Superconductivity above a quantum critical point in a metal -- gap closing vs gap filling, Fermi arcs, and pseudogap behavior

TL;DR

This paper explores the interplay of superconductivity and non-Fermi liquid behavior near a quantum critical point, revealing how gap filling and closing phenomena relate to Fermi arcs and pseudogap in cuprates.

Contribution

It introduces a theoretical framework distinguishing gap filling and closing in a quantum-critical metal, connecting these phenomena to Fermi arcs and pseudogap behavior.

Findings

Superconductivity emerges below a finite temperature T_p with a lower crossover T_{cross}.

DOS and SF show gap closing below T_{cross} and gap filling above it.

Fermi arcs and pseudogap behaviors are explained by phase fluctuations and the competition between pairing and non-Fermi liquid effects.

Abstract

We consider a quantum-critical metal with interaction mediated by fluctuations of a critical order parameter. This interaction gives rise to two competing tendencies -- pairing and non-Fermi liquid behavior. Due to competition, the pairing develops below a finite $T_p $, however its prominent feedback on the fermionic self-energy develops only at a lower $ T_{cross}$. At $T<T_{cross}$ the system behavior is similar to that of a BCS supercoductor -- the density of states (DOS) and the spectral function (SF) have sharp gaps which close as $T$ increases. At higher $T_{cross}<T<T_{p}$ the DOS has a dip, which {\it fills in} with increasing $T$. The SF in this region shows either the same behavior as the DOS, or has a peak at $\omega =0$ (the Fermi arc), depending on the position on the Fermi surface. We argue that phase fluctuations are strong in this $T$ range, and the actual $T_c \sim T_{cross}$, while $T_p$ marks the onset of pseugogap behavior. We compare our theory with the behavior of optimally doped cuprates.

Figures

c has been found in ARPES measurements closer to zone diagonals. Our reasoning for the difference between Figs 1b and c is valid if the pairing boson predominantly couples fermions near $(\pi,0)$ and related points, like a $(\pi,\pi)$ spin fluctuation does Monthoux *et al.* ([2007](#bib.bib1)); *scal2; Abanov *et al.* ([2001](#bib.bib3)); *acs

Equations

$$\Phi ({\omega }_m)=\pi T\sum _{m^{\prime }}\frac{\Phi ({\omega }_{m^{\prime }})\chi ({\omega }_m-{\omega }_{m^{\prime }})}{\sqrt{{\tilde{\Sigma }}^2({\omega }_{m^{\prime }})+{\Phi }^2({\omega }_{m^{\prime }})}}$$

$$\tilde{\Sigma }({\omega }_m)={\omega }_m+g^{\gamma }\pi T\sum _{m^{\prime }}\frac{\tilde{\Sigma }({\omega }_m)\chi ({\omega }_m-{\omega }_{m^{\prime }})}{\sqrt{{\tilde{\Sigma }}^2({\omega }_{m^{\prime }})+{\Phi }^2({\omega }_{m^{\prime }})}},$$

$${\Phi }^{\ast }(\omega )={\left(\frac{2}{N}\right)}^{3/2}\pi T{\left(\frac{g}{\pi T}\right)}^{\gamma }{\left[1-{\left(\frac{T}{T_p}\right)}^{\gamma }\right]}^{1/2}{F}_{\Phi }\left(\frac{\omega }{\pi T}\right)$$

$${\tilde{\Sigma }}^{\ast }(\omega )=\pi T{\left(\frac{g}{\pi T}\right)}^{\gamma }{F}_{\Sigma }\left(\frac{\omega }{\pi T}\right).$$

$${\rho }_s(T)\approx \frac{T}{N}\left(1-{\left(\frac{T}{T_p}\right)}^{\gamma }\right)\frac{E_F}{\pi T\chi (0)}\left(1+O\left(\frac{1}{N}\right)\right),$$

Full text

Introduction.  Pairing near a quantum-critical point (QCP) in a metal and its intriguing interplay with the concurring non-Fermi liquid (NFL) physics continue to attract strong attention of the physics community Monthoux et al. (2007); *scal2; Abanov et al. (2001); *acs; boo ; Fratino et al. (2016); rev ; Fradkin et al. (2010); Bok et al. (2016); Shibauchi et al. (2014). An incoherence associated with the NFL form of the self-energy, acts against pairing, while the latter reduces fermionic incoherence by gapping out low-energy states. The competition between these two opposite tendencies has been analyzed analytically, using field-theoretical methods for effective low-energy models  Combescot (1995); *Bergmann; *Bergmann2; *ad; *Marsiglio_88; *Marsiglio_91; *Karakozov_91; Bonesteel et al. (1996); Abanov et al. (2001); *acs; Son (1999); Lee (2009); Moon and Sachdev (2009); *moon_2; Metlitski and Sachdev (2010); Chowdhury and Sachdev (2014); *wang23; Metlitski et al. (2015); Raghu et al. (2015); Fradkin et al. (2010); Monthoux et al. (2007); *scal2; Efetov et al. (2013); Lederer et al. (2015); Wang et al. (2016); Tsvelik (2017); Vojta and Sachdev (1999); Khveshchenko and Shively (2006); Lee et al. (2018); *we_last_D, and numerically, by, e.g., FRG, QMC and DMFT techniques Vilardi et al. (2018); Gerlach et al. (2017); *berg_2; *berg_3; Haule and Kotliar (2007); *kotliar2; Fratino et al. (2016); *tremblay_2; Georges et al. (2013); *georges2. Earlier studies have found Combescot (1995); *Bergmann; *Bergmann2; *ad; *Marsiglio_88; *Marsiglio_91; *Karakozov_91; Abanov et al. (2001); *acs; Metlitski et al. (2015); Raghu et al. (2015); Wang et al. (2016); Lee et al. (2018); *we_last_D that the onset temperature for the pairing, $T_{p}$, is finite, i.e., a QCP is surrounded by a superconducting done.

The issue which we discuss here is the feedback on the fermions from the pairing in the quantum critical (QC) regime, specifically the behavior of the DOS $N(\omega)$ and the spectral function $A_{{\bf k}_{F}}(\omega)$. We argue that there are two distinct regimes below $T_{p}$, which differ by the strength of the feedback from the pairing on the electrons. At low $T<T_{cross}<T_{p}$ (regime I) the feedback is strong, and both $N(\omega)$ and $A_{{\bf k}_{F}}(\omega)$ have sharp quasiparticle peaks at $\omega=\Delta(T)$. At higher $T_{cross}<T<T_{p}$ (regime II) the feedback is weak, and $N(\omega)$ has a dip at $\omega=0$ and a hump at a frequency which scales with $T$ rather than $\Delta(T)$ and remains finite at $T=T_{p}$. The behavior of the spectral function (SF) $A_{{\bf k}_{F}}(\omega)$ in this region depends on the strength of thermal contribution to the self-energy and varies along the Fermi surface (FS). For ${\bf k}_{F}$ points where it is strong enough, $A_{{\bf k}_{F}}(\omega)$ displays the same behavior as $N(\omega)$, for other ${\bf k}_{F}$ points, $A_{{\bf k}_{F}}(\omega)$ shows a peak at $\omega=0$ instead of a dip (the Fermi arc). We summarize the results in Fig. 1

A very similar evolution of both DOS and SF has been observed in the cuprates around optimal doping Fischer et al. (2007); Reber et al. (2012); Kondo et al. (2013); Kanigel et al. (2006); *kanigel2; *kanigel3; nor ; Damascelli et al. (2003); *shen2; *shen3; *shen4; He et al. (2014) and dubbed as transformation from “gap closing” at low $T$ to “gap filling in” at higher $T$. The behavior similar to Figs. 1a and 1b has been detected in tunneling measurements of $N(\omega)$ and ARPES measurements of $A_{{\bf k}_{F}}(\omega)$ at ${\bf k}$ near $(\pi,0)$ and related points in the Brillouin zone (BZ). The behavior similar to Fig. 1c has been found in ARPES measurements closer to zone diagonals. Our reasoning for the difference between Figs 1b and c is valid if the pairing boson predominantly couples fermions near $(\pi,0)$ and related points, like a $(\pi,\pi)$ spin fluctuation does Monthoux et al. (2007); *scal2; Abanov et al. (2001); *acs

We further analyze the superfluid phase stiffness $\rho_{s}(T)$ and argue that in regime II, $\rho_{s}(T)<T$, that is, phase fluctuations are strong and likely destroy long range superconducting order Emery and Kivelson (1994); *mohit_1. Then the actual $T_{c}\sim T_{cross}$, while between $T_{cross}$ and $T_{p}$ the system displays pseudogap behavior. This also agrees with the experiments, which found that the crossover from gap closing to gap filling occurs at around $T_{c}$ (Ref. Reber et al. (2012)), while the dip in $N(\omega)$ at $\omega=0$ disappears at a higher $T$. It is tempting to associate the onset of the pseudogap behavior in the cuprates at optimal doping with our $T_{p}$. We caution that we do not associate the whole pseudogap region in the cuprates with the pairing state with no phase coherence and with weak feedback on the fermionic self-energy. There are other features, like apparent FS reconstruction tai ; sub , charge and nematic orders Fujita et al. (2014); *cdw1; *cdw2; *cdw3; *cdw4; *cdw5; *cdw6; *cdw7; Perali et al. (1996); *sachdev2013; *wang_ch, and time-reversal symmetry breaking Karapetyan et al. (2012); Baledent et al. (2010), which we do not address in this study. Still, we believe that this work presents microscopic understanding of “weak” pseudogap behavior — the crossover from gap closing to gap filling with increasing $T$ in the DOS and the SF in the anti-nodal regions of the FS, and the development of Fermi arcs in the SF in near-nodal regions.

We also note that the issue we consider here is different from peak-dip-hump in the SF and related phenomena in the DOS, optical conductivity, and other measurements Fink et al. (2006). That phenomenon has been associated with the emergence of quasiparticle scattering in a superconductor at frequencies somewhat below $3\Delta$, well separated from the peak at $\omega=\Delta$. The peak-dip-hump phenomenon was explained in terms of coupling to a propagating boson, either a phonon Johnston et al. (2010), or a spin fluctuation – the latter becomes propagating below $T_{p}$ due to the development of a resonance mode below $2\Delta$ (Ref. Eschrig (2006); *abanov_ch). The phenomenon we discuss here is the destruction of the peak at $\Delta$ due to the existence, in regime II, of strong quasiparticle scattering down to the lowest energies. This phenomenon has been described phenomenologically in the past Balatsky et al. (2006); *imp_general2; *imp_general3; Norman et al. (1998a); *imp_cuprates2; *imp_cuprates3; *imp_cuprates4; *imp_cuprates5; Chubukov et al. (2007), by introducing frequency independent fermionic damping $\gamma(T)$ and allowing it to be comparable to the gap $\Delta(T)$. Our work presents the microscopic theory of the existence of $\text{Im}\Sigma(\omega\to 0,T)$ at $T>T_{cross}$, despite the fact that the pairing gap is non-zero.

The model. We consider the model of itinerant fermions minimally coupled to fluctuations of the order parameter field, which condenses at a QCP. Within Eliashberg-type approximation, which we adopt, the effective 4-fermion interaction is proportional to the susceptibility of an order parameter integrated along the FS, $\chi(\Omega_{m})$. At a QCP, $\chi(\Omega_{m})=(g/|\Omega_{m}|)^{\gamma}$ is a singular function of frequency (the exponent $\gamma$ is small near 3D, and in 2D equals to $1/3$ at a nematic QCP and to $1/2$ at QCP towards a density-wave order Bonesteel et al. (1996); Metlitski et al. (2015); Abanov et al. (2001); *acs; Raghu et al. (2015); Wang et al. (2016)). The regimes I and II exist for all $\gamma<1$, and below we do not specify the value of the exponent. The singular $\chi(\Omega_{m})$ gives rise to an attraction in at least one pairing channel and also gives rise to NFL behavior in the normal state, setting the competition between the pairing and the NFL behavior. We consider spin-singlet pairing and solve the set of non-linear equations for the pairing vertex and fermionic self-energy on the Matsubara axis, and then convert the results to real frequencies Combescot (1995); *Bergmann; *Bergmann2; *ad; *Marsiglio_88; *Marsiglio_91; *Karakozov_91 and find the DOS and the SF. We present the details of the calculations in las and here show the results.

Along the Matsubara axis, the coupled equations for the pairing vertex $\Phi(\omega_{m})$ and fermionic self-energy $\Sigma(\omega_{m})$ are Wang et al. (2016) (${\tilde{\Sigma}}(\omega_{m})=\omega_{m}+\Sigma(\omega_{m})$)

[TABLE]

In principle, one should also include the equation for bosonic self-energy, which describes the feedback from $\Phi(\omega_{m})$ on $\chi(\Omega_{m})$ (Refs. Eschrig (2006); *abanov_ch). This feedback effectively makes the exponent $\gamma$ temperature dependent below $T_{p}$. However, because regimes I and II exist for all $\gamma$, this will only affect the location of $T_{cross}$. Below we neglect this complication and treat the exponent $\gamma$ as temperature independent.

The thermal contributions to $\Phi(\omega_{m})$ and ${\tilde{\Sigma}}(\omega_{m})$ come from $m^{\prime}=m$ terms in the sums. These contributions are essential for the SF, but can be excluded from the Eliashberg set by analogy with non-magnetic impurities agd ; Millis et al. (1988); Abanov et al. (2008), by re-expressing $\Phi(\omega_{m})=\Phi^{\ast}(\omega_{m})\left(1+Q^{\ast}(\omega_{m})\right),~{}{\tilde{\Sigma}}(\omega_{m})={\tilde{\Sigma}}^{\ast}(\omega_{m})\left(1+Q^{\ast}(\omega_{m})\right)$, where $Q^{\ast}(\omega_{m})=\pi T\chi(0)/\sqrt{({\tilde{\Sigma}}^{\ast})^{2}(\omega_{m})+(\Phi^{\ast}(\omega_{m}))^{2}}$. The equations for $\Phi^{\ast}$ and ${\tilde{\Sigma}}^{\ast}$ are the same as in (1) but without $m=m^{\prime}$ term in the sum. We solve these two equations and then obtain $\Phi^{\ast}(\omega)$ and ${\tilde{\Sigma}}^{\ast}(\omega)$ using spectral decomposition method and analytical continuation. In the normal state (at $\Phi^{\ast}=0$) the self-energy has a NFL form $\Sigma^{\ast}(\omega_{m})\propto\omega^{1-\gamma}_{m}$.

The onset temperature for the pairing $T_{p}=T_{p}(\gamma)$ has been obtained in Raghu et al. (2015); Wang et al. (2016). It is finite, of order $g$, and scales as $\gamma^{-1/\gamma}$ at small $\gamma$. The pairing gap $\Delta(\omega)$ is defined as $\Delta(\omega)=\Phi(\omega)\omega/{\tilde{\Sigma}}(\omega)$ and is equally expressed as $\Delta(\omega)=\Phi^{\ast}(\omega)\omega_{m}/{\tilde{\Sigma}}^{\ast}(\omega)$. The DOS $N(\omega)$ is expressed only via $\Delta(\omega)$: $N(\omega)=N_{0}\operatorname{Im}\omega/\sqrt{\Delta^{2}(\omega)-\omega^{2}}$ and therefore has no contribution from thermal fluctuations. The SF $A(\omega)$ for a fermion on the FS does depend on the thermal contribution: $A(\omega>0)=(-1/\pi)\operatorname{Im}L(\omega)\omega/\sqrt{\Delta^{2}(\omega)-\omega^{2}}$, where $L(\omega)=1/(\pi T\chi(0)-({\tilde{\Sigma}}(\omega)/\omega)\sqrt{\Delta^{2}(\omega)-\omega^{2}})$.

The two regimes below $T_{p}$.  The existence of the two different regimes below $T_{p}$ can be understood by analyzing the gap equation along the Matsubara axis. We argue that the existence of the regimes I and II is associated with the special role of fermions with Matsubara frequencies $\omega_{m}=\pm\pi T$. Namely, fermionic self-energy $\Sigma^{\ast}(\omega_{m})$, which acts against the pairing, is strong and singular at a QCP for a generic $\omega_{m}$ but vanishes at $\omega_{m}=\pm\pi T$, i.e. fermions with these two frequencies can be treated as free particles for the purposes of the pairing Wang et al. (2016); Chubukov and Maslov (2012). Meanwhile, the pairing interaction between fermions with $\omega_{m}=\pi T$ and $\omega_{m}=-\pi T$, $\chi(2\pi T)=(g/(2\pi T))^{\gamma}$ is strong. As the consequence, fermions with $\pm\pi T$ form a bound pair at $T_{p}$, and in some range below $T_{p}$ (in region II) act as the source for the pairing gap for fermions with other Matsubara frequencies. At a smaller $T<T_{cross}$ (regime I) fermions with other Matsubara frequencies become capable to pair on their own, without an input from fermions with $\omega_{m}=\pm\pi T$. We verify this by solving the gap equation with and without fermions with $\omega_{m}=\pm\pi T$. We find (see the inset to Fig. 3) that in the first case the critical temperature is $T_{p}$, and in the second it is a smaller $T_{cross}$. The special role of fermions with $\omega_{m}=\pm\pi T$ becomes more transparent if we modify the original model and reduce the the interaction in the pairing channel by $1/N$ compared to that for the self-energy. This can be achieved by extending the model to $SU(N)$ global symmetry Raghu et al. (2015). At $N$ larger than some $N_{cr}(\gamma)\equiv(1-\gamma)\Gamma(\gamma/2)\left(\frac{\Gamma(\gamma/2)}{2\Gamma(\gamma)}+\frac{\Gamma(1-\gamma)}{\Gamma(1-\gamma/2)}\right)>1$, $T_{p}$ is finite only due to fermions with $\omega_{m}=\pm\pi T$, i.e., the regime II extends down to $T=0$ (the line $T_{cross}(N)$ terminates at $N=N_{cr}$), see Fig. 3). This extension does not change the physics as the regime II exists already for the original $N=1$, but it allows us to analyze the system behavior analytically, in $1/N$ expansion. We find that $\Phi^{\ast}(\omega_{m})$ and $\Sigma^{\ast}(\omega_{m})$ are proportional to $T^{1-\gamma}$, i.e., the self-energy retains its NFL form. The pairing gap $\Delta(\pi T)=\pi T(2/N)^{1/2}\left(1-(T/T_{p})^{\gamma}\right)^{1/2}$ is small in $1/N$, and for other $\omega_{m}$, $\Delta(\omega_{m})\propto 1/N^{3/2}$ is even smaller. For all $\omega_{m}$, the gap $\Delta(\omega_{m})$ emerges at $T_{p}$ and vanishes at $T=0$, consistent with the fact that it is induced by fermions with $\omega_{m}=\pm\pi T$ and wouldn’t exist without them. We show $\Delta(\omega_{m})$ in Fig. 2(a),(e).

The large-$N$ expressions for $\Phi^{\ast}(\omega_{m})$ and $\Sigma^{\ast}(\omega_{m})$ can be analytically converted to real frequencies. We obtain

[TABLE]

where $F_{\Phi}$ and $F_{\Sigma}$ are two scaling functions which only depend on $\omega/(\pi T)$las . We see that the frequency dependence of both $\Phi^{*}(\omega)$ and ${\tilde{\Sigma}}^{*}(\omega)$ (and hence of $\Delta(\omega)$ and the DOS and the SF in the anti-nodal region) is set by $T$ rather than by the gap, hence the position of the maximum scales with $T$ and remains finite at $T_{p}$. As the consequence, the gap fills in as $T\to T_{p}$, but does not close (Fig. 2(b),(c)). At the smallest frequencies $\Sigma^{\ast}(\omega)\approx i\operatorname{Im}\Sigma^{\ast}(0)$, and hence $\Delta(\omega)\propto i\omega$, like for gapless superconductivity. Then the DOS $N(\omega)$ is then non-zero down to the lowest frequencies (Fig. 2(c)).

At smaller $N<N_{cr}$, including physical $N=1$, below $T_{cross}$ fermions with $|\omega_{m}|\neq\pi T$ can pair on their own, without a push from fermions with $|\omega_{m}|=\pi T$. This can be checked by solving the Eliashberg equations without fermions with the first Matsubata frequencies (see the insert to Fig. 3). Once other fermions get paired below $T_{cross}$, the system recovers a conventional superconducting behavior, i.e., $\Delta(\pi T)$ tends to a finite value $\Delta$ at $T=0$, and at low $T$, the gap $\Delta(\omega)$ along the real axis is purely real at $\omega\leq\Delta$ (see Fig. 2(e),(f)). This is region I in our notations. Because the scattering at small $\omega$ is destroyed by a feedback from the pairing, the DOS and the SF have sharp peaks at $\omega=\Delta$ (see the lowest $T$ data in Fig. 1). As temperature increases but remains smaller than $T_{cross}$, the position of the maximum follows $\Delta(T)$ and decreases, i.e., the gap start closing. However, once $T$ exceeds $T_{cross}$, the system crosses over to the region II, where the pairing would not be possible without fermions with $\omega_{m}=\pm\pi T$, and the position of the hump in the DOS and the SF is set by $T$ rather than $\Delta(T)$. In this region, the gap progressively fills in as $T$ approaches $T_{p}$, but does not close. We show the results in Fig. 1).

Superfluid stiffness.  Our next goal is to verify whether in region II superconducting order may be destroyed by phase fluctuations. Our earlier results do not rely on phase coherence and are applicable even in the absence of long-range phase coherence. Moreover, the feedback effect from the pairing on fermionic self-energy is weak in region II already in the absence of phase fluctuations, and phase decoherence can only reduce the feedback even further. Still, for comparison with the cuprates it is important to understand whether the region II corresponds to superconducting or pseudogap phase.

To compute the stiffness we follow Ref. Chubukov et al. (2016), set $\Delta(\omega_{m},r)=\Delta(\omega_{m})e^{i{\bf\nabla}\phi{\bf r}}$, i.e., $\Delta(\omega_{m},q)=\Delta(\omega_{m})\delta({\bf q}-{\bf\nabla}\phi)$, and identify $\rho_{s}(T)$ with the prefactor for the $q^{2}$ term in the condensation energy. For a BCS superconductor $\rho_{s}(T<T_{c})\approx E_{F}/(4\pi)$. Because $E_{F}$ is assumed to be much larger than $T_{c}$, phase fluctuations are weak. In our case, we found at large $N$, when region II extends to $T=0$,

[TABLE]

where, we remind, $\chi(0)$ is a static susceptibility of a critical bosonic field. It diverges at a QCP, so formally $\rho_{s}(T)\to 0$. However, Eliashberg theory is only valid when $E_{F}\geq\pi T\chi(0)$ because the integration over fermionic dispersion only holds up to $E_{F}$. This restricts $\chi(0)$ to $\pi T\chi_{0}\leq E_{F}$ and $\rho_{s}$ to $\rho_{s}(T)\geq(T/N)\left(1-\left(T/T_{p}\right)^{\gamma}\right)$. Still, at large $N$, $\rho_{s}(T)\ll T$. In this situation, phase fluctuations are strong and should destroy long range phase coherence  Emery and Kivelson (1994); *mohit_1; Seibold et al. (2012). Then region II becomes the pseudogap phase. In region I the same calculation yields $\rho_{s}\geq\Delta(T=0)$, i.e., phase fluctuations are at most moderate and phase coherence survives.

Summary.  In this paper we analyzed the feedback on the fermions from the pairing in the QC regime, specifically the behavior of the DOS $N(\omega)$ and the SF $A_{{\bf k}_{F}}(\omega)$ on the FS. We considered the model of 2D fermions with singular interaction mediated by the dynamical susceptibility of a critical boson $\chi(\Omega_{m})=(g/|\Omega_{m}|)^{\gamma}$. This interaction gives rise to pairing and also to NFL behavior, which competes with the pairing. To separate between the two tendencies, we extended the model in such a way that the pairing interaction gets smaller by $1/N$.

Our results are summarized in Fig. 3. We found two distinct regimes below the onset temperature for the pairing $T_{p}$. They differ in the strength of the feedback from superconductivity on the electrons. At low $T<T_{cross}<T_{p}$ (regime I) the feedback is strong, and both $N(\omega)$ and $A_{{\bf k}_{F}}(\omega)$ have sharp quasiparticle peaks at $\omega=\Delta(T)$. At higher $T_{cross}<T<T_{p}$ (regime II) the feedback is weak, and $N(\omega)$ has a dip at $\omega=0$ and a hump at a frequency, which scales with $T$ rather than $\Delta(T)$ and remains finite at $T=T_{p}$, i.e., near $T_{p}$ the gap fills in but does not close. The SF either has the same structure as the DOS or a peak at $\omega=0$ (the Fermi arc), depending on the location on the FS. We computed superfluid stiffness and estimated the strength of phase fluctuations. We found that in region II phase fluctuations are parameterically strong, at least at large $N$, and destroy phase coherence. Then the actual $T_{c}\sim T_{cross}$, while in between $T_{cross}$ and $T_{p}$ the system is in phase-disordered pseudogap state. A very similar behavior has been detected in tunneling and ARPES studies of the cuprates near optimal doping, and we propose our theory as a microscopic explanation of the observed behavior. From theoretical perspective, we associate the existence of the region II with special role of fermions with Matsubara frequencies $\pm\pi T$, which appear in the pairing channel without self-energy and form a Cooper pair even when the pairing interaction is reduced by $1/N$. These fermions then induce pairing for fermions with other Matsubara frequencies.