral phases that can induce DQPT by providing different sets of (p,q)𝑝𝑞(p,q)( italic_p , italic_q ).