Anomalous Enhancement of Entanglement Entropy in Nonequilibrium Steady States Driven by Zero-Temperature Reservoirs

TL;DR

This paper studies how entanglement entropy in a one-dimensional quantum wire with impurities grows unexpectedly fast in nonequilibrium steady states driven by zero-temperature reservoirs, revealing a quasi volume law.

Contribution

It uncovers an anomalous growth of entanglement entropy in nonequilibrium conditions, deviating from the typical logarithmic law at equilibrium.

Findings

Entanglement entropy exhibits a quasi volume law in NESS.

Anomalous EE growth is caused by far from equilibrium conditions and impurity scatterings.

EE growth is faster than at equilibrium, indicating enhanced quantum correlations.

Abstract

We investigate the size scaling of the entanglement entropy (EE) in nonequilibrium steady states (NESSs) of a one-dimensional open quantum system with a random potential. It models a mesoscopic conductor, composed of a long quantum wire (QWR) with impurities and two electron reservoirs at zero temperature. The EE at equilibrium obeys the logarithmic law. However, in NESSs far from equilibrium the EE grows anomalously fast, obeying the `quasi volume law,' although the conductor is driven by the zero-temperature reservoirs. This anomalous behavior arises from both the far from equilibrium condition and multiple scatterings due to impurities.

Tables

Table 1. Table 1: Typical value of R L W superscript subscript 𝑅 𝐿 𝑊 R_{L}^{W} (and R ~ L W superscript subscript ~ 𝑅 𝐿 𝑊 \tilde{R}_{L}^{W} )

	$W=0$	$W>0$
$q=O\left(1/L\right)$	$O\left(L^2\right)$	$O\left(L^2\right)$
$p=O\left(1/L\right)$	$O\left(1\right)$	$O\left(L^2\right)$

Equations

$$S_L=\eta (L)L|\Delta k_F|+O(\ln L)\text{mbox}for1\ll L\le L_{\mathrm{C}}.$$

$$a\le \eta (L)\lesssim 2a\text{mbox}for1\ll L\le L_{\mathrm{C}},$$

$${\hat{H}}_{\mathrm{tot}}:=-\sum _x({\hat{c}}_x^{†}{\hat{c}}_{x+1}+h.c.)+\sum _{|x|\le L_{\mathrm{C}}/2}{v}_x{\hat{c}}_x^{†}{\hat{c}}_x.$$

$$\Delta k_F=O(1)\gg \Delta k_{\mathrm{peak}}$$

$$\delta N_L^2\le S_L\le 1+c(\ln L)\delta N_L^2,$$

$$\delta N_L^2={\iint }_{{\Omega }_{\Delta \mu }}d{k}_{1}d{k}_2{R}_L^W(k_1,k_2).$$

$$R_{L}^{W}(k_{1},k_{2}):=|\Delta J_{L}^{pq}|^{2}\big{/}[16\sin^{2}\left(p/2\right)\sin^{2}\left(q/2\right)],$$

$$\delta N_L^2={\int }_{-ϵ}^{ϵ}{\tilde{R}}_L^W|q|dq+{\int }_{-ϵ}^{ϵ}{\tilde{R}}_L^W|p|dp+O(1).$$

$$\delta N_L^2={\int }_{-ϵ}^{ϵ}{\tilde{R}}_L^W|q|dq+{\int }_{-ϵ}^{ϵ}{\tilde{R}}_L^W|p-\Delta k_F|dp+O(1).$$

$$\eta (L)\simeq [S_L(\Delta \mu )-S_L(\Delta {\mu }^{\prime })]/(\Delta k_F-\Delta k_F^{\prime })L,$$

$$S_{L}=\eta(L_{\rm C})L_{\rm C}|\Delta k_{F}|+O(\ln L)\ \mbox{ for $L>L_{\rm C}$},$$

Full text

These numerous works have studied the EE of the ground states or other energy eigenstates, almost all of which are equilibrium states according to the eigenstate-thermalization hypothesis Jensen and Shankar (1985); Deutsch (1991); Srednicki (1994); Rigol et al. (2008); Kim et al. (2014); D’Alessio et al. (2016); Tasaki (2016); Iyoda et al. (2017). A natural question is how the EE behaves in nonequilibrium states. With regard to the thermalization processes in isolated systems, the time evolution of the EE was studied in terms of condensed matter D’Alessio et al. (2016); Nakagawa et al. (2018); Kim and Huse (2013); Bardarson et al. (2012); Bauer and Nayak (2013); Serbyn et al. (2013a); Nandkishore and Huse (2015); Luitz et al. (2015); Serbyn et al. (2013b), quantum field theories and quantum gravity Calabrese and Cardy (2005); Bhattacharya et al. (2013). In these systems, $S_{L}\leq O(S_{L}^{\rm eq}(E))$ throughout the evolution, where $S_{L}^{\rm eq}(E)$ is the equilibrium entropy at energy $E$ of the system. Regarding NESSs, which are fundamental states in nonequilibrium physics Lax (1960); Simmons and Taylor (1971); Zubarev (1974); Jou et al. (1996); Oono and Paniconi (1998); Shimizu and Yuge (2010); Shimizu (2010); Sagawa and Hayakawa (2011); Lieb and Yngvason (2013); Sasa (2014); Tasaki (2001), their EE was studied for certain systems Eisler and Zimborás (2005); Aschbacher (2007); Hoogeveen and Doyon (2015). As in the case of thermalization processes, it was shown that $S_{L}=O(S_{L}^{\rm eq}(E))$ or $S_{L}=O(\sum_{\nu}S_{L}^{\rm eq}(T^{\nu}_{\rm res}))$, where $T^{\nu}_{\rm res}$ is the temperature of the $\nu$th reservoir. However, this is because the systems of these NESSs are invariant under spatial translation, and consequently the NESSs are basically the boosts of equilibrium states. In contrast, the NESSs observed in common experiments are those of systems with symmetry-breaking scatterings, e.g., by impurities, rough walls, or phonons, which define a particular rest frame. Multiple scatterings by such scatterers make NESSs nontrivial, i.e., much different from the boosts of equilibrium states.

In this letter, we study $S_{L}$ of NESSs in a one-dimensional mesoscopic conductor with impurities Sakaki (1989); Imry (2002); Tilke et al. (2003); Datta (1997); Shimizu and Miyadera (1998); Shimizu and Kato (2000); Blanter and Büttiker (2000); Tasaki (2001), which is a long quantum wire (QWR) connected to two electron reservoirs of zero temperature ($T^{\nu}_{\rm res}=0$). The difference $\Delta\mu:=\mu^{+}-\mu^{-}$ in the chemical potentials $\mu^{\pm}$ of the reservoirs induces a steady current $J$ in the QWR, and a NESS is realized. While $S_{L}^{\rm eq}=O(\ln L)$ at equilibrium, we find that, in nontrivial NESSs far from equilibrium (as defined by (4) below),

[TABLE]

Here, $\Delta k_{F}$ is the difference in the Fermi wavenumbers of the reservoirs, $L_{\rm C}$ is the length of the QWR, and $\eta(L)$ is a function of $L$ with the following properties: When $1\ll L\leq L_{\rm C}$, (i) $\eta(L)$ is independent of $\Delta k_{F}$, (ii) gradually decreases with increase in $L$, (iii) and

[TABLE]

where $a$ is a positive constant independent of $L$ or $\Delta k_{F}$. Since $S_{L}\geq aL|\Delta k_{F}|+O(\ln L)$, we call Eq. (1) the quasi volume law. Consequently, $S_{L}>O(\sum_{\nu}S_{L}^{\rm eq})$ in contrast to $S_{L}\leq O(\sum_{\nu}S_{L}^{\rm eq})$ in the previous cases D’Alessio et al. (2016); Nakagawa et al. (2018); Kim and Huse (2013); Bardarson et al. (2012); Bauer and Nayak (2013); Serbyn et al. (2013a); Nandkishore and Huse (2015); Luitz et al. (2015); Serbyn et al. (2013b); Calabrese and Cardy (2005); Bhattacharya et al. (2013); Eisler and Zimborás (2005); Aschbacher (2007); Hoogeveen and Doyon (2015). Both the far from equilibrium condition and multiple scatterings that break the translational symmetry are necessary for this anomalous enhancement of $S_{L}$.

Setup.— We consider a long QWR (conductor) Sakaki (1989); Imry (2002); Tilke et al. (2003); Datta (1997); Shimizu and Miyadera (1998); Shimizu and Kato (2000); Blanter and Büttiker (2000); Tasaki (2001) connected to two electron reservoirs of zero temperature. Although real reservoirs are usually two-dimensional, the total system can be mapped to a one-dimensional system Shimizu and Miyadera (1998); Shimizu and Kato (2000). If many-body interactions are negligible, its effective Hamiltonian is given by

[TABLE]

Here, $\hat{c}^{\dagger}_{x}$ and $\hat{c}_{x}$ are the creation and annihilation operators of an electron at site $x$ ($\in\mathbb{Z}$), respectively. In the QWR of length $L_{\rm C}$ centered at $x=0$ (see Fig. 1 and Fig. S1 of SM for details), a Gaussian random potential $v_{x}$ of impurities exists (with a vanishing average SM ). Its strength is characterized by the standard deviation $W$ of $v_{x}$.

We require that a single-particle state should be either a scattering state $\varphi_{k}(x)$ (Fig. 1) with incoming wavenumber $k$ ($-\pi<k\leq\pi$) and energy $\varepsilon_{k}=-2\cos{k}$, or a bound state $\varphi_{b}(x)$ with a quantum number $b$ and energy $\varepsilon_{b}$ ($|\varepsilon_{b}|>2$). According to the standard model of mesoscopic conductors Datta (1997); Imry (2002); Tilke et al. (2003); Lesovik (1989); Büttiker (1990); Li et al. (1990); Shimizu and Ueda (1992); Shimizu et al. (1993); Blanter and Büttiker (2000); Landauer (1957, 1987); Büttiker et al. (1985); Shimizu and Miyadera (1998); Shimizu and Kato (2000); SM , the quantum state at zero temperature of the total system is a pure quantum state $|\Psi_{\rm tot}\rangle$ such that $\varphi_{k}(x)$ with $-k_{F}^{-}\leq k\leq k_{F}^{+}$ and $\varphi_{b}(x)$ with $\varepsilon_{b}<-2$ are occupied by electrons SM . Here, $k_{F}^{+}$ ($k_{F}^{-}$) is the Fermi wavenumber of the left (right) reservoir, i.e. $\varepsilon_{k_{F}^{\pm}}=\mu^{\pm}=\overline{\mu}\pm\Delta\mu/2,$ where $\overline{\mu}:=(\mu^{+}+\mu^{-})/2$. Without loss of generality, we assume that $\Delta\mu\geq 0$, and hence $\Delta k_{F}:=k_{F}^{+}-k_{F}^{-}\geq 0$. A NESS is realized when $\Delta\mu>0$.

Entropy.— Assuming $|\Psi_{\rm tot}\rangle$, we explore its EE. We take a subsystem A $:=[-L/2,L/2]$ of length $L$ at the center of the QWR (see Fig. 1 and Fig. S1 of SM for details), and consider the von Neumann entropy $S_{L}:=-\mathrm{Tr}[\hat{\rho}_{L}\ln\hat{\rho}_{L}]$ of the reduced density operator $\hat{\rho}_{L}$ of A. It is known that $S_{L}$ at equilibrium agrees with the thermodynamics entropy 111 This point is a direct consequence of the following facts: (a) If the total system is in an equilibrium state then its subsystem, which is much smaller than the total system, is in the (grand)canonical Gibbs state. (b) The von Neumann entropy of the (grand)canonical Gibbs state agrees with its thermodynamic entropy; see, e.g., L.D Landau, E.M Lifshitz Statistical Physics, Part I (Pergamon Press, 1980). . Since $|\Psi_{\rm tot}\rangle$ is a pure quantum state, $S_{L}$ is also the EE that quantifies the entanglement between A and the rest of the system, either at equilibrium or nonequilibrium.

For each $|\Psi_{\rm tot}\rangle$, which is determined by $k_{F}^{\pm}$ and the impurities, we examine the $L$ dependence of $S_{L}$. We are most interested in $S_{L}$ in the QWR (i.e. $S_{L}$ for $L\leq L_{\rm C}$), in which the quantum state in a NESS differs significantly from that in an equilibrium state.

Nontrivial NESSs.— From the electron-hole symmetry, we can limit ourselves, without loss of generality, to the lower half of the band, $-2\leq\varepsilon_{k}\leq 0$. Furthermore, since we are not interested in any specific effect of the band edge $\varepsilon_{k}=-2$ or the band center $\varepsilon_{k}=0$, we take $-1.7\leq\mu^{\pm}\leq-0.8$.

We exclude the case of a short QWR, $L_{\rm C}\sim 1$, because such a QWR is actually a quantum dot, for which we cannot discuss the $L$ dependence of $S_{L}$ for $L\leq L_{\rm C}$. We therefore study the case of $L_{\rm C}\gg 1$. We take $L_{C}=401$ in the numerical calculations SM .

Since we assume zero temperature, the dimensionless conductance Imry (2002) $G:=(J/\Delta\mu)/(e^{2}/2\pi\hbar)$ (which is a nonlinear one; see below) is simply the average value $G={1\over\Delta\mu}\int_{\mu^{-}}^{\mu^{+}}|t_{k}|^{2}d\varepsilon_{k}$ of the transmittance $|t_{k}|^{2}$ in $\mu^{-}\leq\varepsilon_{k}\leq\mu^{+}$. Obviously, $0\leq G\leq 1$. Since $L_{\rm C}$ is finite and impurities are absent in the reservoirs, the Anderson localization Anderson (1958); Abrahams et al. (1979); Anderson et al. (1980); Lee and Ramakrishnan (1985) is incomplete, i.e., the localization length $\xi$ (defined for the hypothetical case $L_{\rm C}=\infty$) can exceed $L_{\rm C}$.

As $W$ is increased, $\xi$ and $G$ decrease on the whole Anderson (1958); Abrahams et al. (1979); Anderson et al. (1980); Lee and Ramakrishnan (1985). When $L_{\rm C}\gg\xi$, the system would be almost an insulator, and $G\simeq 0$. Hence, almost no current would flow even when finite $\Delta\mu$ is applied. On the other hand, when $L_{\rm C}$ is much shorter than the mean free path $(<\xi)$, the electrons would not suffer scatterings, and $G\simeq 1$. The NESS in this case would be almost a boost of an equilibrium state (even in the presence of interactions between electrons Shimizu (1996)). In these cases, it is obvious that $S_{L}$ scales as in equilibrium. We therefore focus on the intermediate regime where $L_{\rm C}$ is comparable to $\xi$, for which $G$ takes an intermediate value, such as $0.3\lesssim G\lesssim 0.7$. We call the NESSs in this case nontrivial, and take $W$ so as to satisfy this condition.

For such NESSs, multiple scatterings by impurities are crucial, and the wavefunctions $\varphi_{k}(x)$ have complicated shapes as shown in Fig. 1. Since $k$ takes a continuous value, there are infinitely many states, including those nearly localized in the QWR (such as the red one) and those penetrating into the QWR (such as the blue one). Consequently, $|t_{k}|^{2}$ varies rapidly as a function of $k$, as shown in Fig. S2 of SM , where each peak indicates the resonant tunneling through a nearly-localized state.

Far from equilibrium.— Since $-\pi<k\leq\pi$ and the number of resonant states $\simeq L_{\rm C}$, the average distance $\Delta k_{\rm peak}$ between the peaks of $|t_{k}|^{2}$ is roughly $\Delta k_{\rm peak}\simeq 2\pi/L_{\rm C}$. When $\Delta k_{F}\ll\Delta k_{\rm peak}$, the current-voltage characteristic is linear, i.e. $G$ is independent of $\Delta\mu$ (while $G$ depends on $\overline{\mu}$). In this regime, the NESS is close to equilibrium. When $\Delta k_{F}$ is increased to $\Delta k_{F}\sim\Delta k_{\rm peak}$, $G$ depends sensitively on $\Delta\mu$ (and $\overline{\mu}$), as shown in Fig. S3 of SM , because only a small number of peaks in $k_{F}^{-}\leq k\leq k_{F}^{+}$ contribute to the conduction, which reflects the characteristics of the individual peaks. When $\Delta k_{F}$ is further increased to $\Delta k_{F}\gg\Delta k_{\rm peak}$, the dependence on $\Delta\mu$ (and $\overline{\mu}$) becomes weak because many peaks and dips of $|t_{k}|^{2}$ contribute to the conduction. When $\Delta k_{F}=O(1)$, such a regime is always achieved for sufficiently large $L_{\rm C}$ because $\Delta k_{\rm peak}\simeq 2\pi/L_{\rm C}$. We call this regime in which

[TABLE]

far from equilibrium. We will show the quasi volume law (1) for nontrivial NESSs far from equilibrium.

Relation to number and current.— Let $\delta N_{L}^{2}:=\braket{(\hat{N}_{L}-\braket{\hat{N}_{L}})^{2}}$, which is the fluctuation of the particle number $\hat{N}_{L}:=\sum_{|x|\leq L/2}\hat{c}^{\dagger}_{x}\hat{c}_{x}$ in A, where $\braket{\bullet}:=\braket{\Psi_{\rm tot}}{\bullet}{\Psi_{\rm tot}}$. In SM , we prove the inequality

[TABLE]

where $c$ is a positive constant independent of $L$. From this inequality, it is sufficient to show the quasi-volume law for $\delta N_{L}^{2}$ instead of $S_{L}$. We will analyze $\delta N_{L}^{2}$ to see the mechanism and the order of magnitude, and calculate $S_{L}$ numerically to find the magnitude, of the anomalous enhancement.

To calculate $\delta N_{L}^{2}$, we neglect small contributions from the bound states, and use the identity SM :

[TABLE]

Here, the region of the integral $\Omega_{\Delta\mu}$ is such that $k_{1}$ is occupied by an electron while $k_{2}$ is empty. Furthermore,

[TABLE]

where $p:=k_{1}+k_{2}$ and $q:=k_{1}-k_{2}$. Here, $\Delta J_{L}^{pq}:=J_{k_{1}k_{2}}(L/2)-J_{k_{1}k_{2}}(-L/2)$, where $J_{k_{1}k_{2}}(x+1/2)$ is the $k$ representation of the current on the bond at $x+1/2$; $J_{k_{1}k_{2}}(x+1/2)=i[\varphi^{*}_{k_{1}}(x+1)\varphi_{k_{2}}(x)-\varphi^{*}_{k_{1}}(x)\varphi_{k_{2}}(x+1)].$ When $p=0$ or $q=0$, where the denominator of $R_{L}^{W}$ vanishes, the numerator $|\Delta J_{L}^{pq}|^{2}$ also vanishes SM . Consequently, $R_{L}^{W}$ is finite everywhere in $\Omega_{\Delta\mu}$. Identity (6) means that $\delta N_{L}^{2}$ in A is determined by the net current flowing into A through its edges, $x=\pm L/2$.

In the following discussion, identity (6) plays a crucial role. It decomposes the parameter dependence into two, $\Omega_{\Delta\mu}$ and $R_{L}^{W}$, which depend on $(\Delta\mu,\overline{\mu})$ and $(W,L,L_{\rm C})$, respectively. We will show that $R_{L}^{W}$ takes large values of $O(L^{2})$ in certain regions in the $k_{1}$-$k_{2}$ plane, whereas $\Omega_{\Delta\mu}$ determines which parts are extracted from such regions.

Behavior of $R_{L}^{W}$ (summarized in Table 3).— For $q=O(1/L)$, the denominator of Eq. (7) becomes as small as $O(1/L^{2})$, and consequently the typical value of $R_{L}^{W}$ becomes as large as $O(L^{2})$. In contrast, for $p=O(1/L)$, $R_{L}^{W}=O(1)$ when $W=0$, because then the numerator also becomes $O(1/L^{2})$.

When $W>0$, however, the difference between the forward-scattering part (small $|q|$) and the backward-scattering part (small $|p|$) is obscured because the impurities scatter the electrons back and forth. As a result, $R_{L}^{W}$ becomes $O(L^{2})$ not only at $q=O(1/L)$ but also at $p=O(1/L)$, as plotted in Figs. S5 and S6 of SM .

Since $R_{L}^{W}=O(1)$ in the other regions in $\Omega_{\Delta\mu}$, the $L$ dependence of $\delta N_{L}^{2}$ is determined by the integral around $q\simeq 0$ and $p\simeq 0$. We therefore focus on the regions $|q|\leq\epsilon$ and $|p|\leq\epsilon$, where $\epsilon$ is a positive constant of $O(1)$.

Behavior of $\Omega_{\Delta\mu}$ (Table 3).— For an equilibrium state, the region of the integral $\Omega_{\Delta\mu}$ is shown in Fig. 2(a). As discussed above, we focus on the regions $q\simeq 0$ and $p\simeq 0$. Then, the area of the portion of $\Omega_{\Delta\mu}$ between $q$ and $q+dq$ ($p$ and $p+dp$) is $|q|dq$ ($|p|dp$).

For a NESS, $\Omega_{\Delta\mu}$ shifts toward the direction of $+45^{\circ}$, as shown in Fig. 2(b). As a result, the area between $p$ and $p+dp$ becomes $|p-\Delta k_{F}|dp$, while that between $q$ and $q+dq$ remains the same as in the equilibrium case. This means that in nonequilibrium, more contributions are extracted from $R_{L}^{W}$ of small $|p|$.

Clean case (Table 3 [A] and [B]).— When impurities are absent ($W=0$), we evaluate Eq. (6) using Tables 3 and 3, as $\delta N_{L}^{2}=\int_{-\epsilon}^{\epsilon}R_{L}^{W}|q|dq+O(1)$ for either equilibrium ($\Delta\mu=0$) or nonequilibrium ($\Delta\mu>0$). A simple power-counting argument estimates the integral as $O(1/L)$ [interval where $q=O(1/L)$] $\times$ $O(1/L)$ [from $|q|$] $\times$ $O(L^{2})$ [from $R_{L}^{W}$, Table 3] $=O(L^{0})$. Actually, the rigorous argument in Sec.IV of SM gives $\delta N_{L}^{2}=O(\ln L)$. From inequality (5), this indicates the logarithmic law $S_{L}=O(\ln L)$, which is confirmed numerically (Fig. S4 of SM ). This agrees with the previous results for $\Delta\mu=0$ Gioev and Klich (2006); Wolf (2006). The same result holds for $\Delta\mu>0$ (case [B]) because a NESS for $W=0$ is basically the boost of an equilibrium state SM .

Equilibrium with random potential (Table 3 [C]).— When the random potential is introduced ($W>0$), $R_{L}^{W}=O(L^{2})$ not only at $q\simeq 0$ but also at $p\simeq 0$ (Table 3). Hence, unlike the clean case, both the regions contribute to the $L$ dependence of $\delta N_{L}^{2}$. Then, for $\Delta\mu=0$, we evaluate Eq. (6) using Table 3, as

[TABLE]

Here, $\tilde{R}_{L}^{W}$ is the average of $R_{L}^{W}$ over $k_{1}$ around $q,p\simeq 0$, which has the same typical value as $R_{L}^{W}$ (Table 3). Since both the integrals give $O(\ln L)$ for the same reason as that of the clean case, we again have $\delta N_{L}^{2}=O(\ln L)$ and $S_{L}=O(\ln L)$ (see SM for details). This is demonstrated in Fig. 3 (red symbols) and Fig. S4 of SM .

Nontrivial NESSs (Table 3 [D]).— Figure 3 also shows that, with increasing $\Delta\mu$, $S_{L}$ grows considerably, and it cannot be fitted by a linear function of $\ln L$. This happens by the following mechanism.

When $\Delta\mu>0$, the area between $p$ and $p+dp$ becomes $|p-\Delta k_{F}|dp$ (Table 3). Hence, Eq, (8) changes to

[TABLE]

The first integral gives $O(\ln L)$ for the same reason as those of cases [A]-[C]. For the second integral, we can rewrite it, by taking $\epsilon=O(1)<\Delta k_{F}$, as $\Delta k_{F}\int_{-\epsilon}^{\epsilon}\tilde{R}_{L}^{W}dp+\int_{-\epsilon}^{0}\tilde{R}_{L}^{W}|p|dp-\int_{0}^{\epsilon}\tilde{R}_{L}^{W}|p|dp.$ The last two terms give $O(\ln L)$ as before (though expected to cancel each other out after the random average). In contrast, the integral $\int_{-\epsilon}^{\epsilon}\tilde{R}_{L}^{W}dp$ basically gives $O(L)$, because $O(1/L)$ [interval where $p=O(1/L)$] $\times$ $O(L^{2})$ [from $\tilde{R}_{L}^{W}$, Table 3] $=O(L)$.

This indicates that $S_{L}=O(L)$. However, there is a slight correction because the backward scatterings become weaker with increasing $L$, i.e. as the edges ($x=\pm L/2$) of A approach the edges ($x=\pm L_{\rm C}/2$) of the QWR. Consequently, the typical value of $R_{L}^{W}$ grows slightly slower than $O(L^{2})$ with increasing $L$, as shown in Figs. S5 and S6 of SM . By representing this effect with a gradually decreasing function $\eta(L)$, we arrive at Eq. (1).

To confirm the quasi volume law, we investigate whether $\eta(L)$ has properties (i)-(iii), mentioned earlier around inequality (2), for nontrivial NESSs far from equilibrium. We note that the $O(\ln L)$ term in the r.h.s. of Eq. (1) should be relatively insensitive to $\Delta k_{F}$. Hence, by subtracting Eq. (1) at $\Delta\mu$ from that at $\Delta\mu^{\prime}$, where $|\Delta\mu-\Delta\mu^{\prime}|\ll 1$, we expect

[TABLE]

for $1\ll L\leq L_{\rm C}$. We plot the r.h.s. of Eq, (10) for various values of $(\Delta\mu,\Delta\mu^{\prime})$ in Fig. 4. When $(\Delta\mu,\Delta\mu^{\prime})$ is small (close to equilibrium), none of the properties (i)-(iii) is satisfied 222 When $\Delta k_{F}\lesssim\Delta k_{\rm peak}$, the integral $\int\tilde{R}_{L}^{W}dp\propto(\Delta k_{F})^{2}$ as $\Delta k_{F}\to 0$, and hence the r.h.s. of Eq. (10) should decrease with decreasing $\Delta k_{F}$, as seen from Fig. 4. . However, when $(\Delta\mu,\Delta\mu^{\prime})$ is large, all the properties are satisfied. For example, property (iii) is satisfied with $a\simeq 0.1$ SM . We have thus confirmed the quasi volume law for nontrivial NESSs far from equilibrium.

These results are summarized in Table 3. It clearly shows that the quasi volume law is peculiar to nontrivial NESSs inv .

Scaling in reservoirs.— When $L$ is increased to $L>L_{\rm C}$, the $O(L^{2})$ components of $R_{L}^{W}$ at small $|p|$ cease to grow with increasing $L$, as shown in Figs. S8 and S9 of SM , because impurities are absent in the reservoir regions, whereas those at small $|q|$ continue to grow. Consequently, the logarithmic law is recovered with an offset value:

[TABLE]

as demonstrated in Fig. S10 of SM .

In summary, at equilibrium, $S_{L}$ obeys the logarithmic law. In nontrivial NESSs far from equilibrium, $S_{L}$ is enhanced anomalously to obey the quasi volume law (1). Consequently, $S_{L}>O(S_{L}^{\rm eq})$ in contrast to the finding in previous works Eisler and Zimborás (2005); Aschbacher (2007); Hoogeveen and Doyon (2015) that $S_{L}\leq O(S_{L}^{\rm eq})$ in NESSs. This anomalous behavior arises from the far from equilibrium property and multiple scatterings due to impurities that break the translational symmetry of the system. This suggests that similar results may be obtained for other models with certain symmetry-breaking scatterers, although we have studied only one such model in this paper.