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Physical review. B, Condensed matter

Pairing fluctuations in a one-dimensional copper oxide model

Ellen B. Stechel, Asle Sudbø, Thierry Giamarchi, C. M. Varma

Published Jan 1, 1995• 17 citations
Publisher

Abstract

The exact ground-state energy ${\mathit{E}}_{0}$(\ensuremath{\Phi}) is calculated for a one-dimensional copper-oxide model defined on a ring of finite length L, threaded by an external flux \ensuremath{\Phi}. It includes on-site and nearest-neighbor intersite Coulomb repulsion V, as well as a nonzero bare ``oxide'' gap ${\mathrm{\ensuremath{\Delta}}}_{0}$ between the orbitals in the unit cell. The correlation exponent ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$ of the corresponding low-energy Hamiltonian shows that singlet (extended s-wave) superconductivity has the most rapidly diverging susceptibility as T\ensuremath{\rightarrow}0 (${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$>1). This superconducting phase is located close to a phase-separation instability of the model, indicated by ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$\ensuremath{\rightarrow}\ensuremath{\infty}. Longer range of the Coulomb interaction tends to frustrate phase separation, enhances the range of V for which ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$>1, and slightly broadens the regime of V where 1${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$\ensuremath{\infty}.

DOI
10.1103/physrevb.51.553

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The exact ground-state energy E0{\mathit{E}}_{0}E0​(\ensuremath{\Phi}) is calculated for a one-dimensional copper-oxide model defined on a ring of finite length L, threaded by an external flux \ensuremath{\Phi}. It includes on-site and nearest-neighbor intersite Coulomb repulsion V, as well as a nonzero bare ``oxide'' gap \ensuremathΔ0{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremathΔ0​ between the orbitals in the unit cell. The correlation exponent K\ensuremathρ{\mathit{K}}_{\mathrm{\ensuremath{\rho}}}K\ensuremathρ​ of the corresponding low-energy Hamiltonian shows that singlet (extended s-wave) superconductivity has the most rapidly diverging susceptibility as T\ensuremath{\rightarrow}0 (K\ensuremathρ{\mathit{K}}_{\mathrm{\ensuremath{\rho}}}K\ensuremathρ​>1). This superconducting phase is located close to a phase-separation instability of the model, indicated by K\ensuremathρ{\mathit{K}}_{\mathrm{\ensuremath{\rho}}}K\ensuremathρ​\ensuremath{\rightarrow}\ensuremath{\infty}. Longer range of the Coulomb interaction tends to frustrate phase separation, enhances the range of V for which K\ensuremathρ{\mathit{K}}_{\mathrm{\ensuremath{\rho}}}K\ensuremathρ​>1, and slightly broadens the regime of V where 1K\ensuremathρ{\mathit{K}}_{\mathrm{\ensuremath{\rho}}}K\ensuremathρ​\ensuremath{\infty}.
We obtain the ground-state energy E0{\mathit{E}}_{0}E0​(\ensuremath{\Phi}) exactly as a function of flux \ensuremath{\Phi} in a one-dimensional ``copper-oxide model'' on rings of finite circumference L, including on-site interactions and a nearest-neighbor interaction V. For V of the order of the charge-transfer gap or larger, the model extrapolated to large L exhibits flux quantization with charge 2e, and a slow algebraic decay of the singlet superconducting correlation function on oxygen sites. The extrapolated superfluid stiffness appears, however, finite only for not too large V. These results suggest a superconductive state at V of order the charge-transfer gap of the model, but a paired and phase-separated state at larger V.
A calculation of the ground state energy E0(Φ) as a function of flux Φ in a one-dimensional `copper-oxide model' on rings of finite circumference L, including on-site interactions, and a nearest neighbor interaction V, is presented. For V of the order of the charge-transfer gap or larger, the model extrapolated to large L exhibits flux-quantization with charge 2e. The extrapolated superfluid stiffness appears, however, finite only for not too large V. These results suggest a superconductive state at V of order the charge-transfer gap of the model, but a paired and phase-separated state at larger V.
The phase diagram of the copper-oxide chain as a function of density and the nearest-neighbour Coulomb interaction, V, is determined. Phase separation takes place above a critical value of VVV when the Cu2+ to Cu+ valence fluctuations dominate. In the proximity of the phase separation boundary the superconducting correlations are the most divergent. We identify the parameter regions where the Luttinger Liquid theory applies and calculate the contours of the charge-charge correlation exponent K_ρ. We show that anomalous flux quantization occurs as K_ρ diverges. At n = 1, for V > t, a gap opens in the spectrum and the ground state is a charge-density wave.
The phase diagram of the copper-oxide chain as a function of density and the nearest-neighbour Coulomb interaction, V , is determined. Phase separation takes place above a critical value of V when the Cu 2+ → Cu + valence fluctuations dominate. In the proximity of the phase separation boundary the superconducting correlations are the most divergent. We identify the parameter regions where the Luttinger-liquid theory applies and calculate the contours of the charge-charge correlation exponent K ρ . We show that anomalous flux quantization occurs as K ρ diverges. At n = 1, for V > t , a gap opens in the spectrum and the ground state is a charge density wave.
The phase diagram of the copper-oxide chain as a function of density and the nearest-neighbour Coulomb interaction, V, is determined. Phase separation takes place above a critical value of V when the Cu 2+ → Cu + valence fluctuations dominate. In the proximity of the phase separation boundary the superconducting correlations are the most divergent. We identify the parameter regions where the Luttinger Liquid theory applies and calculate the contours of the charge-charge correlation exponent Kρ. We show that anomalous flux quantization occurs as Kρ diverges. At n = 1, for V> t, a gap opens in the spectrum and the ground state is a charge-density wave.
The leading superconducting instabilities of the two-dimensional extended repulsive one-band Hubbard model within spin-fluctuation pairing theory depend sensitively on electron density, band, and interaction parameters. We map out the phase diagrams within a random-phase-approximation spin- and charge-fluctuation approach, and find that while B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) and B2g{B}_{2g}B2g​ (dxy{d}_{xy}dxy​) pairing dominates in the absence of repulsive longer-range Coulomb interactions VNN{V}_{\mathrm{NN}}VNN​, the latter induces pairing in other symmetry channels, including, e.g., A2g{A}_{2g}A2g​ (ggg-wave), nodal A1g{A}_{1g}A1g​ (extended sss-wave), or nodal Eu{E}_{u}Eu​ (ppp-wave) spin-triplet superconductivity. At the lowest temperatures, transition boundaries in the phase diagrams between symmetry-distinct spin-singlet orders generate complex time-reversal symmetry broken superpositions. By contrast, we find that boundaries between singlet and triplet regions are characterized by first-order transitions. Finally, motivated by recent photoemission experiments, we have determined the influence of an additional explicitly attractive nearest-neighbor interaction, VNN<0{V}_{\mathrm{NN}}<0VNN​<0, on the superconducting gap structure. Depending on the electronic filling, such an attraction boosts Eu{E}_{u}Eu​ (ppp-wave) spin-triplet or B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) spin-singlet ordering.