The hierarchy recurrences in local relaxation

One useful way to look through this problem is to study the system relaxation contacted with a finite bath, namely, the bath contains a finite number of degrees of freedom (DoF), and then consider its transition to the thermodynamics limit (Cramer et al., 2008; Flesch et al., 2008; Eisert et al., 2015; Xu et al., 2014). The open system and the bath as a whole isolate system always follows the unitary evolution and keeps a constant entropy as the initial state, while the open system itself seems relaxing towards a certain steady state, thus such a relaxation behavior of the open system itself is called the local relaxation (Cramer et al., 2008; Eisert et al., 2015; Flesch et al., 2008).

Due to the finite-size effect of the bath, the local relaxation of the open system would come across a recurrence after a certain time: at first the system dynamics shows a well-ordered oscillatory decay behavior, but then suddenly becomes random (Hänggi et al., 1990; Zwanzig, 2001; Cramer et al., 2008; Flesch et al., 2008; Eisert et al., 2015). With the increase of the DoF number in the bath, such a recurrence time appears much later, thus it does not show up in practice.

In this paper, we find that, in the region after such a recurrence, indeed there exists a hierarchy structure hiding in the randomness: similar recurrences appear in a periodical way, and the later recurrence brings in stronger randomness than the previous one, therefore, we call them hierarchy recurrences.

Here we study the dynamics of a chain of n two-level systems (TLSs). One of the TLSs is treated as the open system, and all the other $(\text{{n}}-1)$ TLSs make up a finite bath. We obtain a Bessel function expansion for the system dynamics, which well explains the appearance of such hierarchy recurrences. Further, we also find the physical reason for the appearance of such hierarchy recurrences: with the time increases, the population of the open system diffuses out and propagates in the finite bath (the periodic TLS chain); once the population regathers back to the open system, the system dynamics exhibits such a recurrence, and this process happens again and again, which gives rise to the hierarchy recurrences.

We also study the dynamics of the total correlation entropy of the n-body system, which sums up the entropy of all the n TLSs (Watanabe, 1960; Groisman et al., 2005; Zhou, 2008; Anza et al., 2020). It turns out the total correlation approximately exhibits a monotonic increasing behavior, and the increasing curve becomes more and more “smooth” with the increase of the bath size. Thus, the total correlation exhibits a quite similar behavior as the irreversible entropy increase in the standard thermodynamics (Lebowitz, 1993; Li, 2017; You and Li, 2018; Li, 2019). In contrast, the whole n-body system always keeps a constant due to the unitary evolution, and the entropy of each single TLS increases and decreases from time to time.

Here site-0 is regarded as an open “system”, while all the other $(\text{{n}}-1)$ TLSs build up a finite “bath”. Initially, the “system” (site-0) starts from the excited state as its initial state, and all the TLSs in the “bath” start from the ground state. Thus effectively the “bath” has a temperature $T\rightarrow 0^{+}$. And now we study the dynamics of the open “system”.

The Hamiltonian (1) is a quantum XX model (Sachdev, 2011), and the dynamics of the whole chain is exactly solvable. Applying the Jordan-Wigner transform, the Hamiltonian (1) becomes a fermionic one,

	$\displaystyle\sigma_{n}^{z}$	$\displaystyle=2\hat{c}_{n}^{\dagger}\hat{c}_{n}-1,\quad\hat{\sigma}_{n}^{+}=\hat{c}_{n}^{\dagger}\,\prod_{i=0}^{n-1}(-\hat{\sigma}_{i}^{z}),$
	$\displaystyle\hat{\mathcal{H}}$	$\displaystyle=\sum_{n=0}^{\text{{n}}-1}\omega\hat{c}_{n}^{\dagger}\hat{c}_{n}+g(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+\hat{c}_{n+1}^{\dagger}\hat{c}_{n}).$		(2)

Under the periodic boundary condition, it can be further diagonalized by the Fourier transform $\hat{c}_{n}=\sum_{k=0}^{\text{{n}}-1}\exp(i\frac{2\pi}{\text{{n}}}nk)\,\hat{b}_{k}\big{/}\sqrt{\text{{n}}}$, which reads $\hat{\mathcal{H}}=\sum\varepsilon_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}$, with the eigen mode energy $\varepsilon_{k}=\omega+2g\cos\frac{2\pi k}{\text{{n}}}$.

The n-body chain as a whole isolated system follows the unitary evolution. From the above transformations, the above initial condition gives $\langle\hat{c}_{0}^{\dagger}\hat{c}_{0}\rangle_{(t=0)}=1$, and $\langle\hat{c}_{m}^{\dagger}\hat{c}_{n}\rangle_{(t=0)}=0$ for the other $m,n$, and that gives the following dynamics

	$\displaystyle\langle\hat{c}_{m}^{\dagger}\hat{c}_{n}\rangle_{(t)}$	$\displaystyle=\sum_{k,q=0}^{\text{{n}}-1}\frac{1}{\text{{n}}}e^{i\frac{2\pi}{\text{{n}}}nq-i\frac{2\pi}{\text{{n}}}mk}\big{\langle}\hat{b}_{k}^{\dagger}(0)e^{i\varepsilon_{k}t}\cdot\hat{b}_{q}(0)e^{-i\varepsilon_{q}t}\big{\rangle}$
		$\displaystyle=\sum_{kq,xy}\frac{\langle\hat{c}_{x}^{\dagger}\hat{c}_{y}\rangle_{(0)}}{\text{{n}}^{2}}e^{i\frac{2\pi q}{\text{{n}}}(n-y)-i\frac{2\pi k}{\text{{n}}}(m-x)+i(\varepsilon_{k}-\varepsilon_{q})t}$
		$\displaystyle:=\big{[}\Phi_{m}^{(\text{{n}})}(2gt)\big{]}^{*}\,\Phi_{n}^{(\text{{n}})}(2gt),$		(3)

where we call $\Phi_{n}^{(\text{{n}})}(2gt:=\tau)$ as the coherence function, and¹¹1Utilizing $\exp[-i\tau\cos x]=\sum_{n=-\infty}^{\infty}(-i)^{n}J_{n}(\tau)e^{\pm inx}$.

	$\displaystyle\Phi_{n}^{(\text{{n}})}(\tau):=$	$\displaystyle\frac{1}{\text{{n}}}\sum_{k=0}^{\text{{n}}-1}\exp\big{[}-i\tau\,\cos\frac{2\pi k}{\text{{n}}}+i\frac{2\pi}{\text{{n}}}kn\big{]}$		(4a)
	$\displaystyle\stackrel{{\scriptstyle\text{{n}}\rightarrow\infty}}{{\longrightarrow}}$	$\displaystyle\int_{0}^{2\pi}\frac{dx}{2\pi}\,e^{-i\tau\cos x+inx}=(-i)^{n}J_{n}(\tau).$		(4b)

In the thermodynamics limit $\text{{n}}\rightarrow\infty$, $\Phi_{n}^{(\text{{n}})}(\tau)$ becomes the Bessel function $J_{n}(\tau)$, which approaches zero when $\tau\rightarrow\infty$ (Hänggi et al., 1990; Zwanzig, 2001; Cramer et al., 2008; Flesch et al., 2008; Eisert et al., 2015).

It can be seen from Eqs. (2, 3) that, each site always keeps a diagonal density state $\hat{\rho}_{n}(t)=p_{n,\mathtt{e}}(t)|\mathtt{e}\rangle_{n}\langle\mathtt{e}|+p_{n,\mathtt{g}}(t)|\mathtt{g}\rangle_{n}\langle\mathtt{g}|$, where $p_{n,\mathtt{e}}(t):=\langle\hat{\sigma}_{n}^{+}\hat{\sigma}_{n}^{-}\rangle_{(t)}=\langle\hat{c}_{n}^{\dagger}\hat{c}_{n}\rangle_{(t)}$ is the excited population of site-$n$. Therefore, if the “bath” is infinitely large ($\text{{n}}\rightarrow\infty$), the “system” would reach and stay at the ground state after long time relaxation, namely, $p_{0,\mathtt{e}}(t)=|J_{0}(2gt)|^{2}\rightarrow 0$ (here the limit $\text{{n}}\rightarrow\infty$ is taken before $t\rightarrow\infty$).

Therefore, based on the local observation within a finite time smaller than $t_{\text{rec}}$, we may conclude the open “system” itself is relaxing towards a certain steady state, but indeed the full n-body state always keeps a pure state during the unitary evolution. With the increase of the size n, the recurrence time $t_{\text{rec}}\equiv\text{{n}}/2g$ becomes larger and larger, thus such a recurrence behavior does not show up in practice.

In Fig. 1(c), the scaling behavior of $\mathrm{Re}\big{[}\Phi_{0}^{(\text{{n}})}(\tau)\big{]}$ for different sizes n is shown. Besides the above recurrence appearing around $t\simeq t_{\text{rec}}\equiv\text{{n}}/2g$, it is worth noting that some well-organized recurrence patterns also appear in the region $t\apprge t_{\text{rec}}$. It can be seen similar recurrences also appear periodically around $t\simeq\mathtt{q}\,t_{\text{rec}}$ for $\mathtt{q}=2,3,4,\dots$ [see the arrows in Fig. 1(c)]. Moreover, each recurrence seems bringing in stronger randomness to $\Phi_{0}^{(\text{{n}})}(\tau)$ than the previous one, which forms a hierarchy structure, thus we call them hierarchy recurrences.

We find that the appearance of such hierarchy recurrences can be explained by the following expansion of $\Phi_{n}^{(\text{{n}})}(\tau)$ [Eq. (4a)], that is${}^{\ref{foot-Bessel}}$,

	$\displaystyle\Phi_{n}^{(\text{{n}})}(\tau)=$	$\displaystyle\frac{1}{\text{{n}}}\sum_{k=0}^{\text{{n}}-1}\big{[}\sum_{m=-\infty}^{\infty}(-i)^{m}J_{m}(\tau)e^{-im\cdot\frac{2\pi k}{\text{{n}}}}\big{]}\cdot e^{i\frac{2\pi}{\text{{n}}}kn}$
	$\displaystyle=$	$\displaystyle\sum_{m=-\infty}^{\infty}(-i)^{m}J_{m}(\tau)\cdot\frac{1}{\text{{n}}}\sum_{k=0}^{\text{{n}}-1}e^{i\frac{2\pi k}{\text{{n}}}(n-m)}$
	$\displaystyle=$	$\displaystyle\sum_{\mathtt{q}=-\infty}^{\infty}(-i)^{n+\mathtt{q}\text{{n}}}J_{n+\mathtt{q}\text{{n}}}(\tau).$		(5)

Here we used the relation $\sum_{k=0}^{\text{{n}}-1}e^{i\frac{2\pi k}{\text{{n}}}(n-m)}=\text{{n}}\delta_{n-m,\,\mathtt{q}\text{{n}}}$, with $\mathtt{q}$ as an arbitrary integer.

For example, site-0 ($n=0$) gives a simple Bessel function series [using $J_{-n}(\tau)=(-1)^{n}J_{n}(\tau)$]³³3When $\text{{n}}\rightarrow\infty$, the function series $\big{\{}\Phi_{0}^{(\text{{n}})}(\tau)\big{\}}$ converges pointwise to $J_{0}(\tau)$ but not uniformly.

	$\displaystyle\Phi_{0}^{(\text{{n}})}(\tau)=$	$\displaystyle J_{0}(\tau)+(-i)^{\text{{n}}}[1+(-1)^{\text{{n}}}]J_{\text{{n}}}(\tau)$
		$\displaystyle+(-i)^{2\text{{n}}}[1+(-1)^{2\text{{n}}}]J_{2\text{{n}}}(\tau)+\dots$		(6)

For a large n, the Bessel function $J_{\text{{n}}}(\tau)\simeq 0$ in the area $0\leq\tau\apprle\text{{n}}$, and starts to exhibit significant oscillations after $\tau\simeq\text{{n}}$ [see $J_{100}(\tau)$ in Fig. 1(b)]. Therefore, in the above expansion of $\Phi_{0}^{(\text{{n}})}(\tau)$, each term $J_{\mathtt{q}\text{{n}}}(\tau)$ contributes a “sudden bump” around $\tau\simeq\mathtt{q}\text{{n}}$, and this is just why the above recurrences appear around $t\simeq\mathtt{q}\,t_{\text{rec}}$ ($\mathtt{q}=1,2,3,\dots$).

The two-side propagations would meet each other at the periodic boundaries at $n\sim\pm\text{{n}}/2$, and then regathers back to site-0 again. Notice that this is just the moment that $\Phi_{0}^{(\text{{n}})}(2gt)$ exhibits its first recurrence ($t\simeq t_{\text{rec}}\equiv\text{{n}}/2g$, see the dashed vertical lines in Fig. 2). The propagation regathered back would be superposed with the original one, and that makes the system dynamics appear more random. Clearly, since such propagation and regathering happens again and again, the “system” (site-0) experiences the above hierarchy recurrences periodically around $t\simeq\mathtt{q}\,t_{\text{rec}}$.

Notice that, in practical observations, indeed the full n-body state is usually not directly accessible for local measurements, and it is the few-body observables that can be directly measured (Swendsen, 2008; Li, 2019; Strasberg, 2019). Therefore, here we consider the dynamics of the total correlation entropy of the n-body state $\hat{\boldsymbol{\rho}}(t)$, that is (Watanabe, 1960; Groisman et al., 2005; Zhou, 2008; Anza et al., 2020),

$$\mathbb{C}_{\text{{t}}}[\hat{\boldsymbol{\rho}}(t)]:=\sum_{n=0}^{\text{{n}}-1}S[\hat{\rho}_{n}(t)]-S[\hat{\boldsymbol{\rho}}(t)],$$

(7)

where $\hat{\rho}_{n}$ are the reduced 1-body states. For bipartite systems ($\text{{n}}=2$), it just returns the mutual information (Nielsen and Chuang, 2000; Li, 2017, 2019; You and Li, 2018).

It turns out, the entropy of each single TLS is increasing and decreasing from time to time, and they also have the above recurrence behavior [Fig. 3(c, d)]. In contrast, their summation as the total correlation $\mathbb{C}_{\text{{t}}}(t)$ approximately exhibits a monotonic increasing behavior [except still carrying small fluctuations, see Fig. 2(e)]. Moreover, with the increase of the chain size n, the increasing curve of $\mathbb{C}_{\text{{t}}}(t)$ appears more and more “smooth” [Fig. 3(a-c)]. Clearly, this is quite similar with the behavior of the irreversible entropy production during the relaxation process in the standard thermodynamics (Spohn, 1978; de Groot and Mazur, 1962; Kondepudi and Prigogine, 2014; Nicolis and Prigogine, 1977).

Now we consider the correlation maximum that $\mathbb{C}_{\text{{t}}}(t)$ might achieve (Jaynes, 1957). With the help of Lagrangian multipliers, under the constraints (1) $p_{n,\mathtt{g}}+p_{n,\mathtt{e}}=1$ (probability normalization) and (2) $\sum_{n}p_{n,\mathtt{e}}=1$ (excitation number conservation), the maximum of $\mathbb{C}_{\text{{t}}}\big{[}\{p_{n,\mathtt{e}(\mathtt{g})}\}\big{]}=\sum_{n}-p_{n,\mathtt{e}}\ln p_{n,\mathtt{e}}-p_{n,\mathtt{g}}\ln p_{n,\mathtt{g}}$ is obtained as

$$\mathbb{C}_{\text{max}}^{(\text{{n}})}=\text{{n}}\ln\frac{\text{{n}}}{\text{{n}}-1}+\ln(\text{{n}}-1).$$

(8)

The maximum is achieved when all the n TLSs have the same populations $\tilde{p}_{n,\mathtt{e}}=1/\text{{n}}$.

Under the scaled time $2gt/\text{{n}}$, the correlation evolutions $\mathbb{C}_{\text{{t}}}(t)/\text{{n}}$ for different sizes n appear quite similar to each other [Fig. 3(a-c)]. They all approach their upper bound $\mathbb{C}_{\text{max}}^{(\text{{n}})}/\text{{n}}$ closely, and come across a “sudden bump” around half of the recurrence time $t\simeq t_{\text{rec}}/2$ (indeed this is just the moment the two-side propagations meet each other at $n\sim\pm\text{{n}}/2$).

We denote $\eta_{\text{{n}}}:=1-\text{max}\{\mathbb{C}_{\text{{t}}}(t)\}\big{/}\mathbb{C}_{\text{max}}^{(\text{{n}})}$ as the relative error between $\text{max}\{\mathbb{C}_{\text{{t}}}(t)\}$ and the correlation maximum $\mathbb{C}_{\text{max}}^{(\text{{n}})}$. With the increase of the size n, the error $\eta_{\text{{n}}}$ decays slowly towards zero [approximately $\eta_{\text{{n}}}\propto\text{{n}}^{-\alpha}$ with $\alpha\simeq 0.062$, see Fig. 3(d)]. In the thermodynamic limit $\text{{n}}\rightarrow\infty$, we may expect $\mathbb{C}_{\text{{t}}}(t)$ could reach the maximum $\mathbb{C}_{\text{max}}^{(\text{{n}})}$.

In this sense, the above correlation maximization effectively gives a pseudo-equilibrium state $\tilde{\boldsymbol{\rho}}_{\text{eq}}\equiv{\textstyle\bigotimes_{n}}\,\tilde{\varrho}_{n}$, where $\tilde{\varrho}_{n}:=\frac{1}{\text{{n}}}|\mathtt{e}\rangle_{n}\langle\mathtt{e}|+(1-\frac{1}{\text{{n}}})|\mathtt{g}\rangle_{n}\langle\mathtt{g}|$, and the whole n-body state $\hat{\boldsymbol{\rho}}(t)$ “looks” like approaching this pseudo-equilibrium state during the unitary evolution (Cramer et al., 2008). But we emphasize indeed $\hat{\boldsymbol{\rho}}(t)$ and $\hat{\rho}_{n}(t)$ never have any steady states when $t\rightarrow\infty$, and $\hat{\boldsymbol{\rho}}(t)$ always keeps a pure state.

The increasing rate of the above total correlation (7) also can be rewritten in the form of relative entropy (Spohn, 1978; Esposito et al., 2010; Manzano et al., 2016; Ptaszyński and Esposito, 2019)

$$\partial_{t}\mathbb{C}_{\text{{t}}}(t)=\partial_{t}D[\hat{\boldsymbol{\rho}}(t)\parallel{\textstyle\bigotimes_{n}}\,\hat{\rho}_{n}(t)],$$

(9)

where $D[\rho\|\varrho]=\mathrm{tr}[\rho(\ln\rho-\ln\varrho)]$ is the relative entropy. Approximately, the reference state ${\textstyle\bigotimes_{n}}\,\hat{\rho}_{n}(t)$ here can be replaced by the pseudo-equilibrium state $\tilde{\boldsymbol{\rho}}_{\text{eq}}\equiv{\textstyle\bigotimes_{n}}\,\tilde{\varrho}_{n}$.

We emphasize that the pseudo-equilibrium state $\tilde{\boldsymbol{\rho}}_{\text{eq}}$ here is determined by the above correlation maximization, but irrelevant with the on-site energy $\omega$ and the interaction strength $g$, thus it is different from the canonical state like $\hat{\boldsymbol{\rho}}_{\text{th}}\sim\exp[-\hat{\mathcal{H}}/T]$. If the on-site energy $\omega\leq 0$, all the above results for the system dynamics still remains the same.

Moreover, when $\omega<0$, the status of $|\mathtt{g}\rangle_{n}$ and $|\mathtt{e}\rangle_{n}$ are indeed reversed: initially, the open “system” starts from the ground state while the TLSs in the “bath” start from the excited state. Therefore, effectively the “bath” has a negative temperature $T\rightarrow 0^{-}$ (Landau and Lifshitz, 1980; Ramsey, 1956). Notice that all the above results of the total correlation entropy still applies in this situation.

Acknowledgment – S.-W. Li appreciates for the helpful discussions with J. Chen, B. Garraway, N. Wu, D. Z. Xu, Y.-N. You. This study is supported by NSF of China (Grant No. 11905007), Beijing Institute of Technology Research Fund Program for Young Scholars.