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Physical review. B, Condensed matter

Superconductivity in ladders and coupled planes

Elbio R. Dagotto, José A. Riera, Douglas J. Scalapino

Published Mar 1, 1992• 758 citations
Publisher

Abstract

We discuss an electronic model consisting of two chains or planes, each described by a t-J model, coupled by t'-J' interactions between them. For J'\ensuremath{\sim}J or larger we show the presence of a spin-gap and hole-pair formation upon doping. The model exhibits superconducting pairing correlations away from half filling. We support our claims by numerical studies of the spin gap, the binding energy of holes, and the pairing correlations on finite clusters. The hole-pairing operator is a spin singlet with one member of the pair in each chain or plane. Our model belongs to the universality class of the U0 Hubbard model with \ensuremath{\Vert}U\ensuremath{\Vert}\ensuremath{\gg}t. We argue that this model may be physically realized by doping the orthorhombic compound (VO${)}_{2}$${\mathrm{P}}_{2}$${\mathrm{O}}_{7}$.

DOI
10.1103/physrevb.45.5744

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We discuss an electronic model consisting of two chains or planes, each described by a t-J model, coupled by t'-J' interactions between them. For J'\ensuremath{\sim}J or larger we show the presence of a spin-gap and hole-pair formation upon doping. The model exhibits superconducting pairing correlations away from half filling. We support our claims by numerical studies of the spin gap, the binding energy of holes, and the pairing correlations on finite clusters. The hole-pairing operator is a spin singlet with one member of the pair in each chain or plane. Our model belongs to the universality class of the U0 Hubbard model with \ensuremath{\Vert}U\ensuremath{\Vert}\ensuremath{\gg}t. We argue that this model may be physically realized by doping the orthorhombic compound (VO{)}_{2}$${\mathrm{P}}_{2}$${\mathrm{O}}_{7}.
The recent discovery of superconductivity under high pressure in the ladder compound BaFe2S3{\mathrm{BaFe}}_{2}{\mathrm{S}}_{3}BaFe2​S3​ has opened a new field of research in iron-based superconductors with focus on quasi-one-dimensional geometries. In this publication, using the density matrix renormalization group technique, we study a two-orbital Hubbard model defined in one-dimensional chains. Our main result is the presence of hole binding tendencies at intermediate Hubbard UUU repulsion and robust Hund coupling JH/U=0.25{J}_{H}/U=0.25JH​/U=0.25. Binding does not occur either in weak coupling or at very strong coupling. The pair-pair correlations that are dominant near half-filling, or of similar strength as the charge and spin correlation channels, involve hole-pair operators that are spin singlets, use nearest-neighbor sites, and employ different orbitals for each hole. The Hund coupling strength, presence of robust magnetic moments, and antiferromagnetic correlations among them are important for the binding tendencies found here.
We study the phase diagram of a one-dimensional Hubbard model where, in addition to the standard nearest-neighbor hopping t, we also include a next-to-nearest-neighbor hopping ${\mathit{t}}^{\ensuremath{'}}$. For strong enough on-site repulsion, this model has a transition at half filling from a magnetic insulator with gapless spin excitations at small ${\mathit{t}}^{\ensuremath{'}}$/t to a dimerized insulator with a spin gap at larger ${\mathit{t}}^{\ensuremath{'}}$/t. We show that upon doping this model exhibits quite interesting features, which include the presence of a metallic phase with a spin gap and dominant superconducting fluctuations, in spite of the repulsive interaction. More interestingly, we find that this superconducting phase can be reached upon hole doping the magnetic insulator. The connections between this model and the two chain models, recently object of intensive investigations, are also discussed. \textcopyright{} 1996 The American Physical Society.
The low-energy charge excitations of a doped antiferromagnetic ladder are modeled by a system of interacting spinless fermions that live on the same ladder. A relatively large spin gap is assumed to ``freeze out'' all spin fluctuations. We find that the formation of rung hole pairs coincides with the opening of a single-particle gap for charge excitations along chains and with the absence of coherent tunneling in between chains. We also find that such hole pairs condense into either a crystalline or superconducting state as a function of the binding energy.
Abstract We introduce a variational state for one-dimensional two-orbital Hubbard models that intuitively explains the recent computational discovery of pairing in these systems when hole doped. Our ansatz is an optimized linear superposition of Affleck–Kennedy–Lieb–Tasaki valence-bond states, rendering the combination a valence-bond liquid dubbed orbital resonant valence bond. We show that the undoped (one-electron/orbital) quantum state of two sites coupled into a global spin singlet is exactly written employing only spin-1/2 singlets linking orbitals at nearest-neighbor sites. Generalizing to longer chains defines our variational state visualized geometrically expressing our chain as a two-leg ladder, with one orbital per leg. As in Anderson’s resonating valence-bond state, our undoped variational state contains preformed singlet pairs that via doping become mobile, leading to superconductivity. Doped real materials with one-dimensional substructures, two near-degenerate orbitals, and intermediate Hubbard U/W strengths—W the carrier’s bandwidth—could realize spin-singlet pairing if on-site anisotropies are small. If these anisotropies are robust, spin-triplet pairing emerges.
We study the effect of doping away from half-filling in weakly (but finitely) interacting N-leg Hubbard ladders using renormalization-group and bosonization techniques. For a small on-site repulsion U, the N-leg Hubbard ladders are equivalent to an N-band model, where at half-filling the Fermi velocities are ${v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.$ We then obtain a hierarchy of energy scales, where the band pairs $(j,N+1\ensuremath{-}j)$ are successively frozen out. The low-energy Hamiltonian is then the sum of N/2N/2N/2 [or $(N\ensuremath{-}1)/2$ for N odd] two-leg ladder Hamiltonians without gapless excitations (plus a single chain for N odd with one gapless spin mode)---similar to the N-leg Heisenberg spin ladders. The energy scales lead to a hierarchy of gaps. Upon doping away from half-filling, the holes first enter the band(s) with the smallest gap: For odd N, the holes enter first the nonbonding band (N+1)/2(N+1)/2(N+1)/2 and the phase is a Luttinger liquid, while for even N, the holes enter first the band pair (N/2,N/2+1)(N/2,N/2+1)(N/2,N/2+1) and the phase is a Luther-Emery liquid, similar to numerical treatments of the $t\ensuremath{-}J$ model, i.e., at and close to half-filling, the phases of the Hubbard ladders for small and large U are the same. For increasing doping, hole pairs subsequently enter at critical dopings the other band pairs $(j,N+1\ensuremath{-}j)$ (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector $(\ensuremath{\pi}/2,\ensuremath{\pi}/2).$ Explicit calculations are given for the cases N=3N=3N=3 and 4.
Short-range antiferromagnetic correlations are known to open a spin gap in the repulsive Hubbard model on ladders with \boldsymbol{M} 𝐌 legs, when \boldsymbol{M} 𝐌 is even. We show that the spin gap originates from the formation of correlated pairs of electrons with opposite spin, captured by the hidden ordering of a spin-parity operator. Since both spin gap and parity vanish in the two-dimensional limit, we introduce the fractional generalization of spin parity and prove that it remains finite in the thermodynamic limit. Our results are based upon variational wave functions and Monte Carlo calculations: performing a finite size-scaling analysis with growing \boldsymbol{M} 𝐌 , we show that the doping region where the parity is finite coincides with the range in which superconductivity is observed in two spatial dimensions. Our observations support the idea that superconductivity emerges out of spin gapped phases on ladders, driven by a spin-pairing mechanism, in which the ordering is conveniently captured by the finiteness of the fractional spin-parity operator.