Superconducting order parameter symmetry for the extended Hubbard model below T c Joel Hutchinson 1, ∗ and Frank Marsiglio 1, 2, † 1 Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada 2 Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2E1, Canada (Dated: October 26, 2021) The extended Hubbard model is known to host s-wave, d-wave and p-wave superconducting phases depending on the values of the on-site and nearest-neighbour interactions. By examining the free energy functional of the gap in this model, we find that these symmetries are often dependent on temperature. The critical points of this functional are highly constrained by symmetry and allow us to formulate stringent conditions on the temperature profile of the gap function, applicable to other models as well. We discuss the finite temperature phase diagram of the extended Hubbard model, and point out the existence of first and second order symmetry transitions below Tc. Understanding the nature of these transitions may be important for assessing the symmetry of some unconventional superconductors such as UPt3. PACS numbers: Introduction. – As more and more unconventional su- perconductors are discovered, the question of the symme- try of the superconducting order parameter has moved to the forefront as one of the most immediate and important questions to answer about any new material, especially given its close connection to the (often unknown) pairing mechanism. Experimentally, it is a difficult question to answer, as the arduous history of the cuprates provides testament for [1]. The resolution of this question has been aided, in part, by phase-sensitive tunnelling mea- surements, which have been particularly useful in uncov- ering the gap symmetry of the heavy-fermion compound UPt 3 [2]. Like 3 He, UPt 3 has an A and a B phase over different temperature ranges differentiated by different symmetries of the order parameter [3]. A variety of mod- els have been proposed to describe the nature of this gap and its symmetry transition [4]. It is possible that the separation of these phases is due to magnetic moments lifting the degeneracy of the hexagonal lattice, but this is not the only explanation. As we point out in this paper, many exotic supercon- ducting symmetries can emerge from a normal state that retains the point group symmetry of the lattice. Irrespec- tive of its origin, the lesson to take from UPt 3 is that in general, the superconducting order parameter does not necessarily retain a fixed symmetry below T c . Similarly, recent observations in LaAlO 3 /SrTiO 3 suggest that a second component of the gap function develops below T c [5]. One might consider these to be unusual circum- stances. Indeed, LaAlO 3 /SrTiO 3 has strong Rashba cou- pling; UPt 3 has time-reversal symmetry breaking, signif- icant spin-orbit coupling and a complex order parameter in the B phase. However, the purpose of our paper is point out that such exotic conditions are not a require- ment for a superconductor to have a rich phase diagram * electronic address: jhutchin@ualberta.ca † electronic address: fm3@ualberta.ca below T c . In fact we illustrate that symmetry transi- tions occur as a function of temperature in one of the simplest and most studied models for superconductivity, the extended Hubbard model. It is surprising that this possibility has escaped notice until now. With this expanding collection of unconventional su- perconductors, one might ask what symmetries can ex- ist in a generic superconductor. The answer is well es- tablished within Landau-Ginzburg theory [6]. The gap function is segmented into pieces that transform under irreducible representations of the normal state symme- try group. Any of these individual pieces could form the superconducting state at T c , but cannot be mixed at this temperature. However, at lower temperatures, when the magnitude of the order parameter is no longer small, higher order terms in the Landau free energy be- come important, and mixing can occur. As we will see, the phases below T c are simply described by bifurcations of critical points of the free energy. Such mixing and bi- furcations have been predicted before in the context of anisotropic tight binding models [7–9]. In this paper, we illustrate these ideas within a case study of the simplest model that has competing symmetry phases: the two- dimensional (2D) extended Hubbard model on a square lattice, relevant for some layered high-temperature su- perconductors. We will see that even in this very ba- sic model, the temperature-dependent superconducting phase diagram is quite rich. Mean Field Solution of the Extended Hubbard Model. –We consider the extended Hubbard model on a square lattice with unit lattice spacing, nearest neighbour hop- ping t, on-site interaction U , nearest-neighbour interac- tion V , and chemical potential µ H = −t  〈i,j〉 σ (c † iσ c jσ + c † jσ c iσ )+ U  i n i↑ n i↓ +V  〈ij〉 σ,σ ′ n iσ n jσ ′ − µ  i,σ n iσ , (1) arXiv:1902.08316v1 [cond-mat.supr-con] 22 Feb 2019