Kibble–Zurek scaling in the quantum Ising chain with a time-periodic perturbation

The LZSM model is described by the two-level Hamiltonian

$\displaystyle H_{\mathrm{LZSM}}(t)$

$\displaystyle=\frac{1}{2}vt\sigma^{z}+\Delta\sigma^{x}.$

(1)

In this model, if a state was an instantaneous eigenstate in the infinite past, the probability of transitioning to another instantaneous eigenstate in the infinite future is given by

$\displaystyle P_{\mathrm{LZSM}}=\exp\quantity(-\frac{2\pi\Delta^{2}}{v}),$

(2)

where the natural units are used. When $\Delta$ is significantly larger than $\sqrt{v}$, the system is considered adiabatic, resulting in a small transition probability. Conversely, when $\Delta$ is significantly smaller than $\sqrt{v}$, the system is characterized as non-adiabatic, leading to a large transition probability. Recent studies of the LZSM model have investigated the dynamics under various conditions, including the presence of external oscillating perturbations Mullen et al. (1989); Malla and Raikh (2018); Wubs et al. (2005, 2006); Saito et al. (2006, 2007b); Zueco et al. (2008); Ashhab (2014, 2016); Sinitsyn and Li (2016); Sun and Sinitsyn (2016); Kayanuma and Mizumoto (2000). The perturbation approach derives the approximate formula for the LZSM model when the diagonal elements of the Hamiltonian are small Mullen et al. (1989).

In this section, we consider the two-level time-dependent Hamiltonian

	$\displaystyle H(t)$	$\displaystyle=H_{z}(t)+H_{x}(t),$		(3)
	$\displaystyle H_{z}(t)$	$\displaystyle=\frac{1}{2}(vt+\varepsilon-A\cos(\omega t))\sigma^{z},$		(4)
	$\displaystyle H_{x}(t)$	$\displaystyle=\quantity(\Delta+\frac{B}{2}\cos\omega t)\sigma^{x},$		(5)

which is the LZSM model with oscillations of magnitude $A$ and $B$ in the diagonal and off-diagonal elements, respectively. The initial state is assumed to be $\ket{\psi(-\infty)}\propto\ket{\uparrow}$, where $\sigma_{z}\ket{\uparrow}=\ket{\uparrow}$ holds. The goal is to obtain the transition probability at the final time $p(\infty)=|\bra{\psi(\infty)}\ket{\uparrow}|^{2}$. We note that the transition probabilities were obtained approximately when either $A$ or $B$ is $0$ with perturbation theory Mullen et al. (1989). Changing the frame with the unitary operator

$\displaystyle U_{z}(t)=\exp\quantity(-i\int_{0}^{t}dt^{\prime}H_{z}(t^{\prime})),$

(6)

the Schrödinger equation $i\dot{U}(t)=H(t)U(t)$ becomes

$\displaystyle i\frac{d}{dt}\hat{U}_{x}(t)$

$\displaystyle=\hat{H}_{x}(t)\hat{U}_{x}(t),$

(7)

where we define $\hat{U}_{x}(t)=U_{z}^{\dagger}(t)U(t)$ and

			(8)

Here, we used the formula

$\displaystyle e^{ix\sin\tau}=\sum_{n=-\infty}^{\infty}J_{n}(x)e^{in\tau},$

(9)

where $J_{n}(x)$ is the Bessel function of the first kind and the basis of the matrix is $\{\ket{\uparrow},\ket{\downarrow}\}$, where $\sigma_{z}\ket{\downarrow}=-\ket{\downarrow}$ holds. We express the state in this basis as $\ket{\tilde{\psi}(t)}=\matrixquantity(C_{\uparrow}(t)&C_{\downarrow}(t))^{\mathrm{T}}$, and these variables satisfy

	$\displaystyle i\dot{C}_{\uparrow}(t)$	$\displaystyle=\quantity(\Delta+\frac{B}{2}\cos\omega t)\sum_{n=-\infty}^{\infty}J_{n}\quantity(\frac{A}{\omega})e^{i\frac{v}{2}t^{2}+i\varepsilon t-in\omega t}C_{\downarrow}(t),$		(10)
	$\displaystyle i\dot{C}_{\downarrow}(t)$	$\displaystyle=\quantity(\Delta+\frac{B}{2}\cos\omega t)\sum_{n=-\infty}^{\infty}J_{n}\quantity(\frac{A}{\omega})e^{-i\frac{v}{2}t^{2}-i\varepsilon t+in\omega t}C_{\uparrow}(t).$		(11)

The initial conditions on these variables can be regarded as $C_{\uparrow}(-\infty)=1$ and $C_{\downarrow}(-\infty)=0$, and the transition probability $p(\infty)$ can be expressed as $|C_{\uparrow}(\infty)|^{2}$. We introduce the dimensionless parameters $\tau=\sqrt{v}t$ and $\eta=A/\omega$, and we define $\tilde{\circ}=\circ/\sqrt{v}$. By successive substitutions, we obtain the following result

	$\displaystyle C_{\uparrow}(\infty)-1$	$\displaystyle\simeq-\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}J_{n}(\eta)J_{m}(\eta)$		(12)
		$\displaystyle\qquad\times\int^{\infty}_{-\infty}d{\tau}\int^{\tau}_{-\infty}d{\tau^{\prime}}\ \quantity(\tilde{\Delta}+\frac{\tilde{B}}{2}\cos\tilde{\omega}\tau)\quantity(\tilde{\Delta}+\frac{\tilde{B}}{2}\cos\tilde{\omega}\tau^{\prime})e^{\frac{i}{2}({\tau}^{2}-\tau^{\prime 2})+i\tilde{\varepsilon}(\tau-\tau^{\prime})-i\tilde{\omega}(n\tau-m\tau^{\prime})}$		(13)
		$\displaystyle=-2\pi\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}\quantity(\tilde{\Delta}J_{n}(\eta)+\frac{\tilde{B}}{4}J_{n+1}(\eta)+\frac{\tilde{B}}{4}J_{n-1}(\eta))\quantity(\tilde{\Delta}J_{m}(\eta)+\frac{\tilde{B}}{4}J_{m+1}(\eta)+\frac{\tilde{B}}{4}J_{m-1}(\eta))$		(14)
		$\displaystyle\qquad\times\exp\quantity(-\frac{i}{2}\tilde{\omega}(n^{2}\tilde{\omega}-2n\tilde{\varepsilon})+\frac{i}{2}\tilde{\omega}(m^{2}\tilde{\omega}-2m\tilde{\varepsilon}))\theta(n-m),$		(15)

where we define the step function

$\displaystyle\theta(x)=\left\{\begin{array}[]{l}1\quad(x>0)\\ 1/2\quad(x=0)\\ 0\quad(x<0)\end{array}\right..$

(19)

Here, we assume $\tilde{\Delta}$ and $\tilde{B}$ are small enough that this approximation is valid in the non-adiabatic region. Finally, the transition probability is approximately

	$\displaystyle p(\infty)$	$\displaystyle\simeq$
		$\displaystyle=:P_{\mathrm{PT}}.$		(20)

We note that the transition probability $p(\infty)$ must be periodic with $\varepsilon$ because

$\displaystyle H\quantity(t,\varepsilon-2\pi n\frac{v}{\omega})$

$\displaystyle=H\quantity(t-n\frac{2\pi}{\omega},\varepsilon),\quad\forall n\in\mathbb{Z}$

(21)

holds and the period is $2\pi/\tilde{\omega}$.

Next, we decompose the Hamiltonian (3) as in

	$\displaystyle H(t)$	$\displaystyle=H_{0}(t)+H_{1}(t),$		(22)
	$\displaystyle H_{0}(t)$	$\displaystyle=\frac{1}{2}(vt+\varepsilon)\sigma_{z}+\Delta\sigma_{x},$		(23)
	$\displaystyle H_{1}(t)$	$\displaystyle=\frac{1}{2}\cos\omega t(-A\sigma_{z}+B\sigma_{x}),$		(24)

where $H_{0}(t)$ is the Hamiltonian of the LZSM model. Let $U_{0}(t)$ be the time-evolution operator of $H_{0}(t)$ and we define $\hat{H}_{1}(t)=U_{0}^{\dagger}(t)H_{1}(t)U_{0}(t)$. If the $\hat{H}_{1}(t)$ is sufficiently small, we can approximate the time-evolution operator by first-order:

$\displaystyle U(t)$

$\displaystyle\simeq U_{0}(t)\quantity(I-i\int_{t_{0}}^{t}dt^{\prime}\,\hat{H}_{1}(t^{\prime})).$

(25)

Then, the transition probability becomes

	$\displaystyle p(\infty)$	$\displaystyle\simeq\quantity|\bra{\uparrow}U_{0}(\infty)\ket{\uparrow}-i\bra{\uparrow}U_{0}(\infty)\int_{-\infty}^{\infty}dt^{\prime}\,\hat{H}_{1}(t^{\prime})\ket{\uparrow}|^{2}$		(26)
		$\displaystyle=:P_{\textrm{FP}}.$		(27)

We note that we need to consider up to the second-order perturbation if we approximate the transition probability by the second-order of $\hat{H}_{1}$:

	$\displaystyle p(\infty)$	$\displaystyle\simeq P_{\textrm{FP}}-2\operatorname{Re}\biggl{(}\left\langle\uparrow\left|U_{0}(\infty)\right|\uparrow\right\rangle^{*}$		(28)
		$\displaystyle\quad\times\langle\uparrow|U_{0}(\infty)\int_{-\infty}^{\infty}dt\int_{-\infty}^{t}dt^{\prime}\hat{H}_{1}(t)\hat{H}_{1}\left(t^{\prime}\right)|\uparrow\rangle\biggr{)}.$		(29)

However, we assume that the last term is negligible. This assumption is justified in the adiabatic limit because the term contains an exponentially small term in the limit. The above method is called the Furry picture formalism Furry (1951).

With this formalism, the transition probability in the adiabatic limit is

	$\displaystyle P_{\mathrm{FP}}$	$\displaystyle\simeq\left|\bra{\downarrow}\int_{-\infty}^{\infty}dt\hat{H}_{1}(t)\ket{\uparrow}\right|^{2}$		(30)
		$\displaystyle\simeq\pi^{2}e^{-2\pi\tilde{\Delta}^{2}}\biggr{|}\eta{}_{1}\tilde{F}_{1}\quantity(-i\tilde{\Delta}^{2},0,i\tilde{\omega}^{2})$		(31)
		$\displaystyle\quad-i\tilde{B}\frac{|\tilde{\Delta}|}{2}\biggl{(}{}_{1}\tilde{F}_{1}\left(-i\tilde{\Delta}^{2},1,i\tilde{\omega}^{2}\right)$		(32)
		$\displaystyle\quad+{}_{1}\tilde{F}_{1}\left(-i\tilde{\Delta}^{2}+1,1,i\tilde{\omega}^{2}\right)\biggr{)}\biggr{|}^{2},$		(33)

where ${}_{1}\tilde{F}_{1}(a,b,x)$ is the regularized confluent hypergeometric function of the first kind. We derive the probability in Appendix.A and the probability without the adiabatic limit is (106). The first-order approximation in this method is valid in the region where both $\eta,\tilde{B}$ are small. Unlike perturbation theory, this method does not treat $\tilde{\Delta}$ as small values, but rather assumes that it takes on large values. In this way, this method calculates non-perturbative effects on $\tilde{\Delta}$.

In this subsection, we compare the approximate formula (20) and (106) or (33) with the results of numerical solution of the Schrödinger equation.

First, consider the case of the non-adiabatic limit and the case $\tilde{A}=0$. In this case, only the off-diagonal component has an oscillating term and the transition probability (20) can be expressed as

$\displaystyle P_{\mathrm{PT}}$

$\displaystyle=$.

The result (II.4) is also derived in the previous study Mullen et al. (1989). We note, however, that a factor of $1/4$ is missing in the third term of equation (7) in Mullen et al. (1989).

The numerical results in this case are shown in Fig. 1(a). In the region where $\tilde{B}$ is small, the numerical calculation and the approximate expression (II.4) are in good agreement. On the other hand, as $\tilde{B}$ increases, the contribution of $O(\tilde{B}^{3})$, which is ignored in the approximate expression (II.4), increases, resulting in deviation from the numerical calculation.

Next, consider the case $\tilde{B}=0$. From (20), the transition probability can be expressed as

$\displaystyle P_{\mathrm{PT}}$

$\displaystyle=$.

The numerical results in this case are shown in Fig. 1(b). Here, the sum was calculated in the range of $-10\leq n,m\leq 10$. In this case, the sum is large enough that even if $\eta=A/\omega$ is not small, the numerical calculation and the approximate expression (II.4) are in good agreement. In Fig. 1(b), the transition probabilities show a simple behavior as $\tilde{\omega}$ increases. This corresponds to the region where $\eta$ is sufficiently small. In this limit, (II.4) yields

$\displaystyle P_{\mathrm{PT}}$

$\displaystyle\simeq\exp\biggl(-2\pi\tilde{\Delta}^{2}\biggl{(}1+2\eta\sin(\tilde{\omega}\tilde{\varepsilon})\sin\frac{\tilde{\omega}^{2}}{2}\biggr{)}\biggr{missing}).$

(34)

This result is also derived in the previous study Mullen et al. (1989).

Finally, consider the case $\tilde{A}\neq 0$ and $\tilde{B}\neq 0$. The numerical results for this case are shown in Fig. 1(c). In this case, the numerical calculation and the approximate formula (20) agree well even for large values of $\eta$ because the sums are sufficiently large.

Next, we show the validity of (106) in the adiabatic process. First, in the case of $\tilde{A}=0$, the results of the numerical calculations are compared with those of the expression (106) in Fig. 2(a). It can be seen that in the region where $\tilde{B}$ is small, the results are in good agreement with the numerical calculations. The dashed line in the figure represents the LZSM transition probability (2). Although this probability is sufficiently small in the adiabatic limit, it can be seen that there are parameter regions where the transition probabilities are much larger than the LZSM transition probability for $\tilde{B}\neq 0$ due to the effects of the oscillations.

Next, in the case of $\tilde{B}=0$, the results of the numerical calculations are compared with those of (106) in Fig. 2(b). It can be seen that in the region where $\eta$ is sufficiently small, the results are in good agreement with the numerical calculations. In this case, as in the previous case, there are parameter regions where the transition probabilities are much larger than the LZSM transition probability due to the oscillations.

In addition, Fig. 2(c) compares the results of the numerical calculations with those of (106) in the case of $A\neq 0$ and $B\neq 0$. In this case, we can see that (106) is in good agreement with the numerical calculation in the region where $\eta$ is sufficiently small due to the small value of $\tilde{B}$.

Finally, we check that (33) is consistent with (106) in the adiabatic limit. The results are shown in Fig. 3. In this case, it can be seen that (106) is consistent with (33), especially in regions where $\eta$ is sufficiently small.

We consider the time-dependent Hamiltonian

	$\displaystyle H_{D}(t)$	$\displaystyle=-\sum^{N}_{j=1}\quantity(\frac{J}{2}\sigma^{x}_{j}\sigma^{x}_{j+1}+g(t)\sigma^{z}_{j}),$		(35)
	$\displaystyle g(t)$	$\displaystyle=\frac{1}{4}\quantity(vt+\varepsilon^{\prime}-A\cos(\omega t)),$		(36)

where we impose the periodic boundary condition

$\displaystyle\sigma^{a}_{N+j}=\sigma^{a}_{j}.$

(37)

This Hamiltonian has $\mathbb{Z}_{2}$ symmetry and only the space to which the ground state belongs will be considered from now on. Here, we introduce the spinless fermion operators $c_{j}$ using the Jordan–Wigner(JW) transformation

$\displaystyle\sigma^{z}_{j}=1-2c^{\dagger}_{j}c_{j},\quad\sigma^{x}_{j}=\quantity(c^{\dagger}_{j}+c_{j})\prod_{l<j}\quantity(-\sigma^{z}_{l}),$

(38)

and we consider the Fourier expansion of the operators

$\displaystyle c_{j}=\frac{1}{\sqrt{N}}e^{-i\frac{\pi}{4}}\sum_{q}e^{iqj}c_{q},$

(39)

where $q=\pm(2n-1)\pi/N,\ n\in\{1,\cdots,N/2\}$. In the Heisenberg picture, these operators satisfy

			(40)
			(41)

The eigenvalues of the Hamiltonian in (III.1) are shown in Fig. 4. It can be seen that when $q\simeq 0,\pm\pi$, the energy gap is small, corresponding to the non-adiabatic region where non-adiabatic transitions occur, while in the other region, the energy gap is large, corresponding to the adiabatic region.

The initial state is set to be the ground state at $t=-\infty$ : $\ket{\psi(-\infty)}=\ket{\downarrow}^{\otimes N}$. The final time is set to $t=t_{F}$ and we calculate the expectation value

	$\displaystyle\expectationvalue{\mathcal{N}}$	$\displaystyle=\frac{1}{2}\bra{\psi(-\infty)}\sum_{j}\quantity(1-\sigma^{z}_{j}(t_{F}))\ket{\psi(-\infty)}$		(42)
		$\displaystyle=\sum_{j=1}^{N}\bra{\psi(-\infty)}c^{\dagger}_{j}(t_{F})c_{j}(t_{F})\ket{\psi(-\infty)}$		(43)
		$\displaystyle=\sum_{q}\bra{1(q)}c_{q}^{\dagger}(t_{F})c_{q}(t_{F})\ket{1(q)},$		(44)

where $c^{\dagger}_{q}(-\infty)c_{q}(-\infty)\ket{1(q)}=\ket{1(q)}$ and we used the fact that the initial state can be written as $\bigotimes_{q}\ket{1(q)}$. The solution of (III.1) can be expressed as

$\displaystyle\matrixquantity(c_{q}(t)\\ c_{-q}^{\dagger}(t))$

$\displaystyle=\begin{pmatrix}u_{q}(t)&-v^{*}_{q}(t)\\ v_{q}(t)&u^{*}_{q}(t)\end{pmatrix}\matrixquantity(c_{q}(-\infty)\\ c_{-q}^{\dagger}(-\infty)),$

(45)

where $|u_{q}(t)|^{2}+|v_{q}(t)|^{2}=1$ holds. Then, the expectation value becomes

	$\displaystyle\expectationvalue{\mathcal{N}}$	$\displaystyle=\sum_{q}\bra{1(q))}c_{q}^{\dagger}(t_{F})c_{q}(t_{F})\ket{1(q)}$		(46)
		$\displaystyle=\sum_{q}\quantity|u_{q}(t_{F})|^{2}.$		(47)

In the thermodynamic limit, the normalized expectation value can be expressed as

	$\displaystyle n(t_{F})$	$\displaystyle=\frac{\expectationvalue{\mathcal{N}(t_{F})}}{N}$		(48)
		$\displaystyle\to\int^{\pi}_{-\pi}\frac{dq}{2\pi}\quantity|u_{q}(t_{F})|^{2}.$		(49)

In the model considered in this study, the phase transition points exist at times satisfying $g(t)=\pm J/2$. However, as shown in (III.1), these phase transition points do not produce interference effects as discussed in previous studies Kou and Li (2022). Therefore, we will assume that $t_{F}=\infty$ for the current discussion.

First, we consider a non-adiabatic region. The non-adiabatic region corresponds to the situation $\kappa_{q}\ll 1$, where $\kappa_{q}=\tilde{J}^{2}\sin^{2}q$. In this region, we obtain from (II.4)

$\displaystyle|u_{q}(\infty)|^{2}$

$\displaystyle\simeq\exp\biggl(-4\pi\kappa_{q}\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}J_{n}\quantity(\eta)J_{m}\quantity(\eta)\cos\quantity(\tilde{\omega}\quantity(\frac{1}{2}(n^{2}-m^{2})\tilde{\omega}-(n-m)(\tilde{\varepsilon}^{\prime}+2\tilde{J}\cos q)))\theta(n-m)\biggr{missing}),$

(50)

where we define $\eta=A/\omega$. In the adiabatic limit where $\tilde{J}$ is sufficiently large, there is a non-adiabatic region only near $q=0,\pm\pi$. In the vicinity of $q=0$, we obtain

	$\displaystyle\quantity|u_{q}(\infty)|^{2}$	$\displaystyle\simeq\exp\quantity(-2\pi\tilde{J}^{2}q^{2}-4\pi\tilde{J}^{2}q^{2}\sum_{n>m}J_{n}\quantity(\eta)J_{m}\quantity(\eta)\cos\quantity(\tilde{\omega}\quantity(\frac{1}{2}(n^{2}-m^{2})\tilde{\omega}-(n-m)(\tilde{\varepsilon}^{\prime}+2\tilde{J}))))$		(51)
		$\displaystyle=e^{-\alpha q^{2}},$		(52)

and in the vicinity of $q=\pm\pi$, we get

	$\displaystyle\quantity|u_{q}(\infty)|^{2}$	$\displaystyle\simeq\exp\biggl(-2\pi\tilde{J}^{2}(q\mp\pi)^{2}-4\pi\tilde{J}^{2}(q\mp\pi)^{2}\sum_{n>m}J_{n}\quantity(\eta)J_{m}\quantity(\eta)\cos\quantity(\tilde{\omega}\quantity(\frac{1}{2}(n^{2}-m^{2})\tilde{\omega}-(n-m)(\tilde{\varepsilon}^{\prime}-2\tilde{J})))\biggr{missing})$		(53)
		$\displaystyle=e^{-\beta(q\mp\pi)^{2}}.$		(54)

We note that $|u_{q}(\infty)|^{2}$ has the finite value near $q=0,\pm\pi$ in this region if $\alpha,\beta\propto\tilde{J}^{2}$ is large enough.

Next, in the adiabatic region $\kappa_{q}\gg 1$, we obtain

	$\displaystyle|u_{q}(\infty)|^{2}$	$\displaystyle\simeq\pi^{2}\eta^{2}e^{-2\pi\kappa_{q}}\biggl{|}{}_{1}\tilde{F}_{1}\quantity(-i\kappa_{q};0;i\tilde{\omega}^{2})\biggr{|}^{2}$		(55)
		$\displaystyle=:P_{\mathrm{FP}}(q)$		(56)

from (33).

The distribution of $|u_{q}(\infty)|^{2}$ is shown in Fig. 5. This figure shows that in addition to the transitions at $q=0,\pi$ predicted by the KZ mechanism, other transitions occur around them, which is the result from the time-periodic perturbation. We note that if $\tilde{J}$ is larger, the transitions in the adiabatic region occur closer to these vicinities as long as $\tilde{\omega}$ and $\eta$ are fixed.

From the above discussion, we obtain the expectation value approximately

	$\displaystyle n(\infty)$	$\displaystyle\simeq\int_{-\infty}^{\infty}\frac{dq}{2\pi}\quantity(e^{-\alpha q^{2}}+e^{-\beta q^{2}})+\int_{-\pi}^{\pi}\frac{dq}{2\pi}P_{\mathrm{FP}}(q)$		(57)
		$\displaystyle=\frac{1}{2\sqrt{\pi}}\quantity(\frac{1}{\sqrt{\alpha}}+\frac{1}{\sqrt{\beta}})+n_{\mathrm{FP}},$		(58)
	$\displaystyle n_{\mathrm{FP}}$	$\displaystyle=\int_{-\pi}^{\pi}\frac{dq}{2\pi}P_{\mathrm{FP}}(q)$		(59)

From (58), we see that the first term is proportional to $\tilde{J}^{-1}$. This corresponds to the part of the QKZM where perturbative oscillatory effects are added to the non-adiabatic transition. The second term $n_{\mathrm{FP}}$ is the contribution from the non-perturbative effect. In fact, this term is also proportional to $\tilde{J}^{-1}$. This can be seen as follows. First, for sufficiently large $\tilde{J}$, $P_{\mathrm{FP}}(q)$ has a value of only $q\simeq 0,\pm\pi$, so $n_{\mathrm{FP}}$ can be transformed to

$\displaystyle n_{\mathrm{FP}}$

$\displaystyle\simeq 2\pi\eta^{2}\int_{0}^{\pi/2}dq\ e^{-2\pi\tilde{J}^{2}q^{2}}\quantity|{}_{1}\tilde{F}_{1}\quantity(-i\tilde{J}^{2}q^{2},0,i\tilde{\omega}^{2})|^{2}.$

(60)

Transforming to $\tilde{J}q=x$ and changing the upper bound of the integral to $\infty$ because the integrand function has no value at $q=\pi/2$, we obtain

$\displaystyle n_{\mathrm{FP}}$

$\displaystyle\simeq\frac{2\pi\eta^{2}}{\tilde{J}}\int^{\infty}_{0}dx\ e^{-2\pi x^{2}}\quantity|{}_{1}\tilde{F}_{1}\quantity(-ix^{2},0,i\tilde{\omega}^{2})|^{2}.$

(61)

This approximation and the original definition (59) are plotted in Fig. 6. The fact that both agree where $\tilde{J}$ is large indicates that this approximation is correct and that the non-perturbative contribution also shows a scaling of $\tilde{J}^{-1}$. It can also be seen numerically that the peak of $n_{\mathrm{FP}}$ appears where $\tilde{\omega}=2\tilde{J}$ is satisfied. This can be interpreted as a result of the resonance phenomenon. Fig. 7 shows the dependence of the coefficient of $\tilde{J}^{-1}$ in (61) on $\tilde{\omega}$. It can be seen that the contribution of this non-perturbative term increases as the frequency and the amplitude increases.

To confirm that the derived equation (58) is correct as an approximation, we finally check it with numerical calculation when $N$ is finite as shown in the Fig. 8. Because the contribution of $n_{\mathrm{FP}}$ is large, it can be seen that the density of defects behaves differently from the case of no oscillations. As discussed, however, the scaling of $\tilde{J}^{-1}\propto\sqrt{v}$ does not change because this is the same in the non-adiabatic and adiabatic regions. This means that the QKZM is robust to the time-periodic perturbations. To verify that the finite $N$ discussed here is sufficiently large, we compared the integral (59) with the sum of (56) and the result is shown in Fig. 9. It can be seen that $N=200$ is sufficient to be regarded as the thermodynamic limit.

Next, we consider another time-dependent Hamiltonian

	$\displaystyle H_{O}(t)$	$\displaystyle=-\sum^{N}_{j=1}\quantity(J(t)\sigma^{x}_{j}\sigma^{x}_{j+1}+g(t)\sigma^{z}_{j}),$		(62)
	$\displaystyle g(t)$	$\displaystyle=\frac{1}{4}\quantity(vt+\varepsilon^{\prime}),$		(63)
	$\displaystyle J(t)$	$\displaystyle=\frac{1}{2}\Delta^{\prime}+\frac{B^{\prime}}{4}\cos\omega t,$		(64)

where we impose the periodic boundary condition.

As before, introducing spinless fermions by the JW transformation yields

			(65)
			(66)

First, we consider a non-adiabatic region: $\kappa_{q}=\tilde{\Delta}^{\prime 2}\sin^{2}q\ll 1$. In this region, the transition amplitude becomes

$\displaystyle|u_{q}(\infty)|^{2}$

$\displaystyle\simeq$,

where we define $\eta_{B}=B^{\prime}/\omega$. We note that this expression becomes the same with (50) in the adiabatic limit if the amplitude is small enough. On the other hand, in the adiabatic region $\kappa_{q}=\tilde{\Delta}^{\prime 2}\sin^{2}q\gg 1$, we obtain

$\displaystyle|u_{q}(\infty)|^{2}$

$\displaystyle\simeq\pi^{2}e^{-2\pi\kappa_{q}}\biggr{|}\eta_{B}\cos q{}_{1}\tilde{F}_{1}\quantity(-i\kappa_{q},0,i\tilde{\omega}^{2})-i\tilde{B}^{\prime}\sin q\frac{\sqrt{\kappa_{q}}}{2}\biggl{(}{}_{1}\tilde{F}_{1}\left(-i\kappa_{q},1,i\tilde{\omega}^{2}\right)+{}_{1}\tilde{F}_{1}\left(-i\kappa_{q}+1,1,i\tilde{\omega}^{2}\right)\biggr{)}\biggr{|}^{2}.$

(67)

These expressions of $|u_{q}(\infty)|^{2}$ have values only at $q\simeq 0,\pm\pi$ if $\tilde{\Delta}^{\prime}$ is sufficiently large (Fig. 10), which is easy to see from the asymptotic expansion of $\kappa_{q}$. Furthermore, the first term in the absolute value in (67) is the largest contribution compared to the others because we focus on the regions at $q\simeq 0,\pm\pi$. This shows that the integral of $|u_{q}(\infty)|^{2}$ scales with $\tilde{\Delta}^{\prime-1}$, as in $H_{\mathrm{D}}(t)$. From the above discussion, the analytical expression for the density of defects $n(\infty)$ can be expressed as in (58).

We check it with a numerical calculation when $N$ is finite. In the Fig. 11, we compared the density of defects obtained by numerically solving Schrödinger equation with an approximate analytical expression. As in the previous subsection, $N=200$ can be regarded as the thermodynamic limit. This figure shows that the density of defects behaves differently compared to the case without oscillations. However, the scaling of the non-perturbative contribution is $\tilde{\Delta}^{\prime-1}\propto\sqrt{v}$, which is not different from the scaling predicted by the QKZM, indicating the robustness of the scaling. In addition, the resonance phenomenon was observed in the diagonal oscillation, but when there was oscillation in the off-diagonal elements, the resonance phenomenon was canceled out by the contribution of the second term in the absolute value of (67).