Pairing mechanism for high temperature superconductivity in the cuprates: what can we learn from the two-dimensional t − J model? Huan-Qiang Zhou 1 $^{1}$Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China More than twenty years have passed since high temperature superconductivity in the copper oxides (cuprates) was discovered by J.G. Bednorz and K.A. M¨uller in 1986 [1]. Although intense theoretical and experimental e ff orts have been devoted to the investigation of this fascinating class of materials, the pairing mechanism re- sponsible for unprecedented high transition temperatures T$_{c}$ remains elusive. Theoretically, the di ffi culty lies in the fact that this class of materials, as doped Mott-Hubbard insulators [2], involve strong electronic correla- tions, which renders conventional theoretical approaches unreliable. Recent progress in numerical simulations of strongly correlated electron systems in the context of tensor network representations [3, 4] makes it possible to get access to information encoded in the ground-state wave functions of the two-dimensional t − J model-a minimal model, as widely believed, to understand electronic properties of doped Mott-Hubbard insulators [5– 8]. In this regard, an intriguing question is whether or not the two-dimensional t − J model holds the key to understanding high temperature superconductivity in the cuprates. As it turns out, such a key lies in a supercon- ducting state with mixed spin-singlet d + s − wave and spin-triplet p$_{x}$ ( p$_{y}$ )-wave symmetries in the presence of an anti-ferromagnetic background [9]. Here, the d + s -wave component in the spin-singlet channel breaks U (1) symmetry in the charge sector, whereas both the anti-ferromagnetic order and the spin-triplet p$_{x}$ ( p$_{y}$ )-wave com- ponent breaks S U (2) symmetry in the spin sector. Therefore, four gapless Goldstone modes occur. However, even if we resort to the Kosterlitz-Thouless transition [10], only the d + s -wave superconducting component sur- vives thermal fluctuations. This turns three gapless Goldst one modes, arising from S U (2) symmetry breaking, into two degenerate soft modes, with twice the spin-triplet p$_{x}$ ( p$_{y}$ )-wave superconducting energy gap as their characteristic energy scale: one is a spin-triplet mode observed as a spin resonance mode in inelastic neutron scattering, the other is a spin-singlet mode observed as a A$_{1}$$_{g}$ peak in electronic Raman scattering. The scenario allows us to predict that pairing is of d + s -wave symmetry, with the two degenerate soft modes as the long- sought key ingredients in determining the transition temperature T$_{c}$ , thus o ff ering a possible way to resolve the controversy regarding the elusive mechanism for high temperature superconductivity in the cuprates. PACS numbers: 74.20.-z, 74.20.Mn, 74.20.Rp Imagine if we would have been able to solve a model system describing doped Mott-Hubbard insulators on a two- dimensional square lattice, whose ground-state wave function is a superconducting state with mixed spin-singlet d + s − wave and spin-triplet p$_{x}$ ( p$_{y}$ ) − wave symmetries in the presence of an anti-ferromagnetic background, with the order parameters for the s -wave, d -wave, and p$_{x}$ ( p$_{y}$ )-wave superconducting com- ponents, together with the anti-ferromagnetic order parame- ter, shown in Fig. 1, in a proper doping range. Note that Δ$_{d}$ and Δ$_{s}$ are, respectively, the spin-singlet d -wave and s -wave superconducting energy gaps, whereas Δ$_{p}$ is the spin-triplet p$_{x}$ ( p$_{y}$ )-wave superconducting energy gap and N is the anti- ferromagnetic N´eel order parameter. A few peculiar features of this state are: (i) Both the 90 degree (four-fold) rotation symmetry and the translation symmetry under one-site shifts are spontaneously broken on the square lattice. (ii) Spin- rotation symmetry S U (2) is spontaneously broken, due to the simultaneous occurrence of both the p$_{x}$ ( p$_{y}$ ) − wave super- conducting component and the anti-ferromagnetic order. (iii) U (1) symmetry in the charge sector is spontaneously broken, due to pairing in both spin-singlet and spin-triplet channels. Here, we emphasize that the symmetry mixing of the spin- singlet and spin-triplet channels arises from the spin-rotation symmetry breaking, simply because spin is not a good quan- tum number . (iv) All superconducting components are homo- geneous, in the sense that their superconducting order param- eters are independent of sites on the lattice. Now let us switch on thermal fluctuations. Suppose we restrict ourselves to a strict two dimensional system. Then, even if the Kosterlitz-Thouless transition [10] is invoked, only spin-singlet d + s -wave superconducting component survives thermal fluctuations. However, the non-abelian S U (2) sym- metry is not allowed to be broken at any finite tempera- ture [11, 12]. This immediately implies that the Goldstone modes arising from the spontaneous symmetry breaking of S U (2) in the spin sector have to be turned into degenerate soft modes, with twice the spin-triplet p$_{x}$ ( p$_{y}$ )-wave supercon- ducting energy gap as their characteristic energy scale: one is a spin-triplet mode associated with the anti-ferromagnetic or- der, with the momentum transfer ( π, π ), and the other is a spin- singlet mode associated with the spin-triplet p$_{x}$ ( p$_{y}$ )-wave su- perconducting component, with the momentum transfer (0 , 0). On the other hand, there is nothing to prevent from the break- ing of the discrete four-fold rotation symmetry on the square lattice. Actually, this broken symmetry not only manifests it- self in the admixture of a small s -wave component to the dom- inant d -wave superconducting state (see Fig. 1, left panel), but also protects the spin-singlet soft mode that is unidirectional as it arises from the p$_{x}$ ( p$_{y}$ )-wave superconducting component. Our argument leads to a scenario that, at any finite tem- perature, the pairing is of d + s -wave symmetry, with two degenerate soft modes acting as the key ingredients in deter- arXiv:1001.3358v1[cond-mat.supr-con]19 Jan 2010