Coulomb repulsion, phase stiffnesses, and doping-induced superconductivity from the Mott insulator in the $t\text{−}{t}^{\prime }\text{−}U\text{−}J$ model of high-$T_c$ cuprates

The electronic $t\text{−}{t}^{\prime }\text{−}U\text{−}J$ model for the $\mathrm{Cu}{\mathrm{O}}_2$ plane in high-$T_c$ superconducting cuprates is elaborated. It incorporates both charge transfer $U$ and exchange correlation $J$ with direct $t$ and diagonal $t^{\prime }$ hopping. Making use of the Yang concept of the off diagonal long range order (ODLRO) in the reduced density matrix of the electronic system we study the loss of quantum wave coherence (and thereby superconductivity) due to the Coulomb interaction. To this aim the second-quantized Hamiltonian of the model is translated to the phase representation with the help of the topologically constrained path integral formalism. As a result the electrons emerge as composite particles consisting of of spin-carrying neutral fermions and charged topological bosons in a form of “flux tubes” with the quantum phase variable dual to the local electron density. The charge carrying bose field forms a collective variable representing the correlated “sliding” motion of the entire electronic system. We found that the spin gap $\Delta$ which appears in one-particle spectrum is the result of valence bond pairing of neutral fermions without ODLRO and deduce the solution for $\Delta$ with “$d$-wave” symmetry. In the bosonic sector we derive the effective quantum rotor model along with microscopic phase stiffnesses setting the energy scale for superconductivity and point out the special role played by the diagonal hopping $t^{\prime }$. We explore the relationship of the quantum rotor phase model and the Bose-Hubbard system emphasizing the ring topology of quantum phase variable. Furthermore, we obtain the ground state phase diagrams which facilitate the Mott insulator-superconductor transition scenario. We elucidate the role of the chemical potential (doping) and the huge degeneracy of the effective bosonic system in achieving ODLRO with long-range phase coherence in the presence of large Coulomb repulsion. Finally, we discuss a number of relevant issues pertaining to the the present work which include: theoretical aspects of the hierarchy of the energy scales in cuprates, “spin-charge separation” concepts, “local-pair” and “phase-fluctuation” scenarios as well as experimental findings related to the role of the “$d$-wave” symmetry of the spin gap and diagonal hopping $t^{\prime }$ in cuprates.

Figure 1

(Color online) Chemical potential dependence of the microscopic phase stiffness parameter $\mathcal{J}\left(\Delta \right)$ for $t=0.486\mathrm{eV}$, $t^{\prime }=0.086\mathrm{eV}$, $J=0.15\mathrm{eV}$, and $U=7\mathrm{eV}$. Upper inset: the density plot of the electronic dispersion relation in Eq. (4). Lower inset: Spin gap parameter as a function of the chemical potential $\overline{\mu }$ for several values of the “superexchange” parameter $J$; note the monotonic behavior of $\Delta$ as a function of $\overline{\mu }$.

Figure 2

Chemical potential dependence of the microscopic phase stiffness parameter ${\mathcal{J}}^{\prime }\left(\Delta \right)$ associated with the next-nearest neighbor hopping $t^{\prime }$ for several values of the “superexchange” $J$ as indicated on the plot. The plot was calculated for $t=0.486\mathrm{eV}$, $t^{\prime }=0.086\mathrm{eV}$, and $U=7\mathrm{eV}$. Note that the maximum of ${\mathcal{J}}^{\prime }\left(\Delta \right)$ is insensitive to the value of the parameter $J$.

Figure 3

Plot of the microscopic phase stiffness ${\mathcal{J}}^{\prime }\left(\Delta \right)$ (at $T=0$) associated with the next nearest neighbor hopping $t^{\prime }$ showing the sensitivity of this parameter to the admixture of the “$s$-wave” component for the spin gap: $\Delta \to (1-\xi ){\Delta }_d\left(\mathbf{k}\right)+\xi {\Delta }_s\left(\mathbf{k}\right)$. The limits $\xi =0$ and $\xi =1$ correspond to the pure “$d$-wave” and “$s$-wave” symmetry of the spin gap, respectively. For establishing of the global phase coherence one requires ${\mathcal{J}}^{\prime }\left(\Delta \right)>0$.

Figure 4

The bosonic occupation number $n_b$ as a function of the chemical potential ${\nu }_b=2\mu$ calculated from Eq. (77) for the system of free quantum rotors for $\beta U=\infty ,20,10$ (from the top to the bottom). The parameter $n_b$ for $T=0$ is pinned to the integer values for $n-1∕2<{\mu }_b∕U<n+1∕2$ $(n=1,2,\dots )$. For ${\mu }_b∕U=n+1∕2$ there are two degenerate states with boson numbers differing by one. For electron occupation number $0<n_e<1$ the relevant range of the bosonic density is $0<n_b<1$.

Figure 5

Phase diagram of the effective phase model defined by the action of interacting quantum rotors, Eq. (81), with the bare stiffness ${\mathcal{J}}^{\prime }$ as the input parameter. The picture shows the periodic arranged series of Mott-insulating lobes with the phase coherent superconducting ground states between them. In the large $U$ limit ${\mu }_b∕U\approx 1$ and departure from the electronic density $n_e=1$ by hole doping corresponds to the movement from the center of the MI.1 towards MI.0 lobe as indicated by arrow. For small ${\mathcal{J}}^{\prime }∕U$ values (large $U$!) the superconductivity is only possible in the vicinity of the degeneracy point ${\mu }_b∕U=1∕2$. The inset shows the neighborhood of the degeneracy where the horizontal line indicates the value of ${\mathcal{J}}^{\prime }∕U$ for $t=0.486\mathrm{eV}$, $t^{\prime }=0.086\mathrm{eV}$, and $U=7\mathrm{eV}$.

Figure 6

Ground state phase diagrams in Figs. 5, 8, respectively, calculated in terms of the original electronic model parameter instead of bare phase stiffnesses. Main plot: Mott-insulating and superconducting phases in the $t^{\prime }∕U\text{−}\mathrm{vs}\text{−}{\mu }_b∕U$ in the vicinity of the charge degeneracy point at ${\mu }_b∕U=1∕2$ (see, Fig. 5). Inset: $t∕U\text{−}\mathrm{vs}\text{−}{\mu }_b∕U$ plot for the phase action Eq. (96) for the $2e$ charged bosonic system (cf. Fig. 8). The phase coherence for the latter is possible for ${(t∕U)}_c>0.25$ at ${\mu }_b∕u=1∕2$ which, for a given $t$, set an upper limit on $U$ (of the order of bandwidth) for the occurrence of superconductivity.

Figure 7

The ODLRO order parameter ${\Psi }_e$ at $T=0$ as a function of the chemical potential ${\mu }_b=2\mu$ for $t=0.486\mathrm{eV}$, $t^{\prime }=0.086\mathrm{eV}$ $J=0.15\mathrm{eV}$, and $U=7\mathrm{eV}$. In contrast to the spin gap parameter $\Delta$ the SC order parameter ${\Psi }_e$ shows a nonmonotonic lobelike chemical potential dependence. The maximum of ${\Psi }_e$ occurs at the charge degeneracy point ${\mu }_b∕U=1∕2$ which defines the “optimal doping.” Although the superconducting phase extends over a narrow range of ${\mu }_b$ (on the scale set by $U$) when plotting of ${\Psi }_e$ against electronic occupation number $n_e$ the SC lobe will expand over a finite range of doping.

Figure 8

The phase diagram as in Fig. 5 but for the phase action Eq. (96) of the $2e$ charged bosonic system with the bare stiffness $\mathcal{J}$ as an input parameter. In contrast to Fig. 5 the superconducting ground state does not extend down to $\mathcal{J}∕U=0$, but there is a critical value of $\mathcal{J}∕U$ below which the system remains in the Mott insulating state, even for ${\mu }_b∕U=n+1∕2$ $(n=1,2,\dots )$.