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arXiv:0708.3888 · Physical Review B

Superconductivity in zigzag CuO chains

Erez Berg, T. H. Geballe, Steven Allan Kivelson

Published Dec 12, 2007• 14 citations

Abstract

Superconductivity has recently been discovered in ${\mathrm{Pr}}_{2}{\mathrm{Ba}}_{4}{\mathrm{Cu}}_{7}{\mathrm{O}}_{15\ensuremath{-}\ensuremath{\delta}}$ with a maximum ${T}_{c}$ of about $15\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. Since the CuO planes in this material are believed to be insulating, it has been proposed that the superconductivity occurs in the double (or zigzag) CuO chain layer. On phenomenological grounds, we propose a theoretical interpretation of the experimental results in terms of a phase for the zigzag chain, labeled by ${\mathrm{C}}_{1}{\mathrm{S}}_{3∕2}$. This phase has a gap in the relative charge mode and a partial gap in the relative spin mode. It has gapless uniform charge and spin excitations and can have a divergent superconducting susceptibility, even for repulsive interactions. A microscopic model for the zigzag CuO chain is proposed, and on the basis of density matrix renormalization group and bosonization studies, we adduce evidence that supports our proposal.

DOI: 10.1103/physrevb.76.214505

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Superconductivity has recently been discovered in

{\mathrm{Pr}}_{2}{\mathrm{Ba}}_{4}{\mathrm{Cu}}_{7}{\mathrm{O}}_{15\ensuremath{-}\ensuremath{\delta}}

with a maximum

{T}_{c}

of about

15\phantom{\rule{0.3em}{0ex}}\mathrm{K}

. Since the CuO planes in this material are believed to be insulating, it has been proposed that the superconductivity occurs in the double (or zigzag) CuO chain layer. On phenomenological grounds, we propose a theoretical interpretation of the experimental results in terms of a phase for the zigzag chain, labeled by

{\mathrm{C}}_{1}{\mathrm{S}}_{3∕2}

. This phase has a gap in the relative charge mode and a partial gap in the relative spin mode. It has gapless uniform charge and spin excitations and can have a divergent superconducting susceptibility, even for repulsive interactions. A microscopic model for the zigzag CuO chain is proposed, and on the basis of density matrix renormalization group and bosonization studies, we adduce evidence that supports our proposal.

Using the numerical diagonalization method, we examine the one-dimensional t_1-t_2-J_1-J_2 model (zigzag chain t-J model) which represents an effective model for metallic CuO double chain in the superconductor Pr_2Ba_4Cu_7O_15-δ. Based on the Tomonaga-Luttinger liquid theory, we calculate the Luttinger-liquid parameter K_ρas a function of electron density n. It is found that superconductivity is realized in parameter region corresponding to the experimental result. We show phase diagram of spin gap on the t_2/|t_1|-n plane by analyzing the expectation value of twist-operator Z_σin the spin sector. The spin gap appears in the region with large t_2/|t_1|, where the phase boundary at half-filling is consistent with that of the known frustrated quantum spin system. The analysis also suggests that the estimated value of the spin gap reaches 100K in the realistic parameter region of Pr_2Ba_4Cu_7O_15-delta.

Using the numerical diagonalization method, we examine the one-dimensional t1–t2–J1–J2 model (zigzag-chain t–J model) which is an effective model for metallic CuO double chains in the superconductor Pr2Ba4Cu7O15-δ. Based on the Tomonaga–Luttinger liquid theory, we calculate the Luttinger liquid parameter Kρ as a function of the electron density n. It is found that superconductivity is realized in the parameter region, which is in accordance with experimental results. We show the phase diagram of a spin gap in the t2/|t1|–n plane by analyzing the expectation value of the twist operator Zσ in the spin sector. The spin gap appears in the region with a large t2/|t1|, where the phase boundary at half filling is consistent with that of the known frustrated quantum spin system. The analysis also suggests that the estimated value of the spin gap reaches ∼100 K in the realistic parameter region of Pr2Ba4Cu7O15-δ.

The density-matrix renormalization-group method is used to study the electronic states of a two-chain Hubbard model for CuO double chains of

{\mathrm{PrBa}}_{2}{\mathrm{Cu}}_{4}{\mathrm{O}}_{8}.

We show that the model at quarter filling has charge-ordered phases with stripe-type and in-line--type patterns in the parameter space, and in between, there appears a wide region of vanishing charge gap; the latter phase is characteristic of either a Tomonaga-Luttinger liquid or a metallic state with a spin gap. We argue that the low-energy electronic state of the CuO double chains of

{\mathrm{PrBa}}_{2}{\mathrm{Cu}}_{4}{\mathrm{O}}_{8}

should be in the metallic state with a possibly small spin gap.

We have studied the one-dimensional Kondo chain at half filling using a density matrix formulation of the numerical renormalization group. The charge gap is larger than the spin gap for all antiferromagnetic values of the exhange coupling J. A new type of excitation, a neutral spin singlet, consists of a particle and a hole which are repulsive for J\ensuremath{\gtrsim}5t and attractive for J\ensuremath{\lesssim}5t. As J\ensuremath{\rightarrow}0, RKKY interactions become more important and the staggered susceptibility \ensuremath{\chi}(q=2

{\mathit{k}}_{\mathit{F}}

) diverges. We have also studied \ensuremath{\chi}(q) and the dispersion of the low-lying spin excitations as a function of wave vector q.

A two-band Hubbard model describing a CuO chain is studied using quantum Monte Carlo simulation. The Luttinger-liquid exponent K ρ is extracted as a function of the nearest-neighbor repulsion V , at hole fillings ⟨ n ⟩ between 1 (half-filled lower band) and 2. Evidence is presented for the existence of a regime with dominant singlet superconducting fluctuations and a gap in the spin excitation spectrum at ⟨ n ⟩ = 1.75, in the neighborhood of the V for which the system phase separates. Closer to half-filling (⟨ n ⟩≲1.6), spin-density-wave fluctuations always dominate.