Anisotropic dressing of electrons in electron-doped cuprate superconductors

In ARPES experiments Damascelli03 ; Campuzano04 ; Fink07 , the weight of ARPES spectrum at zero energy is used to map out EFS. In other words, EFS is determined by the single-particle spectral function $A({\bf k},\omega=0)$ in Eq. (5) at zero energy to map out the locus of the maximum in the spectral weight of $A({\bf k},\omega=0)$. For the understanding of the essential physics of an interacting system, the study of the nature of EFS is a starting point Mahan81 . This follows from a fact that the EFS topology is a fundamental ground-state property, and influences directly the low-energy transport properties. In particular, the shape of EFS in cuprate superconductors has been central to addressing the strong electron correlation effect and multiple nearly-degenerate electronic orders Neto14 ; Fujita14 ; Vishik18 ; Comin16 ; Comin14 ; Neto15 ; Neto16 . In this case, we first characterize the EFS topology. In the case without considering the electron interaction, a large EFS is characterized as a closed contour of the gapless quasiparticle excitations in momentum space, where the peaks of the quasiparticle excitation spectrum with the same weight distribute uniformly along with the EFS contour, and then the quasiparticle lifetime is infinitely long on this EFS contour. However, in the presence of the strong interaction of the electrons with a strongly dispersive spin excitation, which is manifested itself by the energy and momentum dependence of the total self-energy $\Sigma_{\rm tot}({\bf k},\omega)$, the closed EFS contour is broken up into the disconnected Fermi arcs. To see this EFS reconstruction more clearly, we plot the underlying EFS map in the $[k_{x},k_{y}]$ plane at the electron doping $\delta=0.15$ with $T=0.002J$ for $t/J=-2.5$ and $t^{\prime}/t=0.3$ in Fig. 3a. For comparison, the experimental result Song17 obtained from the ARPES measurement on Pr_1-xLaCe_xCuO_4-δ under the proper annealing at around the electron doping $\delta=0.15$ is also shown in Fig. 3b. In a qualitative analogy to the hole-doped counterparts Feng15a ; Gao18 , EFS has been reconstructed due to the strong redistribution of the spectral weight, where there are two typical regions: (a) the antinodal region, where the result shows a heavy suppression of the spectral weight, leading to that EFS at around the antinodal region is invisible. Moreover, the AF pseudogap at around the crossing points is totally absent; (b) the nodal region, where the result indicates a modest reduction of the spectral weight, and then EFS is clearly visible as the reminiscence of the EFS contour in the case without considering the electron interaction to form the Fermi arcs centered around the nodal region. However, the dressing from the quasiparticle scattering further moves the spectral weight from the Fermi arcs to the tips of the Fermi arcs, which leads to that although a quasiparticle peak emerges in the entire EFS, the spectral weight exhibits a largest value at around the tips of the Fermi arcs. This EFS reconstruction is also qualitatively consistent with the recent experimental observations on the electron-doped cuprate superconductors Horio16 ; Song17 ; Lin20 ; Horio20b , where upon the proper annealing, (a) the weight of the ARPES spectrum around the antinodal region is suppressed, and then EFS is truncated to form the Fermi arcs located around the nodal region; (b) the AF pseudogap is closed on the whole EFS, and then the quasiparticle peak is observed on the whole EFS. Moreover, as in the hole-doped case Gao19 ; Chatterjee06 ; He14 , these tips of the Fermi arcs linked up by the scattering wave vectors ${\bf q}_{i}$ in the electron-doped side also can construct an octet scattering model Tan20 , and then different ordered electronic states determined by the quasiparticle scattering processes with the corresponding scattering wave vectors ${\bf q}_{i}$ therefore are driven by this intrinsic EFS instability. This is also why a exotic feature in the electron-doped cuprate superconductor is the coexistence and competition between different ordered electronic states and superconductivity Armitage10 ; Neto15 ; Neto16 . In particular, it has been shown that the charge-order wave vector ${\bf q}_{1}={\bf Q}_{\rm co}$ connecting the straight tips of the Fermi arcs smoothly increases with the increase of the electron doping Mou17 , which is consistent with the experimental results Neto16 . However, this is unlike the case in the hole-doped counterparts Gao18 ; Comin16 ; Comin14 , where the charge-order wave vector smoothly decreases with the increase of the hole doping, which is another evidence of the electron-hole asymmetry. Furthermore, we have also studied the doping dependence of the EFS reconstruction, and the results show that the EFS reconstruction can persist into the overdoped regime, also in agreement with the experimental observations Horio16 ; Song17 ; Lin20 ; Horio20b .

The physical origin of the EFS reconstruction in the electron-doped side is the same as that in the hole-doped case Gao18 , and can be also attributed to the highly anisotropic momentum dependence of the quasiparticle scattering rate (6). The EFS contour in momentum space is determined directly by the poles of the single-particle diagonal propagator (3a) at zero energy: $\varepsilon_{\bf k}+{\rm Re}\Sigma_{\rm tot}({\bf k},0)=0$, and then the spectral weight of the single-particle spectral function $A({\bf k},0)$ in Eq. (5) at EFS is governed by the inverse of the quasiparticle scattering rate $1/\Gamma({\bf k},0)$. To reveal this highly anisotropic $\Gamma({\bf k},0)$ in momentum space clearly, we plot the angular dependence of $\Gamma({\bf k}_{\rm F},0)$ along EFS from the antinode to the node at $\delta=0.15$ with $T=0.002J$ for $t/J=-2.5$ and $t^{\prime}/t=0.3$ in Fig. 4 in comparison with the corresponding ARPES result Horio16 observed on Pr_1.3-xLa_0.7Ce_xCuO₄ under the proper annealing condition (inset). Apparently, the main feature of the quasiparticle scattering rate along EFS observed from the ARPES experiment Horio16 is qualitatively reproduced, where the quasiparticle scattering rate increases as the momentum approaches the antinode, and then the strongest scattering emerges at the antinode, leading to a heavy suppression of the spectral weight at around the antinode. Moreover, the weak quasiparticle scattering occurs around the nodal region, and then the formation of the Fermi arcs is due to a modest reduction of the spectral weight around the nodal region. This strong redistribution of the spectral weight on EFS therefore induces the EFS reconstruction or instability. It is very interesting to note that the similar angular dependence of the quasiparticle scattering rate has been also observed experimentally in the hole-doped case Vishik09 , indicating a common quasiparticle scattering mechanism both in the hole- and electron-doped cuprate superconductors.

However, as seen by comparison with the experimental data Horio16 , there is also a substantial difference between theory and experiment, namely, the weakest quasiparticle scattering occurs at the crossing points in experiment Horio16 , while the calculation in the present parameters $t/J=-2.5$ and $t^{\prime}/t=0.3$ anticipates it at the tips of the Fermi arcs. In the electron-doped cuprate superconductors, (a) as we have mentioned above in Section III, the values of $t$, $t^{\prime}$, and $J$ are different between different families Park07 ; Nekrasov09 ; Ikeda09 ; Ikeda09a ; Harter12 ; Harter15 ; (b) however, the positions of the tips of the Fermi arcs are strongly depend on the curvature of EFS, while this curvature Park07 ; Nekrasov09 ; Ikeda09 ; Ikeda09a ; Harter12 ; Harter15 is dominated by the next NN hopping $t^{\prime}$. In other words, the difference in $t^{\prime}$ among the electron-doped cuprate superconductors Park07 ; Nekrasov09 ; Ikeda09 ; Ikeda09a ; Harter12 ; Harter15 leads to the difference of the positions of the tips of the Fermi arcs. In this case, we have also made a series of calculations for the different value of $t^{\prime}$, and found that the positions of the tips of the Fermi arcs can locate exactly on the corresponding positions of the crossing points by the proper choice or adjustment of the parameter $t^{\prime}$.

The dressing of the electrons affects the quasiparticle excitation spectrum and the momentum and energy dependence of the quasiparticle scattering rate, which can be obtained respectively from the energy distribution curves and the widths of the corresponding peaks in ARPES experiments Damascelli03 ; Campuzano04 ; Fink07 . In the hole-doped cuprate superconductors, a hallmark of the spectral line-shape of the ARPES spectrum is the well-known PDH structure Dessau91 ; Norman97 ; Campuzano99 ; Wei08 ; DMou17 , which is closely associated with the EFS reconstruction, and now has been identified experimentally along the entire EFS. Since the strong coupling of the electrons with bosonic excitations has been observed experimentally in whole families of the hole-doped cuprate superconductors within a wide hole doping range, it is thus believed that the strong coupling of the electrons with bosonic excitations gives rise to the PDH structure Dessau91 ; Norman97 ; Campuzano99 ; Wei08 ; DMou17 . In particular, we Gao18 ; Gao18a have shown based on the kinetic-energy driven SC mechanism that this strong coupling of the electrons with bosonic excitations can be identified as the strong electron’s coupling to a strongly dispersive spin excitation. In this subsection, we discuss the spectral line-shape in the quasiparticle excitation spectrum of the electron-doped cuprate superconductors. We have performed a calculation for the single-particle spectral function (5), and the results of $A({\bf k},\omega)$ as a function of energy at (a) the antinode and (b) node for $\delta=0.15$ with $T=0.002J$ for $t/J=-2.5$ and $t^{\prime}/t=0.3$ are plotted in Fig. 5, where we find that the main features of the PDH structure in the electron-doped side are in a striking similarity to those obtained in the hole-doped case Dessau91 ; Norman97 ; Campuzano99 ; Wei08 ; DMou17 ; Gao18 ; Gao18a . In particular, the position of the dip in the PDH structure is shifted to the lower energy when one moves the momentum from the antinode to the node, while the coupling strength of the electrons with a strongly dispersive spin excitation appears to be weaker at around the nodal region than at around the antinodal region. Moreover, the results in Fig. 5 also show that the PDH structure is an intrinsic feature of the quasiparticle excitation spectrum in the electron-doped cuprate superconductors, and is found to be present all around EFS. Although the experimental data of the PDH structure in the quasiparticle excitation spectrum of the electron-doped cuprate superconductors under the proper annealing condition are still lacking to date, the similar PDH structure has been observed experimentally on the electron-doped cuprate superconductors under the improper annealing condition Armitage10 ; Armitage01 ; Matsui05 ; Matsui05a ; Matsui07 , while our present results in Fig. 5 are also qualitatively consistent with these experimental observations Armitage10 ; Armitage01 ; Matsui05 ; Matsui05a ; Matsui07 .

The mechanism of the PDH structure in the quasiparticle excitation spectrum of the electron-doped cuprate superconductors is also the same as in the hole-doped counterparts DMou17 ; Gao18 ; Gao18a , and is directly associated with the emergence of the corresponding sharp peak in the quasiparticle scattering rate. Following the single-particle spectral function in Eq. (5), the position of the peak in the momentum distribution curve, plotted as a function of energy in Fig. 5, is determined self-consistently by the quasiparticle dispersion,

while the weight of the peak (then the lifetime of the quasiparticle excitation) is dominated by the inverse of the quasiparticle scattering rate $\Gamma({\bf k},\omega)$. In this case, the appearance of the dip in the PDH structure of the quasiparticle excitation spectrum along EFS is closely related to the emergence of the corresponding sharp peak in $\Gamma({\bf k},\omega)$. To see this point more clearly, we plot $\Gamma({\bf k},\omega)$ as a function of energy at (a) the antinode and (b) the node for $\delta=0.15$ with $T=0.002J$ for $t/J=-2.5$ and $t^{\prime}/t=0.3$ in Fig. 6, where the sharp peaks emerge at the antinode and node, respectively. More specifically, we find that the position of the peak in $\Gamma({\bf k},\omega)$ at the antinode (node) in Fig. 6 is exactly corresponding to the position of the dip in the PDH structure in the quasiparticle excitation spectrum at the antinode (node) shown in Fig. 5, and then the spectral weight at around the dip is suppressed heavily by the corresponding sharp peak in $\Gamma({\bf k},\omega)$, which leads to form the PDH structure in the quasiparticle excitation spectrum. In other words, the striking PDH structure in the quasiparticle excitation spectrum shown in Fig. 5 is caused directly by the the emergence of the sharp peak in $\Gamma({\bf k},\omega)$ shown in Fig. 6. Furthermore, we have also discussed the spectral line-shape in the electron-doped cuprate superconductors in the normal-state, and found the sharp peak in $\Gamma({\bf k},\omega)$ that can persist into the normal-state, indicating that the PDH structure is totally unrelated to superconductivity, as the PDH structure in the hole-doped counterparts Dessau91 ; Norman97 ; Campuzano99 ; Wei08 ; DMou17 .

The strong coupling of the electrons and various bosonic excitations in cuprate superconductors manifested itself as a slope change (or a kink) in the quasiparticle dispersion Damascelli03 ; Campuzano04 ; Fink07 . In the hole-doped cuprate superconductors, the quasiparticle dispersion displays a kink at all around EFS Kaminski01 ; Lanzara01 ; Kordyuk06 ; Iwasawa08 ; Plumb13 , with the energy range $50\sim 80$ meV at which the kink appears. In spite of the extensive studies, the controversy still exists on what bosonic mode causes the dispersion kink. In the recent studies for the hole-doped case Liu20 , we have shown within the framework of the kinetic-energy driven superconductivity that the quasiparticle dispersion is affected by a strongly dispersive spin excitation, and then the dispersion kink associated with the dressing of the electrons is electronic in nature. On the electron-doped side, although the experimental results of the PDH structure in the quasiparticle excitation spectrum under the proper annealing condition are still lacking to date as we have mentioned above in Section I, the dispersion kinks in the electron-doped cuprate superconductor Pr_1.3-xLa_0.7Ce_xCuO₄ under the proper annealing condition along the nodal and antinodal directions have been detected very recently from the ARPES spectra Horio20b , where the bare band structure Horio20b ; Horio18a has been well fitted in terms of the single-band tight-binding model with $t^{\prime}/t=0.15\sim 0.19$. In this subsection, we discuss the nature of the dispersion kink in the electron-doped cuprate superconductors, and adopt the parameters $t/J=-2.5$ and $t^{\prime}/t=0.16$ in the $t$-$J$ model (1) for a quantitative or semiquantitative comparison between the theory and experiment.

In Fig. 7, we plot the intensity map of the single-particle spectral function as a function of binding-energy along (a) the nodal cut and (b) the antinodal cut at $\delta=0.15$ with $T=0.002J$ for $t/J=-2.5$ and $t^{\prime}/t=0.16$ in the upper panel . In the lower panel, we plot the corresponding quasiparticle dispersions along (c) the nodal cut and (d) the antinodal cut extracted from the positions of the lowest-energy quasiparticle excitation peaks in (a) and (b), respectively. The arrow in Fig. 7 marks the kink where the the linear quasiparticle dispersion is separated as the low-energy and high-energy ranges with different slopes. It is especially interesting to note that these results of the dispersion kink in the electron-doped cuprate superconductors shown in Fig. 7 are quite similar to those obtained in the hole-doped counterparts Kaminski01 ; Lanzara01 ; Kordyuk06 ; Iwasawa08 ; Plumb13 ; Liu20 . In particular, with a reasonably estimative value of $J\sim 100$ meV, the quasiparticle dispersion deviates from the low-energy linear dispersion at around the kink energy $\omega_{\rm kink}\sim 0.79J=79$ meV along the nodal cut, while this kink energy occurs at around $\omega_{\rm kink}\sim 0.40J=40$ meV along the antinodal cut, which are semiquantitatively consistent with the corresponding results Horio20b observed on the electron-doped cuprate superconductor Pr_1.3-xLa_0.7Ce_xCuO₄ under the proper annealing condition along the nodal and antinodal directions, respectively. The above these results in Fig. 7 also indicate that although the kink in the quasiparticle dispersion is present all around EFS, the kink energy decreases when the cut moves from the nodal region to the antinodal region. Moreover, it should be noted that the coupling of the electrons with bosonic excitations has been also observed early in all the electron-doped cuprate superconductors under the improper annealing condition Santander-Syro09 ; Schmitt08 ; Park08a ; Armitage03 , where the quasiparticle dispersions present the kinks at around $50\sim 70$ meV along both the antinodal and nodal cuts, which are also semiquantitatively consistent with our present results in Fig. 7.

A natural and important question is which bosonic mode causes the dispersion kink in the electron-doped cuprate superconductors? Within the framework of the kinetic-energy driven superconductivity, the dispersion kink in the electron-doped cuprate superconductors arises from the strong coupling between the electrons and a strongly dispersive spin excitation. This follows a fact that from the quasiparticle dispersion in Eq. (7), the dispersion kink shown in Fig. 7 does not originate from the single-electron band energy $\varepsilon_{\bf k}$, but it is due to the slope change in the real part of the total self-energy ${\rm Re}\Sigma_{\rm tot}({\bf k},\omega)$ , while the drop seen at around the kink is directly associated with the corresponding peak in the quasiparticle scattering rate $\Gamma({\bf k},\omega)$ in Eq. (6) [then the imaginary part of the total self-energy]. To further explore how the total self-energy generates the dispersion kink, we plot the real part of the total self-energy ${\rm Re}\Sigma_{\rm tot}({\bf k},\omega)$ as a function of binding-energy along (a) the nodal dispersion and (b) the antinodal dispersion as shown in Fig. 7c and Fig. 7d, respectively, at $\delta=0.15$ with $T=0.002J$ for $t/J=-2.5$ and $t^{\prime}/t=0.16$ in the upper panel of Fig. 8, where the red arrow indicates the inflection point (then the point of the slope change in the quasiparticle dispersion). In the lower panel, we plot the corresponding quasiparticle scattering rate $\Gamma({\bf k},\omega)$ as a function of binding-energy along (c) the nodal dispersion and (d) the antinodal dispersion, where the blue arrow denotes the peak position (then the point of the intensity depletion in the quasiparticle dispersion). It is obvious that a well-pronounced peak in $\Gamma({\bf k},\omega)$ is clearly visible with a corresponding inflection point in ${\rm Re}\Sigma_{\rm tot}({\bf k},\omega)$ that appears at the exactly same energy in $\Gamma({\bf k},\omega)$. This sharp peak in $\Gamma({\bf k},\omega)$ suppresses heavily the spectral weight at around the inflection point. More importantly, we find that the position of the dispersion kink in Fig. 7 matches exactly with the position of the inflection point in ${\rm Re}\Sigma_{\rm tot}({\bf k},\omega)$ [then the position of the peak in $\Gamma({\bf k},\omega)$] in Fig. 8. In other words, the emergence of the dispersion kink is determined by both the inflection point in ${\rm Re}\Sigma_{\rm tot}({\bf k},\omega)$ and the well-pronounced peak in $\Gamma({\bf k},\omega)$. This is why the dispersion kink marks the crossover between two different slopes, while the weak spectral intensity appears always at around the kink position Santander-Syro09 ; Schmitt08 ; Park08a ; Armitage03 .

In ARPES experiments Damascelli03 ; Campuzano04 ; Fink07 , the energy distribution curves are observed when the photoemission intensity is measured at constant momentum, while the momentum distribution curves are observed when the photoemission intensity is measured at constant energy. The information of the energy and momentum dependence of the electron self-energy then can be extracted in terms of the single-particle spectral function (5) by analyzing the spectral intensity of the energy and momentum distribution curves. However, as shown in the single-particle spectral function (5), only the total self-energy can be extracted directly from ARPES experiments Damascelli03 ; Campuzano04 ; Fink07 . In particular, for our exploration of the strongly dispersive spin excitation coupling in the kinetic-energy driven SC mechanism that is how the normal and anomalous self-energy effects compete in the SC-state, it thus is crucial to extract the normal and the anomalous self-energies separately. This also follows a basic fact that in the kinetic-energy driven superconductivity Feng15a ; Feng0306 ; Feng12 ; Feng15 , the SC-state is controlled by both the electron pair gap and single-particle coherence, where the single-particle coherence is dominated by the normal self-energy $\Sigma_{\rm ph}({\bf k},\omega)$, and therefore antagonizes superconductivity, while the energy and momentum dependent electron pair gap is determined completely by the anomalous self-energy $\Sigma_{\rm pp}({\bf k},\omega)$, and therefore is corresponding to the energy for breaking an electron pair. In this case, if both the normal and anomalous self-energies are deduced from the experimental data, it can be used to examine the microscopic theory of the kinetic-energy driven superconductivity and understand the essential physics of cuprate superconductors.

In the hole-doped cuprate superconductors, the machine learning technique has been applied recently to deduce both the normal and anomalous self-energies from the experimental data of the ARPES spectra observed in Bi₂Sr₂CaCu₂O_8+δ at the optimum doping and Bi₂Sr₂CuO_6+δ in the underdoped regime Yamaji19 , and the deduced results show clearly that both the normal and anomalous self-energies exhibit the sharp low-energy peak-structures. Moreover, these machine learning ’experimental’ results confirm that the peak in the anomalous self-energy makes a dominant contribution to the SC gap, and therefore provide a decisive testimony for the origin of superconductivity Yamaji19 . In the very recent studies Liu20a , we have made a comparison of these deduced normal and anomalous self-energies in Ref. Yamaji19, with those obtained based on the kinetic-energy driven SC mechanism, and then show explicitly that the interaction between electrons by the exchange of spin excitations generates the sharp low-energy peak-structures both in the normal and anomalous self-energies, in striking agreement with the corresponding results in the normal and anomalous self-energies deduced via machine learning ’experiments’ Yamaji19 . In this subsection, we analyze the normal and anomalous self-energies in the electron-doped cuprate superconductors, and then compare these normal and anomalous self-energies with those obtained in the hole-doped counterparts Yamaji19 ; Liu20a .

In Fig. 9, we plot (a) the real (blue line) and imaginary (red line) parts of the normal self-energy and (b) the real (blue line) and imaginary (red line) parts of the anomalous self-energy at the antinode as a function of energy in the electron doping $\delta=0.15$ with $T=0.002J$ for $t/J=-2.5$ and $t^{\prime}/t=0.3$. For a better comparison, the corresponding results Liu20a of (c) the real and imaginary parts of the normal self-energy and (d) the real and imaginary parts of the anomalous self-energy at the antinode as a function of energy in the hole doping $\delta=0.15$ with $T=0.002J$ for $t/J=2.5$ and $t^{\prime}/t=0.3$ are also shown in Fig. 9. Surprisedly, both the normal and anomalous self-energies in the electron-doped cuprate superconductors also exhibit the sharp low-energy peak-structures. More specifically, the main features of these low-energy peak-structures are in a stark similarity with the corresponding low-energy peak-structures in the normal and anomalous self-energies of the hole-doped cuprate counterparts obtained based on the kinetic energy driven superconductivity Liu20a , and deduced from the ARPES spectra via machine learning Yamaji19 . This is why the absence of the disparity between the phase diagrams of the electron- and hole-doped cuprate superconductors can be observed experimentally Song17 ; Lin20 ; Horio20b . In the SC-state, the pairing force and electron pair order parameter have been incorporated into the anomalous self-energy $\Sigma_{\rm pp}({\bf k},\omega)$, which is identified as the SC gap, and therefore describes the strength of the electron pair. In this case, the dominant contribution to the strength of the electron pair arises from the sharp low-energy peaks both in ${\rm Re}\Sigma_{\rm pp}({\bf k}_{\rm AN},\omega)$ and ${\rm Im}\Sigma_{\rm pp}({\bf k}_{\rm AN},\omega)$ Yamaji19 ; Liu20a . On the other hand, from the total self-energy in Eq. (4), the dressing of the electrons are mainly dominated by the normal self-energy $\Sigma_{\rm ph}({\bf k}_{\rm AN},\omega)$, indicating that the sharp low-energy peak in ${\rm Re}\Sigma_{\rm ph}({\bf k}_{\rm AN},\omega)$ offsets mainly the single-electron band energy, while the sharp low-energy peak in ${\rm Im}\Sigma_{\rm ph}({\bf k}_{\rm AN},\omega)$ governs mainly the lifetime of the quasiparticle excitation Liu20a . However, the sharp low-energy peak in the anomalous self-energy disappears in the normal-state, while the sharp low-energy peak in the normal self-energy can persist into the normal-state. This is also why the exotic features, including the intrinsic EFS reconstruction, the PDH structure in the quasiparticle excitation spectrum, the dispersion kink, and the ARPES autocorrelation and its connection with the quasiparticle scattering interference Tan20 , arisen from the dressing of the electrons that emerge in the SC-state also appear in the normal-state.

For a further understanding of the essential physics of the electron-doped cuprate superconductors, we have also studied the doping and momentum dependence of the low-energy peak structures both in the normal and anomalous self-energies, and all the obtained results are consistent with the corresponding results obtained in the hole-doped counterparts Yamaji19 ; Liu20a . We therefore complete the picture of the dressing of the electrons for the electron-doped cuprate superconductors, and show the existence of the common origin of the anisotropic dressing of the electrons both in the electron- and hole-doped cuprate superconductors, i.e., the dressing of the electrons arises from the interaction of the electrons with a strongly dispersive spin excitation. Since the remarkable low-energy peak-structures both in the normal and anomalous self-energies of the hole-doped cuprate superconductors Liu20a are well consistent with those deduced via machine learning ’experiments’ Yamaji19 , we therefore predict that the similar low-energy peak-structures both in the normal and anomalous self-energies can be also deduced from the experimental data of the ARPES spectra observed on the electron-doped cuprate superconductors in the proper annealing condition via machine learning.