Low-temperature T² resistivity in the underdoped pseudogap phase versus T-linear resistivity in the overdoped strange-metal phase of cuprate superconductors

The common element in the layered crystal structure of cuprate superconductors is the square-lattice copper-oxide layerBednorz86 , and then the unusual features mainly come from the strongly correlated motion of the electrons in the copper-oxide layerCooper94 ; Nakamura93 ; Hou94 ; Takenaka94 . Quickly after the discovery of superconductivity in cuprate superconductors, AndersonAnderson87 recognized that the essential physics of the doped copper-oxide layer can be modeled with the low-energy effective $t$-$J$ model,

where $C^{\dagger}_{l\sigma}$ ($C_{l\sigma}$) is electron operator that creates (annihilates) an electron on site $l$ with spin $\sigma$, ${\bf S}_{l}$ is the spin operator with its components $S_{l}^{x}$, $S_{l}^{y}$, and $S_{l}^{z}$, and $\mu$ is the chemical potential. This $t$-$J$ model (1) describes a competition between the kinetic energy and magnetic energy, and the kinetic energy includes the nearest-neighbor (NN) hopping with the hopping integral $t$ and next NN hopping with the hopping integral $t^{\prime}$, while the magnetic energy is described by an AF Heisenberg term with the NN coupling $J$. $\langle l\hat{\eta}\rangle$ ($\langle l\hat{\tau}\rangle$) indicates that $l$ runs over all sites, and for each $l$, over its NN sites $\hat{\eta}$ (next NN sites $\hat{\tau}$). The parameters in this paper are chosen as $t/J=2.5$ and $t^{\prime}/t=0.3$ as the recent discussions Ma23 . Moreover, we use a notation in which the magnetic coupling $J$ and the lattice constant of the square lattice are set to the energy and length units, respectively. However, when necessary to compare with the experimental data, we take $J=100$meV, which is the typical value of cuprate superconductors.

The $t$-$J$ model (1) is supplemented by a crucial on-site local constraint $\sum_{\sigma}C^{\dagger}_{l\sigma}C_{l\sigma}\leq 1$ that the double occupancy of a site by two electrons of opposite spins is not allowed Anderson87 , while the strong electron correlation in the system manifests itself by this no double electron occupancy local constraint Yu92 ; Lee06 ; Edegger07 ; Spalek22 ; Zhang93 . However, the no double electron occupancy also leads to the difficulty for the studying of the $t$-$J$ model (1). For a proper treatment of this no double electron occupancy local constraint, the fermion-spin transformationFeng0494 ; Feng15 has been established, where the physics of no double electron occupancy is taken into account by representing the constrained electron as,

where the $U(1)$ gauge invariant spin-raising (spin-lowering) operator $S^{+}_{l}$ ($S^{-}_{l}$) carries spin index of the constrained electron, and therefore the collective mode from this spin degree of freedom of the constrained electron is interpreted as the spin excitation responsible for the spin dynamics of the system, and the $U(1)$ gauge invariant spinful fermion operator $h^{\dagger}_{l\sigma}=e^{i\Phi_{l\sigma}}h^{\dagger}_{l}$ ($h_{l\sigma}=e^{-i\Phi_{l\sigma}}h_{l}$) creates (annihilates) a charge carrier on site $l$, and therefore represents the charge degree of freedom of the constrained electron together with some effects of spin configuration rearrangements due to the presence of the doped charge carrier itself, while the charge carrier and localized spin recombine to form the physical electron responsible for the electronic properties.

In the past three decades, a series of experiments from NMR, NQR, and the inelastic neutron scattering (INS) measurements Fujita12 ; Birgeneau89 ; Fong95 ; Yamada98 ; Arai99 ; Bourges00 ; He01 ; Tranquada04 ; Bourges05 has provided an intrinsic connection between the electron pairing mechanism and spin excitations in cuprate superconductors, where a key question is whether the spin excitation can mediate electron pairing in analogy to the phonon-mediate pairing mechanism in the conventional superconductorsSchrieffer64 . Starting from the $t$-$J$ model in the fermion-spin representation, the kinetic-energy-driven superconductivityFeng15 ; Feng0306 ; Feng12 ; Feng15a has been developed, where the d-wave charge-carrier pairing state is generated by the charge-carrier interaction directly from the kinetic energy of the $t$-$J$ model by the exchange of the spin excitation. However, the d-wave electron pairs originated from this d-wave charge-carrier pairing state are due to the charge-spin recombinationFeng15a , and these electron pairs condensation reveals the d-wave SC-state. This kinetic-energy-driven SC mechanism is purely electronic without phonon, since the glue to hold the constrained electron pairs together is the spin excitation, the collective mode from the spin degree of freedom of the constrained electron itself. Moreover, this electron paring state mediated by the spin excitation in a way is in turn strongly influenced by the electron coherence, which leads to that the maximal $T_{\rm c}$ occurs at the optimal doping, and then decreases in both the underdoped and overdoped regimes. Following these previous discussionsFeng15 ; Feng0306 ; Feng12 ; Feng15a , the full charge-carrier propagator in the normal-state can be derived as [see Appendix A],

where $\xi_{\bf k}=4t\chi_{1}\gamma_{{\bf k}}-4t^{\prime}\chi_{2}\gamma_{{\bf k}}^{\prime}-\mu_{\rm h}$ is the mean-filed (MF) charge-carrier energy dispersion, with $\gamma_{\bf k}=({\rm cos}k_{x}+{\rm cos}k_{y})/2$, $\gamma_{\bf k}^{\prime}={\rm cos}k_{x}{\rm cos}k_{y}$, and the spin correlation functions $\chi_{1}=\langle S^{+}_{l}S^{-}_{l+\hat{\eta}}\rangle$ and $\chi_{2}=\langle S_{l}^{+}S_{l+\hat{\tau}}^{-}\rangle$, while the charge-carrier normal self-energy $\Sigma^{({\rm h})}_{\rm ph}({\bf k},\omega)$ is obtained explicitly in Eq. (43) in Appendix A.

In the framework of the kinetic-energy-driven superconductivity Feng15 ; Feng0306 ; Feng12 ; Feng15a , the charge-carrier pseudogap forms due to the strong coupling between charge and spin degrees of freedom of the constrained electron. To explore the nature of this charge-carrier pseudogap more clearly, we rewrite the charge-carrier normal self-energy in Eq. (3) as,

where $\xi_{0{\bf k}}=L^{({\rm h})}_{2}({\bf k})/L^{({\rm h})}_{1}({\bf k})$ is the energy spectrum of $\Sigma^{({\rm h})}_{\rm ph}(\bf k,\omega)$, and $\bar{\Delta}^{({\rm h})}_{\rm pg}({\bf k})=L^{({\rm h})}_{2}({\bf k})/\sqrt{L^{({\rm h})}_{1}({\bf k})}$ is refereed to as the momentum dependence of the charge-carrier pseudogap, since it plays a role of the suppression of the low-energy spectral weight of the charge-carrier excitation spectrum, with the functions $L^{({\rm h})}_{1}({\bf k})=-\Sigma^{({\rm h})}_{\rm pho}({\bf k},\omega=0)$ and $L^{({\rm h})}_{2}({\bf k})=-\Sigma^{({\rm h})}_{\rm ph}({\bf k},\omega=0)$, while $\Sigma^{({\rm h})}_{\rm ph}({\bf k},\omega=0)$ and the antisymmetric part $\Sigma^{({\rm h})}_{\rm pho}({\bf k},\omega)$ of the charge-carrier normal self-energy are obtained directly from $\Sigma^{({\rm h})}_{\rm ph}({\bf k},\omega)$ in Eq. (43).

Moreover, the sharp peaks appearing at low-temperature in $\Sigma^{({\rm h})}_{\rm ph}(\bf k,\omega)$ and the related quantities are actually a $\delta$-function, which are broadened by a small damping employed in the numerical calculation for a finite latticeBrinckmann01 ; Restrepo23 . As the same approach of the numerical calculation carried out in Ref. Ma23, , the calculation in this paper for $\Sigma^{({\rm h})}_{\rm ph}(\bf k,\omega)$ and the related quantities is performed numerically on a $160\times 160$ lattice in momentum space, where the infinitesimal $i0_{+}\rightarrow i\Gamma$ is replaced by a small damping $\Gamma=0.05J$.

For a convenience in the following discussions, the charge-carrier pseudogap Feng15 ; Feng12 $\bar{\Delta}^{({\rm h})}_{\rm pg}$ at temperature $T=0.002J$ as a function of doping is replotted in Fig. 1a, where the relatively large $\bar{\Delta}^{({\rm h})}_{\rm pg}$ appears in the underdoped regime, and then it weakens as the optimal doping is approached. However, a quite weak $\bar{\Delta}^{({\rm h})}_{\rm pg}$ is still present at around the optimal doping, but it disappears at the heavily overdoped region. Moreover, at a given doping concentration, this $\bar{\Delta}^{({\rm h})}_{\rm pg}$ is identified with a crossover with a charge-carrier pseudogap crossover temperature $T^{*}_{\rm h}$ rather than a phase transition. To see the evolution of $T^{*}_{\rm h}$ with doping more clearly, $T^{*}_{\rm h}$ as a function of doping is also replotted in Fig. 1b, where in conformity with the doping dependence of $\bar{\Delta}^{({\rm h})}_{\rm pg}$ in Fig. 1a, $T^{*}_{\rm h}$ is relatively high at the slight underdoping, and then it smoothly decreases with the increase of doping in the underdoped regime, eventually terminating at the heavily overdoped regionFeng12 ; Feng15 . More importantly, as the electron pairing state originated from the charge-carrier pairing state are due to the charge-spin recombination, the normal-state pseudogap state originated from the charge-carrier pseudogap state is also due to the charge-spin recombinationFeng15a , and then the anomalous properties associated with the formation of the normal-state pseudogap are explained in a natural way Damascelli03 ; Campuzano04 ; Fink07 ; Deutscher05 ; Devereaux07 ; Fischer07 ; Hussey08 ; Kastner98 ; Timusk99 ; Hufner08 ; Vishik18 ; Fujita12 . We will return to the discussion of the normal-state pseudogap towards next subsection II.3.

On the other hand, in the fermion-spin theoryFeng0494 ; Feng15 , the scattering of spins due to the charge-carrier fluctuation dominates the spin dynamics. In this case, the dynamical spin response of cuprate superconductors has been discussed from the SC-state to the normal-stateKuang15 ; Yuan01 ; Feng98 , where the full spin propagator in the normal-state has been derived as [see Appendix A],

with the MF spin excitation energy dispersion $\omega_{\bf k}$ and the corresponding weight function $B_{\bf k}$ of the MF spin excitation spectrum that have been obtained explicitly in the previous worksFeng15 ; Kuang15 , while the spin self-energy in the normal-state obtained in terms of the collective charge-carrier mode in the particle-hole channel that is given in Eq. (A.2) in Appendix A.

For a better understanding of the nonconventional features of the spin pseudogap, the above spin self-energy in Eq. (5) can be also rewritten as,

where $\omega^{2}_{0{\bf k}}=L^{({\rm s})}_{1{\bf k}}/L^{({\rm s})}_{2{\bf k}}$ is the energy spectrum of $\Sigma^{({\rm s})}_{\rm ph}({\bf k},\omega)$, while $[\bar{\Delta}^{({\rm s})}_{\rm pg}({\bf k})]^{2}=[L^{({\rm s})}_{1{\bf k}}]^{2}/B_{\bf k}L^{({\rm s})}_{2{\bf k}}$ is identified as being a region of the spin self-energy in which $\bar{\Delta}^{({\rm s})}_{\rm pg}({\bf k})$ anisotropically reduces the density of the spin excitation, and in this sense, $\bar{\Delta}^{({\rm s})}_{\rm pg}({\bf k})$ is refereed to as the spin pseudogap. The functions $L^{({\rm s})}_{1{\bf k}}$ and $L^{({\rm s})}_{2{\bf k}}$ is derived directly from the spin self-energy $\Sigma^{({\rm s})}_{\rm ph}({\bf k},\omega)$ in Eq. (A.2) as,

respectively, where the renormalized charge-carrier energy dispersion $\bar{\xi}_{\bf k}=Z^{({\rm h})}_{\rm F}\xi_{\bf k}$, with the charge-carrier coherent weight $Z^{({\rm h})-1}_{\rm F}=1-{\rm Re}\Sigma^{({\rm h})}_{\rm pho}({\bf k},0)\mid_{{\bf k}=[\pi,0]}$, while the function $F^{\rm(s)}({\bf k},{\bf p},{\bf q})$ and vertex function $\Omega^{({\rm s})}_{{\bf k}{\bf p}{\bf q}}$ are presented in Appendix A. Substituting the above spin self-energy (6) into Eq. (5), the full spin propagator in the normal-state can be derived as,

with the renormalized spin excitation energy dispersions,

and the corresponding weight functions of the full spin excitation spectrum,

It should be emphasized that the equation (6) is an identity only in the case of $\omega=0$, however, as in the case of the charge-carrier pseudogap Feng15 ; Feng12 , it is a proper approximation for the low-energy case of $\omega\neq 0$.

The momentum dependence of the spin pseudogap $\bar{\Delta}^{({\rm s})}_{\rm pg}({\bf k})$ in Eq. (6) evolves strongly with doping and temperature, as that in the charge-carrier pseudogap Feng15 ; Feng12 . On the other hand, weMa23 have shown that the umklapp scattering between electrons by the exchange of the MF effective spin propagator can give a consistent description of the low-temperature T-linear resistivity in the overdoped strange-metal phase, where the dominant contribution to the scattering rate mainly comes from the antinodal umklapp scattering. In this case, we focus on the exotic features of the doping and temperature dependent spin pseudogap $\bar{\Delta}^{({\rm s})}_{\rm pg}({\bf k})$ at around the antinodal region for a convenience in the following discussions of the low-temperature resistivity in the underdoped pseudogap phase. In Fig. 2a, we plot $\bar{\Delta}^{({\rm s})}_{\rm pg}({\bf k}_{\rm AN})$ at the antinode as a function of doping with temperature $T=0.002J$, where ${\bf k}_{\rm AN}$ is the wave vector at the antinode of the Brillouin zone (BZ). Apparently, this antinodal spin pseudogap is insensitive to the doping in the slightly underdoped region, and then it falls off rapidly as the doping is grown in the heavily underdoped region. To our big surprise, this antinodal spin pseudogap abruptly disappears at around the optimal doping. This antinodal spin pseudogap formation in the underdoped regime lowers the density of the spin excitations in response to the intense electron umklapp scattering from the spin excitations associated with the antinodes. However, in the overdoped regime, the main properties of the antinodal spin excitations can be well described by the MF spin propagator. Furthermore, for a given doping, this antinodal spin pseudogap vanishes when temperature reaches the antinodal spin-pseudogap crossover temperature $T^{*}_{\rm s}$. To see this doping dependence of $T^{*}_{\rm s}$ more clearly, we plot $T^{*}_{\rm s}$ as a function of doping in Fig. 2b, where in corresponding to the result of $\bar{\Delta}^{({\rm s})}_{\rm pg}({\bf k}_{\rm AN})$ in Fig. 2a, $T^{*}_{\rm s}$ is relatively high in the slightly underdoped region, and then it decreases rapidly when the doping is increased in the heavily underdoped region, eventually disappearing at around the optimal doping. It should be noted that the disappearance of the antinodal spin pseudogap at around the optimal doping does not indicate the existence of QCP at around the optimal doping. This follows from a basic fact that the experimental observations in the optimally doped and overdoped regimes show that although the strength of the T-linear resistivity gradually diminishes as a function of doping, the low-temperature T-linear resistivity retains a finite value up to the edge of the SC dome Keimer15 ; Hussey23 ; Legros19 ; Ayres21 ; Grisso21 , which can be described by the momentum relaxation due to the umklapp scattering between electrons by the exchange of the MF effective spin propagatorMa23 , however, it may be a challenging issue for the usual picture of a single QCP doping associated with the strange-metal phaseVarma20 ; Damle97 ; Sachdev11 .

Based on the full spin propagator (8), the dynamical spin response of cuprate superconductors in the normal-state has been investigatedKuang15 ; Yuan01 ; Feng98 , where the obtained results show the existence of damped but well-defined dispersive spin excitations in the whole doping phase diagram. In particular, the spectral weight of the spin excitation spectrum at around the antinodal region is strongly suppressed by the antinodal spin pseudogapKuang15 . Moreover, the low-energy spin fluctuation is dominated by the process from the mobile charge carriers, while the high-energy spin excitation on the other hand retains roughly constant energy as a function of doping, with spectral weight and dispersion relation comparable to those in the corresponding SC-state. All these results are qualitatively consistent with the experimental observations Fujita12 ; Birgeneau89 ; Fong95 ; Yamada98 ; Arai99 ; Bourges00 ; He01 ; Tranquada04 ; Bourges05 .

For the understanding of the nature of the electronic structure in cuprate superconductors, it is needed to derive the electron propagator, which is characterized by the full charge-spin recombinationFeng15a . Following the previous discussionsFeng15a , the full electron propagator in the normal-state can be derived as [see Appendix A],

where $\varepsilon_{\bf k}=-4t\gamma_{\bf k}+4t^{\prime}\gamma_{\bf k}^{\prime}+\mu$ is the electron energy dispersion in the tight-binding approximation, while the electron normal self-energy $\Sigma_{\rm ph}({\bf k},\omega)$ is given in Eq. (51) in Appendix A.

The electron normal self-energy in Eq. (11) originated from the charge-carrier normal self-energy in Eq. (3) is due to the charge-spin recombination Feng15a , indicating that the normal-state pseudogap state originated from the charge-carrier pseudogap state is due to the charge-spin recombination. To see the nature of the normal-state pseudogap more clearly, the above electron normal self-energy in Eq. (11) can be reexpressed as,

where $\varepsilon_{0{\bf k}}=L_{2{\bf k}}/L_{1{\bf k}}$ is the energy spectrum of $\Sigma_{\rm ph}({\bf k},\omega)$, and $\bar{\Delta}^{2}_{\rm PG}({\bf k})=L^{2}_{2{\bf k}}/L_{1{\bf k}}$ is the normal-state pseudogap, with the functions $L_{1}({\bf k})=-\Sigma_{\rm pho}({\bf k},\omega=0)$ and $L_{2}({\bf k})=-\Sigma_{\rm ph}({\bf k},\omega=0)$, while the antisymmetric part $\Sigma_{\rm pho}({\bf k},\omega)$ of the electron normal self-energy can be obtained directly from the electron normal self-energy $\Sigma_{\rm ph}({\bf k},\omega)$ in Eq. (51). This normal-state pseudogap $\bar{\Delta}_{\rm PG}({\bf k})$ is also identified as being a region of the electron normal self-energy in which the normal-state pseudogap anisotropically suppresses the electronic density of states on EFS. Since the normal-state pseudogap state is induced by the charge-carrier pseudogap state, the normal-state pseudogap $\bar{\Delta}_{\rm PG}({\bf k})$ [the normal-state pseudogap crossover temperature $T^{*}$] as a function of doping presents a similar behavior of the charge-carrier pseudogap $\bar{\Delta}^{({\rm h})}_{\rm pg}({\bf k})$ [the charge-carrier pseudogap crossover temperature $T^{*}_{\rm h}$]Feng15 ; Feng12 . This normal-state pseudogap crossover temperature $T^{*}$ is actually a crossover line below which a novel electronic state emerges, as exemplified by the presence of the Fermi arcs, the competing electronic orders, etc., and then the unconventional features of this novel electronic state can be well interpreted in terms of the formation of the normal-state pseudogap Damascelli03 ; Campuzano04 ; Fink07 ; Deutscher05 ; Devereaux07 ; Fischer07 ; Hussey08 ; Kastner98 ; Timusk99 ; Hufner08 ; Vishik18 .

The measured energy and momentum distribution curves can be depicted by the electron excitation spectrum Dessau91 ; Randeria95 ; Fedorov99 ; Campuzano99 ; DMou17 ; Bogdanov00 ; Kaminski01 ; Johnson01 ; Iwasawa08 ; Plumb13 ,

with the electron spectral function,

where ${\rm Re}\Sigma_{\rm ph}({\bf k},\omega)$ and ${\rm Im}\Sigma_{\rm ph}({\bf k},\omega)$ are the real and imaginary parts of $\Sigma_{\rm ph}({\bf k},\omega)$, respectively. The electrons are renormalzed due to the electron scattering mediated by the spin excitation and then they acquire a finite lifetime.

The EFS topology is known to be crucial in the understanding of the unconventional superconductivity in cuprate superconductors Damascelli03 ; Campuzano04 ; Fink07 ; Deutscher05 ; Devereaux07 ; Fischer07 and the related anomalous normal-state properties Hussey08 ; Kastner98 ; Timusk99 ; Hufner08 ; Vishik18 . The ARPES measured EFS can be described theoretically by the intensity map of the electron excitation spectrum (13) at zero energy $\omega=0$, where the locations of the EFS continuous contour in momentum space is determined directly by the poles of the electron spectral function (14), $\varepsilon_{\bf k}+{\rm Re}\Sigma_{\rm ph}({\bf k},0)=\bar{\varepsilon}_{\bf k}=0$, with the renormalized electron energy dispersion $\bar{\varepsilon}_{\bf k}=Z_{\rm F}\varepsilon_{\bf k}$ and the single-particle coherent weight $Z^{-1}_{\rm F}=1-{\rm Re}\Sigma_{\rm pho}({\bf k},0)\mid_{{\bf k}=[\pi,0]}$. However, the strong redistribution of the spectral weight on EFS is mainly governed by the momentum dependence of the normal-state pseudogap $\bar{\Delta}_{\rm PG}({\bf k})$ [then the single-electron scattering rate $\Gamma_{\bf k}(\omega)={\rm Im}\Sigma_{\rm ph}({\bf k},\omega)=\pi[\bar{\Delta}_{\rm PG}({\bf k})]^{2}\delta(\omega+\varepsilon_{0{\bf k}})]$.

For a convenience in the following discussions of the resistivity in the underdoped pseudogap phase, the EFS mapFeng16 ; Liu21 ; Cao21 ; Zeng22 at doping $\delta=0.09$ with temperature $T=0.002J$ is replotted in Fig. 3, where the BZ center has been shifted by [$\pi,\pi$], and AN, TFA, and ND indicate the antinode, tip of the Fermi arc, and node, respectively. It thus shows clearly that the antinodal region of EFS becomes partially gapped, leading to that EFS consists, not of a closed contour, but only of four disconnected Fermi arcs centered around the nodes, in qualitative agreement with the experimental results Loeser96 ; Norman98 ; Chatterjee06 ; Shi08 ; Sassa11 ; Kaminski15 ; Loret17 ; Loret18 ; He14 ; Comin14 ; Comin16 . However, the most of the spectral weight on the Fermi arcs locates at around the tips of the Fermi arcs, indicating that the electrons at around the tips of the Fermi arcs have a largest density of states Chatterjee06 ; Shi08 ; Sassa11 ; Kaminski15 ; Loret17 ; Loret18 ; He14 ; Comin14 ; Comin16 , and then charge order is driven by this EFS instability, with a characteristic wave vector corresponding to the tips of the Fermi arcsFeng16 ; Comin14 ; Comin16 .

The elementary excitations are parameterized by the electron spectral function (14), which has given a consistent description of the renormalization of the electrons in the underdoped pseudogap phaseFeng16 ; Liu21 ; Cao21 ; Zeng22 , where the obtained results show that the single-electron scattering rate has a well-pronounced peak structure at around the antinodal and nodal regions, which leads to the remarkable peak-dip-hump structure in the energy distribution curve. Moreover, the dispersion kink is induced by the inflection point in the single-electron scattering rate, while the spectral weight at around the dispersion kink is reduced highly by the corresponding peak in the real part of the electron normal self-energy. All these results are in qualitative agreement with the corresponding experimental results Dessau91 ; Randeria95 ; Fedorov99 ; Campuzano99 ; DMou17 ; Bogdanov00 ; Kaminski01 ; Johnson01 ; Iwasawa08 ; Plumb13 . More importantly, the anisotropic suppression of the electronic density of states on EFS by the normal-state pseudogap can affect the electrical transport in two ways Timusk99 : (i) through the reduction of the number of electron current-carrying states as a normal-state pseudogap forms; and (ii) since the electron current carriers are scattered by the spin excitation, through the reduction in the density of spin excitations.

We now turn to discuss the low-temperature electrical transport in the underdoped pseudogap phase. Theoretically, the Boltzmann transport equation is the cornerstone for the discussions of the electrical transportAbrikosov88 ; Mahan81 , since the Boltzmann transport equation is valid in the case of either the existence of the well-defined quasiparticles or the treatment of the electron interaction mediated by different bosonic modes within the Eliashberg approach. This builds on the following pioneering works: (i) in the early days of the electrical transport research in the conventional superconductorsPrange64 , Prange and Kadanoff demonstrated that in an electron-phonon system, a set of transport equations can be derived in the Migdal’s approximation, where the electron interaction mediated by the phonon leads to the electron self-energy and vertex correction. In particular, this coupled set of transport equations for the electron and phonon distribution functions is correct even in the case of the absence of the well-defined quasiparticlesPrange64 . Nevertheless, one of the forms of the electrical transport equation,

is identical to the electrical Boltzmann transport equation suggested by Landau for the case in which the quasiparticle is well-definedAbrikosov88 ; Mahan81 , with the charge of an electron $e$, the electron distribution function in a homogeneous system $f({\bf k},t)$, and the electron-electron collision term $I_{\rm e-e}$. For a convenience in the following discussions, the external magnetic field ${\bf H}$ has been dropped, and only an external electric field ${\bf E}$ is applied to the system; (ii) however, this Boltzmann transport equation (15) is not specific to the electron interaction mediated by the phonon in the conventional superconductors, and has been confirmed recently that it is also valid for the system with the electron interaction mediated by other bosonic excitations Lee21 ; Ma23 . With the advantage of their insightLee21 ; Ma23 ; Prange64 , the electrical transport due to the electron scattering mediated by various kinds of bosonic modes can be evaluated in a simple way even in the case of the break down of the quasiparticle picture.

In the following discussions, we study the the low-temperature resistivity in the underdoped pseudogap phase based on the Boltzmann transport equation (15). For the calculation of this Boltzmann transport equation (15), the linear perturbation from the equilibrium in terms of the distribution function has been introduced Lee21 ; Ma23 ; Prange64 ,

with the fermion distribution functions $n_{\rm F}(\omega)$, and the local shift of the chemical potential at a given patch of EFS $\tilde{\Phi}({\bf k})$, which satisfies the antisymmetric relation $\tilde{\Phi}(-{\bf k})=-\tilde{\Phi}({\bf k})$. We substitute the above result in Eq. (16) into Eq. (15), and then linearize the Boltzmann equation (15) as,

where ${\bf v}_{\bf k}=\nabla_{\bf k}{\bar{\varepsilon}_{\bf k}}$ is the electron velocity.

The electron-electron collision term $I_{\rm e-e}$ in the Boltzmann equation is directly connected with the electrical scattering mechanism Abrikosov88 ; Mahan81 . Although there is no consensus on the origin of the electrical transport in the underdoped regime to date, it is widely believed that the opening of the spin pseudogap induces a deviation from the T-linear behaviour of the resistivityGurvitch87 ; Bucher93 ; Ito93 ; Nakano94 ; Ando01 . In particular, it has been shown clearly that the electron umklapp scattering is the origin of the low-temperature T-linear resistivity in the strange-metal phase Lee21 ; Honerkamp01 ; Hartnoll12 . Moreover, weMa23 have also shown very recently that the low-temperature T-linear resistivity in the overdoped strange-metal phase originates from the umklapp scattering between electrons by the exchange of the MF effective spin propagator. In the following discussions, we will show that when this electron umklapp scattering mediated by the spin excitation in the overdoped regimeMa23 flows to the underdoped regime, the opening of the momentum dependence of the spin pseudogap naturally leads to a low-temperature T-quadratic behaviour of the resistivity. To see the electron umklapp scattering process more clearly, we show a schematic picture of the electron umklapp scattering processLee21 ; Ma23 in Fig. 4, where an electron on a circular EFS (left) is scattered by its partner on the umklapp EFS (right). It should be noted that the intensity map of EFS in Fig. 4 is the same as the EFS map in Fig. 3, while the Fermi wave vector of the tips of the Fermi arcs ${\rm k}^{\rm TFA}_{\rm F}$ is the radius of the EFS circle (red). Moreover, this circle EFS (red) connects all tips of the Fermi arcs, and then the most of the electronic density of states is concentred on this circular EFS.

Following these recent discussionsLee21 ; Ma23 , we can derive the electron-electron collision $I_{\rm e-e}$ in Eq. (17) in the underdoped pseudogap phase as,

where ${\bf G}$ represents a set of reciprocal lattice vectors. It should be emphasized that the above electron umklapp scattering (18) is described as a scattering between electrons by the exchange of the effective spin propagator,

rather than the scattering between electrons via the emission and absorption of the spin excitationLee21 ; Ma23 , where $\Lambda_{{\bf k}}=4t\gamma_{\bf k}-4t^{\prime}\gamma_{\bf k}^{\prime}$ is the bare vertex function.

In cuprate superconductors, a small density of charge carriers is sufficient to destroy the AF long-range order (AFLRO). However, the doped charge carriers and the coupled spins organize themselves in a cooperative way to enhance both the electrons mobility and the AF short-range order (AFSRO) correlation, and then the spin excitations in the spin liquid state with AFSRO appear to survive from the underdoped regime to the overdoped regimeFujita12 . Moreover, it has been shown that the coupling strength of the electrons with the spin excitations gradually weakens with the increase of doping from a strong-coupling case in the underdoped regime to a weak-coupling side in the overdoped regime Feng12 ; Johnson01 ; Kordyuk10 , reflecting a reduction of the strength of the magnetic fluctuation with the increase of doping. In other words, (i) the effect of the magnetic fluctuation in the overdoped regime is less dramatic than in the underdoped regimeKastner98 ; Timusk99 ; Hufner08 ; Vishik18 ; Fujita12 ; (ii) on the other hand, as shown in Fig. 2, the effect from antinodal spin pseudogap is absent in the overdoped regime, and then the main properties of the antinodal spin excitations can be well described by the MF spin propagator. These are reasons why the interpretation of the low-temperature T-linear resistivity in the overdoped strange-metal phase can be well made in terms of the umklapp scattering between electrons by the exchange of the MF effective spin propagatorMa23 .

However, in the strong-coupling side (then in the underdoped regime), (i) the effect of the magnetic fluctuation is much dramatic Kastner98 ; Timusk99 ; Hufner08 ; Vishik18 ; Fujita12 ; and (ii) as shown in Fig. 2, the effect from antinodal spin pseudogap is particularly notable, where the antinodal spin pseudogap lowers the density of the spin excitationsKuang15 in response to the intense electron umklapp scattering, which reduces the strength of the electron umklapp scattering from the spin excitation states into the antinodal region. In this case, the umklapp scattering between electrons should be mediated by the exchange of the full effective spin propagator for a proper description of the low-temperature resistivity in the underdoped pseudogap phase.

Now our aim is to obtain the full effective spin propagator. The full spin bubble $\bar{\Pi}({\bf p},{\bf q},\omega)$ in Eq. (19) is a convolution of two full spin propagators, and can be derived directly from the full spin propagator (8) as,

where the spin excitation energy dispersions $\bar{\omega}^{(1)}_{\alpha\alpha^{\prime}{\bf p}{\bf q}}$ and $\bar{\omega}^{(2)}_{\alpha\alpha^{\prime}{\bf p}{\bf q}}$ are given by,

respectively, and the corresponding functions,

where $n_{\rm B}(\omega)$ is the boson distribution function. From the above spin bubble (20), the full effective spin propagator (19) now can be derived as,

with the weight functions,

The electron umklapp scattering in Eq. (18) shows that the electron-electron collision $I_{\rm e-e}$ is both functions of momentum and energy. Howerver, at low temperatures, everything happens near EFSAbrikosov88 ; Mahan81 . In this case, an any given patch on the circular EFS shown in Fig. 4 is represented via the Fermi angle $\theta$ with the Fermi angle range $\theta\in[0,2\pi]$, and then the momentum integration along the perpendicular momentum can be replaced by the integrationLee21 ; Ma23 ; Prange64 over $\bar{\varepsilon}_{\bf k}$. For the umklapp scattering between electrons by the exchange of the full effective spin propagator in Eq. (18), an electron on the circular EFS parametrized by the Fermi angle $\theta$ is scattered to a point parametrized by the Fermi angle $\theta^{\prime}$ on the umklapp EFS via the spin excitation carrying momentum ${\rm p}(\theta,\theta^{\prime})$ as shown in Fig. 4. Following the above treatment and the calculation process in the recent worksLee21 ; Ma23 for the low-temperature resistivity in the overdoped strange-metal phase, the electron-electron collision $I_{\rm e-e}$ in Eq. (18) in the underdoped pseudogap phase can be obtained straightforwardlyMa23 , and then the Boltzmann transport equation (17) can be derived as,

with $\Phi(\theta)=\tilde{\Phi}[{\rm k}(\theta)]$, the Fermi velocity ${\bf v}_{\rm F}(\theta)$ at the Fermi angle $\theta$, the density of states factor $\zeta(\theta^{\prime})={\rm k}^{2}_{\rm F}/[4\pi^{2}{\rm v}^{3}_{\rm F}]$ at angle $\theta^{\prime}$, the Fermi wave vector ${\rm k}_{\rm F}$, and the Fermi velocity ${\rm v}_{\rm F}$. In particular, the antisymmetric relation $\tilde{\Phi}(-{\bf k})=-\tilde{\Phi}({\bf k})$ for $\tilde{\Phi}({\bf k})$ in Eq. (16) has been converted into $\Phi(\theta)=-\Phi(\theta+\pi)$ for $\Phi(\theta)$ in the above Eq. (26). Moreover, the coefficient of $\Phi(\theta)$ in the first term of the right-hand side of Eq. (26),

can be referred to as the angular (momentum) dependence of the umklapp scattering rateMa23 ; Lee21 , with the kernel function $F(\theta,\theta^{\prime})$ that connects the point $\theta$ on the circular EFS with the point $\theta^{\prime}$ on the umklapp EFS via the amplitude of the momentum transfer ${\rm p}(\theta,\theta^{\prime})$ as shown in Fig. 4, which can be derived as,

where the full effective spin propagator $\bar{P}[{\rm k}(\theta),{\rm p}(\theta,\theta^{\prime}),\omega]$ has been rewritten explicitly in terms of the Fermi angles $\theta$ and $\theta^{\prime}$ asMa23 ,

where $\varpi^{(1)}_{\alpha\alpha^{\prime}}(\theta,\theta^{\prime},{\bf q})=\varpi^{(1)}_{\alpha\alpha^{\prime}}[{\rm k}(\theta),{\rm p}(\theta,\theta^{\prime}),{\bf k}^{\prime}_{\rm F},{\bf q}]$, $\varpi^{(2)}_{\alpha\alpha^{\prime}}(\theta,\theta^{\prime},{\bf q})=\varpi^{(2)}_{\alpha\alpha^{\prime}}[{\rm k}(\theta),{\rm p}(\theta,\theta^{\prime}),{\bf k}^{\prime}_{\rm F},{\bf q}]$, $\bar{\omega}^{(1)}_{\alpha\alpha^{\prime}\theta,\theta^{\prime}}({\bf q})=\bar{\omega}^{(1)}_{\alpha\alpha^{\prime}{\rm p}(\theta,\theta^{\prime}){\bf q}}$, and $\bar{\omega}^{(2)}_{\alpha\alpha^{\prime}\theta,\theta^{\prime}}({\bf q})=\bar{\omega}^{(2)}_{\alpha\alpha^{\prime}{\rm p}(\theta,\theta^{\prime}){\bf q}}$.

At low temperatures, the electron elastic scattering occurs on EFS, while the electron inelastic scattering occurs near EFSKohsaka08 ; Abrikosov88 ; Mahan81 . In this case, the above energy integration in the kernel function (28) actually includes the inelastic scattering process of momentum and energy exchange between electronsLee21 ; Ma23 ; Prange64 . This follows from a fact that the umklapp scattering rate $\gamma(\theta)$ in Eq. (27) is directly associated with the kernel function $F(\theta,\theta^{\prime})$ in Eq. (28), while this kernel function is obtained by the integration of the spin excitation energy $\omega$, leading to the occurrence of the inelastic scattering process near EFS in the present umklapp scattering. In particular, in the case of the absence of the spin pseudogap, this electron umklapp scattering induces a low-temperature T-linear resistivity in the overdoped regimeMa23 , in agreement with the experimental observations Legros19 ; Ayres21 ; Grisso21 , where the experimental analyses seem to indicate that the T-linear resistivity is due to the inelastic scattering.

For the discussions of the low-temperature resistivity, we need to obtain the electron current density, which can be derived in terms of the local shift of the chemical potential $\Phi(\theta)$ asMa23 ,

with the momentum relaxation that is generated by the action of the electric field on the mobile electrons at EFS with the density $n_{0}$. However, this local shift of the chemical potential $\Phi(\theta)$ can be evaluated directly in the relaxation-time approximation asLee21 ; Ma23 , $\Phi(\theta)=-e{\rm v}_{\rm F}{\rm cos}(\theta)E_{\hat{x}}/[2\gamma(\theta)]$, where the electric field ${\bf E}$ has been selected along the $\hat{x}$-axis. With the help of the above treatment, the dc conductivity can be derived as Lee21 ; Ma23 ,

and then the resistivity can be expressed directly in terms of the above dc conductivity as,

It thus shows that the electrical resistance originates from the umklapp scattering between electrons by the exchange of the full effective spin propagator.