# Anomalous superconductivity and superfluidity in repulsive fermion systems

## Abstract

We discuss the mechanisms of unconventional
superconductivity and superfluidity in 3D and 2D fermionic systems
with purely repulsive interaction at low densities. We construct
phase diagrams of these systems and find the areas of the
superconducting state in free space, as well as on the lattice in
the framework of the Fermi-gas model with hard-core repulsion, the
Hubbard model, the Shubin-Vonsovsky model, and the model. We
demonstrate that the critical superconducting temperature can be
greatly increased in the spin-polarized case or in a two-band
situation already at low densities. The proposed theory is based
on the Kohn-Luttinger mechanism or its generalizations and
explains or predicts anomalous -, -, and -wave pairing in
various materials, such as high-temperature superconductors, the
idealized monolayer and bilayer of doped graphene, heavy-fermion
systems, layered organic superconductors, superfluid He,
spin-polarized He mixtures in He, ultracold quantum gases
in magnetic traps, and optical lattices.

Keywords: anomalous superconductivity, Kohn-Luttinger mechanism,
superfluidity, Fermi gas with repulsion, Hubbard and *t—J*
model,
Shubin-Vonsovsky model, graphene monolayer, graphene bilayer

PACS Numbers: 67.85.-d, 74.20.-z, 74.20.Mn, 74.20.Rp, 74.25.Dw,
74.78.Fk, 81.05.ue

Anomalous superconductivity and superfluidity in repulsive fermion systems \rauthorM. Yu. Kagan, V. A. Mitskan, M. M. Korovushkin

###### Contents:

- 1 Introduction
- 2 Superconductivity in the model of a Fermi gas with repulsion
- 3 Superconductivity in the 3D and 2D Hubbard model with repulsion
- 4 Enhancement of the critical temperature in the two-band Hubbard model and in a spin-polarized Fermi gas
- 5 Nontrivial corrections to the Landau Fermi liquid theory in 2D low-density systems
- 6 Superconductivity in the Shubin-Vonsovsky model
- 7 Superconductivity in the 2D t-J model
- 8 Superconductivity in an idealized graphene monolayer
- 9 Enhancement of superconductivity in an idealized graphene bilayer
- 10 New promising systems with anomalous pairing
- 11 Conclusions

## 1 Introduction

The recent discovery of Cooper pairing at a recordly high temperature of 190 K in metallic hydrogen sulfide HS [1, 2] under pressure of the order of 1 Mbar raises our hopes to move forward from ‘high-temperature’ to ‘room-temperature’ superconductors. At the same time, there are many interesting low-temperature superconducting and superfluid systems with anomalous types of pairing and a nontrivial structure of the order parameter. In this review, we consider systems with a low density of fermions in the framework of the nonphonon mechanisms of superconductivity, such as the famous Kohn-Luttinger mechanism and its generalizations, and the exchange mechanisms connected with the antiferromagnetic attraction of spins on neighboring sites, which are topical, in particular, for the model, in which both low and high critical temperatures of the superconductive transition and anomalous -, -, and -wave Cooper pairing can appear.

As is known from textbooks, the conduction electrons in metals, together with the positively charged ions, form a solid-state plasma, which determines the combination of their electric, galvanomagnetic, kinetic, and superconducting properties. The coupling between the subsystem of massive positive ions and the subsystem of light fermions leads to the appearance of the electron-phonon interaction, which affects the properties of the electron subsystem. In particular, the effective interaction between electrons in a solid-state plasma can differ significantly from the Coulomb interaction of electrons in the vacuum and can even change sign. This most important effect is the basis of the electron-phonon mechanism of Cooper instability in standard superconductors [3].

It is obvious that the role of the mediator (coupling to which initiates the renormalization of the Coulomb interaction) can be played by any other subsystem. It is only necessary that the interaction of the electron gas with this subsystem lead to polarization effects that cause the generation of electrons and holes in the vicinity of the Fermi surface. Notably, in many theoretical studies on high-temperature superconductors, collective excitations of the subsystem of localized spins of copper ions serve as such a mediator. This, in particular, determines the spin-fluctuation mechanism of Cooper instability, which leads to the formation of a superconducting phase with type symmetry of the order parameter.

Within the formalism of the secondary quantization of fermions, the operator of Coulomb interaction of electrons contains terms that in the higher orders of the perturbation theory initiate polarization contributions to the ground-state energy, which also leads to the renormalization of the Coulomb interaction of electrons. Therefore, the effective interaction of electrons in such a metal can differ significantly from the electron-electron interaction in the vacuum. This makes topical the problem (first formulated by Anderson [4]) of such a renormalization of Coulomb interaction under which the effective electron-electron interaction in a substance would have an attractive rather than repulsive nature. In other words, the problem consists in searching for conditions under which the above-mentioned polarization effects in the electron plasma of a metal would lead to a change in the sign of the resulting interaction between the electrons. From the mathematical standpoint, the problem reduces to calculating the effective pairwise interaction of electrons with multiparticle effects in the electron ensemble taken into account. No less important, according to Anderson, is the problem of explaining the unconventional properties of the normal state of many strongly correlated electronic systems at temperatures higher than the critical temperature, especially in the pseudogap state.

In recent decades, considerable progress has been achieved in experimental and theoretical studies of superconductive systems with a nonphonon nature of the Cooper pairing and with a complex, nontrivial structure of the order parameter. The first experimentally discovered systems with unconventional triplet wave pairing (the total spin of the Cooper pair and the orbital momentum of the relative motion of the pair ) were the superfluid A and B phases of He with low critical temperatures, . Another example of systems in which the wave pairing is realized are Li and K molecules in magnetic traps in the regime of the wave Feshbach resonance with ultralow critical temperatures: [5, 6]. It is assumed that unconventional wave pairing with critical temperatures is realized in some heavy-fermion intermetallic compounds, such as UThBe and UNlAl, with large effective masses [7, 8]. Frequently, the wave pairing is mentioned in connection with organic superconductors, such as (BEDT-TTF)I with [9]. Finally, wave pairing with is apparently realized in the ruthenates SrRuO [10, 11], and it cannot be excluded in the layered dichalcogenides CuS–CuSe or the semimetals and semimetallic superlattices InAs–GaSb,PbTe–SnTe [12], The heavy-fermionic intermetallic compound UPt with and , as well as a large class of high-temperature cuprate superconductors with critical temperatures from (for the lanthanum-based compounds) to (obtained in mercury-based superconductors under pressure), are related to unconventional superconductors with the singlet wave pairing (). Finally, in connection with the problems of applied superconductivity, it is also necessary to mention new multiband superconductors with a more conventional wave pairing, such as MgB [13], and the recently discovered superconductors based on iron arsenide [14] and the HS and PoH metallic compounds already noted above [15].

Along with the problems of Cooper pairing in the above-mentioned electron systems, also of significant interest is the search for fermionic superfluidity in three-dimensional (3D) and two-dimensional (2D) (thin films, submonolayers) solutions of He in He [16, 17, 18] and for superconductivity in doped graphene [19], which are problems that have still not been solved experimentally. These systems are among the most promising ones from the standpoint of the experimental and theoretical description of a wide class of physical phenomena and of the nature of multiparticle correlations in them.

Notably, submonolayers of He adsorbed on different substrates, such as a solid substrate or the free surface of superfluid He, with the variation of the particle density in wide ranges, make it possible to realize different regimes in the system – from the ultrararefied Fermi gas to strongly correlated Fermi systems [20]. This makes the solutions ideal objects for the development and verification of different methods of the Fermi liquid theory. The unbalanced (spin-polarized) ultracold Fermi gases in 3D and especially in 2D magnetic traps are also very promising [21, 22].

Of significant interest from both the fundamental and applied standpoints is graphene, because of its unique electronic properties [23, 24]. Near the Fermi level, the electrons in graphene have a linear dispersion, and the energy gap between the valence band and the conduction band is absent; therefore, the electrons in graphene can be described by a 2D Dirac equation for massless charged quasiparticles [25]. The properties of these quasiparticles, such as their two-dimensionality, the spinor nature of their spectrum, the zero mass, and the absence of the gap in the spectrum, lead to a number of phenomena that have no counterparts in other physical systems [26].

The above-mentioned studies have stimulated an intensive search for alternative mechanisms of pairing based on strong correlations in the Fermi liquid. The most promising in this respect are the Kohn-Luttinger mechanism [27], proposed in 1965, and its generalizations (see, e.g., [28]). The Kohn-Luttinger mechanism assumes the transformation of the pure repulsive interaction of two particles in the vacuum in the presence of a fermionic background into an effective attraction in the substance in the channel with a nonzero orbital angular momentum of the pair.

This review is devoted to the description of basic results obtained in recent decades concerning Kohn-Luttinger superconductivity in repulsive Fermi systems and its generalizations as well as the exchange mechanisms of superconductivity in the generalized model.

## 2 Superconductivity in the model of a Fermi gas with repulsion

The basic model for studying the nonphonon mechanisms of superconductivity in low-density electron systems is the model of a Fermi gas. In the case of a Fermi gas with attraction, the scattering length is negative (), which results in a traditional -wave pairing (total spin , orbital angular momentum ) with the critical temperature

(1) |

where is the Fermi energy and is the Fermi momentum.

This result was obtained in Ref. [29] soon after the appearance of the Bardeen-Cooper-Schrieffer (BCS) theory [3]. Result (1) differs from the classical formula given in [3] by the presence of a quantity in the preexponential factor instead of the Debye frequency typical for the phonon models in conventional superconductors.

In the model of a Fermi gas with repulsion, the scattering length is positive () and the superconductivity corresponds to the Kohn-Luttinger mechanism in the low-temperature region. The physical reason for this consists in the effective interaction of quasiparticles, which arises as a result of the polarization of the fermionic background. Due to a sharp boundary existing in the momentum space which is equal to the diameter of the Fermi sphere and separates the occupied states from the empty ones, the effective interaction of quasiparticles that are located on the Fermi level does not decrease exponentially, but has an oscillating form (Friedel oscillations [30, 31])

(2) |

If the distance between two electrons in a Cooper pair is relatively large, effective interaction (2) in the coordinate space has a large number of maxima and minima (Fig. 1). Then the integral effect determined by the averaging over the potential relief of Friedel oscillations can, in principle, lead to an effective attraction and the appearance of superconductivity in the system.

The first to advocate this mechanism of superconductivity were Kohn and Luttinger [27], who considered 3D repulsive Fermi systems. They showed that the effective interaction in the first two orders of the perturbation theory in the gas parameter (more precisely, in the scattering length ) is described by the sum of the five diagrams shown in Fig. 2. The first diagram corresponds to the bare interaction of two electrons in the Cooper channel. The next four diagrams (Kohn-Luttinger diagrams) are due to second-order processes and take into account the polarization effects of the filled Fermi sphere. In the case of a short-range potential, the contribution to the effective interaction is determined only by the fourth exchange diagram, and in the first two orders of the perturbation theory, the expression for can be written as

(3) |

where is the pseudopotential, which corresponds to the wavy line in Fig. 2, and is the static polarization operator, which is described by the standard Lindhard function [34, 35]

(4) |

This operator is responsible for the charge screening in the electron plasma in metals. The plus sign in the argument of the polarization operator is due to the so-called crossing, which, in the case of short-range repulsion, distinguishes the exchange diagram from the true polarization loop, which contains the argument . In the absence of a lattice, is the energy spectrum,

is the Fermi-Dirac distribution function, and is the chemical potential.

It was noted in the early work of Migdal [36] and Kohn [37] that at low temperatures (), the polarization operator contains, apart from a regular part, a singular part – the so-called Kohn anomaly, which in the 3D case has the form

(5) |

where we have in the cross-channel. In the coordinate space, the singular part of leads to Friedel oscillations (2) in the effective interaction (see Fig. 1).

Thus, the purely repulsive short-range potential between two particles in vacuum induces an effective interaction in the electron gas in a metal with the competition between repulsion and attraction. It turns out that the singular part in favors attraction, ensuring a contribution that always exceeds the repulsive contribution caused by the regular part of . At large orbital momenta , this leads to a superconducting instability with the critical temperature . In this case, the conventional singlet pairing in the -wave channel () is suppressed by the short-range Coulomb repulsion caused by the main maximum in (see Fig. 1), and superconductivity is realized at large orbital momenta, . We note that at , the role of the main maximum is weakened by the centrifugal potential, which improves the conditions for the appearance of superconductivity in channels with anomalous pairing.

From Ref. [27], a nontrivial conclusion followed that no Fermi systems exist in the normal state at a zero temperature; any 3D electronic system with a purely repulsive interaction between particles is unstable with respect to the transition to the superconducting state with a large orbital angular momentum of the relative motion of a Cooper pair (). However, the estimates carried out in [27] for the critical temperature in electronic systems in metals with realistic parameters and for superfluid helium at gave very low values of the critical temperature: for He and for metallic plasma. The low value of was one of the reasons why the Kohn-Luttinger mechanism was not popular among researchers for a sufficiently long period and was unjustly forgotten.

Later on, in Refs [38, 39], it was shown that the temperature of the superconducting transition in [27] was underestimated because of the utilization of an asymptotic expression for large values of the orbital angular momentum, . In reality, at , an exact analytic calculation shows that the contributions to that correspond to the attraction of quasiparticles dominate over the repulsive contributions. As a result, the repulsive 3D Fermi gas is unstable with respect to the superconducting transition with the triplet -wave pairing at the critical temperature [38, 39, 40, 41]

(6) | |||||

where is the effective 3D Galitskii gas parameter [42], We note that for , the contribution from the Kohn anomaly only increases the value of , but does not play a decisive role in the appearance of the triplet superconductivity itself.

It was demonstrated in Ref. [43] the critical temperature of a superfluid transition can be substantially increased already at low fermionic densities by placing the system of neutral Fermi particles into a magnetic field or by creating spin polarization (). This occurs because the paramagnetic suppression of the superconductivity (which takes place for -wave pairing) is absent in the -wave channel for the so-called phase and the increase in is possible due to the enhancement of the effective interaction and the changes in the character of the Kohn anomaly. The highest critical temperatures then correspond to the phase, where the Cooper pair is formed by two spins up, and the effective interaction for them is prepared by two spins down. In this case, is a function of the ratio of the density of spin-up particles to the density of spin-down particles, , or more precisely of the spin polarization .

In the case of a repulsive 2D electron gas, in the first two orders of the perturbation theory in the gas parameter, the effective interaction takes the form [44, 45]

(7) |

where is the Bloom 2D gas parameter [46] and is the 2D polarization operator in the cross channel.

In the 2D situation, the effective interaction in the coordinate space also contains Friedel oscillations:

(8) |

which are even much stronger than in the 3D case. But in the momentum space, the 2D Kohn anomaly has one-sided character [47],

(9) |

for , and, therefore, is ineffective for the problem of superconductivity (in which ). Thus, the 2D Fermi gas with repulsion remains in the normal state at least in the first two orders of the perturbation theory in the gas parameter . Nevertheless, it was shown in [44] that the superconducting -wave pairing appears in the next (third) order of the perturbation theory in , in which, for the singular contribution to the effective interaction, the expression under the square root in (9) reverses sign:

(10) |

In this case, an exact calculation [48] of the critical temperature taking all irreducible third-order diagrams into account, yields

(11) |

For the submonolayers of He on the surface of superfluid films of He [49], the temperature of the superconducting transition is estimated as K [44, 48] for the maximal densities at which the Fermi-gas description is still applicable, and this estimate is quite reasonable for experimental observation.

## 3 Superconductivity in the 3D and 2D Hubbard model with repulsion

In connection with the discovery of high-temperature superconductivity [50], the Hubbard model [51] acquired substantial popularity as one of the basic models for describing the anomalous properties of cuprates. The Hubbard model is a particular case of the general model of interacting electrons, whose band structure is described within the strong-coupling approximation and is the minimal model that accounts for band electron motion in the metal as well as the strong electron-electron interaction [52, 53, 54, 55, 56]. This model is of special importance in the description of narrow-band metals [48]. The Hamiltonian of the Hubbard model on a lattice has the form

where is the operator of creation (annihilation) of an electron with a spin projection at a site ; is the on-site energy of the electron; is the chemical potential of the system; is the operator of the number of particles on the site ; the matrix element stands for electron hoppings from site to site ; and is the parameter of the Coulomb interaction of two electrons that are located on the same site and have opposite spin projections (Hubbard repulsion).

Since extensive experimental data have indicated that the basic dynamics of Fermi excitations in cuprates is developed in the CuO planes, the nonphonon mechanisms of superconductivity were mainly based on the 2D Hubbard model on a simple square lattice. In the momentum space, the Hamiltonian of the model has the form

where the energy of an electron, including distant hoppings, which are determined by the parameters and , is given by

(14) | |||||

where is the lattice constant (intersite distance).

We note that when simulating electron spectrum (14) and constructing the phase diagram of the superconducting state in the Hubbard model, going beyond the framework of the nearest-neighbor approximation () becomes essential. This is because the leading contribution to the effective coupling constant comes from the interaction of electrons that are located near the Fermi surface, whose geometry depends on the structure of the energy spectrum. An important role is also played by the fact that account for distant hoppings shifts the Van Hove singularity in the density of electronic states from the position at half-filling () into the region of smaller or higher electronic densities (Fig. 3). We note that the introduction of hoppings to the third coordination sphere of the square lattice, , can lead to a qualitative change in the geometry of the Fermi surface, which connected with the formation of a multisheet Fermi contour (see the inset in Fig. 3).

Thus, an account for distant hoppings can lead to a modification of the phase diagram that determines the regions of the realization of the superconducting states with different types of the order parameter symmetry.

In the Hubbard model, the perturbation theory can be constructed in two limiting cases: (1) the Born weak-coupling approximation, ( is the bandwidth; is the number of nearest neighbors) and an arbitrary electron density, ; and (2) the strong-coupling approximation at low electron density, . The utilization of the weak-coupling approximation in the analysis of the feasibility of Kohn-Luttinger superconducting pairing allows us to calculate for the Cooper channel at all the densities by restricting ourselves to the second order diagrams in the interaction (see Fig. 2). In the opposite limit of strong coupling, , the restriction to first- and second-order diagrams is justified only in the region of low electron density , where the Galitskii-Bloom Fermi-gas expansion is valid [42, 46].

In one of the first studies [57], the authors analyzed the conditions for the realization of the Kohn-Luttinger superconductivity in the 2D Hubbard model with Hamiltonian (3) in the weak-coupling limit , in the nearest-neighbor approximation () at low electron densities (). In this case, the following expansion is valid for the electron spectrum:

(15) | |||||

where is the band mass. It can be seen that in the chosen approximation, the bare spectrum of electrons at almost coincides with the spectrum of a free Fermi gas, and the Hubbard Hamiltonian is equivalent to the Hamiltonian of a weakly nonideal Fermi gas with short-range repulsion between the particles [58]. To determine the possibility of the superconducting pairing in this approximation, the effective bare vertex for the Cooper channel was calculated in first two orders of perturbation theory:

(16) |

where is polarization operator (4). To solve the problem of superconducting pairing, the function was expanded in a series with eigenfunctions of the irreducible representations of the symmetry group of the square lattice (see Section 6 ), and then the sign of the expressions for was analyzed for each representation . As a result, it was shown that the 2D electron system described by the Hubbard model for small filling and is unstable towards the superconducting pairing with the -type symmetry of the order parameter , where the integer satisfies the condition .

The weak-coupling limit in the 3D and 2D Hubbard model near the half-filling, , was analyzed in [59, 60, 61]. In the 2D case [61], in the nearest-neighbor approximation at , the electron spectrum becomes quasihyperbolic

(17) |

near the corner points and at which the Fermi surface almost touches the Brillouin zone (Fig. 4). As it is well known, the density of electronic states has a logarithmic singularity in these regions near the Van Hove filling, namely , where is the modulus of the chemical potential near the half-filling. It can be seen from Fig. 4 that there are almost flat regions of the Fermi surface, which satisfy the condition for ideal nesting in the exactly half-filled case ():

(18) |

where is the nesting vector for the 2D square lattice. In these regions, the polarization operator reads [61, 62], where one logarithmic factor comes from the density of states and the other one is due to the Kohn anomaly. The following quantity serves here as the parameter of the perturbation theory in the 2D weak-coupling limit:

(19) |

and in the second order of the perturbation theory in , the expression for the effective interaction takes the form

(20) |

Since the expression for the Cooper loop at , apart from the usual Cooper logarithm, also contains a logarithm due to the Van Hove singularity, we have

(21) |

where . Therefore, the expression for the critical temperature with the order parameter of the typical for cuprates -wave symmetry obtained in Ref. [61] in the leading logarithmic approximation has the form

(22) |

or

(23) |

It can be seen from expression (23) that the denominator in the right-hand side, in spite of the low value of at , increases substantially due to the large value of .

The results in [57] on the realization of -wave pairing at and on the -wave pairing at [59, 60, 61] in the weak-coupling limit were subsequently confirmed by other authors. In [63], a phase diagram of the superconducting state of the 2D Hubbard model was constructed at small and intermediate occupation numbers, which shows how the results of the competition of different types of the order parameter symmetry to depend on the value of the parameter of the electron hoppings to the next-nearest-neighbor sites. The phase diagram obtained in the second order of the perturbation theory shows that at in the region of the low electron densities, , superconductivity with the -type symmetry of the order parameter is realized. In the range of , the ground state corresponds to the phase with the -wave pairing, and at the -wave pairing appears. Similar results were obtained in [64] in the framework of the renormalization group approach.

In vicinity of the half-filling, , where strong competition between superconductivity and antiferromagnetism is observed, the problem of Cooper instability was examined in Refs [65, 62, 66]. In these studies, the so-called parquet diagrams were summed up and at the relation

(24) |

was obtained. This yields an elegant estimate for the maximal critical temperature:

(25) |

The maximal critical temperature of the superconducting transition in the 2D Hubbard model was obtained in Ref. [67] in the regime at optimal electron concentrations . According to the estimate obtained in [67], the critical temperature can reach the values , which is quite reasonable for optimally doped cuprate superconductors.

## 4 Enhancement of the critical temperature in the two-band Hubbard model and in a spin-polarized Fermi gas

Along with the possibility to enhance the anomalous superconductivity by applying a magnetic field to a system of neutral Fermi particles as described in Section 2 , there is also another possibility to increase significantly already at a low electron density. It is related to the two-band situation [68] or a multilayer system with geometrically separated layers. In this case, the role of spins-up is played by the electrons of the first band (or layer), and the role of spins-down by the electrons of the second band (or layer). The coupling between electrons of two bands is achieved by means of the interband Coulomb interaction . As a result, the following exciton-type mechanism of superconductivity becomes possible: the electrons of one band form a Cooper pair via polarization of the electrons of the other band [68, 69, 70]. In this case, the role of spin polarization is played by the relative filling of two bands .

We now examine the two-band Hubbard model with one broad band and one narrow band [69, 70], which accordingly contains ‘heavy’ () and ‘light’ () electrons. This model is a natural generalization of the well-known Falikov-Kimball model [71] for systems with a mixed valence; however, it exhibits a richer physics in view of the presence of a finite bandwidth of heavy electrons instead of a localized level. In the Hubbard model with one narrow band, the effective interaction, as it was shown in Refs [69, 70, 72, 73], is determined mainly by the interband Coulomb repulsion of heavy and light electrons . The corresponding critical temperature of the superconducting transition depends nonmonotonically on the relative filling of the bands and exhibits a wide and clearly pronounced maximum at in the 2D case. The maximal critical temperature can be expressed as

(26) |

and corresponds to the triplet -wave pairing of heavy particles via the polarization of light particles. In the Born weak-coupling approximation, the effective gas parameter depends linearly on the interband Coulomb repulsion and on the square root of the product of the masses [68]:

In the opposite limit of strong coupling [69, 70],

In the so-called unitary limit of the strongly screened Coulomb interaction () and of the strong electron-polaron effect, . Correspondingly, the maximal critical temperature in this case yields [69, 70]

(27) |

where is the renormalized (strongly narrowed) Fermi energy of heavy particles

Let us stress that in the unitary limit the sharp increase in the effective mass of heavy particles up to is caused by the many-body electron-polaron effect [72, 73]. As a result, quite reasonable critical temperatures are obtained for the superconducting -wave pairing: , typical for uranium-based heavy-fermion compounds, mentioned in the Introduction, UThBe and UNlAl with large effective masses [7, 8] and for organic superconductors.

We note that the electron-polaron effect, which leads to a significant increase in the effective mass in the model, is connected with the nonadiabatic part of the wave function that describes a heavy electron surrounded by a cloud of virtual electron-hole pairs that belong to the band of light electrons [72, 73].

The discussed mechanism of superconductivity possibly can be realized in the bismuth- and thallium-based cuprate superconductors, as well as in the PbTe-SnTe superlattices [12] and dichalcogenides CuS and CuSe with geometrically separated layers. We note that, in general, the two bands can belong to the same layer or to different layers. It is also reasonable to assume that this mechanism can be fulfilled in the ruthenates SrRuO [10] and in the ultracold Fermi gas of Li atoms in magnetic traps with a strong imbalance of the hyperfine components (see Section 10 ).

We note again that in the presence of the band of heavy and light electrons with strongly different masses, , and different densities, , the critical temperature is determined mainly by the pairing of the heavy electrons via the polarization of the light electrons. However, taking into account even an infinitely small Geilikman-Moskalenko-Suhl term [74, 75, 76, 77, 78] (where is the parameter of interaction corresponding to the rescattering of the Cooper pairs between the ‘heavy’ and ‘light’ bands), leads to the opening of superconducting gaps in both bands at the same critical temperature close to one in (26).

Let us consider the application of this theory to low-temperature physics in more details. We emphasize that the ultracold quantum gases in magneto-dipole traps, as well as with the spin-polarized solutions of He in He, especially in the quasi-two-dimensional situation (in which the largest increase in the temperature of the triplet -wave pairing occurs), are the excellent systems for verifying the theoretical predictions of [20, 41, 43, 68, 69, 70, 79]. Good experimental opportunities for the realization of the ‘high-temperature’ superfluidity in spin-polarized (imbalanced) Fermi gases in quasi-two-dimensional magnetic traps, in particular, has G E Thomas’s group in North Carolina [21]. We also note that in the 1990s G Frossati’s group in Leyden experimentally obtained a 20% increase (from 2.5 up to 3.14 mK) in the critical temperature of the superfluid transition in the phase of the superfluid He in magnetic fields up to T (at a spin polarization of 7%) [80, 81]. In this case, at the maximum for the critical temperature of the triplet wave pairing (reached at the spin polarization %), the theory of [20, 43] predicts an increase in by a factor of 6.4 in comparison with the nonpolarized case. A similar value for at the maximum (with the maximum ), but for a 35% spin polarization, was also predicted in the metamagnetic model in Refs [82, 83] in the framework of the so-called approximation for the Landau Fermi liquid theory [84]. We note, however, that the approach in [20, 43] in the framework of the enhanced Kohn-Luttinger mechanism of superfluidity is characterized by the only one fitting parameter, the gas parameter , and therefore this approach is more model-independent than the one in [82], which uses two fitting parameters that are not connected with each other, namely, the and harmonics of the scattering amplitude of quasiparticles on the Fermi surface.

We also note that for solutions of He in He, the theory of [20] and the results in [85, 86] predict a phase diagram for the fermionic superfluidity of He in the 3D case with the regions of -wave pairing at small concentration of He in the solution () and the regions of wave pairing at larger concentrations of He (). The critical temperatures of the -wave pairing in the solutions are maximal at a zero magnetic fields and at . According to the estimates in [85, 86], the -wave critical temperature is in the range .

The temperature of the triplet -wave pairing grows sharply in a magnetic field, and at the maximal possible concentration of He in the solution in the field T, we have . In 2D solutions of He in He, that is for the submonolayers of He at the Andreev levels [87, 88] (appearing similarly to Tamm levels on the free surfaces of thin films of superfluid He [89, 90]) or on the grafoils [91, 92], the phase diagram of the solution also contains the regions of - and -wave pairings.

We emphasize that for -wave pairing in 2D systems, usually two phenomena are realized simultaneously: the pairing of two particles in the real space (with the formation of a molecule or a dimer) and the Cooper pairing in the momentum space. The maximal for the -wave pairing, according to [93, 94, 95], is in the range of at the 2D density of a monolayer [85, 86]. At the same time, the temperature of the triplet -wave pairing can be increased in magnetic fields, and at T and the 2D density of a monolayer, it can become quite accessible experimentally (, according to [20]). The experimental observation of fermionic superfluidity in the 3D solutions and submonolayers of He remains a challenge for the ‘low-temperature community’ [96]. A similar phase diagram with the regions of - and -wave pairings was theoretically predicted in [41] and in [97] for the fermionic isotope of lithium Li on the Cooper (BCS) side of the crossover with a region of the Bose-Einstein condensation (BEC) in the regime of the -wave Feshbach resonance.

Let us recall that in the fermionic Li, the quasiresonance scattering length, which is very large in absolute value (), corresponds to attraction. As a result, in the balanced case (), a singlet -wave pairing with the critical temperature determined by formula (1) is realized for the two hyperfine components of the nuclear spin that are captured in a magnetic trap. The maximal temperature in the 3D case, according to [97], is of the order of at . However, if the imbalance between the hyperfine components is sufficiently large, such that , then, according to the Landau criterion of superfluidity [41], the -wave pairing is totally suppressed. Nevertheless, in this case a -wave pairing can arise if the Cooper pairs (as is the case of the phase of superfluid He) are formed by Fermi particles of one hyperfine component while the effective interaction for them is prepared by Fermi particles of another (or others) hyperfine component. In this situation, the maximal critical temperature of the -wave pairing for the optimal ratio of the densities of the hyperfine components, according to [41], can reach at and .

The effect of enhancement in total analogy with the solutions of He in He, for the -wave pairing in the imbalanced gases manifests itself much more strongly and clearly in the quasi-two-dimensional situation [68]. Therefore, the experimental achievements obtained in [21], which make it possible to create the quasi-two-dimensional traps and to control their essential parameters (such as the density, temperature, and number of particles on layer-by-layer basis) are very important.

We finally consider one more promising prediction of this theory. It was shown in Ref. [79], that in quasi-two-dimensional (layered) materials in a magnetic field that is strictly parallel to the layer, the appearing vector potential (, with and being the coordinates in the layer) does not change the motion of the electrons and Cooper pairs in the plane of the layer. Therefore, the diamagnetic Meissner effect is completely suppressed. As a result, the electronic monolayer (or the layered system) becomes equivalent to an uncharged (neutral) fermionic layer of He.

Thus, for low-density quasi-two-dimensional systems, the phase diagram of the superconducting state in a magnetic field parallel to the electronic layer can contain again the regions of conventional -wave pairing in the absence of a magnetic field and the regions of triplet wave pairing (similar to the phase of the superfluid He) in strong magnetic fields, when the -wave pairing is totally suppressed paramagnetically. Moreover, in the magnetic fields T and at low Fermi energies () for sufficiently large degrees of spin polarization of electrons () the reasonable critical temperatures () can be obtained. Of course, in this case, as in the case of graphene (see Sections 8 and 9), the experimentalists should be very careful analyzing the role of the structural disorder and nonmagnetic impurities, which lead to the isotropization of the order parameter and therefore suppress the nonspherical -wave pairing [99, 98]. Furthermore, it is necessary to ensure a high degree of the parallelism of the magnetic field to the plane of the layer, because the presence of even a relatively small perpendicular component would lead to the diamagnetic suppression of superconductivity [96]. Nevertheless, the proposed mechanism is very interesting for the possible realization of superconductivity in very pure heterostructures (see Section 10 ).

## 5 Nontrivial corrections to the Landau Fermi liquid theory in 2D low-density systems

It is well known that a high temperature of the superconducting transition in cuprate superconductors is connected with very unusual properties of these systems in the normal (nonsuperconducting) state. Among the unconventional properties of the normal phase, the small jump in the distribution function of the interacting particles on the Fermi surface and a linear temperature dependence of the resistivity at temperatures much lower than Debye temperature in optimally doped cuprate superconductors are of interest. To explain these facts, Anderson advanced a concept of a Luttinger-type Fermi liquid with a zero jump of the distribution function on the Fermi surface [100]. A similar idea of a marginal Fermi liquid, which is a special case of a Luttinger liquid, was proposed by the authors of [101] based on the analysis of the experimental data.

Later on, Anderson advanced an even more nontrivial idea that not only a high-density strongly interacting 2D Fermi system but even a weakly interacting low-density 2D Fermi gas should also described by a Luttinger Fermi liquid [102] rather than by the Landau Fermi-liquid theory with a finite jump of the distribution function. In Refs [100, 102], Anderson formulated three important points, which led to his doubts regarding the applicability of the standard Galitskii-Bloom Fermi-gas approach [46, 42] in the 2D case. These are, firstly, the problem of the finite scattering phase-shift for the particles with almost parallel spins, which leads to the vanishing of the -factor (Migdal jump) on the Fermi surface; secondly, the problem (connected with the first one) concerning the essential role of the upper Hubbard band in the lattice models already in the case of a low electron density; and, finally, the problem of the possible existence of a strong singularity in the Landau -function for the quasiparticles interaction, which arises in a 2D Fermi gas even in the absence of a lattice.

Many theorists participated in the discussion developed after the publication of Anderson’s work; most of them [103, 104, 105, 106, 107] supported the Fermi-gas ideology and attempted to prove its consistency in the 2D case using ladder and parquet approximations in terms of the diagrammatic technique. Anderson continued to insist on his point of view, assuming that the diagrammatic technique is inapplicable to the 2D systems even at the level of summing up an infinite series of parquet diagrams. In fact, the dispute considered the problem of the choice of a correct ground state, which would allow us to construct a regular procedure of successive approximations in the interaction (or, to be more precise, in its part that was not taken into account in choosing the ground state). According to Anderson’s qualitative considerations, the Landau function of the interaction of quasiparticles with almost parallel momenta p and and opposite spins of the colliding particles in the 2D case, when there is a small deviation from the Fermi surface for p and , contains a singular part of the form

(28) |

The existence of such a strong singularity leads to a logarithmic divergence of all Landau harmonics , and, thus, to the complete crush of the Fermi-liquid theory. The accurate calculation of the Landau quasiparticles interaction function carried out in the second order of the perturbation theory in Ref. [106] and, independently, in [107], leads to a significantly weaker singularity in of the form , which, in addition to that, exists only in a small window of angles near the parallel orientation of momenta p and . As a result, this singularity leads only to nontrivial temperature corrections to the -function rather than to the destruction of the Fermi-liquid picture as whole.

Concerning the second point of the discussion raised by Anderson, the authors of [108] examined the 2D Hubbard model in the limit of strong coupling () and low electron density () in the Kanamori matrix approximation [109]. In the low-energy region , in the framework of this description, the 2D Hubbard model becomes equivalent to a 2D electron gas with a quadratic spectrum and short-range repulsion (see Section 2 ). This model can be characterized by the 2D Bloom gas parameter [46], which allows to conduct a controlled diagrammatical expansion (here, the electron density in the 2D case for both spin projections). For the first iteration of the self-consistent matrix approximation, the authors of [108] found the contribution from the matrix pole corresponding to the upper Hubbard band. As a result, a dressed one-particle Green’s function was obtained with a double-pole structure [108], which resembles the Green function in the Hubbard-I approximation [51]:

(29) |

where is the notation for infinitesimally small imaginary part The first term in the right-hand side of (5) corresponds to the contribution from the lower Hubbard band, and the second term corresponds to the contribution from the upper Hubbard band. We note that the second iteration of the self-consistent -matrix approximation does not change the principal properties of expression (5). Thus, the presence of the upper Hubbard band leads to nontrivial corrections to the Landau Fermi liquid picture at low electron densities without total destruction of this picture in the 2D case. More specifically, they produce only small Hartree-Fock corrections to the thermodynamic potential.

We note that all the results concerning superconductivity in the Hubbard model obtained in the single-pole approximation for the one-particle Green function remain valid at and low electron density (up to small corrections , where is the bandwidth), when the second pole is taken into account. Thus, this result concerning the two-pole structure of the Green function plays the role of a very interesting bridge connecting the exact results of Galitskii and the Hubbard-I approximation (the Gutzwiller approximation) in the Hubbard model. At the same time, it does not affect the type of pairing or the estimate of the critical temperature at low electron densities.

## 6 Superconductivity in the Shubin-Vonsovsky model

The authors of [110] raised an important problem of the role of full (not reduced) Coulomb interaction in the nonphonon superconductivity mechanisms, which in real metals does not limited to the short-range Hubbard repulsion. The authors of [110] examined the 3D jelly model with realistic values of the electron density, when the Wigner-Seitz correlation radius is not very large ,

(30) |

where is the Bohr radius of the electron (). In calculation of the effective interaction, the contributions from the first and second order caused by all diagrams presented in Fig. 2 were taken into account. The authors of [110] noted that the previous studies of Kohn-Luttinger superconductivity were mainly limited to the calculation of only the short-range Hubbard interaction of electrons , in view of the computational difficulties connected with taking into account the first and higher orders of the Fourier transform of the long-range Coulomb repulsion (depending on the wave vector q) in the diagrams. As a result, the strong long-range Coulomb repulsion in the first order of the perturbation theory (the first diagram in Fig. 2) was ignored, and the contribution of the electrons to the effective interaction in the Cooper channel, which was determined only by the last second-order (exchange) diagram in Fig. 2, had an attractive nature and ensured -wave pairing in the 3D case [38, 39] and -wave pairing in the 2D case [40, 67].

In Ref. [110], the long-range Coulomb interaction was chosen in the form of the Fourier transform of the Yukawa potential , which in the 3D case takes the standard form

(31) |

where is the reciprocal Debye screening length. The authors of [110] concluded, based on the results of calculations, that the low and intermediate values of the Hubbard repulsion in the presence of the long-range part of Coulomb interaction (31) do not lead to the Cooper instability both in 3D and 2D Fermi systems in the -wave and -wave channels, irrespective of how small the screening length is. The pairing appearing at large orbital momenta () leads to the almost zero values of the critical temperature at any reasonable value of the Fermi energy. According to the authors of [110], the anomalous pairing caused by strong Coulomb repulsion cannot be measured experimentally in practice, since the corresponding condensation energy (if it exists) is several times lower than the condensation energy caused by the electron-phonon interaction.

The growth of interest in the problem of account for the long-range part of Coulomb correlations in the description of the phase diagram of high-temperature superconductors raised the popularity of the extended Hubbard model that includes the interaction between the electrons located on different sites of the crystal lattice (in the Russian literature, this model is often called the Shubin-Vonsovsky model [111, 112, 113]).

In the historical aspect, the Shubin-Vonsovsky model, which was formulated almost immediately after the creation of quantum mechanics, is a predecessor of some important models in the theory of strongly correlated electronic systems, in particular, the model and the Hubbard model itself. The Shubin-Vonsovsky model was actively used in studies of polar states in solids [114, 115], for describing the metal-insulator transition [116], and also in the study of the influence of the intersite Coulomb repulsion on the effective band structure and superconducting properties of strongly correlated systems [117, 118, 119].

In the Wannier representation, the Hamiltonian of the Shubin-Vonsovsky model can be written as

(32) | |||||

where the last term in the right-hand side corresponds to the energy of the Coulomb interaction of electrons that are located on different sites of the crystalline lattice. The last three terms together in the right-hand side of (32) reflect the fact that the screening radius in the systems in question is equal to several lattice spacings [116]. This determines the efficiency of the Shubin-Vonsovsky model, in which the intersite Coulomb interaction is taken into account within several coordination spheres. In the momentum representation, the Hamiltonian (32) takes the form

(33) | |||||

where the Fourier transform of the Coulomb interaction between the electrons located on the nearest-neighbor sites, , and on the next-nearest sites, , of the square lattice in the 2D case is written as

(34) |

The authors of [120] made a contribution to the discussion in [67, 110] by investigating the conditions of the appearance of the Kohn-Luttinger superconducting pairing in the 3D and 2D Shubin-Vonsovsky models with a Coulomb repulsion of the electrons located on neighboring sites (). As for the interaction, they considered the maximally strong Coulomb repulsion on both the same and neighboring sites: ( is the bandwidth; for the 3D cubic lattice and for the 2D square lattice).

On the cubic lattice in the 3D case, we have the following expressions for the bare interaction of electrons in vacuum in the -wave and -wave channels:

(35) |

In this case, the -matrices in the appropriate channels in the strong-coupling limit are determined as

(36) |

where and for the scattering lengths in the -wave and -wave channels. As a result, the dimensionless gas parameter in the -wave channel takes the form , just as in the Hubbard model, whereas the bare gas parameter in the -wave channel is proportional to , in accordance with the general quantum-mechanical results for slow particles () in vacuum [121].

Thus, even in the maximally repulsive 3D Shubin-Vonsovsky model, which is the most unfavorable for the appearance of effective attraction and superconductivity, the normal state in the strong-coupling regime with low electron density is unstable with respect to the triplet -wave pairing. Notably, the effective interaction of electrons at in the substance takes the form [120]

(37) |

where is the density of states in the 3D Fermi gas. As was mentioned above, the contribution from the -wave harmonic of the polarization operator in substance, , favors attraction, and it cannot be compensated by the contribution from the intersite Coulomb repulsion in the -wave channel, which is proportional to .

Similarly, in the 2D case, in the regime of strong coupling and low electron density, the dimensionless gas parameter in the -wave channel is , just as in the 2D Hubbard model, whereas the dimensionless gas parameter in the -wave channel is , again in agreement with the results for slow particles in vacuum. The effective interaction in the 2D case in substance takes the form [120]

(38) |

where is the density of states of a 2D Fermi gas. Since for , we obtain , as in the case (see Section 2 ).

Thus, the previous results concerning the realization of superconducting -wave pairing in both 3D and 2D repulsive Hubbard models at strong coupling () with low electron density remain valid even when we take the strong Coulomb repulsion of electrons at the nearest sites into account in the framework of the Shubin-Vonsovsky model. As a result, the same expressions for the main exponent (which determines the critical temperature of -wave pairing (6) and (11)), are obtained just as in the absence of the intersite Coulomb repulsion () in both three-dimensional and two-dimensional cases. Account for changes only the preexponential factor [48], which means that the superconducting -wave pairing can be developed in Fermi systems with pure repulsion [120] (in the absence of electron-phonon interaction) even in the presence of the long-range Coulomb repulsion.

The authors of [122] carried out a similar analysis for the extended Hubbard model in the Born weak-coupling approximation and came to the same conclusions as the authors of [120]. Moreover, it was noted in [122] that in the weak-coupling regime , the effect of the long-range Coulomb interactions is also suppressed, and does not impair the conditions for the development of Cooper instability. This is explained by the fact that the long-range interactions in the lattice models usually contribute only to some specific channels of pairing and do not affect the other channels. At the same time, the polarization contributions that are described by the diagrams shown in Fig. 2 have components in all the channels and usually more than one of them favors attraction. In this situation, it turns out that the long-range interactions either do not influence the principal components of the effective interaction which lead to the pairing or suppress the main components but do not affect the secondary ones [see the discussion after expression (49)].

In this connection, a phase diagram was constructed in Ref. [122] based on the extended Hubbard model in the framework of the Kohn-Luttinger mechanism, which clearly reflects the result of the competition of the superconducting phases with different types of the symmetry of order parameter. In the calculations of the effective coupling constant, an expression for the renormalized scattering amplitude in the Cooper channel was used in the form

(39) |

where is the Fourier transform of the intersite Coulomb repulsion of electrons, Eqn (34), and is the Lindhard function (4). Thus, the intersite Coulomb interaction was taken into account only in the first order of the perturbation theory, and the polarization contributions were determined only by the term of the order of . It was shown in [122] that although the long-range interactions have a tendency to suppress the anomalous pairing in some channels, the Kohn-Luttinger superconductivity survives in the entire region of electron concentrations and for all relations of the model parameters.

It was noted in Refs [123, 124] that the effective interaction is characterized by a dependence that is quadratic in the quasimomentum only in the region of . Outside this region, the dependence of on the momentum is determined by periodic functions. As a result, the behavior of is modified significantly in comparison with the behavior of the momentum dependence of the Fourier transform of the Yukawa potential. These factors substantially affect the conditions of the realization of Cooper instability at large electron densities, when the Fermi surfaces do not have the spherical symmetry. Therefore, it can be expected that the conditions for the realization of superconducting pairing in the framework of the Kohn-Luttinger mechanism are determined not only by the dynamic effects caused by the Coulomb interactions but also by the effects related to the Brillouin zone.

The authors of [123, 124] discussed the influence of the Coulomb interaction of electrons located in the first and second coordination spheres on the development of Cooper instability in the Born weak-coupling approximation, . Accordingly, they used the effective interaction , which is determined in the graphic form by the sum of five diagrams (see Fig. 2) and for the Shubin-Vonsovsky model has the following analytic form

(40) | |||

(41) |

The presence of the renormalized expression for the effective interaction allows us to analyze the conditions for the realization of the Cooper instability. Taking into account the fact that the leading contribution to the total scattering amplitude of two electrons with opposite momenta and spin projections (the total amplitude in the Cooper channel) is determined by electron scattering near the Fermi surface, the dependence of on the Matsubara frequency can be neglected in the Bethe-Salpeter integral equation. As a result, this equation is simplified taking the form

(42) |

where is the standard expression for the kernel of the Cooper loop.