Spin-polarized superconducting phase in semiconducting system with next-nearest-neighbor hopping on the honeycomb lattice

The particles in the honeycomb lattice with on-site $s$-wave pairing exhibit many interesting behaviors, which can be described in the framework of the Hubbard model. Among others, at the half-filling, some critical value $|U_c|$ of pairing interaction $U<0$ exists that, for $U<U_c$, the superconducting phase becomes unstable. Introduction of the nonzero hopping $t'$ between next-nearest-neighbor sites strongly modifies the physical properties of the system. Here, we discuss the behavior of the system for $t' \neq 0$ (at the ground state), where the hopping between next-nearest neighbors leads to change of the order of phase transition between superconducting and normal phases from discontinuous to continuous one in the external magnetic field $h$. We show that this behavior is strongly dependent on $t'$ and associated with the Dirac cones in the non-interacting band structure of the system. For non-zero magnetic field and for some range of $t'$, a spin-polarized superconducting phase occurs in the ground state phase diagram (only at the half-filling and for $h\neq 0$).


Introduction
The honeycomb lattice, containing two atoms in a primitive unit cell, is characterized by several properties. For instance, the electronic structure is formed by two bands, which allows to realize the Dirac cones at the K-points of the Brillouin zone. The most famous and the simplest example of realization of such a lattice in the nature is graphene [1]. Observation of the Dirac physics in that simple lattice of carbon atoms attracted a huge attention not only in a context of fundamental studies, but also in potential applications. From experimental point of view, the graphene-like lattice exhibits several interesting features. For example, the following phenomena are worth to mention: a realization of the edge-dependent electronic edge mode in the nanoribbons geometry [2][3][4], the quantum spin Hall effect (originally formulated for graphene) [5], or experimentally observed the quantum Hall effect [6]. Such unique phenomena open various possibilities of the graphene applications in, e.g., spinotronics [7] or valleytronics [8].
Also a single honeycomb layer exhibits extraordinary properties in a context of the superconducting states. For example, the fermionic particles in the atomic Fermi gas on the honeycomb lattice undergo a crossover from the Bardeen-Cooper-Schrieffer (BCS) state to the Bose-Einstein condensation (BEC) of diatomic molecules, shorter named as the BCS-BEC crossover [24]. Additionally, in the presence of the external magnetic field, the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase can occur [25].
Motivation. One of parameters which strongly affects the electronic band structure of the honeycomb lattice is the hopping integral between next-nearest-neighbor (NNN) sites [26]. In the absence of the hopping between NNN sites, the behavior of fermions on the honeycomb lattice at half-filling is characterized by a critical pairing interaction strength | |, below which the system is semiconducting. Let us underline that, in this case, the semiconducting (sometimes refereed as semimetallic) behavior is related to: (i) vanishing density of states at the Fermi level and (ii) two bands (the conduction and valence bands), which touch each other at some points in momentum space (i.e., the Fermi surface shrinks to the Dirac points). However, for the pairing interaction above | |, the ground states becomes superconducting. In this work, we investigate the phase transitions between semiconducting and superconducting phases as well as an influence of the NNN hopping on them.
The paper is organized as follows. First, the model used in the current study is formulated in Sec. 2. Next, in Sec. 3 we present and discuss obtained numerical results in the ground state. Finally, the main conclusions summarize the work in Sec. 4.

Model and approximation
In this work, we investigate the fermionic particles on the honeycomb lattice in the frame of the Hubbard model: wherê † (̂) is the creation (annihilation) operator of a fermion with spin ∈ {↑, ↓} at -th site, and̂=̂ †î s the particle number operator. Parameters , , ℎ denotes the chemical potential, the on-site Coulomb interaction, and the external magnetic field, respectively. The first term describes the kinetic part (cf. Ref. [26]). Here, we consider that hopping only between nearest neighbors (with ≡ > 0 as energy unit) and between NNN (with ≡ ′ ). In the work, we restricts our investigation to the case of attractive < 0, which is a source of superconductivity in the system. The interaction term in such a case, after the mean-field decoupling, leads to the BCS-like term [27][28][29][30][31]: which describes -wave superconductivity. The following order parameter Δ = ⟨̂↓̂↑⟩ is a superconducting order parameter (here, the spatially homogeneous system is assumed, i.e., Δ ≡ Δ). To find the ground state solutions we minimize the grand canonical potential defined as: with respect to Δ and fixing other model parameters ( , ′ , and ). This procedure allows us to find order parameter Δ, total number of particles , and magnetization (a difference between number of spin-↑ and spin-↓ particles) from the following equations: respectively. More details can be found in Ref. [32].

Results
Let us start our investigation with the non-interacting electronic density of states (DOS), which is presented in Fig. 1. For convenient comparison of the DOSs for different ′ , we present them in such a way, that the half-filling condition corresponds to = 0. The honeycomb lattice without NNN hopping ( ′ = 0) possess the DOS, which is symmetric with respect to the half-filling. For = ± , the Van Hove singularities (VHS) occur, while for ± → 0 the DOS disappear due to the Dirac cones existence. For finite ′ , the DOS lost its symmetric form, and the VHSs are splitted up. The increase of the hopping between next nearest neighbours sites ′ , initially leads to modification of the DOS around the "oryginal" VHS (for small, | ′ | < ∕3). For ′ large enough (i.e., | ′ | > ∕3), modification of the DOS is well visible also around = 0 (e.g. red line in Fig. 1). Transition from zero to finite value of DOS at = 0 occurs directly for | ′ | = ∕3. What is interesting, the (non)zero DOS at = 0 is related to the type of the phase transition from semiconducting to superconducting phase realized in the system. In the next paragraph, we show that this value of ′ has crucial role on physical properties of the honeycomb lattice.
The type of the phase transition from semiconducting to the superconducting one at the ground state, can be clearly seen in the -dependence superconducting order parameter Δ (Fig. 2). Let us start discussions from a case, when the external magnetic field is absent (ℎ = 0). In the case of the half-filing ( = 1, Fig. 2(b)), a continuous phase transition from the superconducting (the BCS phase) to the normal state is related to the almost linear dependence of Δ. This type of behavior was earlier reported for the pure honeycomb lattice [24], and preserved as long as | ′ | < ∕3. In fact, the existence of this phase transition is restricted to the cases where the DOS for = 0 disappears. For such cases, the superconducting phase can be realized for pairing interaction stronger than some finite critical one | |. Indeed, for | ′ | > ∕3 or away from half-filling (Figs. 2(a), and 2(c)), the superconducting phase is stable for any < 0 (which corresponds to the exponential decay of Δ for → 0). For ′ = 0 in the superconducting phase (with Δ ≠ 0), the magnetization is zero ( = 0). Note also that presented curves in Fig. 2 clearly shows that the investigated model exhibits asymmetry with respect to the half-filling (results for < 1 and ′ = 2 − > 1 are different) as well as for asymmetry with respect to change ′ into − ′ . However, the asymmetry ↔ ′ = 2 − is less pronounced for larger ′ (cf. Fig. 2(c)). Now, we discuss the role of the magnetic field ℎ in described system. In the presence of the magnetic field, the gap equation (the left expression in Eq. (4)) can have two nonequivalent solutions with Δ ≠ 0. One of them, related to the (non-polarized magnetically) BCS-like phase, corresponds to Δ, which is independent of ℎ. The other solution, related to phase called Sarma phase [26,33], corresponds to the spin-polarized phase, for which Δ exhibit strong ℎ dependence. For a typical situation, in the weak coupling region, the BCS phase has usually lower energy then than Sarma phase. However, introduced ′ ≠ 0 can lead to stabilization of the spin-polarized superconducting state (in the calculations, only the Cooper pairs with zero total momentum are considered, i.e., we do not introduce the FFLO phase, where the Cooper pairs have non-zero total momentum, e.g., Refs. [25,32,34]). Indeed, the results presented below clearly show that the Sarma phase can be stable in some range of model parameters.

The presence of the external magnetic field
The resulting phase diagram at the half-filing for ′ = 0 is presented in Fig. 3(a). The BSC-like superconducting phase (SC 0 ) is the only superconducting phase existing in the diagram. It can exist for | |∕ > | |∕ ≈ 2.23. The value of magnetic field destroying superconductivity increases with | |∕ . The transition to the normal phase is a discontinuous one for ℎ ≠ 0, whereas it is continuous for ℎ = 0.
In Fig. 3(b), the phase diagram for ∕ = −2.0 and = 1 is presented. In the center of the diagram, for small ′ ∕ , only the normal phase exists. With further increase of | ′ |∕ , both superconducting phases appear (the BCS phase for low ℎ∕ and the Sarma phase for larger ℎ∕ ). For higher | ′ |∕ the Sarma phase is destroyed and between the BCS phase and the NO phase the regions of (macroscopic) phase separation exist (coexistence of the NO phase and the BCS phase in two domains).
The exemplary curves for dependence of Δ as a function of ℎ at the ground state, are presented in Fig. 4. It is clearly visible that increasing of the magnetic field destabilizes superconducting state (cf. also [35,36]). Additionally, for ′ ≠ 0 and with increasing ℎ, the discontinuous transition between two different superconducting phases occurs. For low ℎ the phase with = 0 (i.e., the BCS phase) is stable with Δ independent of ℎ, whereas above the transition point, the spin-polarized superconducting phase (with ≠ 0, the Sarma phase [26,33]) is stable, cf. Fig. 4. The transition between two superconducting phases occurring in non-zero field is discontinuous. Further increasing of ℎ leads to a continuous transition from spin-polarized superconducting phase to the normal state. One notices that increasing of | ′ | extends the regions of superconducting phases occurrence ( Fig. 4(a)). From Fig. 4(b), it is clearly visible that for ′ < 0 these regions are wider that for ′ of the opposite sign, which is also in agreement with the results from Fig. 3(b) (at least for ∕ = −2; for ∕ = −2.5 differences are smaller, cf. also Fig. 3 from Ref. [26]). To have overall picture of the behavior of studied model as a function of the total concentration , the ℎ∕ vs. phase diagram for ′ ∕ = −0.1 and ∕ = −2.5 is presented in Fig. 5. As mentioned previously, the Sarma phase occurs in a very narrow region only at = 1 for ℎ ≠ 0. In the absence of magnetic field, for these model parameters, the ground state is the BSC-like superconductor. The magnetic field destroys the BCS-like phase, however, the critical field depends on .

Summary
In this work, we investigated the role of the hopping between next-nearest-neighbor sites on the type of phase transition between semiconducting and superconducting phases at close vicinity of the half-filling in the honeycomb lattice (at the ground state). First, we showed that the occurrence of the phase transition from semiconducting (semimetallic) to superconducting states depends on the next nearest-neighbor hopping. For the hopping integral larger than third part of nearest-neighbor hopping, the phase transition between the mentioned phases does not occur (the superconducting phase is stable for any ), whereas for | ′ | < ∕3 some critical value of exists and the transition is a continuous one. Additionally, we investigated the dependence of the superconducting order parameter Δ as a function of the external magnetic field. Independently of the hopping between next-nearest-neighbor sites, we observed the discontinuous phase transition from the BCS superconducting phase (with Δ independent of magnetic field) to the Sarma phase (with Δ dependent on magnetic field). The range of magnetic field, for which the Sarma phase exists, strongly depends on the system parameters. However, the Sarma phase cannot occur for | ′ | > ∕3 and ≠ 1 (away from the half-filling).
The occurrence of the Sarma phase is associated to the semiconducting features of non-interacting band structure for the honeycomb lattice, particularly with the existence of the Dirac cones at the Fermi level for half-filling.