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arXiv:1803.05224 · Journal of the Physical Society of Japan

Excitonic Order and Superconductivity in the Two-Orbital Hubbard Model: Variational Cluster Approach

Ryo Fujiuchi, K. Sugimoto, Y. Ohta

Published May 15, 2018• 7 citations
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Abstract

Using the variational cluster approach based on the self-energy functional theory, we study the possible occurrence of excitonic order and superconductivity in the two-orbital Hubbard model with intra- and inter-orbital Coulomb interactions. It is known that an antiferromagnetic Mott insulator state appears in the regime of strong intra-orbital interaction, a band insulator state appears in the regime of strong inter-orbital interaction, and an excitonic insulator state appears between them. In addition to these states, we find that the $s^{\pm}$-wave superconducting state appears in the small-correlation regime, and the $d_{x^2 - y^2}$-wave superconducting state appears on the boundary of the antiferromagnetic Mott insulator state. We calculate the single-particle spectral function of the model and compare the band gap formation due to the superconducting and excitonic orders.

DOI
10.7566/jpsj.87.063705

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Using the variational cluster approach based on the self-energy functional theory, we study the possible occurrence of excitonic order and superconductivity in the two-orbital Hubbard model with intra- and inter-orbital Coulomb interactions. It is known that an antiferromagnetic Mott insulator state appears in the regime of strong intra-orbital interaction, a band insulator state appears in the regime of strong inter-orbital interaction, and an excitonic insulator state appears between them. In addition to these states, we find that the s±s^{\pm}s±-wave superconducting state appears in the small-correlation regime, and the dx2−y2d_{x^2 - y^2}dx2−y2​-wave superconducting state appears on the boundary of the antiferromagnetic Mott insulator state. We calculate the single-particle spectral function of the model and compare the band gap formation due to the superconducting and excitonic orders.
The variational cluster approximation is used to study the spontaneous symmetry breaking of the excitonic insulator (EI) state in the two-orbital Hubbard model defined on the two-dimensional square lattice. Evaluating the order parameter of the EI state, staggered magnetization, and number of particles on each orbital, we obtain the ground-state phase diagram of the model in a wide parameter space of the intra and interorbital Coulomb interactions. We also calculate the single-particle and anomalous Green's functions. We show that the normal metallic phase is unstable to the formation of the EI state and the EI phase appears in a wide parameter region between the band-insulator and Mott-insulator phases.
Optical responses in an excitonic insulating (EI) system with strong electron correlation are studied. We adopt the two-orbital Hubbard model with a finite energy difference between the two orbitals where the spin state degree of freedom exists. This model is analyzed by the variational cluster approach. In order to include the local electron correlation effect, the vertex correction is taken into account in the formulation of the optical conductivity spectra. We calculate a finite-temperature phase diagram in which an EI phase appears between a low-spin band insulating state and a high-spin Mott insulating state. Characteristic components of the optical conductivity spectra consisting of a sharp peak and continuum appear in the EI phase. Integrated intensity almost follows the order parameter of the EI state, suggesting that this component is available to identify the EI phases and transitions.
We investigate a minimal two-orbital Hubbard model with intra- and inter-orbital nearest-neighbor hopping ttt and t~\tilde{t}t~, as well as intra- and inter-orbital density-density interactions UUU and U′U'U′ by means of variational ansätze based on Jastrow-Slater wave functions within quantum Monte Carlo techniques. To focus on the electronic mechanisms of superconductivity, we restrict the variational ansätze to uniform nonmagnetic states with an explicit pairing amplitude and compute the pairing correlations as a function of filling and model parameters. At U/t=10U/t=10U/t=10, superconducting correlations are highly enhanced by the presence of inter-orbital terms, U′U'U′ and t~\tilde{t}t~. For t~=0\tilde{t}=0t~=0, a finite value of U′U'U′ effectively screens the intra-orbital repulsion UUU, producing a shift in the superconducting dome. Consequently, inter-orbital repulsion yields a sizable increase in electron pairing compared to the single-orbital baseline. Furthermore, introducing a finite t~\tilde{t}t~ provides an additional boost to superconducting correlations, an effect driven by the simultaneous presence of flat and broad bands in the electronic structure.
A phase diagram of two Mott-Hubbard planes interacting with a short-range Coulomb repulsion is presented. Considering the case of equal amount of doping by holes in one layer as electrons in the other, a holon-doublon inter-layer exciton formation is shown to be a natural consequence of Coulomb attraction. Quasiparticle spectrum is gapped and incoherent below a critical doping δcδ_cδc​ due to the formation of excitons. A spin liquid insulator (SLI) phase is thus realized without the lattice frustration. The critical value δcδ_{c}δc​ sensitively depends on the inter-layer interaction strength. In the tJtJtJ model description of each layer with the d-wave pairing, δcδ_{c}δc​ marks the crossover between SLI and d-wave superconductor. The SLI phase, despite being non-superconducting and charge-gapped, still shows electromagnetic response similar to that of a superfluid due to the exciton transport. Including antiferromagnetic order in the tJtJtJ model introduces magnetically ordered phases at low doping and pushes the spin liquid phase to a larger inter-layer interaction strength and higher doping concentrations.
Cooperation and competition between the antiferromagnetic, ddd-wave superconducting, and Mott-insulating states are explored for the two-dimensional Hubbard model including nearest and next-nearest-neighbor hoppings at zero temperature. Using the variational cluster approach with clusters of different shapes and sizes up to 10 sites, it is found that the doping-driven transition from a phase with microscopic coexistence of antiferromagnetism and superconductivity to a purely superconducting phase is discontinuous for strong interaction and accompanied by phase separation. At half-filling the system is in an antiferromagnetic Mott-insulating state with vanishing charge compressibility. Upon decreasing the interaction strength UUU below a certain critical value of roughly ${U}_{c}\ensuremath{\sim}4$ (in units of the nearest-neighbor hopping), however, the filling-dependent magnetic transition changes its character and becomes continuous. Phase separation or, more carefully, the tendency towards the formation of inhomogeneous states disappears. This critical value is in contrast to previous studies, where a much larger value was obtained. Moreover, we find that the system at half-filling undergoes the Mott transition from an insulator to a state with a finite charge compressibility at essentially the same value. The weakly correlated state at half-filling exhibits superconductivity microscopically admixed to the antiferromagnetic order. This scenario suggests a close relation between phase separation and the Mott-insulator physics.
We study a two-orbital attractive Hubbard model with a repulsive Hund's exchange coupling JJJ as an idealized model for a two-band superconductor. This framework is motivated by a system where a large isotropic electron-phonon coupling drives the on-site Hubbard repulsion UUU negative, while leaving the exchange term unaffected. We focus on the intra-orbital (intra-band) singlet superconducting phase and we solve the model at zero temperature and half-filling using Dynamical Mean-Field Theory. Already at J=0J=0J=0, the two-orbital model features a superconductor-insulator transition as ∣U∣\vert U\vert∣U∣ grows, as opposed to the single-orbital case which remains superconducting for any U<0U < 0U<0. We find that a finite JJJ strengthens the effect of the attractive UUU, both in the normal state and, more significantly, in the superconducting state. However, this pushes the system towards an effectively stronger coupling, hence to a faster transition to the insulating state. Remarkably, the transition from a superconductor to an insulator occurs with a vanishing quasi-particle weight ZZZ and in a scenario that recalls strongly correlated superconductivity close to a Mott transition, even though the present model is dominated by attractive interactions.