Three studies in high-Tc cuprates

There are clearly three main actors playing a central role in the physics of high-Tc cuprates, which unfolds on the stage of the CuO₂-planes: the 3d electrons of the Cu⁺⁺ ions, and the 2p_x and 2p_y electrons of the O⁻⁻ ions. Let us call ${d}_{\sigma }^{{\dagger}}({\mathcal{R}}_{I})$, ${p}_{x,\sigma }^{{\dagger}}({\mathbf{R}}_{i})$ and ${p}_{y,\sigma }^{{\dagger}}({\mathbf{R}}_{i}+{\mathbf{d}}_{j})$ the creation operators of electrons with spin σ = ↑, ↓ belonging to these orbitals. The electronic density of electrons belonging to orbital a = d, p_x , p_y is denoted by ${n}_{\sigma }^{a}$. It is quite plausible that the essential features of the behavior of these electrons should be properly described by the so-called three bands Hubbard model (3BHM) [6–13], which corresponds to the Hamiltonian

$$\begin{align}\hfill {H}_{3\text{BH}}& =-{t}_{pp}\sum\limits _{\mathbf{R},{\mathbf{d}}_{i}}\sum\limits _{\sigma }{p}_{x}^{{\dagger}}(\mathbf{R},\sigma ){p}_{y}(\mathbf{R}+{\mathbf{d}}_{i},\sigma )+\text{hc}\hfill & \hfill \\ \hfill & \quad -{t}_{pd}\sum\limits _{\mathcal{R}}\sum\limits _{\sigma }{d}^{{\dagger}}(\mathcal{R},\sigma )\left[{p}_{x}\left(\mathcal{R}+\frac{a}{2}\hat{x},\sigma \right)+{p}_{x}\left(\mathcal{R}-\frac{a}{2}\hat{x},\sigma \right)\right.\hfill & \hfill \\ \hfill & \quad \left.+{p}_{y}\left(\mathcal{R}+\frac{a}{2}\hat{y},\sigma \right)+{p}_{y}\left(\mathcal{R}-\frac{a}{2}\hat{y},\sigma \right)\right]+\text{hc}{U}_{p}\sum\limits _{\mathbf{R}}{n}_{{\uparrow}}^{{p}_{x}}{n}_{{\downarrow}}^{{p}_{x}}+{U}_{p}\sum\limits _{\mathbf{R}+\mathbf{d}}{n}_{{\uparrow}}^{{p}_{y}}{n}_{{\downarrow}}^{{p}_{y}}\hfill & \hfill \\ \hfill & \quad +{U}_{d}\sum\limits _{\mathcal{R}}{n}_{{\uparrow}}^{d}{n}_{{\downarrow}}^{d}{U}_{pd}\sum\limits _{\mathcal{R},\mathbf{R}}{n}_{{\uparrow}}^{d}{n}_{{\downarrow}}^{{p}_{x}}+{U}_{pd}\sum\limits _{\mathcal{R},\mathbf{R}+\mathbf{d}}{n}_{{\uparrow}}^{d}{n}_{{\downarrow}}^{{p}_{y}},\hfill \end{align} \tag{ 1 }$$

where $\mathcal{R}$ represents the copper ions lattice positions, R denotes the positions of the oxygen atoms in one sublattice and d_i , i = 1, ..., 4 its four nearest neighbors located in the opposite sublattice. This the first generation model.

As we dope the system by the introduction of holes, these become the main actors on the stage of CuO₂-planes. Then, a model is required for the description of the physics of the doped holes in such planes. A convenient starting point is a version of the 3BHM that is formulated in terms of holes [11]. This, however, is complicated to such an extent that, in order to extract physical information from it, the model was either approached by numerical methods or by considering simplified versions that could, yet, capture the essential physics.

The simplest example of a model describing the doped holes is the t–J model [14], which assumes that the doped holes form singlet states (Zhang–Rice singlets) with the localized copper spins. Its associated RVB solution [15–17], served as the base for one of the first proposed mechanisms for SC in cuprates [18–20]. It seems, however, that the t–J model does not capture the whole complexity of the high-Tc cuprates.

The spin-fermion (SF) model, which is a natural evolution of the 3BHM [21], offers an appealing framework for the description of the doped holes in cuprates.

The SF-model has been derived by integrating out the electron degrees of freedom in the oxygen orbitals [20, 21], thereby yielding a model that contains just the holes in such oxygen orbitals. These will interact with the localized spins of the copper ions in a Kondo-like way.

The SF Hamiltonian comprises the three terms below [1, 22]: the hopping term, the AF Heisenberg term and the Kondo-like term, namely

$$\begin{align}\hfill {H}_{0}& =-{t}_{p}\sum\limits _{\mathbf{R},{\mathbf{d}}_{i}}\sum\limits _{\sigma }{\psi }_{A,\sigma }^{{\dagger}}(\mathbf{R}){\psi }_{B,\sigma }(\mathbf{R}+{\mathbf{d}}_{i})+\text{hc}\hfill \\ \hfill {H}_{\text{AF}}& ={J}_{\text{AF}}\sum\limits _{\langle IJ\rangle }{\mathbf{S}}_{I}\cdot {\mathbf{S}}_{J}\hfill \\ \hfill {H}_{\text{K}}& ={J}_{\text{K}}\sum\limits _{I}{\mathbf{S}}_{I}\cdot \left[\sum\limits _{\mathbf{R}\in I}{\eta }_{A}{\eta }_{C}\ {\mathcal{S}}_{A}+\sum\limits _{\mathbf{R}+\mathbf{d}\in I}{\eta }_{B}{\eta }_{C}^{\prime }\ {\mathcal{S}}_{B}\right],\hfill \end{align} \tag{ 2 }$$

where ${\psi }_{A}^{{\dagger}}(\mathbf{R})$, ${\psi }_{B}^{{\dagger}}(\mathbf{R}+\mathbf{d})$ are the hole creation operators on sites R and R + d, respectively, of the A, B oxygen sub-lattices and S_I are the spin operators of the localized Cu⁺⁺ ions.

In the equation above,

$$\begin{equation}{\mathcal{S}}_{A,B}=\frac{1}{2}{\psi }_{(A,B)\alpha }^{{\dagger}}{\vec{\sigma }}_{\alpha \beta }{\psi }_{(A,B)\beta }\ \ \end{equation} \tag{ 3 }$$

which is the spin operator of holes belonging to the p_x , p_y oxygen orbitals, which are associated, respectively, to the A and B sub-lattices. The sign factors η_A , η_B = ±1 originate in the exchange integrals involving the atomic orbitals and are determined by the sign of the half-portion of the p_x and p_y oxygen orbitals that overlaps with the one-fourth portion of the ${d}_{{x}^{2}-{y}^{2}}$ copper orbitals, whose sign we denote by either η_C = ±1 or ${\eta }_{C}^{\prime }=\pm 1$.

In what follows, we focus on two possible configurations of sign factors ${\eta }_{A},{\eta }_{B},{\eta }_{C},{\eta }_{C}^{\prime }$, which are explicitly displayed in the figures below (red and green): figures 3 and 1. These are quantum states with properties determined by the dynamics of oxygen orbitals. The ground-state, in particular can be either of them or a linear combination thereof. It turns out that the ground-state corresponding to the configuration depicted in figure 3, leads, according to equation (15), to an attractive interaction between holes, which will ultimately generate a SC phase. This is an important difference between our model, which is derived from the 3BHM with the orbital phases convention depicted in figure 3, and the usual version of the 3BHM, which is found in the literature, and uses the phases arrangement depicted in figure 1.

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According to figures 2 and 3 (red configuration), we see that the oxygen orbitals can arrange themselves in such a way that η_A η_C and ${\eta }_{B}{\eta }_{C}^{\prime }$ have opposite signs for all nearest neighbor (A, B) pairs.

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This fact will be crucial for the magnetic interaction existing between the itinerant oxygen holes and the localized copper ions to produce a net attractive interaction between the holes belonging, respectively, to the A and B sub-lattices. At the same time this arrangement of the signs of the oxygen orbitals guarantees the super-exchange coupling that governs the magnetic interaction among the Cu-ion spins to be antiferromagnetic. We can see clearly, therefore, that a close relation exists between the occurrence of a Néel state in the parent compound and the emergence of a superconducting state upon doping. The interaction between the copper and oxygen spins resembles the one found in Kondo systems, in the sense that it couples the spins of itinerant holes with the spins of localized copper ions. Notice, however, the profound difference that exists between the physics of the Kondo effect and that of the high-Tc cuprates: in the first case the itinerant electrons are essentially free, whereas in the present case the holes belong to the p_x , p_y oxygen orbitals.

In the case of the cuprates, the holes, rather than being free, belong to a highly organized network of atomic p-orbitals of the oxygen ions. We shall see that there is a peculiar organization of the signs of such oxygen orbitals, that generates an effective attractive interaction between the holes belonging to the two (A, B) oxygen sub-lattices thereby leading to a superconducting state. Below Tc, such arrangement, which leads to a superconductor state is, energetically, most favorable.

Just for completeness, we mention that the t–J model can be obtained from the SF model by means of a perturbative 1/J_K approach [14].

The SF Hamiltonian, given by (2), does not include the description of the Coulomb repulsion between the holes. Given the relative magnitude of such interaction, when compared with the magnetic ones (for LSCO U_p = 5.5 eV, J_AF = 0.43 eV, J_K = 1.17 eV [1]), however, it seems quite natural to include Hubbard repulsive terms for the holes in the CuO₂ planes. This leads us to a third generation Hamiltonian, namely, the SF-Hubbard model [1, 6], whose Hamiltonian possesses the four terms below:

$$\begin{align}\hfill {H}_{\text{SFH}}& ={H}_{0}+{H}_{\text{AF}}+{H}_{\text{K}}+{H}_{\text{U}}\hfill \\ \hfill {H}_{\text{U}}& ={U}_{p}\sum\limits _{\mathbf{R}}{n}_{{\uparrow}}^{A}{n}_{{\downarrow}}^{A}+{U}_{p}\sum\limits _{\mathbf{R}+\mathbf{d}}{n}_{{\uparrow}}^{B}{n}_{{\downarrow}}^{B},\hfill \end{align} \tag{ 4 }$$

where the first three terms are given by (2) and ${n}_{\sigma }^{A,B}$ are the hole densities in sub-lattices A, B, respectively.

The SF-Hubbard model is the starting point for our proposed theory for the cuprates. In order to derive the effective Hamiltonian we propose for describing the interaction among the doped holes, we are going to perform two operations on the SFH Hamiltonian: (1) we are going to trace over the ferromagnetic fluctuations component of localized spins S_I in H_AF and H_K; (2) we are going to do a perturbation expansion on the hopping term H₀ with respect to the unperturbed Hubbard term H_U. These two operations will lead to our effective Hamiltonian for the cuprates, which is, therefore, a fourth generation Hamiltonian.

We start by tracing out the ferromagnetic fluctuations of the localized spins [22]. This will produce an effective interaction Hamiltonian, H₁[ψ], for the itinerant holes.

We have

$$\begin{equation}Z={\mathrm{Tr}}_{\psi }\,{\mathrm{Tr}}_{{\mathbf{S}}_{I}}\,{\mathit{\text{e}}}^{-\beta \left[{H}_{0}[\psi ]+{H}_{\text{U}}[\psi ]+H[{\mathbf{S}}_{I},\psi ]\right]}.\end{equation} \tag{ 5 }$$

By using an appropriate base of coherent spin states, namely {|N⟩}, with the property

$$\begin{equation}\langle \mathbf{N}\vert {\mathbf{S}}_{I}\vert \mathbf{N}\rangle =s{\mathbf{N}}_{I}\end{equation} \tag{ 6 }$$

(s = 1/2 is the spin quantum number) we can express the trace over such localized spins S_I as a functional integral over the classical vector field N of unit length (see, for instance [1]) from which, we have [22]

$$\begin{equation}{\mathrm{Tr}}_{{\mathbf{S}}_{I}}\,{\mathit{\text{e}}}^{-\beta H[{\mathbf{S}}_{I},\psi ]}=\int D\mathbf{N}\delta (\vert \mathbf{N}{\vert }^{2}-1){\mathit{\text{e}}}^{-\beta H[s{\mathbf{N}}_{I},\psi ]}.\end{equation} \tag{ 7 }$$

We now consider separately the antiferromagnetic and ferromagnetic components of N, denoted, respectively, by n and L. We then write, expanding up to the second order in the lattice parameter a

$$\begin{equation}{\mathbf{N}}_{I}={(-1)}^{I}{\mathbf{n}}_{I}+{a}^{2}{\mathbf{L}}_{I}+O({a}^{4})\end{equation} \tag{ 8 }$$

such that |N|² = |n|² = 1 and L ⋅ n = 0.

We can re-write the trace over S_I as a double functional integral on n and L [1]:

$$\begin{equation}{\mathrm{Tr}}_{{\mathbf{S}}_{I}}=\int D\mathbf{n}D\mathbf{L}\delta (\vert \mathbf{n}{\vert }^{2}-1).\end{equation} \tag{ 9 }$$

Then, inserting (8) in (7) and expanding in a, we obtain

$$\begin{align}\hfill {\mathrm{Tr}}_{{\mathbf{S}}_{I}}\,{\mathit{\text{e}}}^{-\beta H[{\mathbf{S}}_{I},\psi ]}& =\int D\mathbf{n}D\mathbf{L}\delta (\vert \mathbf{n}{\vert }^{2}-1)\mathrm{exp}\left\{-\frac{1}{2}\int {\mathrm{d}}^{2}r{\int }_{0}^{\beta }\mathrm{d}\tau \left[{J}_{\text{AF}}{s}^{2}{\nabla }_{i}\mathbf{n}\cdot {\nabla }_{i}\mathbf{n}\right.+4{J}_{\text{AF}}{s}^{2}{a}^{2}\vert \mathbf{L}{\vert }^{2}\right.\hfill \\ \hfill & \quad \left.+\mathbf{L}\cdot \left[{J}_{\text{K}}\left[{\eta }_{A}{\eta }_{C}\ {\mathcal{S}}_{A}+{\eta }_{B}{\eta }_{C}^{\prime }\ {\mathcal{S}}_{B}\right]-is\mathbf{n}\times \frac{\partial \mathbf{n}}{\partial \tau }\right]\right\}.\hfill \end{align} \tag{ 10 }$$

We are going to integrate out the ferromagnetic fluctuations by performing the quadratic functional integral on L. This will produce three terms: the 2nd term squared, which provides a kinetic term for n [1], the 1st term squared, which produces an effective interaction among the itinerant doped holes (which is the 3rd term in the exponent below) and the crossed term, which vanishes [22, 23]

$$\begin{align}\hfill {\mathrm{Tr}}_{{\mathbf{S}}_{I}}\,{\mathit{\text{e}}}^{-\beta H[{\mathbf{S}}_{I},\psi ]}& =\int D\mathbf{n}\delta (\vert \mathbf{n}{\vert }^{2}-1)\times \mathrm{exp}\left\{-\int {\mathrm{d}}^{2}r{\int }_{0}^{\beta }\mathrm{d}\tau \,\frac{{\rho }_{\text{s}}}{2}\left[{\nabla }_{i}\mathbf{n}\cdot {\nabla }_{i}\mathbf{n}+\frac{1}{{c}^{2}}{\partial }_{\tau }\mathbf{n}\cdot {\partial }_{\tau }\mathbf{n}\right]\right.\hfill \\ \hfill & \quad \left.+\frac{{J}_{\text{K}}^{2}}{8{J}_{\text{AF}}{a}^{2}}\left[\sum\limits _{\mathbf{R}\in I}{\eta }_{A}{\eta }_{C}\ {\mathcal{S}}_{A}+\sum\limits _{\mathbf{R}+\mathbf{d}\in I}{\eta }_{B}{\eta }_{C}^{\prime }\ {\mathcal{S}}_{B}\right]\cdot \left[\sum\limits _{\mathbf{R}\in I}{\eta }_{A}{\eta }_{C}\ {\mathcal{S}}_{A}+\sum\limits _{\mathbf{R}+\mathbf{d}\in I}{\eta }_{B}{\eta }_{C}^{\prime }\ {\mathcal{S}}_{B}\right]\right\},\hfill \end{align} \tag{ 11 }$$

where ${\rho }_{\text{s}}=\frac{{J}_{\text{AF}}}{4}$ is the spin stiffness and c = J_AF a is the spin-waves velocity.

Using the fact that the continuum limit involves the replacement a² ∑_k ↔ ∫d² r, we conclude that

$$\begin{align}\hfill {\mathrm{Tr}}_{{\mathbf{S}}_{I}}\,{e}^{-\beta \left[{H}_{\text{AF}}[{\mathbf{S}}_{I}]+{H}_{\text{K}}[{\mathbf{S}}_{I},\psi ]\right]}& ={Z}_{\text{NL}\sigma \text{M}}{e}^{-\beta {H}_{1}[\psi ]}={Z}_{\text{NL}\sigma \text{M}}\mathrm{exp}\left\{-{\int }_{0}^{\beta }\mathrm{d}\tau \sum\limits _{\mathbf{R},\mathbf{R}+\mathbf{d}}\left[\frac{{J}_{\text{K}}^{2}}{8{J}_{\text{AF}}}{\left[{\eta }_{A}{\eta }_{C}{\mathcal{S}}_{A}+{\eta }_{B}{\eta }_{C}^{\prime }{\mathcal{S}}_{B}\right]}^{2}\right]\right\},\hfill \end{align} \tag{ 12 }$$

where Z_NLσM is the partition function of the nonlinear sigma model (see, for instance [22]).

From the last term in (12) we see that, indeed,

$$\begin{equation}{H}_{1}[\psi ]=\frac{{J}_{\text{K}}^{2}}{8{J}_{\text{AF}}}\sum\limits _{\mathbf{R},\mathbf{R}+\mathbf{d}}{[{\eta }_{A}{\eta }_{C}{\mathcal{S}}_{A}+{\eta }_{B}{\eta }_{C}^{\prime }{\mathcal{S}}_{B}]}^{2}.\end{equation} \tag{ 13 }$$

Inserting the expressions for ${\mathcal{S}}_{A}$ and ${\mathcal{S}}_{B}$ in (13), we obtain, up to a constant,

$$\begin{align}\hfill {H}_{1}[\psi ]& =\frac{{J}_{\text{K}}^{2}}{8{J}_{\text{AF}}}{\eta }_{A}{\eta }_{B}{\eta }_{C}{\eta }_{C}^{\prime }\sum\limits _{\mathbf{R},\mathbf{R}+{\mathbf{d}}_{i}}\left[{\psi }_{A{\uparrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\downarrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i})+{\psi }_{A{\downarrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\uparrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i})\right]\hfill \\ \hfill & \quad \times \left[{\psi }_{B{\downarrow}}(\mathbf{R}+{\mathbf{d}}_{i}){\psi }_{A{\uparrow}}(\mathbf{R})+{\psi }_{B{\uparrow}}(\mathbf{R}+{\mathbf{d}}_{i}){\psi }_{A{\downarrow}}(\mathbf{R})\right].\hfill \end{align} \tag{ 14 }$$

From figure 4 we see that

$$\begin{equation}{\eta }_{A}{\eta }_{C}{\eta }_{B}{\eta }_{C}^{\prime }=-1.\end{equation} \tag{ 15 }$$

For this reason, the above interaction is always attractive between nearest neighbor holes, which belong to different sub-lattices.

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We conclude that a crucial ingredient of the mechanism that leads to an attractive interactions between neighboring holes in cuprates, being, consequently, responsible for the SC in these materials, is the oxygen p-orbitals organization depicted in figures 2 and 3.

By examining figures 5 and 6, where the oxygen p-orbitals configurations corresponding to the two states introduced in section 2.1 was made more explicit, we immediately conclude that the orbital arrangement depicted in red, which is responsible for the hole attractive interaction that leads to the superconducting phase of cuprates, spontaneously breaks the 90° rotation symmetry. This explains the peculiar symmetry of the SC order parameter in cuprates. This study reveals in a clearcut way the interplay between the symmetry of the SC order parameter and the pairing mechanism itself.

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The complete effective Hamiltonian we use for describing the cuprates is obtained by carrying on a second order perturbation theory on the H₀ + H_U terms [1]. Including this result, we obtain the following effective Hamiltonian:

$$\begin{align}\hfill {H}_{\text{eff}}[\psi ]& =-t\sum\limits _{\mathbf{R},{\mathbf{d}}_{i}}{\psi }_{A\sigma }^{{\dagger}}(\mathbf{R}){\psi }_{B\sigma }(\mathbf{R}+{\mathbf{d}}_{i})+\text{hc}-{g}_{S}\sum\limits _{\mathbf{R},{\mathbf{d}}_{i}}\left[{\psi }_{A{\uparrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\downarrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i})+{\psi }_{B{\uparrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i}){\psi }_{A{\downarrow}}^{{\dagger}}(\mathbf{R})\right]\hfill \\ \hfill & \quad \times \left[{\psi }_{B{\downarrow}}(\mathbf{R}+{\mathbf{d}}_{i}){\psi }_{A{\uparrow}}(\mathbf{R})+{\psi }_{A{\downarrow}}(\mathbf{R}){\psi }_{B{\uparrow}}(\mathbf{R}+{\mathbf{d}}_{i})\right]\hfill \\ \hfill & \quad -{g}_{P}\sum\limits _{\mathbf{R},{\mathbf{d}}_{i}}\left[{\psi }_{A{\uparrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\uparrow}}(\mathbf{R}+{\mathbf{d}}_{i})+{\psi }_{A{\downarrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\downarrow}}(\mathbf{R}+{\mathbf{d}}_{i})\right]\hfill \\ \hfill & \quad \times \left[{\psi }_{B{\uparrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i}){\psi }_{A{\uparrow}}(\mathbf{R})+{\psi }_{B{\downarrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i}){\psi }_{A{\downarrow}}(\mathbf{R})\right],\hfill \end{align} \tag{ 16 }$$

where g_S , is the hole-attractive interaction coupling parameter and g_P , the hole-repulsive one, given, respectively, by

$$\begin{equation}{g}_{S}=\frac{{J}_{\text{K}}^{2}}{8{J}_{\text{AF}}}\ \ \ \,\ \ \ {g}_{P}=\frac{2{t}_{p}^{2}}{{U}_{p}}.\end{equation} \tag{ 17 }$$

This is the effective Hamiltonian of our theory for high-Tc cuprates. In the next section, we present it, in the form containing just tri-linear interactions.

We can write our effective Hamiltonian in terms of the Hubbard–Stratonovitch fields Φ and χ, as [1]

$$\begin{align}\hfill {H}_{\text{eff}}& =-{t}_{p}\sum\limits _{\mathbf{R},{\mathbf{d}}_{i}}\sum\limits _{\sigma }{\psi }_{A,\sigma }^{{\dagger}}(\mathbf{R}){\psi }_{B,\sigma }(\mathbf{R}+{\mathbf{d}}_{i})+\text{hc}+\sum\limits _{\mathbf{R},{\mathbf{d}}_{i}}{\Phi}(\mathbf{R},{\mathbf{d}}_{i})\hfill \\ \hfill & \quad \times \left[{\psi }_{A{\uparrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\downarrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i})+{\psi }_{B{\uparrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i}){\psi }_{A{\downarrow}}^{{\dagger}}(\mathbf{R})\right]+\text{hc}\hfill \\ \hfill & \quad +\sum\limits _{\mathbf{R},{\mathbf{d}}_{i}}\chi (\mathbf{R},{\mathbf{d}}_{i})\left[{\psi }_{A{\uparrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\uparrow}}(\mathbf{R}+{\mathbf{d}}_{i})+{\psi }_{A{\downarrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\downarrow}}(\mathbf{R}+{\mathbf{d}}_{i})\right]\hfill \\ \hfill & \quad +\text{hc}+\frac{1}{{g}_{S}}\sum\limits _{\mathbf{R},{\mathbf{d}}_{i}}{{\Phi}}^{{\dagger}}(\mathbf{R},{\mathbf{d}}_{i}){\Phi}(\mathbf{R},{\mathbf{d}}_{i})+\frac{1}{{g}_{P}}\sum\limits _{\mathbf{R},{\mathbf{d}}_{i}\in \mathbf{R}}{\chi }^{{\dagger}}(\mathbf{R},{\mathbf{d}}_{i})\chi (\mathbf{R},{\mathbf{d}}_{i}),\hfill \end{align} \tag{ 18 }$$

In order to describe the doping process, we add to the above Hamiltonian the chemical potential term

$$\begin{equation}-\mu \left[\sum\limits _{\sigma }({\psi }_{A,\sigma }^{{\dagger}}{\psi }_{A,\sigma }+{\psi }_{B,\sigma }^{{\dagger}}{\psi }_{B,\sigma })-d(x)\right],\end{equation} \tag{ 19 }$$

where d(x) is a function of the stoichiometric doping parameter x, to be determined self-consistently.

From this we derive the field equations

$$\begin{equation}{{\Phi}}^{{\dagger}}(\mathbf{R},{\mathbf{d}}_{i})={g}_{S}\left[{\psi }_{A{\uparrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\downarrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i})+{\psi }_{B{\uparrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i}){\psi }_{A{\downarrow}}^{{\dagger}}(\mathbf{R})\right]\end{equation} \tag{ 20 }$$

and

$$\begin{equation}{\chi }^{{\dagger}}(\mathbf{R},{\mathbf{d}}_{i})={g}_{P}\left[{\psi }_{A{\uparrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\uparrow}}(\mathbf{R}+{\mathbf{d}}_{i})+{\psi }_{A{\downarrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\downarrow}}(\mathbf{R}+{\mathbf{d}}_{i})\right].\end{equation} \tag{ 21 }$$

Observe that Φ^† creates two holes with opposite spins, each one in the neighboring sites A, B, located, respectively, at (R, R + d_i ), where

$$\begin{align}\hfill {\mathbf{d}}_{1}& =\frac{1}{2}[\mathbf{X}-\mathbf{Y}]\ ;\qquad {\mathbf{d}}_{2}=\frac{1}{2}[\mathbf{X}+\mathbf{Y}]\hfill \\ \hfill {\mathbf{d}}_{3}& =\frac{1}{2}[-\mathbf{X}+\mathbf{Y}]\ ;\qquad {\mathbf{d}}_{4}=\frac{1}{2}[-\mathbf{X}-\mathbf{Y}],\hfill \end{align} \tag{ 22 }$$

where $\mathbf{X}=a\hat{x}$ and $\mathbf{Y}=a\hat{y}$ are primitive vectors of the oxygen lattices (see figure 6). It is, therefore, a Cooper pair creation operator. χ^†, conversely, creates an electron and a hole with parallel spins, also in such neighboring sites A, B. It is, therefore, an exciton creation operator.

Let us examine the ground-state expectation value of these operators, namely

$$\begin{equation}{\Delta}({\mathbf{d}}_{i})={g}_{S}\left\langle {\psi }_{A{\uparrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\downarrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i})+{\psi }_{B{\uparrow}}^{{\dagger}}(\mathbf{R}+{\mathbf{d}}_{i}){\psi }_{A{\downarrow}}^{{\dagger}}(\mathbf{R})\right\rangle \end{equation} \tag{ 23 }$$

and

$$\begin{equation}M({\mathbf{d}}_{i})={g}_{P}\left\langle {\psi }_{A{\uparrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\uparrow}}(\mathbf{R}+{\mathbf{d}}_{i})+{\psi }_{A{\downarrow}}^{{\dagger}}(\mathbf{R}){\psi }_{B{\downarrow}}(\mathbf{R}+{\mathbf{d}}_{i})\right\rangle .\end{equation} \tag{ 24 }$$

Notice that, because of the invariance under Bravais lattice translations, it follows that Δ(d_i ) and M(d_i ) do not depend on the Bravais lattice sites' position R, rather, they depend only on d_i .

According to figure 6, we see that

$$\begin{equation}{\Delta}({\mathbf{d}}_{1,3})=-{\Delta}({\mathbf{d}}_{2,4})\end{equation} \tag{ 25 }$$

and

$$\begin{equation}M({\mathbf{d}}_{1,3})=-M({\mathbf{d}}_{2,4}).\end{equation} \tag{ 26 }$$

Now, considering that

$$\begin{equation}{\Delta}(\mathbf{k})=\sum\limits _{{\mathbf{d}}_{i}=1,\dots ,4}{\Delta}({\mathbf{d}}_{i})\mathrm{exp}\left\{i\mathbf{k}\cdot {\mathbf{d}}_{i}\right\}\end{equation} \tag{ 27 }$$

and that

$$\begin{equation}M(\mathbf{k})=\sum\limits _{{\mathbf{d}}_{i}=1,\dots ,4}M({\mathbf{d}}_{i})\mathrm{exp}\left\{i\mathbf{k}\cdot {\mathbf{d}}_{i}\right\},\end{equation} \tag{ 28 }$$

we find, using the fact that $\vert {\mathbf{d}}_{i}\vert ={a}^{\prime }=a/\sqrt{2}$,

$$\begin{equation}{\Delta}(\mathbf{k})={\Delta}\left[\mathrm{cos}\,{k}_{+}{a}^{\prime }-\mathrm{cos}\,{k}_{-}{a}^{\prime }\right]\end{equation} \tag{ 29 }$$

and also that

$$\begin{equation}M(\mathbf{k})=M[\mathrm{c}\mathrm{o}\mathrm{s}\;{k}_{+}{a}^{\mathrm{\prime }}-\mathrm{c}\mathrm{o}\mathrm{s}\;{k}_{-}{a}^{\mathrm{\prime }}],\end{equation} \tag{ 30 }$$

where ${k}_{\pm }=\frac{{k}_{x}\pm {k}_{y}}{\sqrt{2}}$.

We see that the SC and PG order parameters both have a d-wave symmetry, namely, change the sign under a 90° rotation, and have nodal lines along the $\pm \hat{x}$ and $\pm \hat{y}$ direction.

Replacing Φ and χ by their ground-state expectation values Δ and M in the Hamiltonian (18), and using this in the grand-canonical ensemble, after functional integration over the fermionic (holes) degrees of freedom, we obtain the grand-canonical potential

$$\begin{equation}{\Omega}({\Delta},M,\mu )=-{k}_{\text{B}}T\;\mathrm{l}\mathrm{n}\;{Z}_{\text{G}}({\Delta},M,\mu ),\end{equation} \tag{ 31 }$$

where Z_G is the grand-partition function.

The stationary equations obtained by minimizing Ω(Δ, M, μ) with respect to each of its variables leads, respectively, to equations (1), (5) and (9) of the supplementary material, which are our starting point for obtaining the phase diagram of cuprates.

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For the sake of simplicity, in the presentation of the model made so far in the present study, we have considered the case of a single-layered cuprate. The general case of cuprate compounds containing N CuO₂ planes intercepting the unit cell has been considered in detail in [1] (see also the supplementary material) and can be easily introduced. The multi-layered character of a cuprate manifests itself through an enhancement of the SC coupling parameter: g_S → Ng_S (N is the number of layers) [1]. Such observation has allowed for an accurate prediction of the optimal SC transition temperatures for the different-N multi-layered compounds of a given family of cuprates (e.g. Bi, Hg, Tl) [1].

Consider a system with eigen-energies given by $E=\hslash \omega \left(\mathbf{k}\right)$. The spectral weight, for such a system, is defined as [22]

$$\begin{equation}N(\omega )=\sum\limits _{{\mathbf{k}}_{n}}\delta \left(\omega -\omega \left({\mathbf{k}}_{n}\right)\right).\end{equation} \tag{ 32 }$$

For a system with a volume V, we have

$$\begin{equation}\frac{N(\omega )}{V}=\int \frac{{\mathrm{d}}^{2}k}{{(2\pi )}^{2}}\delta \left(\omega -\omega \left(\mathbf{k}\right)\right)\end{equation} \tag{ 33 }$$

or, equivalently,

$$\begin{equation}\frac{N(\omega )}{V}={\int }_{0}^{\infty }\frac{\mathrm{d}k}{(2\pi )}k\delta \left(\omega -\omega \left(\mathbf{k}\right)\right)=g(\omega )\equiv \frac{k}{2\pi \frac{\mathrm{d}\omega \left(\mathbf{k}\right)}{\mathrm{d}k}}\end{equation} \tag{ 34 }$$

from where we conclude that the spectral weight per unit volume coincides with the density of states g(ω).

Also,

$$\begin{equation}\frac{N(\omega )}{V}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{d}\lambda }{2\pi }{\text{e}}^{-\text{i}\lambda \omega }f(\lambda ),\end{equation} \tag{ 35 }$$

where

$$\begin{equation}f(\lambda )=\int \frac{{\mathrm{d}}^{2}k}{{(2\pi )}^{2}}{\text{e}}^{\text{i}\lambda \omega \left(\mathbf{k}\right)}.\end{equation} \tag{ 36 }$$

In the SM phase, we have M = 0, and the dispersion relation is given by [1]

$$\begin{align}\hfill \omega (\mathbf{k})=V\left[\mathrm{cos}\,\frac{{k}_{+}a}{\sqrt{2}}+\mathrm{cos}\,\frac{{k}_{-}a}{\sqrt{2}}\right]=2V\,\mathrm{cos}\,\frac{{k}_{x}a}{\sqrt{2}}\,\mathrm{cos}\,\frac{{k}_{y}a}{\sqrt{2}},\end{align} \tag{ 37 }$$

where V is proportional to the kinetic hopping parameter t. Expanding around the Brillouin zone center k = 0, we have

$$\begin{equation}\omega (\mathbf{k})=2V\left[1-\frac{{a}^{2}}{4}\left({k}_{x}^{2}+{k}_{y}^{2}\right)\right].\end{equation} \tag{ 38 }$$

Inserting this in (36), and using the result

$$\begin{equation}{\int }_{-\infty }^{\infty }\frac{\mathrm{d}\lambda }{2\pi }{\text{e}}^{-\text{i}\lambda \left(\omega -{\omega }_{0}\right)}=i\pi -2\pi i\theta (\omega -{\omega }_{0}),\end{equation} \tag{ 39 }$$

where θ(x) is the Heaviside function, we obtain the spectral weight in the SM phase:

$$\begin{equation}N(\omega )/V=\left(\frac{4}{{a}^{2}}\right)[\theta (\omega )-\theta (\omega -{\omega }_{\text{F}})].\end{equation} \tag{ 40 }$$

In the PG phase, we have M ≠ 0 and considering just the two oxygen sub-lattices, we have

$$\begin{align}\hfill \omega (\mathbf{k})=\sqrt{{V}^{2}{\left[\mathrm{cos}\,{k}_{x}a+\mathrm{cos}\,{k}_{y}a\right]}^{2}+{M}^{2}{\left[\mathrm{cos}\,{k}_{x}a-\mathrm{cos}\,{k}_{y}a\right]}^{2}}.\end{align} \tag{ 41 }$$

Now, making an expansion analogous to that we did before, but now expanding around the center of the elliptical Fermi surface pockets, namely, $\mathbf{K}=(\frac{\pi }{2a},\frac{\pi }{2a})$[1] we obtain

$$\begin{equation}\omega (\mathbf{k})\simeq \sqrt{2}\left[\frac{{V}^{2}+{M}^{2}}{\sqrt{{V}^{2}+{M}^{2}}}\right]\frac{{k}^{2}{a}^{2}}{4}+\left[\frac{{V}^{2}-{M}^{2}}{\sqrt{{V}^{2}+{M}^{2}}}\right]{k}^{2}{a}^{2}\,\mathrm{sin}(2\theta ),\end{equation} \tag{ 42 }$$

where $y\equiv {k}^{2}={k}_{x}^{2}+{k}_{y}^{2}$, k_x = k cos θ and k_y = k sin θ.

Considering that the Bessel function J₀(x) is given by

$$\begin{equation}{J}_{0}(x)=\frac{2}{\pi }{\int }_{0}^{\pi }\mathrm{d}\theta \,\mathrm{cos}\left(x\,\mathrm{sin}\,\theta \right),\end{equation} \tag{ 43 }$$

and using (36), we arrive, after performing the θ-angular integration, at

$$\begin{equation}N(\omega )/V=\frac{1}{4\pi {a}^{2}}{\int }_{0}^{{k}_{F}^{2}}\mathrm{d}y{\int }_{-\infty }^{\infty }\frac{\mathrm{d}\lambda }{2\pi }\,\mathrm{exp}\left[-i\lambda \omega +i\lambda \left[\left[\frac{{V}^{2}+{M}^{2}}{\sqrt{{V}^{2}+{M}^{2}}}\right]\frac{y{a}^{2}}{4}\right]\right]{J}_{0}\left(\lambda \left[\frac{{V}^{2}-{M}^{2}}{\sqrt{{V}^{2}+{M}^{2}}}\right]\frac{y{a}^{2}}{4}\right).\end{equation} \tag{ 44 }$$

Now, integrating over λ [24], we obtain

$$\begin{equation}N(\omega )/V=\frac{1}{4\pi {a}^{2}}{\int }_{0}^{{k}_{\text{F}}^{2}}\mathrm{d}y\,\frac{1}{\sqrt{\left[\frac{2{M}^{2}}{\sqrt{{V}^{2}+{M}^{2}}}y-\omega \right]\left[\frac{2{M}^{2}}{\sqrt{{V}^{2}+{M}^{2}}}y+\omega \right]}}.\end{equation} \tag{ 45 }$$

Finally, integrating on y, we arrive at [24]

$$\begin{equation}N(\omega )/V\propto \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left\{\frac{1}{\vert {A}^{2}-{\omega }^{2}\vert }\right\}-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left\{\frac{1}{\vert \omega \vert }\right\},\end{equation} \tag{ 46 }$$

where $A=\frac{2{M}^{2}{k}_{\text{F}}^{2}}{\sqrt{{V}^{2}+{M}^{2}}}$. This function has sharp peaks at ±A, as we can see in figures 8–11. The peaks occur at frequencies which are determined by the pseudogap order parameter M, being always nonzero for M ≠ 0. The case M → 0, occurs for T → T^∗ and is depicted in figure 7.

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The opening of a region of lesser spectral weight in between the two peaks has been observed in the whole PG phase by means of tunneling experiments and is a convincing experimental proof of the correctness of our prediction about the existence of a PG order parameter in the whole PG phase of the cuprates: the PG order parameter is proportional to the peaks separation.

We emphasize the fact that the double peak structure, with a depletion in between, appearing in the spectral weight is a direct consequence of the asymmetry of the PG order parameter under 90° rotations. Should we have a homogeneous order parameter M, this would imply a real gap (not a pseudogap!) with the spectral weight vanishing in between [−M, M]. It is remarkable that the same asymmetry of the SC order parameter under 90° rotations is responsible for the pairing of holes, as we showed in section 2.4, equations (15) and (16).

We may invert equation (1), of the supplementary material and thereby express the chemical potential as a function of the temperature on the T_c(x)-curve, namely

$$\begin{equation}\begin{cases}\mu ({T}_{\text{c}})=-{T}_{\text{c}}\,\mathrm{ln}\left[{2}^{{T}_{\mathrm{max}}/{T}_{\text{c}}}-1\right],\hspace{14.22636pt}x > {x}_{0}\quad \hfill \\ \mu ({T}_{\text{c}})=2\,\mathrm{ln}\,2\left({T}_{\mathrm{max}}-{T}_{\text{c}}\right)+{T}_{c}\left[1-{\mathit{\text{e}}}^{-\mu ({T}_{\text{c}})/{T}_{\text{c}}}\right],\hspace{8.5359pt}x< {x}_{0}\quad \hfill \\ {T}_{\text{c}}(x)=\frac{\mathrm{ln}\,2\,\ {T}_{\mathrm{max}}}{\mathrm{ln}\,2+\frac{\vert \mu ({T}_{\text{c}})\vert }{2{T}_{\text{c}}(x)}-\frac{1}{2}\left(1-{e}^{-\frac{\vert \mu ({T}_{\text{c}})\vert }{{T}_{\text{c}}(x)}}\right)},\hspace{8.5359pt}\text{LSCO},\forall x,\quad \hfill \end{cases}\end{equation} \tag{ 47 }$$

where the last two equations are implicit for the chemical potential μ(T).

In order to determine the chemical potential right over the PG transition line, we invert the equation (5) of the supplementary material, meant for T* and obtain

$$\begin{equation}\tilde{\mu }({T}^{\ast })=-{T}^{\ast }\;\mathrm{l}\mathrm{n}[{e}^{{\Lambda}\tilde{\eta }/2{T}^{\ast }}-1],\end{equation} \tag{ 48 }$$

where $\tilde{\eta }$, which appears in the equation above, is a function of the g_P coupling parameter, appearing in (17), which is defined in the supplementary material.

In figures 12, 13 and 14 we plot the solutions μ(T) along the T_c transition line, both in the underdoped (blue) and overdoped (red) regions, as well as $\tilde{\mu }(T)$ along the T* transition line (green).

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We then compare these results with the corresponding figures (figures 15, 16 and 17) for the phase diagram T × x. Segments with a given color correspond to the corresponding segment in the phase diagram graph.

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Observe that what determines whether the PG transition curve ends inside the SC dome or not are the values of the chemical potential at T = 0 for the SC and PG curves, namely μ(T = 0) and $\tilde{\mu }(T=0)$. In the examples considered here, such values coincide for LSCO and Hg1201 and differ for Bi2212. This implies a back-bending of the T* line for the latter material, as we can see in figure 16 and has been observed in [5]