A theory of coupled dual dynamics of macroscopic phase coherence and microscopic electronic fluids: effect of dephasing on cuprate superconductivity

Within the framework of Bardeen, Cooper and Schrieffer (BCS) theory of conventional superconductivityBCS , the emergence of the superconducting order parameter by forming Cooper pairs occurs upon cooling at the critical temperature $T_{c}$, below which superconductivity (zero-resistance) phenomenon occurs. The high-$T_{c}$ superconducting materials, like cuprates first discovered in 1986cuprate1 ; cuprate2 , are beyond the BCS mechanismcuprate3 ; cuprate4 ; cuprate5 . Specifically, it is established that the superconducting order parameter shows up as an energy gap in spectroscopic probes of quasiparticle energy spectrum. Whereas in cuprate superconductors, detected most directly by angle-resolved photo-emission spectroscopyARPES1 ; ARPES2 ; ARPES3 and scanning tunneling microscopeSTM1 ; STM2 ; STM3 ; STM4 , a finite normal-state gap (pseudogap) with $d$-wave symmetry opens below a temperature $T^{*}$, far above the superconducting transition temperature $T_{c}$.

The origin of the pseudogap and its relationship to superconductivity have attracted extensive experimental and theoretical interest. Transport properties like electrical resistivityresistivity1 ; resistivity2 , Hall conductivityHall1 ; Hall2 and Nernst coefficientNernst1 ; Nernst2 ; Nernst3 ; Nernst4 ; Nernst5 ; Nernst6 that are remarkably affected by the opening of the pseudogap have been used to detect $T^{*}$. While among various theoretical models, significant superconducting phase fluctuation is suggested to be a possible attributionphase1 ; phase2 ; phase3 . Specifically, the generation of the superconducting order parameter breaks the continuous $U(1)$ symmetry spontaneouslygi0 , and then, according to Goldstone theoremGm1 ; Gm2 , a collective gapless Bosonic excitation that describes the phase fluctuation of the order parameter emergesgi0 ; AK ; Ba0 ; Am0 . This excitation in conventional bulk superconductors is inactiveAK ; Ba0 ; Am0 , since its original low-energy spectrum is lifted to high-frequency plasma energy by Anderson-Higgs mechanismAHM , due to the coupling to longitudinal electromagnetic field and hence long-range Coulomb interactionAK ; Ba0 ; Am0 . Whereas in high-$T_{c}$ superconductors, an active phase fluctuation becomes inevitable because of the low-dimensional layered structurecuprate4 ; cuprate5 . Then, in one view, the pseudogap state is a incoherent precursor of the superconducting statephase2 ; PCP1 ; PCP2 ; PCP3 ; PCP4 ; PCP5 , reflecting a state of preformed Cooper pairs without the phase coherence necessary to achieve superconductivity. The superconducting transition then occurs upon cooling below a lower temperature where the long-range phase coherence is established. $T_{c}$ is therefore determined by the onset of the phase coherence rather than the formation of the Cooper pairsphase2 ; PCP1 ; PCP2 ; PCP3 ; PCP4 ; PCP5 .

The existence of the preformed Cooper pairing above $T_{c}$ received a number of experimental supports by various approaches like diamagnetism probeTM1 ; TM2 ; TM3 ; TM4 , specific heatSH and paraconductivityPC1 measurements, Nernst effectNF1 , ultrafast pump-probe spectroscopiesPPS1 ; PPS2 and angle-resolved photo-emissionAF1 ; AF2 ; AF3 ; AF4 as well as detection of optical conductivity in infraredInfrad1 ; Infrad2 ; Infrad3 , microwaveMS1 ; MS2 ; MS3 and terahertzTHZ1 ; THZ2 ; THZ3 ; THZ4 frequency regimes. The most convincing evidence so far comes from a recent observation of the collective Higgs mode above $T_{c}$DHM1 ; DHM2 ; DHM3 , since this excitation describes the amplitude fluctuation of the order parameterAm0 and thus directly reflects the existence of the pairing. Nevertheless, most of these experimentsTM2 ; TM3 ; TM4 ; SH ; PC1 ; NF1 ; AF1 ; AF2 ; AF3 ; AF4 ; PPS1 ; PPS2 ; Infrad1 ; Infrad2 ; Infrad3 ; MS1 ; MS2 ; MS3 ; THZ1 ; THZ2 ; THZ3 ; THZ4 ; DHM3 realized that the regime of the significant superconducting phase fluctuation is in fact a relatively narrow one that tracks $T_{c}$, whereas the upper onset temperature of this regime $T^{\rm os}$ lies well below the pseudogap temperature $T^{*}$ in underdoped regime and tends to coincide with $T^{*}$ in overdoped regimeTM4 ; SH ; PC1 ; NF1 ; AF4 ; MS1 ; MS2 ; MS3 ; THZ1 ; THZ2 ; THZ3 ; THZ4 ; DHM3 . Then, in an alternative view, the pseudogap state between $T^{\rm os}$ and $T^{*}$ in underdoped regime, exhibiting various intertwined orders (nematicity, charge-density-wave and spin-density-wave orders), possibly represents another state of matter that competes/couples with superconductivitySTM1 ; AF4 ; NF1 . Whereas the precursor of the superconducting state with significant phase fluctuation (incoherent preformed Cooper pair) actually begins upon cooling at $T^{\rm os}$.

The phenomenological preformed Cooper-pair model with significant phase fluctuation is now widely accepted from experimental findings, but its microscopic theoretical description is not yet developed. Interestingly, from earlier thermodynamicNSB1 and recent opticalNSB2 measurements, substantial fraction of the uncondensed normal state, which exhibits $T$-linear specific heat and Drude optical conductivity, persists down to temperatures far below $T_{c}$. This suggests that the phase fluctuation exists not only in the pseudogap state but also in the superconducting one. Therefore, to achieve superconductivity, the phase coherence must exceed a specific nonzero threshold. The theory of determining this threshold and in particular, $T_{c}$, requires coupled dual dynamics with different scales, i.e., in different Hilbert spaces: macroscopic phase-coherence dynamics and microscopic electronic fluid (including superfluid and normal fluidSNT ) dynamics. In the early-stage works, without distinguishing the amplitude and phase fluctuations, Ussishkin et al.GT1 ; GT2 applied the Gaussian approximationGT to calculate the contribution from the fluctuations of order parameter to Nernst signal. After that, within the path-integral method to study thermodynamics, Curty and BeckMT1 treated separately amplitude and phase fluctuations, and used the Monte Carlo procedure and Wolff algorithm for simulations, respectively. But in both approaches, the origin of the fluctuations is unclear. Recently, Li et al.disorder suggested that in cuprate superconductors, the disorder effect at low temperature plays the crucial role in determining the amplitude and phase fluctuations of the order parameter. By using tight-binding model with the random on-site potential (Anderson disorder)disorder , they numerically obtained granular superconducting islands to explain the substantial fraction of the normal state at $T=0$, and then, suggested that the strong phase fluctuation can emerge in regions with small gap as a consequence. But this stationary-state calculation actually does not take account of the phase-coherence dynamics seriously, and is hard to extend for finite temperatures. Most importantly, the microscopic electronic fluid dynamics is decoupled and overlooked in all approaches above, inhibiting the deep insight into the key issue, separation between $T_{c}$ and $T^{\rm os}$.

Actually, a microscopic gauge-invariant kinetic equation (GIKE) approach has been developed in conventional $s$-wave superconductorsGIKE1 ; GIKE2 ; GIKE3 . This approach, as analytically demonstrated, not only involves both superfluid and normal-fluid dynamicsGIKE1 , but also is capable of formulating both phase and amplitude fluctuations of the order parameter (i.e., Nambu-Goldstone and Higgs modes)GIKE2 . The complete microscopic scattering is also constructed in GIKEGIKE1 ; GIKE3 . Very recently, this approach has also been extended into the $d$-wave superconductors for studying the Higgs modesGIKE4 . It is therefore natural to further use this approach to study the phase-coherence dynamics and its influence on electronic fluid dynamics in cuprate superconductors, and then, elucidate the fundamental nature of the incoherent preformed Cooper pairs.

In this work, by using the GIKE approachGIKE1 ; GIKE2 ; GIKE3 ; GIKE4 , we construct the coupled dual dynamics of macroscopic phase coherence and microscopic electronic fluids (consisting of normal fluid and superfluid) in cuprate superconductors. Then, both the dephasing process of macroscopic superconducting phase coherence from long range to short range with the increase of temperature, and the influence of this dephasing on microscopic electronic fluids (including gap, densities of superfluid and normal fluid, and in particular, the transport property to determine superconducting transition temperature $T_{c}$) can be formulated.

Specifically, to develop the macroscopic phase-coherence dynamics, the equation of motion of the superconducting phase fluctuation, in which both disorder and long-range Coulomb interaction effects are considered, is derived analytically. We show that differing from the conventional bulk superconductors with inactive phase fluctuation due to Anderson-Higgs mechanismAHM ; AK ; Am0 ; Ba0 , the phase fluctuation in cuprate superconductors retains gapless energy spectrum after considering the long-range Coulomb interaction, thanks to the layered structurescuprate4 ; cuprate5 , and hence, is active. We also find that the superfluid density determines the superconducting phase stiffness in the phase-coherence dynamics, in consistency with the previous understanding in the literaturephase1 ; phase2 ; phase3 . The phase-coherence dynamics is therefore influenced by electronic fluids. The derived microscopic electronic-fluid dynamics includes two parts: the anomalous correlation to reflect pairing (i.e., distinguish superfluid and normal fluidSNT ) and determine gap and superfluid density; the microscopic scattering of the electronic fluids that is essential for studying the transport property and hence superconductivity. It is established that the spatial fluctuation of superconducting phase can generate a superconducting momentum ${\bf p}_{s}$ gi0 ; Ba0 ; G1 , which drives the Doppler shift ${\bf v}_{\bf k}\!\cdot\!{\bf p}_{s}$ in quasiparticle energy spectraFF4 ; FF5 ; FF6 ; GIKE1 , with ${\bf v}_{\bf k}$ being the group velocity. Then, both anomalous correlation and microscopic scattering in the electronic-fluid dynamics are affected by the superconducting phase fluctuation through this Doppler shift. Therefore, the macroscopic phase-coherence dynamics and microscopic electronic-fluid dynamics are mutually coupled.

It is noted that $T^{\rm os}$ and $T_{c}$ are determined by critical temperatures where the gap and resistivity vanish, respectively. Based on the developed dual dynamics, we present theoretical descriptions of the separation between $T_{c}$ and $T^{\rm os}$ as well as the emerged normal fluid in superconducting state (below $T_{c}$). The quantum effect of disorder, which excites a macroscopic inhomogeneous phase fluctuation through the Josephson effectJosephson , provides the key origin. This excited phase fluctuation drives the Doppler shift ${\bf v}_{\bf k}\cdot{\bf p}_{s}$ mentioned above. Following the idea of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in conventional superconductors, the anomalous correlation vanishes in region where $|{\bf v}_{\bf k}\cdot{\bf p}_{s}|>\Delta_{\bf k}$ with $\Delta_{\bf k}$ being the superconducting gapFF1 ; FF2 ; FF7 ; FF8 ; FF9 ; GIKE1 . Particles then no longer participate in the pairing in this region and behave like the normal ones, leading to the emergence of normal fluidFF1 ; GIKE1 . Interestingly, with the phase fluctuation, we find that the condition of the unpairing region is always satisfied around nodal points in $d$-wave superconductors. One therefore always finds a nonzero fraction of the normal fluid even at low temperature, in consistency with the experimentally observed substantial fraction of normal state at low temperatureNSB1 ; NSB2 .

Particles in regions with nonzero anomalous correlation contribute to the superconducting gap as superfluid. Moreover, we find that there exists the scattering between particles in pairing and unpairing regions in $d$-wave superconductors, and this scattering behaves like the friction between superfluid and normal fluid. We prove analytically that due to this friction, part of superfluid becomes viscous with nonzero momentum-relaxation rate. Consequently, in addition to conventional non-viscous superfluid, there also exist normal fluid and viscous superfluid at small phase fluctuation in cuprate superconductors, similar to the three-fluid model proposed in our previous workGIKE1 in conventional superconductors which is caused by external electromagnetic field. A scheme to detect distinguish these three electronic fluids in cuprate superconductors is then proposed.

Particularly, as shown in Ref. GIKE1, , the increase of Doppler shift leads to the increases of normal fluid and viscous superfluid but the shrinkage of the non-viscous superfluid. When the non-viscous superfluid vanishes at large enough Doppler shift, one can find an exotic state with only normal fluid and viscous superfluid left, showing the nonzero resistivity as well as the finite gap from the viscous superfluid. Following the same idea, we demonstrate that by increasing the temperature in $d$-wave cuprate superconductors, the suppressed superconducting gap and hence superfluid density weaken the phase stiffness, enhancing the phase fluctuation and hence Doppler shift. Once the phase fluctuation (i.e., temperature) exceeds the critical point, the non-viscous superfluid vanishes, leaving only normal fluid and viscous superfluid. The system then enters the pseudogap state with nonzero resistivity and finite gap due to the significant phase fluctuation. It is noted that in this circumstance, the viscous superfluid matches the description of the incoherent preformed Cooper pairs, as they both contribute to gap but experience the scattering. Whereas the existing normal fluid in our description implies the existence of normal particles in pseudogap state, which has been overlooked in previous preformed Cooper-pair model to describe pseudogap statephase2 ; PCP1 ; PCP2 ; PCP3 ; PCP4 ; PCP5 . With a further increase of temperature in pseudogap state, the viscous superfluid starts to shrink, until vanishes at $T^{\rm os}$ where the gap is eventually destroyed.

To confirm the derivation from the GIKE approach, we also apply the standard path-integral approach to recover the equation of motion of the phase fluctuation as well as anomalous correlation, gap equation and superfluid density in the presence of the superconducting momentum, except the microscopic scattering of the electronic fluids which is hard to handle within the path-integral approach. A self-consistent numerical simulation by applying Anderson disorder is also addressed, to verify our theoretical description. Finally, we show that the developed dual dynamics can also be applied similarly to the low-dimensional disordered $s$-wave superconductors.

In this section, from the rigorous analytic derivation within GIKE approach (refer to Sec. IV), we summarize the simplified results to present the coupled dual dynamics of macroscopic phase coherence and microscopic electronic fluids for a generalized order parameter:

Here, $x=(x_{0},{\bf r})$ denotes the space-time vector; $R={(x+x^{\prime})}/{2}=(t,{\bf R})$ stands for the center-of-mass coordinate; $\Delta_{{\bf k}}$ denotes the equilibrium-state gap, independent of the center-of-mass coordinate due to the translational symmetry; $\delta\theta(R)$ and $\delta\Delta_{\bf k}(R)$ represent the phase and amplitude fluctuations, respectively.

Specifically, the macroscopic phase-coherence dynamics involves the generation of the phase fluctuation $\delta\theta(R)$, which along ${\bf e}_{\phi}$ direction is determined by

Here, $p_{s}^{\phi}{\bf e}_{\phi}$ denotes the generated superconducting momentum ${\bf p}_{s}={\bm{\nabla}}_{\bf R}\delta\theta(R)/2$ by phase fluctuation along ${\bf e}_{\phi}$ direction; $U_{\bf q}$ represents the Fourier component of disorder-induced local electric potential; $n_{B}(x)$ stands for the Bose distribution; $\omega_{N}=\sqrt{\omega_{p}^{2}+n_{s}q^{2}/(2Dm)}$ denotes the energy spectrum of the phase fluctuation, where $\omega_{p}=\sqrt{q^{2}V_{q}n_{s}/m}$ represents the plasma frequency and $n_{s}$ stands for the superfluid density, with $V_{q}$ being the Coulomb potential; $m$ and $D$ denote the effective mass and density of states of carriers, respectively; $1/C$ is the normalized factor in frequency-momentum space.

The microscopic electronic-fluid dynamics includes: the anomalous correlation $F_{\bf k}$, used to characterize the pairing and determine gap $\Delta_{\bf k}$ equation and superfluid density $n_{s}$; the microscopic momentum-relaxation rate $\Gamma_{\bf k}$ in superfluid. The expressions of these quantities are written as

Here, $E^{\pm}_{\bf k}$ denotes the quasi-electron and quasi-hole energies; $g_{\bf kk^{\prime}}$ represents the pairing potential, in which we approximately taking $k=k^{\prime}=k_{F}$ around the Fermi surface so that the gap $\Delta_{\bf k}$ only has angular dependence of the momentum; $\sum_{\bf k}^{\prime}$ here and hereafter stands for the summation restricted in the spherical shell ($\xi_{\bf k}\leq\omega_{D}$) with $\omega_{D}$ being the cutoff frequency, following the BCS theoryBCS ; G1 ; $M_{\bf kk^{\prime}}$ denotes the effective matrix element of the electron-impurity scattering. It is established that in the presence of the superconducting momentum ${\bf p}_{s}$, the quasiparticle energy is tilted as $E_{\bf k}^{\pm}=({\bf v}_{\bf k}\cdot{\bf p}_{s})\pm{E_{\bf k}}$, where $E_{\bf k}=\sqrt{\xi_{\bf k}^{2}+\Delta_{\bf k}^{2}}$ is the original Bogoliubov quasiparticle energy and ${\bf v}_{\bf k}\cdot{\bf p}_{s}$ denotes the Doppler shiftFF1 ; FF4 ; FF5 ; FF6 ; GIKE1 , with the group velocity ${\bf v}_{\bf k}=\partial_{\bf k}\xi_{\bf k}$. In the present work, we approximately take the parabolic spectrum, i.e., $\xi_{\bf k}={\bf k}^{2}/(2m)-\mu$ with $\mu$ being the chemical potential.

Particularly, it is noted that the generation of the phase fluctuation in Eq. (2) is coupled with electronic fluids through the superfluid density $n_{s}$ in the energy spectrum $\omega_{N}$, whereas the phase fluctuation is involved in the electronic-fluid dynamics in Eqs. (3)-(6) by Doppler shift. The macroscopic phase-coherence dynamics and microscopic electronic-fluid dynamics are therefore mutually coupled. For phase fluctuation along a certain direction, by self-consistently solving Eqs. (2) and (4) as well as (5), one can uniquely determine the superconducting gap $\Delta_{\bf k}$, superfluid density $n_{s}$ and superconducting phase fluctuation $p_{s}$, and then, the microscopic momentum-relaxation rate $\Gamma_{\bf k}$ and various physical quantities are obtained. Nevertheless, in realistic situation, there exist phase fluctuations along all directions. Hence, the experimentally observed quantity is a statistical average of phase fluctuations in all directions. The expected value of the physical quantity $X$ therefore reads

with $X(\phi)$ being the solved $X$ for phase fluctuation along ${\bf e}_{\phi}$ direction.

Equations (2)-(6) then provide an efficient and simplified way to understand the superconductivity properties in cuprate and disordered $s$-wave superconductors with significant phase fluctuation.

In this section, we apply the developed dual dynamics into the $d$-wave cuprate superconductors. Without losing generality, we choose $d_{x^{2}-y^{2}}$-wave order parameter for analysis, i.e., $\Delta_{{\bf k}}=\Delta\cos(\zeta\theta_{\bf k})$ with $\zeta=2$, and the pairing potential $g_{\bf kk^{\prime}}{\approx}g\cos[\zeta(\theta_{\bf k}-\theta_{\bf k^{\prime}})]$ as a consequence of the translational and time-reversal symmetriesSM1 .

In this section, we present the rigorous derivation of the coupled dual dynamics from the GIKE approach. The recovery from the path-integral approach is also addressed to confirm our derivation.