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arXiv:cond-mat/0309572 · Physical Review B

Mechanism of high-temperature superconductivity in a striped Hubbard model

Enrico Arrigoni, Eduardo Fradkin, Steven Allan Kivelson

Published Jun 25, 2004• 102 citations
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Abstract

It is shown, using asymptotically exact methods, that the two-dimensional repulsive Hubbard model with strongly modulated interactions exhibits ``high-temperature superconductivity.'' Specifically, the explicit modulation, which has the same symmetry as period 4 bond-centered stripes, breaks the system into an alternating array of more and less heavily hole-doped, nearly decoupled two-leg ladders. It is shown that this system exhibits a pairing scale determined by the spin gap of the undoped two-leg ladder, and a phase ordering temperature proportional to a low positive power of the interladder coupling.

DOI
10.1103/physrevb.69.214519

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It is shown, using asymptotically exact methods, that the two-dimensional repulsive Hubbard model with strongly modulated interactions exhibits ``high-temperature superconductivity.'' Specifically, the explicit modulation, which has the same symmetry as period 4 bond-centered stripes, breaks the system into an alternating array of more and less heavily hole-doped, nearly decoupled two-leg ladders. It is shown that this system exhibits a pairing scale determined by the spin gap of the undoped two-leg ladder, and a phase ordering temperature proportional to a low positive power of the interladder coupling.
Unidirectional ("stripe") charge density wave order has now been established as a ubiquitous feature in the phase diagram of the cuprate high-temperature superconductors, where it generally competes with superconductivity. Nonetheless, on theoretical grounds it has been conjectured that stripe order (or other forms of "optimal" inhomogeneity) may play an essential positive role in the mechanism of high-temperature superconductivity. Here, we report density matrix renormalization group studies of the Hubbard model on long four- and six-leg cylinders, where the hopping matrix elements transverse to the long direction are periodically modulated-mimicking the effect of putative period 2 stripe order. We find that even modest amplitude modulations can enhance the long-distance superconducting correlations by many orders of magnitude and drive the system into a phase with a substantial spin gap and superconducting quasi-long-range order with a Luttinger exponent, [Formula: see text].
We study signatures of superconductivity in a two-leg ``checkerboard'' Hubbard ladder model, defined as a one-dimensional (period 2) array of square plaquettes with an intraplaquette hopping ttt and interplaquette hopping ${t}^{\ensuremath{'}}$, using the density-matrix renormalization-group method. The highest pairing scale (characterized by the spin gap or the pair binding energy, extrapolated to the thermodynamic limit) is found for doping levels close to half-filling, $U\ensuremath{\approx}6t$ and ${t}^{\ensuremath{'}}/t\ensuremath{\approx}0.6$. Other forms of modulated hopping parameters, with periods of either one or three lattice constants, are also found to enhance pairing relative to the uniform two-leg ladder, although to a lesser degree. A calculation of the phase stiffness of the ladder reveals that in the regime with the strongest pairing, the energy scale associated with phase ordering is comparable to the pairing scale.
Starting from an array of four-leg Hubbard ladders weakly doped away from half-filling and weakly coupled by interladder tunneling, we derive an effective low-energy model which contains a partially truncated Fermi surface and a well-defined cooperon excitation formed by a bound pair of holes. An attractive interaction in the Cooper channel is generated on the Fermi surface through virtual scattering into the cooperon state. Although the model is derived in the weak coupling limit of a four-leg ladder array, an examination of exact results on finite clusters for the strong coupling $t\text{\ensuremath{-}}J$ model suggests the essential features are also present for a strong coupling Hubbard model on a square lattice near half-filling.
The Hubbard model is an iconic model in quantum many-body physics and has been intensely studied, especially since the discovery of high-temperature cuprate superconductors. Combining the complementary capabilities of two computational methods, we found superconductivity in both the electron- and hole-doped regimes of the two-dimensional Hubbard model with next-nearest-neighbor hopping. In the electron-doped regime, superconductivity was weaker and was accompanied by antiferromagnetic Néel correlations at low doping. The strong superconductivity on the hole-doped side coexisted with stripe order, which persisted into the overdoped region with weaker hole-density modulation. These stripe orders varied in fillings between 0.6 and 0.8. Our results suggest the applicability of the Hubbard model with next-nearest hopping for describing cuprate high–transition temperature ( T c ) superconductivity.
The dualism between superconductivity and charge/spin modulations (the so-called stripes) dominates the phase diagram of many strongly-correlated systems. A prominent example is given by the Hubbard model, where these phases compete and possibly coexist in a wide regime of electron dopings for both weak and strong couplings. Here, we investigate this antagonism within a variational approach that is based upon Jastrow-Slater wave functions, including backflow correlations, which can be treated within a quantum Monte Carlo procedure. We focus on clusters having a ladder geometry with M M legs (with M M ranging from 2 2 to 10 10 ) and a relatively large number of rungs, thus allowing us a detailed analysis in terms of the stripe length. We find that stripe order with periodicity \lambda=8 λ = 8 in the charge and 2\lambda=16 2 λ = 16 in the spin can be stabilized at doping \delta=1/8 δ = 1 / 8 . Here, there are no sizable superconducting correlations and the ground state has an insulating character. A similar situation, with \lambda=6 λ = 6 , appears at \delta=1/6 δ = 1 / 6 . Instead, for smaller values of dopings, stripes can be still stabilized, but they are weakly metallic at \delta=1/12 δ = 1 / 12 and metallic with strong superconducting correlations at \delta=1/10 δ = 1 / 10 , as well as for intermediate (incommensurate) dopings. Remarkably, we observe that spin modulation plays a major role in stripe formation, since it is crucial to obtain a stable striped state upon optimization. The relevance of our calculations for previous density-matrix renormalization group results and for the two-dimensional case is also discussed.
The two-dimensional Hubbard model is widely believed to capture key ingredients of high-T_{c} superconductivity in cuprate materials. However, compelling evidence remains elusive. In particular, various magnetic orders may emerge as strong competitors of superconducting orders. Here, we study the ground state properties of the doped two-dimensional t-t^{'} Hubbard model on a square lattice via the infinite projected entangled-pair state method with U(1) or SU(2) spin symmetry. The former is compatible with antiferromagnetic orders, while the latter forbids them. Therefore, we obtain by comparison a detailed understanding of the magnetic impact on superconductivity. Moreover, an additional t^{'} term accommodates the particle-hole asymmetry, which facilitates studies on the discrepancies between electron- and hole-doped systems. We demonstrate that (i) a positive t^{'}/t significantly amplifies the strength of superconducting orders; (ii) at sufficiently large doping levels, the t-t^{'} Hubbard model favors a uniform state with superconducting orders instead of stripe states with charge and spin modulations; and (iii) the enhancement of magnetic frustration, by increasing either the strength of next-nearest neighbor interactions or the charge doping, impairs stripe orders and helps stabilize superconductivity.