# Pairing mechanism for high temperature superconductivity in the cuprates: what can we learn from the two-dimensional t − J model? 

$^{1}$Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China 

Huan-Qiang Zhou 1 

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. 

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- 

PACS numbers: 74.20.-z, 74.20.Mn, 74.20.Rp 

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- 

