Unveiling quantum entanglement and correlation of sub-Ohmic and Ohmic baths for quantum phase transitions in dissipative systems

In Fig. 1(a), the first derivative of the ground-state energy, $\partial E_{g}/\partial\alpha$, is plotted against the coupling $\alpha$. To investigate the discretization effect, one employs different values of the discretization factors $\Lambda=1.05\sim 8$ with the same lowest frequency $\omega_{\rm min}\approx 10^{-10}$. All curves decrease with the coupling $\alpha$, and have sharp kinks, indicating a high-order singularity expected at the transition point. It confirms that the phase transition is of second order. Moreover, the transition boundary marked by the dash-dotted line displays a linear dependence of $\partial E_{g}/\partial\alpha$ on the critical coupling $\alpha_{c}$. The convergence is reached at $\Lambda=1.05$, corresponding to the number of effective bath modes $M=430$. It is about the same order of magnitude as that in the Davydov work on the quantum dynamics of the SBM Hartmann et al. (2019).

Taking the asymptotic value $\alpha_{c,\Lambda\rightarrow 1}=0.01802$ as input, the shift of the transition point $\Delta\alpha_{c}=\alpha_{c}(\Lambda)-\alpha_{c,\Lambda\rightarrow 1}$ is plotted in Fig. 1(b) as a function of the logarithm of the discretization factor $\ln\Lambda$ on a log-log scale. A power-law increase of $\Delta\alpha_{c}$ is found to provide a good fitting to the numerical data, and the value of the slope yields $d-1/\nu=1.79(2)$ from the scaling arguments and the relation $\ln\Lambda=-(\ln\omega_{\rm min})/M\sim 1/M$ where $M$ is equivalent to the length of the Wilson chain. By that we mean the Hamiltonian of SBM can be exactly mapped to the one describing a bosonic chain, i.e., Wilson chain, with nearest-neighbour interactions via a canonical transformation Chin et al. (2010); Zhou et al. (2014). In addition, quantum phase transition for the system with short-range interactions in $d$ spatial dimensions is generally believed to be equivalent to the classical transition in $d+1$ dimensions under the quantum-classical mapping. Thus taking effective spatial dimension $d_{\rm eff}=1+1$ by assumption, one calculates the correlation length exponent $1/\nu=0.21(2)$, in agreement with the mean-field prediction $1/\nu_{MF}=s=0.2$.

On the other hand, the dependence of the critical coupling $\alpha_{c}$ on the lowest frequency $\omega_{\rm min}$ is also investigated, and the results are shown in the inset where the discretization factor $\Lambda=2.0$ is set. Note the correlation length is given by an inverse energy scale $\xi=1/\omega^{*}$ where $\omega^{*}$ is the frequency above which the quantum critical behavior is established. Therefore, a power-law relation $\Delta\alpha_{c}\propto L^{-1/\nu}=\omega_{\rm min}^{\rm 1/\nu}$ can be derived from the finite-size scaling analysis in frequency space. The exponent $1/\nu=0.22(1)$ is estimated from the slope of the curve in the inset, well consistent with the earlier one $0.21(2)$.

Critical behaviors of quantum entanglement and correlation within the sub-Ohmic bath are investigated in Fig. 2 with the summations of the von-Neumann entropy $S_{\rm b}$, mutual information $\rm I_{\rm b}$, linear entropy $S_{\rm L,b}$, and quantum discord $D_{\rm b}$ on a linear scale. Obviously, they reach a sharp peak right at the phase transition, pointing to a cusp-like singularity. By normalizing the peak value to unity, all the curves have similar shapes, suggesting they obey the same scaling law in the sub-Ohmic regime. The transition point $\alpha_{c}=0.01803(1)$ is determined by the peak position which is indicated by the vertical dash-dotted line. It is in good agreement with the extrapolation result $\alpha_{c,\Lambda\rightarrow 1}=0.01802$ in Fig. 1, showing $\Lambda=1.05$ is sufficient small for the convergence in the continuum limit. Besides, the accuracy of the variational result on the critical coupling is significantly improved, in comparison with previous ones $0.0168$ Chin et al. (2011) and $0.02014$ He et al. (2018). Moreover, it agrees well with numerical results $0.0185(4),0.0175(2)$, and $0.0179(5)$ obtained from the numerical renormalization group (NRG) Bulla et al. (2003), quantum Monte Carlo (QMC) Winter et al. (2009), and variational matrix product Frenzel and Plenio (2013), respectively, thereby lending further support to the validity of variational calculations in this work.

Using the summation of the quantum discord $\sum D_{\rm b}$ over the frequency $\omega_{k}$ as a representative indicator, one investigates the influence of the discretization factor $\Lambda$ in Fig. 3(a). Similar with that in Fig. 1(a), the critical coupling $\alpha_{c}$ determined by the peak position of $\sum D_{\rm b}$ is shifted left with the decreasing discretization factor $\Lambda$. While the slopes $1.00(2)$ and $2.37(3)$ in two sides are almost unchanged, showing that critical exponents are robust. Noting $\ln\Lambda\sim 1/M$, one concludes that the peak value of $\sum D_{\rm b}$ increases slightly with the environmental size $M$, confirming the general assumption that the quantum correlation has a singularity at the transition point. Moreover, the dependence on the low-energy cutoff $\omega_{\rm min}$ is also demonstrated in Fig. 3(b). Interestingly, all the curves of $\sum D_{\rm b}$ for different $\omega_{\rm min}\approx 10^{-9}\sim 10^{-3}$ almost overlap with each other in the delocalized phase, indicating quantum discord $D_{\rm b}$ in the low-frequency regime is negligible. In the localized phase, however, the decay exponent is changed slightly from $3.3(1)$ to $2.37(4)$ with the decreasing $\omega_{\rm min}$, different from that in Fig. 3(a). It indicates the critical behavior of $\sum D_{\rm b}$ is dependent on $\omega_{\rm min}$ rather than $\Lambda$. The slope approaches the asymptotic value $2.37$ at $\omega_{\rm min}\approx 10^{-9}$, confirming that the value of the parameter $\omega_{\rm min}$ used in this work is already sufficiently small.

As mentioned before, quantum phase transition in the deep sub-Ohmic regime $s<0.5$ is mean-field like where the critical exponent obeys $1/\nu=s$. In the shallow sub-Ohmic regime $s>0.5$, however, non-trivial critical behaviors have been found in both the numerical work and analytical analysis Vojta et al. (2005); Orth et al. (2010); Winter et al. (2009). The hyperscaling relations hold, and the exponent $\nu$ is expected to diverge as $1/\sqrt{2(1-s)}$ near the ohmic point $s=1$. In order to demonstrate the validity of our scaling analysis and NVM in such non-mean-filed regime, additional simulations with the spectral exponent $s=0.7$ are performed for different values of $\Lambda$ and $\omega_{\rm min}$.

Taking $\Lambda=1.05$ as an example, the first derivative of the ground-state energy, $\partial E_{g}/\partial\alpha$, is plotted versus the coupling $\alpha$ in Fig. 4(a). Similar to that in Fig. 1, high-order singularity is obtained at the transition point $\alpha_{c}=0.2276(1)$, though the slope difference between two phases ($1.5-0.75=0.75$) becomes far less, inferring that the transition is weakened in the shallow sub-Ohmic regime. In addition, the first derivative of the spin coherence, $\partial\langle\sigma_{x}\rangle/\partial\alpha$, is also presented with stars, and a tiny discontinuity is detected at the critical point, again supporting the transition is of second order.

In Fig. 4(b), the asymptotic behavior of $\Delta\alpha_{c}$ is carefully examined. By taking into account the mass flow corrections, the transition point $\alpha_{c}=0.2430$ is estimated in the inset at $\Lambda=2$, in agreement with QMC result $0.241(2)$ Winter et al. (2009). The asymptotic value of $\alpha_{c}$ is then refined to be $0.22731$ in the continuum limit $\Lambda\rightarrow 1$, corresponding to a high dense sub-Ohmic bath. Moreover, the exponent value $1/\nu=0.43(3)$ is determined, well consistent with the QMC result $0.45(5)$, but far away from the predictions $1/\nu=s=0.7$ and $\sqrt{2(1-s)}\approx 0.775$. Quantum-to-classical correspondence gives that the SBM quantum transition is in the same universality class as the classical Ising chain with long-range interactions decaying as $1/r^{1+s}$ Vojta et al. (2005). Interestingly, our result agrees well with $1/\nu=1/2+1/3\epsilon-2.628\epsilon^{2}+\mathcal{O}(\epsilon^{3})\approx 0.461$ with $\epsilon=s-0.5$ which can be read off from the hyperscaling relation $\gamma=(2-\eta)\nu$ and two-loop renormalization-group results on the exponents $\gamma$ and $\eta$ in the Ising chain Fisher et al. (1972).

In Fig.6(a), the summations of the correlation function $\rm Cor_{X}$, logarithmic negativity $E_{\rm N,b}$, von-Neumann entropy $S_{\rm b}$, mutual information $I_{\rm b}$, linear entropy $S_{\rm L,b}$, and quantum discord $D_{\rm b}$ are plotted on a log-log scale for the quantum entanglement and correlation within the Ohmic bath. Different from the cusp-like singularity in Fig. 2, the discontinuities analogous to the universal jump of the superfluid density in the XY model are found in all curves, again pointing to the emergence of Kosterlitz-Thouless phase transition. The critical value of the coupling $\alpha_{c}=1.01(1)$ is then estimated for $\Delta=0.01$, in good agreement with the prediction $\alpha_{c}=1+\mathcal{O}(\Delta/\omega_{c})$ Le Hur (2008). Furthermore, the asymptotic value $\alpha_{c}=1.0053$ is measured by the extrapolation $\omega_{\rm min}\rightarrow 0$, and the linear coefficient $(\alpha_{c}-1)\omega_{c}/\Delta=5.3$ is excellent consistent with that in Ref. De Filippis et al. (2020). It points out that our variational method is as powerful and efficient as QMC and variational Feynman in providing an accurate description of the physical features of SBM.

In the delocalized phase $\alpha<\alpha_{c}$, the summations of the von-Neumann entropy $S_{\rm b}$ and its linear term $S_{\rm L,b}$ increase with the coupling $\alpha$ as a power-law form with the slope close to $1$, pointing to the linear dependence. A good coincidence is found for the curves of the correlation function $\rm Cor_{X}$, mutual information $I_{\rm b}$, and quantum discord $D_{\rm b}$, indicating the bath embodies pure quantum effect. They behave similarly to the entropy, though the slope $1.25(2)$ is slightly larger than $1$. In contrast, the summation of the logarithmic negativity $E_{\rm N,b}$ exhibits a power-law decay with a larger exponent $6.04(8)$, suggesting the average bipartite entanglement is rapidly erased by the decoherence effect from the environment. The opposite trend and vanishing value of $\sum E_{\rm N,b}$ show that the ground state of the Ohmic bath in the delocalized phase is separable mixed, rather than pure entangled.

Subsequently, the frequency dependence is examined, typified by the quantum discord $D_{\rm b}$. In Fig. 6(b), $D_{\rm b}(\omega_{k})$ is displayed for different couplings $\alpha=0.5,0.6,0.7,0.8$, and $1.0$ on a log-log scale. It increases monotonously with the frequency $\omega_{k}$, and quickly stabilizes at a constant which is coupling dependent. As the coupling $\alpha$ increases, the growth curve flattens out gradually, and tends to be $\omega_{k}$-independent at the transition point. For comparison, the quantum discord $D_{\rm b}(\omega_{k})$ in the sub-Ohmic regime $s=0.2$ is also given in the inset. Different from the Ohmic case, all the curves of $D_{\rm b}(\omega_{k})$ exhibit sharp peaks in the high-frequency region. It is worth noting that the peak value reaches a maximum at the transition $\alpha_{c}=0.018$, while the position of the peak remains practically unchanged. Therefore, two distinct scaling behaviors of the bosonic bath can be found for the Ohmic and sub-Ohmic transitions, lending further support to the claim that they belong to different universality classes.

For better understanding the nature of ground states in the sub-Ohmic and Ohmic cases, the structure of the environmental wave function characterized by average displacement coefficients defined as $\overline{f}_{k}=\langle\Psi_{1}|(b_{k}+b_{k}^{{\dagger}})(1+\sigma_{z})|\Psi_{1}\rangle/2$ and $\overline{g}_{k}=\langle\Psi_{1}|(b_{k}+b_{k}^{{\dagger}})(1-\sigma_{z})|\Psi_{1}\rangle/2$, is demonstrated in Fig. 7. For the sub-Ohmic SBM at $s=0.2$, a perfect antisymmetry relation $\overline{f}_{k}=-\overline{g}_{k}$ is found over the whole range of $\omega_{k}$ in the upper panel of the subfigure (a), supporting the usual assumption concerning the delocalized phase Silbey and Harris (1984); Bera et al. (2014). A huge jump appears in the low-frequency value of the displacement coefficient when the coupling is changed from $\alpha=0.01803$ to $0.01804$, showing a sharp transition. In the middle panel, power-law decays of both $\overline{f}_{k}$ and $\overline{g}_{k}$ are observed with the slope $0.4$ for $\alpha>\alpha_{c}$ at low frequencies, confirming that they follow the same classical displacement $\lambda_{k}/(2\omega_{k})\sim\omega_{k}^{-(1-s)/2}=\omega_{k}^{-0.4}$, and thereby the antisymmetry is broken in the localized phase.

For the Ohmic SBM at $s=1$, the antisymmetry relation as well as the antipolaron which was proposed as an important concept in Ref. Bera et al. (2014) is confirmed in our numerical work, indicating that the picture of the variational parameters in the delocalized phase is almost the same. Spontaneous breakdown of the antisymmetry is also found at the quantum phase transition, as shown in Fig. 7(b). While the deep Kondo regime where $\alpha\rightarrow 1$ and the localized phase were untouched in that work Bera et al. (2014). Further analysis points out that the amount of the jump for the average displacement coefficient at $\omega_{\rm min}$ is four orders of magnitude smaller than the sub-Ohmic one. That is because the value of $\overline{f}_{k}$ or $\overline{g}_{k}$ is independent of $\omega_{k}$ in the localized phase, for the classical displacement $\lambda_{k}/(2\omega_{k})=\rm constant$. Hence, it can be inferred that the bath modes with low and high frequencies may follow the same critical scaling in the Ohmic case, but behave differently in the sub-Ohmic regime.

Quantum fluctuations mainly caused by the effect of the antipolarons are investigated in the lower panels of Fig. 7, which can be measured by the departure from the single-coherent state, $\Delta X_{\rm b}-1/2$. In the sub-Ohmic regime, as expected, quantum fluctuations at low frequencies vanish in both the localized and delocalized phases. In contrast, the curve develops a plateau in the high-frequency region for any coupling $\alpha$, and the plateau value reaches the maximum at the transition point $\alpha_{c}=0.018$, similar with that in the inset of Fig. 6(b). It suggests that the emergences of kinks in Figs. 2 and 3 are only related to the high-frequency bath modes. In the Ohmic case, the quantum fluctuation $\Delta X_{\rm b}-1/2$ for $\alpha=0.5,0.7$, and $1.0$ grows with the frequency $\omega_{k}$, and approaches a $\alpha$-dependent constant value. For a larger coupling, e.g., $\alpha=1.1$, however, it vanishes $\Delta X_{\rm b}-1/2=0$ over the whole frequency range, confirming that bath modes are independent of each other, and behave as a single-coherent state in the localized phase. The picture is quite different from that in the sub-Ohmic regime.

Environmental entanglement, correlation, and entropy in two-spin SBM are also presented in Fig. 8(a), in the presence of the vanishing spin-spin coupling $K=0$ and weak tunneling $\Delta=0.01$. Similar to those in Fig.6(a), they follow power-law behaviors in the delocalized phase, though the values of the growth exponents are slightly larger. The transition point $\alpha_{c}=0.191(2)$ is then determined by sudden drops of the curves, much lower than earlier variational results $\alpha_{c}=0.5$ McCutcheon et al. (2010) and $0.31$ Zhou et al. (2018). It is because that the value of critical coupling can be well refined with the help of the high dense spectrum and broad frequency range $\omega_{c}/\omega_{\rm min}>10^{5}$ Qian et al. (2021). Moreover, it is in good agreement with the NRG one $\alpha_{c}=0.185$ and QMC one $0.210$ estimated from the linear extrapolation $\Delta\rightarrow 0.01$ of numerical results obtained in previous work Orth et al. (2010); Winter and Rieger (2014), further confirming the accuracy of our NVM calculations.

General scaling arguments for the continuous phase transition lead to the scaling form of the quantum discord,

where $\alpha^{\lambda}$ indicates the scaling dimension, $\widetilde{D}_{b}(r=\omega_{k}/\omega_{s})$ represents the scale invariance of the quantum discord in the Ohmic bath, and $\omega_{s}=1/\xi$ denotes the inverse of the correlation length.

The scaling function of the quantum discord $\widetilde{D}_{b}(r)=\rm D_{\rm b}\alpha^{-\lambda}$ defined in Eq. (18) is shown in Fig. 8(b) with respect to the ratio $r=\omega_{k}/\omega_{s}$. With the exponent $\lambda=1$ and energy scale $\omega_{s}(\alpha)$ as inputs, all data of different coupling $\alpha$ nicely collapse to a single curve, fully confirming the scaling form of the quantum discord $D_{\rm b}$. Clearly, $\widetilde{D}_{b}(r)\rightarrow{\rm const}$ when $r\rightarrow\infty$, and $\widetilde{D}_{b}(r)\sim r^{1.60}$ when $r\rightarrow 0$, suggesting it is a non-analytic function. In the inset, the energy scale $\omega_{s}$ extracted from the data collapse decays exponentially with the coupling $\alpha$. Solid line provides a good fit to the numerical data with a power-law correction to scaling, yielding the exponent value $52.7(4)$. It is perfectly consistent with the result $52.1(5)$ measured from the magnetization $\langle\sigma_{z}\rangle$ under a tiny bias $10^{-5}$, and almost twice as large as that from the renormalized tunneling $\Delta_{r}$ Zhou et al. (2018). Therefore, it is concluded that exponentially divergent correlation length $1/\omega_{s}$ plays an essential role in critical behaviors of two-spin SBM.

For comparison, the correlation, entanglement, and entropy of two spins are also investigated in this work. Firstly, the von Neumann entropy $S_{\rm{v-N}}$ that characterizes the entanglement between the spin system and its surrounding bath is introduced, $S_{\rm{v-N}}=-\rm Tr[\rho_{s}log_{2}\rho_{s}]$, where $\rho_{s}=\rm Tr_{b}[\rho_{sb}]$ is a reduced system density matrix given by tracing the total (system + bath) density operator $\rho_{sb}$ over the bosonic bath. The linear entropy of the system is then calculated as $S_{L}=\rm 1-Tr[\rho_{s}\rho_{s}]$. With the reduced density matrix $\rho_{s}$ at hand, the quantum entanglement between two spins can be measured by the concurrence,

where $\lambda_{i}$ ($i=1,2,3,$ or $4$) represents a square root of the eigenvalues of the matrix $\rho_{s}\widetilde{\rho}_{s}$ arranged in a descending order, and $\widetilde{\rho}_{s}=(\sigma_{y}\otimes\sigma_{y})\rho_{s}^{*}(\sigma_{y}\otimes\sigma_{y})$. The entanglement of the formation is then calculated,

with the function $h(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)$.

In addition, the spin-spin correlation function $\rm Cor=\langle\sigma_{z1}\sigma_{z2}\rangle-\langle\sigma_{z1}\rangle\langle\sigma_{z2}\rangle$ and mutual information $I=S_{\rm{v-N}}(\rho_{s1})+S_{\rm{v-N}}(\rho_{s2})-S_{\rm{v-N}}(\rho_{s})$ are investigated for the correlation in the spin system, where the subscripts $1$ and $2$ denote the ranks of the spins. The negativity is a measure of quantum entanglement derived from the PPT criterion for separability, which is an entanglement monotone. The negativity of a subsystem can be defined as $N(\rho_{s})=(||\rho_{s}^{\rm T}||_{1}-1)/2$, where $\rho_{s}^{\rm T}$ is the partial transpose of the reduced system density $\rho_{s}$ with respect to the spin $1$, and $||\hat{X}||_{1}=\text{Tr}|\hat{X}|=\text{Tr}\sqrt{\hat{X}^{\dagger}\hat{X}}$ is the trace norm or the sum of the singular values of the operator $\hat{X}$. The logarithmic negativity is then calculated as $E_{\rm N}=\log_{2}(2N+1)$. It is believed to be an easily computable entanglement measurement and an upper bound to the distillable entanglement Horodecki et al. (2009).

Finally, the quantum discord reflecting the nonclassical part of the total correlation is calculated as $D(\rho_{s2|1})=I(\rho_{s})-C(\rho_{s2}:\rho_{s1})$ where $I$ denotes mutual information between two spins $1$ and $2$, and $C$ represents the classical correlation,

given by a certain projection measurement $\left\{\Pi_{j1}\right\}$ on the spin $1$ with

Thus, the quantum discord can be written as

with the element of the projective measurement $\Pi_{j1}(j=1,2)$ and density matrix in the Bloch representation $\rho_{s}^{\prime}$ defined as

Where $\vec{n}_{1}=-\vec{n}_{2}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ is a three-dimensional unit vector in an arbitrary direction, $\vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ denotes a vector of Pauli matrices, $\vec{a}=\text{Tr}(\rho_{s}\vec{\sigma}_{1}\otimes\mathbbm{1})$ as well as $\vec{b}=\text{Tr}(\rho_{s}\mathbbm{1}\otimes\vec{\sigma}_{2})$ represents a local Bloch vector, and $T_{ij}=\text{Tr}(\rho_{s}\sigma_{i1}\otimes\sigma_{j2})$ denotes one component of the correlation tensor. By scanning all possible measurements with parameters $(\theta,\phi)$, the quantum discord $D$ is obtained by means of the minimization procedure.

The spin-related observables defined in Eqs. (19)-(23) are displayed in Fig. 9, whose behaviors are quite different from those of bath-related ones in Fig. 8(a). In particular, the von Neumann entropy $S_{\rm{v-N}}$ increases monotonically due to the suppression of the renormalized tunneling amplitude, and reaches a plateau at $\alpha_{t}\approx 0.1$ with the maximal system-bath entanglement. It indicates that coherence is lost already before the system becomes localized, and the spin dynamics is incoherent in the plateau. This coherent-to-incoherent crossover in the two-spin model occurs at the Toulouse point $\alpha_{t}=\alpha_{c}/2$, the same as that in the single-spin model. Moreover, the entanglement between the quantum system and bath can also be measured by the linear entropy $S_{L}$ which behaves similarly to the von Neumann entropy $S_{\rm{v-N}}$, although the value of $S_{L}$ is slightly lower when $\alpha<\alpha_{t}$.

Besides the entropy, the spin-spin correlation function $\rm Cor$ and quantum mutual information $I$ are also plotted in Fig. 9(a). Since the mutual information measures the total amount of the correlations in the spin system, the relation $\rm Cor<I$ in the coupling regime $\alpha<\alpha_{t}$ indicates that the correlation function $\rm Cor=\langle\sigma_{z1}\sigma_{z2}\rangle-\langle\sigma_{z1}\rangle\langle\sigma_{z2}\rangle$ is an incomplete measure of the correlation. Further analysis points out that $\rm Cor$ has a similar behavior to $C\left(\rho_{s2}:\rho_{s1}\right)=I-D$, suggesting it belongs to the degree of the classical correlation. Besides, it reaches its maximum $\rm Cor=I=1$ at $\alpha\geq\alpha_{t}$, the same as the system-bath entanglement $S_{\rm{v-N}}$. Therefore, one can conjecture that the spin-spin correlation $\rm Cor$ may come from the effect of the bath-induced ferromagnetic interaction $K_{r}=(-4\alpha\omega_{c}/s)$ Zhou et al. (2018).

In Fig. 9(b), the entanglement between two spins is depicted by the concurrence $C$, entanglement of formation $S_{\rm E}$, and logarithmic negativity $E_{\rm N}$. Exponential decays of them are found in the delocalized phase with large slopes, i.e., $42$ for $E_{\rm N}$ and $C$, and $78$ for $S_{\rm E}$, respectively. It indicates that the entanglement diminishes rapidly with the environmental coupling $\alpha$. Besides, the quantum discord $D$ reflecting the nonclassical correlation is also plotted in this subfigure. In contrast to the mutual information $I$ and spin-spin correlation function $\rm Cor$, quantum discord $D$ exhibits a monotonic smooth decrease in the delocalized phase. Abrupt drops of the curves are found at the transition point $\alpha_{c}=0.191$, analogous to the universal jump of the superfluid density in the XY model, again supporting that the quantum phase transition belongs to the Kosterlitz-Thouless universality class.

Note that the behaviors of the correlation, entanglement, and entropy between two spins significantly differ from those between two bath modes. For example, the classical correlation in the former ${\rm Cor}\sim I-D$ increases with the coupling $\alpha$, and reaches its plateau with the maximal correlation. While the one in the latter $\sum(I_{\rm b}-D_{\rm b})$ vanishes in the delocalized phase, showing the pure quantumness which can be identified as a signature of the bosonic bath. Furthermore, the logarithmic negativity $E_{\rm N,b}$ decreases with the coupling, displaying the opposite trend of the quantum discord $D_{\rm b}$. It indicates that the nonclassical correlation in the Ohmic bath is irrelevant to the entanglement, in contrast to that in quantum spin system where the discord is strongly restricted by the entanglement as shown in Fig. 9(b). Further analysis suggests that this nonclassical correlation is essentially determined by the quantum correlation in the position space $\rm Cor_{X}$, as given in Fig. 8(b). Although the average entanglement in the bath $\sum E_{\rm N,b}/(M-1)$ is negligibly small as compared to that $E_{\rm N}$ between two spins, both of them decay with the coupling $\alpha$ in the delocalized phase. At the transition point, however, there exist an abnormal increase of $\sum E_{\rm N,b}$ and an abrupt drop of $E_{\rm N}$, suggesting different singularities of quantum entanglements.