3.1. -behavior in the small frequency regime

For the Ohmic (s = 1) and the super-Ohmic ( ) baths, the small frequency behavior of C(ω) in the delocalized phase obeys the Shiba relation,^[35] which in our notation reads^[12]

Here is the local spin susceptibility. The proof in Ref. [35] applies also to the case with a finite bias, but not to the localized phase at for the Ohmic bath. This exact relation was used to test the quality of various approximate results for SBM.^{[11,12,19,36–39]}

There have been attempts to generalize the Shiba relation to (i) sub-Ohmic bath, (ii) with finite bias, and (iii) strong coupling regime . Such a generalized relation, if it exists, would imply a universal long time behavior for any s values, coupling strength α (except at the critical point and ε = 0), and spin parameters and ε. Up to now these activities received partial success only. Florens and others^[40] proved Eq. (8) for the sub-Ohmic bath, based on an exact relation from the perturbation theory in Majorana representation. However, their proof applies only to delocalized phase at the symmetry point ε = 0. The numerical results from the approximate perturbation theory built on a unitary transformation fulfils Eq. (8) exactly for the sub-Ohmic bath with^[41,42] or without^[18] bias in the weak to intermediate coupling. In the strong coupling regime the fulfilment is good but not exact. This leaves the question open whether the Shiba relation holds in the biased strong coupling regime of the sub-Ohmic SBM.

As stated above, the inherent truncation errors of NRG hinder it from the quantitative confirmation/falsification of the generalized Shiba relation. However, qualitatively, it is possible to check the factors on the right-hand side of Eq. (8) each for a time. In this section, we will show that holds for the sub-Ohmic SBM with general α and ε. In Fig. 6, data are presented to further support that for general α and ε values.

In Fig. 1, we first present the -dependence of C(ω) obtained from NRG under bias, in order to gauge the choice of in our study. The energy flows (Fig. 1(a)) and C(ω) (Fig. 1(b)) are shown for a sub-Ohmic bath s = 0.8 in the strong coupling regime , with a finite bias . We use a series of values from to with and . In Fig. 1(a), for a given , the excitation energies (i = 1, 2, 3) first flow to an unstable fixed point (with ) in the small N regime. After a crossover , they flow to the stable fixed point (with ) in the large N limit. We found that the unstable fixed point in the small N regime is the same as the weak-coupling fixed point of the SBM in the non-bias case (dashed lines obtained using α = 0 and ε = 0). With increasing geometrically, the crossover increases linearly, showing that the associated energy scale decreases to zero as a negative power of . This supports that the unstable fixed point (dashed lines in Fig. 1(a)) will extend to infinitely large N in the limit and it is the true fixed point of the biased SBM. In this biased fixed point, ε flows to infinity and the spin is effectively decoupled from the bath, leading to the same excitation levels as the free boson chain.

Figure 1(b) shows the corresponding C(ω), which has two different power-law regimes: in the low frequency regime , and in the high frequency regime . NRG data for s = 0.8 gives and . With increasing , the crossover frequency decreases to zero as a negative power of , proving that is the correct result of the biased SBM (dashed line in Fig. 1(b)). Our results for other s values in agree with within an error of 2% (not shown).

For , the NRG results converge with much more difficulty with increasing . In Fig. 2, we show the strong coupling data ( ) for s = 0.3. Both the energy flow and C(ω) have -dependent crossover scales. With increasing to 60, a clear trend can be seen that the -converged levels flow towards the free boson energy levels (Fig. 2(a)). For both and , the section of C(ω) with ω^s behavior extends to smaller ω with increasing (Fig. 2(b)). The shift of C(ω) with has an apparent scaling form and an analysis for could be carried out to extract the correct exponent, as done in Ref. [31]. Here in Fig. 2(b), we only mark out the expected asymptotic ω^s line for guiding the eye. With this understanding of the -dependence of C(ω), in the rest of this paper, we only show the physically correct results obtained using sufficiently large .

In Fig. 3, we investigate the influence of ε on C(ω) for α values ranging from weak coupling (Fig. 3(a)) to strong coupling (Fig. 3(b)) for s = 0.8 and . It is seen that in the small frequency limit for all parameters.In Fig. 3(a), for the small coupling α = 0.1 (dash-dotted lines), C(ω) does not change with ε up to and the three curves for ε = 0, 10⁻³, and 10⁻⁵ overlap. While for a moderate coupling α = 0.4 (dashed lines), a slight downward shift is observed between and and the lines for ε = 0 and 10⁻⁵ overlap. C(ω) is most sensitive to ε when α is close to . The same trend is observed in Fig. 3(b) for the strong coupling regime.

The universal behavior observed so far in Figs. 1, 2, and 3 has a natural understanding. When the parity symmetry is already broken by a finite bias, the ground state of SBM can be tuned continuously on the α–ε plane, going from a delocalized state at , , through a half circle in a finite ε region, to the symmetry-spontaneously-broken state at , , without passing the critical point. Therefore, the ground state has the same nature and no qualitative change is expected in the small ω limit of C(ω). Indeed, our NRG results for different s, α, and ε confirm that is a universal feature of SBM.

In the strong coupling case shown in Fig. 3(b), the same tendency is observed, i.e., with increasing ε, C(ω) shifts downwards and the most prominent change occurs near . The suppression of C(ω) can be understood as the weight transfer: with increasing ε, increases and so does the weight of the zero-frequency δ peak. Due to the sum rule, decreases uniformly. Comparing figs. 3(a) and 3(b), one sees that a broad peak forms for and .

Besides the universal low frequency behavior of C(ω) that we focus on in this paper, the high frequency structure in C(ω) for the deep sub-Ohmic SBM ( ) has also attracted a great deal of attention recently.^[17,18,41] For small s values, a peak at the renormalized tunnelling is observed in the strong coupling regime. It persists even in the localized phase . For α close to and small ε, another pronounced peak is observed at the crossover scale . C(ω) thus has a two-peak structure for small s and close to the critical point. Here, our C(ω) for s = 0.3 and a weak bias does have the two-peak structure, as shown by the dashed lines in Fig. 2(b). The high frequency peak is weak due to the over broadening of the log-Gaussian function. Decreasing the broadening parameter b from 1.2 to 0.8, the high frequency peak gets sharper and more pronounced, being consistent with previous results.^[17,18,40]

3.2. Equilibrium dynamics near QCP

In the above section, we established the universal ω^s-behaviour of C(ω) in the full parameter space of SBM. In this section, we focus on the parameter regime near the QCP and study the critical properties of C(ω) under a bias. For the unbiased SBM, is the only energy scale that controls the crossover between the stable fixed points (localized and delocalized fixed points) and the critical fixed point. Here ν is the critical exponent of the correlation length and z = 1 is the dynamical critical exponent. plays an important role in the temperature dependence of physical quantities close to QCP. At zero temperature, it also appears in the dynamical correlation function in the delocalized phase: for and for , and . is the critical behavior which holds for arbitrarily small frequency at the exact critical point , and is confined to away from the critical point. As α approaches , tends to zero and the behavior will be recovered in the full frequency range.

For the biased case, is another energy scale that influences the crossover between different behaviors of C(ω). Our NRG results for C(ω) at different α and ε can be understood in terms of the competition between ε and . Due to the difficulty of convergence for , here we show NRG data for s = 0.8, representing a typical case for . Our conclusion also applies to , as will be discussed below.

In Fig. 4 and Fig. 5, we show the flow diagrams (Fig. 4) and C(ω) (Fig. 5) for (a), (b), and (c), respectively. For each α, ε varies from zero to 10⁻³. The purpose is to observe the influence of ε in the delocalized, localzied, and critical phases. Figures 4(a) and 5(a) are for . At ε = 0, the energy flow in Fig. 4(a) has a crossover at around (solid circle), from the critical fixed point to the weak-coupling one. As ε increases from zero, the flow does not change for and the crossover (empty squares) begins to decreases only for . The corresponding evolution of C(ω) is shown in Fig. 5(a). For , for and for . The crossover scale of the symmetric SBM is given by . With increasing ε, the crossover frequency increases and occurs only when , corresponding to the occurrence of in the flow diagram. A crossover scale can be defined as such that for , becomes significantly larger than , or equivalently, . For , C(ω) stays same as the symmetric case (ε = 0) and . For , C(ω) is suppressed in the low frequency regime and is set by ε.

Figures 4(b) and 5(b) show the influence of ε in the localized phase. In Fig. 4(b), the excitation energy level at ε = 0 flows from the critical fixed point towards a two-fold degenerate fixed point, with the critical-to-localize crossover around (solid circle). Now we study the change of under bias (empty squares). With a vanishingly small ε, the degeneracy is lifted and the excited energy level has an upturn at an arbitrarily large , showing a crossover from the localized symmetric state to the biased state. We find that the crossover energy scale . With further increasing ε, the upturn moves to the left and for , occurs, which means that is now the crossover from the critical fixed point to the biased fixed point. NRG data give the critical-to-biased crossover . The critical exponent θ will be discussed with Fig. 5(c).

In Fig. 5(b), the evolution of C(ω) with ε is shown, which looks similar to the case . That is, C(ω) does not change much with ε for and begins to be suppressed for . It is noted that for a given , the degenerate-to-biased crossover at in the energy flow has no correspondence in C(ω): the latter has a perfect ω^s behavior at . The crossover in the energy flow will only show up in the temperature dependence of C(ω).

Figures 4(c) and 5(c) show the influence of ε on the flow and C(ω) in the critical regime . Since at this point, ε is the only energy scale that controls the critical-to-biased crossover in the energy flow and C(ω). Especially, Fig. 5(c) shows that for and for . We define a critical exponent θ as

For s = 0.8, the fitted exponent from Fig. 5(c) is . Note that the same dependence of on ε applies to the and cases in the regime , which are shown in panels (a) and (b) of figs. 4 and 5.

The NRG result in Fig. 5(c) shows that for , with an ε-independent factor c. Combining this observation with the assumption that the Shiba relation Eq. (8) holds at and (which will be discussed in Fig. 6 below), we can derive θ by equating the small and the large frequency expression at , giving

Employing the critical behavior , one obtains , giving . From the exact expression δ = 3 for and for , we obtainFor s = 0.8, this expression gives θ = 1.111, which agrees well with the NRG result 1.12.

Having shown that ( ) for general α and ε values in the previous section, we now check the Shiba relation Eq. (8) at s = 0.8 using a fixed small frequency . Figure 6 shows C(ω) and the right-hand side of Eq. (8) as functions of ε, for the same α values as in figs. 4 and 5. We find qualitative agreement between them for a wide range of ε. For ( ), C(ω) shows the crossover from critical (power law dependence on ε) for to the localized-like (delocalized-like) behavior (being constant) for , as expected. For , a power-law dependence is observed, with the fitted exponent −1.88 (for C(ω)) and −1.84 (for ), respectively, in reasonable agreement with the exact . It is notable that although C(ω) and have more than 5 decades of variations in the range , their ratio does not change much. This result supports the relation .

The ratio also depends weakly on ω due to the slight inaccuracy in the exponent of NRG-produced C(ω). The uniform deviation from Eq. (8) observed here is more likely due to the error of NRG data than due to the invalidity of the Shiba relation. After all, a factor of 3 is a reasonable level of error in the NRG calculation of dynamical quantities,^[11,12] considering the logarithmic error, truncation errors, as well as the approximation used to calculate C(ω) and χ.^[33] It is expected that the agreement can be improved if we increase M_s and N_b, and extrapolate to unity, which will not be pursued here. The good agreement of the NRG value of the exponent θ and Eq. (11) is also consistent with , since equation (11) is derived from this assumption. For s = 0.3, it is more difficult to check the Shiba relation quantitatively due to much stronger boson state truncation error. As shown in Fig. 2(b), using as large as 60, one can obtain the expected ω^s behavior only in a narrow frequency window and for certain ε values. For those parameter regimes and frequencies where ω^s behavior can be obtained, NRG results give a ratio for the delocalized, critical, and the localized phases, much larger than the s = 0.8 case. This shows the difficulty of a quantitatively accurate NRG study for the deep sub-Ohmic SBM.

In Fig. 7(a), we fix and plot C(ω) for different α values. For , C(ω) increases with increasing α and the height of the peak reaches a maximum at certain . With further increasing α, C(ω) begins to decrease uniformly, with the peak height suppressed. In Fig. 7(b), we show the maximum of C(ω) as functions of α for various εʼs. From this figure we can extract the α value at which the height of the peak in C(ω) reaches the maximum.

The phase diagram on the α–ε plane shown in Fig. 8 summarizes our results for C(ω) near the QCP. Close to and with very small ε, three regions I, II, and III of different nature are separated by a crossover line (squares with eye-guiding line). Regions I and III are continuously connected to the delocalized ( ) and localized phase ( ) of the symmetric SBM, respectively. They are characterized by . In region I, with χ being the magnetic susceptibility. In region III, close to the saturate value. For each α, the crossover in Fig. 8 is defined as the ε value at which the maximum of C(ω) decreases to 90% of its value at ε = 0.

In region II, vanishes near and becomes the only characteristic energy scale. In this regime, the magnetization shows the quantum critical behavior as , with δ being a critical exponent. The empty circles with eye-guiding line marks out the parameter at which C(ω) has a highest peak. Both the crossover line and the peak position can be extracted from the data in Fig. 7(b). It is observed that the location of the maximum peak resembles that of the maximum of entanglement entropy.^[21] This is not a pure coincident since the crossover peak of C(ω) reflects a strong fluctuation near the critical point and it is therefore naturally related to the entanglement maximum.

In the inset of Fig. 8, we fit the crossover line close to the QCP in a power law

with the obtained critical exponent η = 1.99 for s = 0.8. Note that the asymmetry of in and regimes means different pre-factors c. Besides using C(ω) here, the critical behavior of can also be determined by the scaling form of obtained from the scaling analysis of the NRG data.^[32] These two approaches give consistent results. For the second approach, the scaling analysis of NRG data gives (see Eqs. (31), (33), (35), (36) in Ref. [32])Here and is a two-variable scaling function. By comparing the magnitude of the two variables in Eq. (13), one obtains , giving . Here, β is the critical exponent of the order parameter. It is known that for , β = 1/2 and δ = 3. For , β is a function of s whose explicit expression is unknown yet, and .^{[13,14,31,43,44]} The NRG result is in reasonable agreement with the fitted exponent η = 1.99 in the inset of Fig. 8, showing the consistency of the static and dynamical approaches.

For , NRG calculation is hindered by the slow convergence with and it is difficult to obtain the quantitative data. However, we note that the scaling from Eq. (13) is obtained from the NRG calculation combined with the scaling analysis for the full sub-Ohmic regime . It is stronger than the scaling ansatz of free energy which applies only to the regime .^[43,44] The two exponents introduced above, θ and η, are not independent. We have . This gives , meaning . This is a relation independent of the validity of hyperscaling relation.