Microscopic theory of pseudogap phenomena and unconventional Bose-liquid superconductivity and superfluidity in high-$T_{c}$ cuprates and other systems

We now calculate the ground state energies of strong coupling large polarons and bipolarons in lightly doped cuprates using the values of the parameters entering into Eqs. (5) and (6). The lattice parameter value of the orthorhomic cuprates is about $a_{0}\simeq 5.4{\AA}$. According to the spectroscopy data, the Fermi energy $E_{F}$ of the undoped cuprates is equal to 7 eV 87 . To determine the value of the short-range carrier-phonon coupling constant $g_{s}$, we can estimate the deformation potential as $E_{d}=(2/3)E_{F}$ 88 . For the cuprates, typical values of other parameters are $m^{*}\simeq m_{e}$ 89 (where $m_{e}$ is the free electron mass), $\varepsilon_{\infty}=3-5$ 76 ; 90 , $\varepsilon_{0}\simeq 22-85$ 76 ; 77 ; 90 , $B\approx 1.4\cdot 10^{12}dyn/cm^{2}$ 91 . The minima of $E_{p}(\beta)$ and $E_{B}(\beta,\gamma)$ correspond to the ground state energies of strong coupling large polaron and bipolaron, respectively, which are measured with respect to the top of the oxygen valence band. The basic parameter of such polarons and bipolarons are their binding energies, which are defined as $E_{p}=|E_{p}(\beta_{min})|$ and $E_{bB}=|E_{B}(\beta_{min},\gamma_{min})-2E_{p}(\beta_{min})|$. In 3D systems there is generally a potential barrier that must be overcome to initiate self-trapping, while in 2D systems there is no barrier for self-trapping 76 .

From Eq. (5), we find

The states of large and small polarons are separated by the potential barrier determined as

Using the values of parameters $m^{*}=m_{e}$, $a_{0}=5.4{\AA}$, $E_{d}\simeq 4.67$ eV $\varepsilon_{\infty}=4$ and $B=1.4\cdot 10^{12}dyn/cm^{2}$, we find $E_{a}\simeq 4.21\rm{eV}$ at $\eta=0.08$. It follows that the large and small polaron states are separated by very high potential barrier. Such a high potential barrier prevents the formation of small polarons and bipolarons in the bulk of hole-doped cuprates, where the relevant charge carriers are large polarons and bipolarons. The binding energies of strong-coupling 2D polarons determined using the relation $E^{2D}_{p}=(\pi/8)\hbar\omega_{0}\alpha^{2}_{F}$ (where $\hbar\omega_{0}$ is the optical phonon energy, $\alpha_{F}$ is the Fröhlich polaron coupling constant) 92 would be much larger than those of strong-coupling 3D polarons and such polarons tend to be localized rather than mobile.

There is now experimental evidence that polaronic carriers are present in the doped cuprates 89 ; 93 and they have effective masses $m_{p}\simeq(2-3)m_{e}$ 9 ; 89 and binding energies $E_{p}\simeq(0.06-0.12)$ eV 93 . In lightly doped cuprates, these large polarons tend to form real-space pairs, which are localized large bipolarons. The calculated values of the binding energies of large polarons $E_{p}$ and bipolarons $E_{bB}$ and the ratio $R_{bB}=E_{bB}/2E_{p}$ in 3D lightly doped cuprates for different values of $\varepsilon_{\infty}$ and $\eta$ are given in Table I.

There is now serious problem in describing excitations in lightly doped cuprates. If the insulating state of these materials is considered as the non-conducting state of the Mott insulator with the AF ordering, then it is difficult to describe the insulating behavior of lightly doped cuprates above the Neel temperature $T_{N}$. The puzzling insulating state of lightly doped cuprates both above $T_{N}$ and above some doping level (e.g., at $x\gtrsim 0.02$ in $\rm{La_{2-x}Sr_{x}CuO_{4}}$ (LSCO)) can be described properly on the basis of the above theory of large (bi)polarons. Therefore, it is of interest to compare the above presented results with experimental data on localized in-gap states (or bands) and energy gaps which are precursors to the pseudogaps observed in the metallic state of hole-doped cuprates. The (bi)polaronic states emerge in the CT gap of the cuprates and are manifested as the localized in-gap states (at very low doping) or the narrow in-gap bands (at intermediate doping) in the cuprates, as observed in various experiments 80 ; 94 . The characteristic binding energies of large polarons and bipolarons should be manifested in the excitation spectra of hole-doped cuprates as the low-energy gaps, which are different from the high-energy CT gaps ($\Delta_{CT}\simeq 1.5-2.0$ eV 80 ) of the cuprates. Actually, the values of $E_{bB}$ (see Table I) are close to the observed energy gaps $E_{g}\simeq 0.03-0.05$ eV in the excitation spectra of these materials 8 ; 89 ; 95 . The values of the binding energies of large bipolarons $E_{bB}\simeq 0.01-0.04$ eV obtained at $\varepsilon_{\infty}=3-5$ and $\eta=0.06-0.08$ are also consistent with the energies of the absorption peaks in the far-infrared transmission spectra observed in $\rm{YBa_{2}Cu_{3}O_{7-\delta}}$ (YBCO) at 0.013-0.039 eV 96 . Other experimental observations indicative of the existence of localized in-gap states 82 ; 97 and the well-defined semiconducting gap in the lightly doped LSCO ($x=0.02$) 98 , where the observed energy gap has the value 0.04 eV and is almost temperature independent up to 160 K. The value of this energy gap is close to the binding energies of large bipolarons presented in Table I for $\varepsilon_{\infty}=3$ and $\eta=0.06$. Further, in various experiments the excitation spectra of doped cuprates show gap-like features on the other energy scales of 0.06-0.15 eV 43 ; 81 ; 99 ; 100 , which are consistent with the binding energies of large polarons $E_{p}\simeq 0.06-0.15$ eV at $\varepsilon_{\infty}=3-4$ and $\eta=0.01-0.14$. In particular, the measured mid-infrared (MIR) spectral shape in doped YBCO is similar to the photoinduced polaronic features observed in the insulating phase of the undoped YBCO 99 . Such a characteristic MIR feature led many researchers to a polaronic interpretation of the MIR response and the Raman spectra of YBCO (see Refs. 92 ; 99 ). In the polaronic model, the MIR absorption and the peak in the Raman spectra are expected due to excitations of charge carriers from the polaronic states (or bands) to the delocalized states of quasi-free carriers. The energy gap seen in the angle-integrated photoemission spectra of LSCO at $\sim 0.1$ eV 101 is likely associated with the excitations of carriers from the polaronic state to the quasi-free states. Further, the in-gap band observed in this system at 0.13 eV is attributed to the energy band of polarons 89 . By taking $\varepsilon_{\infty}=3$ and $\eta=0.07$ for LSCO, we obtain the value of $E_{p}\simeq 0.13$ eV (see Table I) in accordance with this experimental observation. Another important experimental observation is that in LSCO the flatband 102 , which is $\sim 0.12$ eV below the Fermi energy for $x=0.05$, moves upwards monotonically with increasing $x$, but the flatband is lowered as $x$ decreases and loses its intensity in the insulating phase. We argue that the flatband observed by angle-resolved photoemission spectroscopy (ARPES) in the lightly doped LSCO ($x\lesssim 0.05$) is the energy band of large polarons, since the effective mass of carriers obtained from analysis of the ARPES spectra is about $2.1m_{e}$ 102 . The existence of the unconventional electron-phonon interactions in doped cuprates, which are responsible for the formation of large (bi)polarons and in-gap states, have been clearly confirmed in the above experiments 89 ; 92 ; 97 ; 99 and other experiments 103 ; 104 . In recent experimental observations 105 ; 106 ; 107 , giant phonon anomalies in underdoped cuprates confirm also a large electron-phonon interaction leading to the complex ionic displacement pattern associated with the charge-density-wave (CDW) formation. These phonon anomalies are reminiscent of anomalous phonon softening and broadening effects, which are caused by the polaron formation. Therefore, the formation of the CDW in doped cuprates is none other than polaron formation in a deformable lattice. Actually, the CDW associated with the lattice distortion is similar to the polaronic picture.

Apparently, two distinct pseudogaps observed in the metallic state of the underdoped high-$T_{c}$ cuprates 35 ; 100 are precursors of the above discussed insulating gaps in the lightly doped cuprates (cf. Refs. 102 ; 108 ). Finally, scanning tunneling microscopy/spectroscopy (STM/STS) studies showed 109 that, as the carrier density decreases, the delocalized carriers in momentum ($k$) space progressively become localized in real ($r$) space, and the pseudogap state develops in poorly understable manner. The possible pseudogap excitations in doped cuprates will be discussed below.

When the energy band of polarons is formed in the CT gap of the cuprates, the states of quasi-free hole carriers in the oxygen valence band become the excited states of a polaron. The Fermi level of polarons $\varepsilon_{F}$ lies inside the CT gap and the threshold energy for photoexcitation of a hole carrier from the polaronic state to a free hole state is given by

where $\varepsilon_{F}^{f}=\hbar^{2}(3\pi^{2}n)^{2/3}/2m^{*}$ is the Fermi energy of quasi-free hole carriers, $m^{*}$ is the effective mass of these carriers.

The polaronic effects are caused by the unconventional electron-phonon interactions and result in lowering the electronic energy (i.e., the Fermi level or chemical potential $\mu_{F}$ is shifted) by an amount $\Delta\varepsilon_{F}$ and the suppression of the density of states at the Fermi surface of quasi-free electrons (or holes). The so-called non-pairing pseudogap is opened on the former Fermi surface due to the polaronic shift of the electronic states of free carriers. As a result, a large Fermi surface of quasi-free carriers transforms into a small polaronic Fermi surface. Therefore, the excitation energy $\Delta\varepsilon_{F}$ of polarons is manifested in the single-particle spectrum of doped high-$T_{c}$ cuprates as the suppression of the density of states (DOS) at the Fermi level $\varepsilon^{f}_{F}$ and as the non-pairing polaronic pseudogap. As the doping (or $n$) is increased, the Coulomb repulsion between polarons increases and the binding energy $E_{p}$ of polarons decreases, so that the polaronic effect weakens and disappears in the overdoped region. Indeed, the binding energies of polarons $E_{p}=0.12$ eV and $E_{p}=0.06$ eV were observed experimentally in the underdoped and optimally doped cuprates, respectively 93 . One can expect that the dissociation of large polarons due to the Coulomb repulsion between them at short distances occurs at some critical doping level $x=x_{p}$. At $x<x_{p}$, the threshold energy for the thermal excitation of a carrier from the polaronic state to a free-carrier state or for the thermal dissociation of a large polaron can be approximately defined as

where $E_{c}=e^{2}/\varepsilon_{0}a_{p}$ is the Coulomb interaction energy between two large polarons, $a_{p}=(3/4\pi n_{a}x)^{1/3}$ is the mean distance between these polarons.

According to the above considerations, depending on the excitation ways, the non-pairing polaronic pseudogap can be determined either from Eq. (11) or from Eq. (12). A better way to define the doping-dependent polaronic pseudogap might be the latter result, Eq. (12), which provides useful information about the characteristic crossover temperature associated with this pseudogap. To evaluate the energy scales of such a pseudogap in underdoped cuprates LSCO, we use Eq. (12) and choose the parameters as $x=0.12$, $\varepsilon_{0}=30$ 89 , $n_{a}\simeq 5.3\cdot 10^{21}\rm{cm^{-3}}$ and $E_{p}\simeq 0.135$ eV. Then we obtain $\Delta_{p}\approx 0.068$ eV, which is in fair agreement with the temperature-independent pseudogap observed experimentally in underdoped LSCO at $500-600\rm{cm^{-1}}$ ($0.06-0.072$ eV) 43 . By taking the parameters $x=0.087$, $\varepsilon_{0}=30$ 89 ; 90 , $n_{a}\approx 10^{22}\rm{cm^{-3}}$ and $E_{p}\simeq 0.143$ eV for YBCO, we find $\Delta_{p}\simeq 0.069$ eV, which is also close to the observed value of the pseudogap $\Delta_{p}\simeq 0.07$ eV in underdoped YBCO 112 . The origin of the large pseudogap ($\Delta_{PG}\sim 0.1$ eV) observed in all underdoped cuprates is most likely associated with the formation of the non-pairing polaronic pseudogap. According to Eq. (12), the polaronic pseudogap decreases with increasing $x$ and disappears at $x=x_{p}$ in accordance with experimental findings 52 ; 53 ; 55 . The pseudogap crossover temperature $T_{p}$ decreases with increasing $x$ and the quantum criticality (quantum phase transition) occurs at some critical doping at which $T_{p}$ goes to zero near a quantum critical point (QCP), $x_{QCP}=x_{p}$ where the breakdown of the usual Fermi-liquid and BCS pairing theories occurs 61 . In the overdoped regime, a large Fermi surface transforms into a small polaronic Fermi surface at $x<x_{QCP}$. This formerly predicted Fermi surface transformation at a QCP 62 was discovered later experimentally 113 . We now consider the doping dependences of the pseudogap crossover temperature $T_{p}(x)=\Delta_{p}(x)/k_{B}$ in various high-$T_{c}$ cuprates. In so doing, we show that the different binding energies $\Delta_{p}$ of polarons determine the different positions of the QCP found from the condition $\Delta_{p}(x)=0$ in LSCO, YBCO and Bi-2212 systems.

Various experiments indicate 52 that the unusual and usual metallic states of underdoped to overdoped cuprates above $T_{c}$ are separated by the pseudogap phase boundary or pseudogap crossover line which intersects the superconducting dome and reaches $T=0$ at some critical doping level (i.e., at a QCP).

In particular, the QCP ($x=x_{p}$) in LSCO lies in the doping range $0.20<x<0.24$ 52 ; 114 . By taking $E_{p}\simeq(0.095-0.10)$ eV, $\varepsilon_{0}=30$ and $n_{a}\simeq 5.3\cdot 10^{21}\rm{cm^{-3}}$ (at $V_{a}=190{\AA}$) for LSCO, we obtain $x_{p}\simeq 0.20-0.24$ in accordance with these experimental findings. In Fig. 2, the calculated doping dependence of the polaronic pseudogap crossover temperature $T_{p}(x)$ is compared with the pseudiogap crossover temperature measured on LSCO 52 . As can be seen in Fig. 2, there is a fair agreement between the calculated curve $T_{p}(x)$ and experimental results for $T_{PG}\simeq E_{g}/k_{B}$ in LSCO, where $E_{g}$ is the energy scale of a pseudogap 52 . The transformation of the large Fermi surface of quasi-free carriers to small Fermi surface of large polarons occurs at the QCP located at $x_{QCP}=x_{p}\lesssim 0.22$ above which the Fermi surface of LSCO is in its pristine large Fermi surface state.

In YBCO the doping-dependent pseudogap determined by using different experimental techniques decreases with increasing $x$ and tends to zero as $x\longrightarrow x_{QCP}\simeq 0.19$ 52 ; 53 or as $x\longrightarrow x_{QCP}\simeq 0.22$ 55 .

If we take $E_{p}\simeq 0.142$ eV, $\varepsilon_{0}=22$ and $n_{a}=1.2\cdot 10^{22}\rm{cm^{-3}}$ for YBCO, we find from the condition $\Delta_{p}(x)=0$ somewhat different position of the QCP at $x=x_{p}\simeq 0.20$ located between the above experimental values of $x_{QCP}\simeq 0.19$ and $x_{QCP}\simeq 0.22$. In this system the transformation of the large Fermi surface of quasi-free carriers to small Fermi surface of large polarons occurs at the QCP located around $x_{p}\simeq 0.20$, which separates two types of Fermi-liquids (i.e., usual Fermi-liquid and polaronic Fermi-liquid).

Experimental results 52 ; 115 provide evidence for the existence of a finite pseudogap at around $x\simeq 0.22$ in the overdoped system Bi-2212. Apparently, the QCP in this system is located at $x\gtrsim 0.22$. By taking the parameters $E_{p}\simeq 0.133$ eV, $\varepsilon_{0}=25$ and $n_{a}=1.3\cdot 10^{22}\rm{cm^{-3}}$ for Bi-2212, we find $x_{QCP}=x_{p}\gtrsim 0.22$ (Fig. 4) in accordance with the experimental results 115 . It follows that the transformation of the large Fermi surface of quasi-free carriers to small Fermi surface of large polarons in Bi-2212 occurs at the QCP located at $x_{p}\gtrsim 0.22$. We see that each of the cuprate superconductor is characterized by the distinct pseudogap phase boundary ending at a specific QCP where Fermi surface reconstruction occurs, somewhere in the overdoped region.

The formation of large strongly overlapping Cooper pairs and the superfluidity of such fermionic Cooper pairs in weak-coupling BCS superconductors are assumed to occur at the same temperature $T_{c}$. However, the situation is completely different in more complex systems, in particular, in high-$T_{c}$ cuprates in which the BCS-type Cooper pairing of fermionic quasiparticles without superconductivity can occur above $T_{c}$ and the superfluidity of preformed Cooper pairs becomes possible only at $T_{c}$. In this case, one expects a significant suppression of the density of states at the Fermi surface above $T_{c}$ in the normal state. This so-called BCS-like pairing pseudogap regime extends up to a crossover temperature $T^{*}=T_{F}>T_{c}$ 21 ; 50 . Because the unconventional and more effective electron-phonon interactions are believed to be responsible for the BCS-like pairing correlation above $T_{c}$ and the formation of incoherent (non-superconducting) Cooper-like polaron pairs in the normal state of underdoped, optimally doped and moderately overdoped cuprates. Actually, these high-$T_{c}$ cuprates are unusual metals and have a well-defined and large Fermi surface as follows from ARPES data 43 ; 45 .

The symmetry of the BCS-like pairing pseudogap is still one of the most controversial issues in the physics of high-$T_{c}$ cuprates. Although some experimental observations advocate in favor of a $d$-wave symmetry (see Ref. 116 ), many other experiments 117 ; 118 ; 119 ; 120 closely trace a $s$-wave pairing gap and are incompatible with a $d$-wave pairing symmetry. Further, the $c$-axis bicrystal twist Josephson junction and natural cross-whisker junction experiments provide strong evidence for a $s$-wave pairing gap in high-$T_{c}$ cuprates 121 ; 122 . Müller has argued 123 that a $s$-wave pairing state may, in fact, exist in the bulk of high-$T_{c}$ cuprates. Therefore, we take the view that for high-$T_{c}$ cuprates, the $s$-wave BCS-like pairing state is favored in the bulk and the $s$-wave BCS-like pairing pseudogap could originate from unconventional electron-phonon interactions and carry the physics of the novel BCS-like pairing effects above $T_{c}$.

The high-$T_{c}$ cuprates with low Fermi energies ($\varepsilon_{F}\simeq 0.1-0.3$ eV 44 ; 124 ) and high-energy optical phonons ($\hbar\omega_{0}=0.05-0.08$ eV 81 ; 89 ; 124 ) are in the nonadiabatic regime (i.e., the ratio $\hbar\omega_{0}/\varepsilon_{F}$ is no longer small). For these reasons, the BCS-Eliashberg theory turned out to be inadequate for the description of the formation of unconventional Cooper pairs in high-$T_{c}$ cuprates, where the polaronic effects seem to be important and control the new physics of these systems. In hole-doped cuprates, the new situation arises when the polaronic effects exist and the attractive interaction mechanism (e.g., due to exchange of static and dynamic phonons) between the carriers operating in the energy range $\{-(E_{p}+\hbar\omega_{0}),(E_{p}+\hbar\omega_{0})\}$ is much more effective than in the simple BCS picture. Therefore, the BCS pairing theory should be modified to include polaronic effects. In the case of high-$T_{c}$ cuprates the unusual form of BCS-like pairing theory can describe the formation of polaronic Cooper pairs above $T_{c}$ naturally.

By applying the modified BCS formalism to the interacting Fermi-gas of polarons, we can write the Hamiltonian of this systems with the pair interaction in the form

where $\xi(k)=\varepsilon(k)-\varepsilon_{F}$ is the energy of polarons measured from the Fermi energy $\varepsilon_{F}$, $\varepsilon(k)={\hbar}^{2}k^{2}/2m_{p}$, $a^{+}_{{\vec{k}\sigma}}(a_{\vec{k}\sigma})$ is the creation (annihilation) operator for a polaron having momentum $\vec{k}$ and spin projection $\sigma$ ($=\uparrow$ or $\downarrow$), $V_{p}(\vec{k},\vec{k^{\prime}})$ is the pair interaction potential (which has both an attractive and a repulsive part) between large polarons.

The ground state energy of the interacting many-polaron system is calculated by using the model Hamiltonian (III.5). One can assume that the deviations of the products of operators $a^{+}_{\vec{k^{\prime}}\uparrow}$$a^{+}_{-\vec{k^{\prime}}\downarrow}$ and $a_{-\vec{k^{\prime}}\downarrow}$$a_{\vec{k}\uparrow}$ in Eq. (III.5) from their average values $<a^{+}_{\vec{k^{\prime}}\uparrow}a^{+}_{-\vec{k^{\prime}}\downarrow}>$ and $<a_{-\vec{k^{\prime}}\downarrow}a_{\vec{k}\uparrow}>$ are small. Then one pair of operators, $a_{-\vec{k^{\prime}}\downarrow}$$a_{\vec{k}\uparrow}$ or $a^{+}_{\vec{k^{\prime}}\uparrow}$$a^{+}_{-\vec{k^{\prime}}\downarrow}$, can be replaced by its average value. We can write further the identity, following Tinkham 125 , in the form

where $F_{\vec{k}}=<a_{-\vec{k}\downarrow}a_{\vec{k}\uparrow}>$. This is a mean-field approximation and the quantity in the bracket in Eq. (14) is a small fluctuation term. Substituting Eq. (14) and its Hermitian conjugate into the Hamiltonian (III.5) and dropping the term $\sum_{\vec{k},\vec{k^{\prime}}}V_{p}(\vec{k},\vec{k^{\prime}})(a^{+}_{\vec{k^{\prime}}\uparrow}a^{+}_{-\vec{k^{\prime}}\downarrow}-F^{*}_{\vec{k^{\prime}}})(a_{-\vec{k}\downarrow}a_{\vec{k}\uparrow}-F_{\vec{k}}),$ which is the second-order in the fluctuations and is assumed to be very small, the model mean-field Hamiltonian can be written as

Now, we introduce the gap function (or order parameter)

The function $\Delta_{F}(\vec{k})$ and the Hermitian conjugate function $\Delta^{*}_{F}(\vec{k})$ can be chosen as the real functions 126 . Substituting these functions into Eq. (III.5), we obtain the following resulting Hamiltonian:

The Hamiltonian (III.5) is diagonalized by using the Bogoliubov transformation:

where $b^{+}_{\vec{k}}(b_{\vec{k}})$ is the new creation (annihilation) operator for a Fermi quasiparticle, $u_{k}$ and $v_{k}$ are real functions satisfying the condition

The new operators $b_{\vec{k}\sigma}$ and $b^{+}_{\vec{k}\sigma}$ just as the old operators $a_{\vec{k}\sigma}$ and $a^{+}_{\vec{k}\sigma}$ satisfy the anticommutation relations of Fermi operators:

Substituting Eq. (III.5) into Eq. (III.5) and taking into account Eq. (19) and Eq.(III.5), we obtain

We now choose $u_{k}$ and $v_{k}$ so that they are satisfied the condition

Then the Hamiltonian (III.5) has the diagonal form and it includes the terms of the ground-state energy $E_{0}$ and the energy $E(\vec{k})$ of quasiparticles

where

As can be seen from Eq. (23), the Hamiltonian (III.5) is reduced to the Hamiltonian of an ideal gas of non-interacting fermionic quasiparticles. Combining Eq. (19) and Eq. (22), and solving the quadratic equation, we have

Substituting Eqs. (24), (25) and (26) into Eq. (23), we obtain

For the unconventional pairing interactions, it is argued 21 (see also Ref. 26 ) that the pseudogap phase has a BCS-like dispersion given by $E(\vec{k})=\sqrt{\xi^{2}(k)+\Delta^{2}_{F}(\vec{k})}$, but the BCS-like gap $\Delta_{F}(\vec{k})$ is no longer superconducting order parameter and appears on the Fermi surface at a characteristic temperature $T^{*}$, which represents the onset temperature of the Cooper pairing of fermionic quasiparticles above $T_{c}$.

We can now determine the BCS-like energy gap $\Delta_{F}(\vec{k})$ and related normal state pseudogap crossover temperature $T^{*}$. After replacing the $a_{\vec{k}\sigma}$ operators by the $b_{\vec{k}\sigma}$ operators and dropping the off-diagonal operators $b_{-\vec{k}^{\prime}_{\downarrow}}b_{\vec{k}^{\prime}_{\uparrow}}$ and $b^{+}_{\vec{k}^{\prime}_{\uparrow}}b^{+}_{-\vec{k}^{\prime}_{\downarrow}}$, which do not contribute to the average value of the product of operators $a_{-\vec{k}^{\prime}_{\downarrow}}a_{\vec{k}^{\prime}_{\uparrow}}$, the gap function or order parameter $\Delta_{F}(\vec{k})$ is given by