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Journal of Physics Condensed Matter

Enhanced d-wave pairing in the two-dimensional Hubbard model with periodically modulated hopping amplitudes

Shuhui Yang, Xingcan Liu, Weiqi Li, Jianqun Yang, Tao Ying, Xingji Li, Xiudong Sun

Published Jul 5, 2022• 0 citations
Publisher

Abstract

Abstract Utilizing determinant quantum Monte Carlo algorithm, the evolution of the d-wave pairing in the Hubbard model on the square lattice tuned by the periodically modulated hopping amplitudes is studied. The hopping amplitudes are homogeneous in the x ˆ -direction, while in the y ˆ -direction the hopping amplitudes are modulated with period P, where t y = t + d t , t y ′ = t − ( P − 1 ) d t , and the modulation period P equals 2, 3 and 4 lattice spacings. The latter two modulation periods are motivated by the observation of period-3 and period-4 stripe order in cuprate superconductors. For all the periods P, we find that the modulated hopping inhomogeneity enhances the d-wave pairing and an optimal inhomogeneity exists.

DOI
10.1088/1361-648x/ac7e9c

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Abstract Utilizing determinant quantum Monte Carlo algorithm, the evolution of the d-wave pairing in the Hubbard model on the square lattice tuned by the periodically modulated hopping amplitudes is studied. The hopping amplitudes are homogeneous in the x ˆ -direction, while in the y ˆ -direction the hopping amplitudes are modulated with period P, where t y = t + d t , t y ′ = t − ( P − 1 ) d t , and the modulation period P equals 2, 3 and 4 lattice spacings. The latter two modulation periods are motivated by the observation of period-3 and period-4 stripe order in cuprate superconductors. For all the periods P, we find that the modulated hopping inhomogeneity enhances the d-wave pairing and an optimal inhomogeneity exists.
Striped phases, in which spin, charge, and pairing correlations vary inhomogeneously in the CuO2{}_{2}2​ planes, are a known experimental feature of cuprate superconductors, and are also found in a variety of numerical treatments of the two-dimensional Hubbard Hamiltonian. In this paper, we use determinant quantum Monte Carlo to show that if a stripe density pattern is imposed on the model, the ddd-wave pairing vertex is significantly enhanced. We attribute this enhancement to an increase in antiferromagnetic order which is caused by the appearance of more nearly half-filled regions when the doped holes are confined to the stripes. We also observe a $\ensuremath{\pi}$-phase shift in the magnetic order.
Determinant quantum Monte Carlo (DQMC) is used to determine the pairing and magnetic response for a Hubbard model built up from four-site clusters---a two-dimensional square lattice consisting of elemental $2\ifmmode\times\else\texttimes\fi{}2$ plaquettes with hopping ttt and on-site repulsion UUU coupled by an interplaquette hopping ${t}^{\ensuremath{'}}\ensuremath{\le}t$. Superconductivity in this geometry has previously been studied by a variety of analytic and numeric methods, with differing conclusions concerning whether the pairing correlations and transition temperature are raised near half filling by the inhomogeneous hopping or not. For U/t=4U/t=4U/t=4, DQMC indicates an optimal ${t}^{\ensuremath{'}}/t\ensuremath{\approx}0.4$ at which the pairing vertex is most attractive. The optimal ${t}^{\ensuremath{'}}/t$ increases with U/tU/tU/t. We then contrast our results for this plaquette model with a hamiltonian which instead involves a regular pattern of site energies whose large site energy limit is the three-band CuO2{\mathrm{CuO}}_{2}CuO2​ model; we show that there the inhomogeneity rapidly, and monotonically, suppresses pairing.
We have performed a variational Monte Carlo simulation on a two-dimensional Hubbard model with first- and second-neighbor hopping terms in order to study the coexistence state of a static stripe state and a modulated d-wave superconductivity in the underdoped cuprates. In addition to a Gutzwiller, a Jastrow and a doublon-holon correlation effects, the band-renormalization effect was considered in the trial wave function. The condensation energies of an 8-period stripe state was computed as a function of a Coulomb energy under the hole-density x=1/8. Our results reveal that the renormalization of higher hopping parameters due to the strong correlation effect enhances the one-dimensional hole motion on a quarter-filled band in the stripe state, and brings quasi-Fermi surface close to the magnetic zone boundary in the coexistence state.
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].
Using dynamic cluster quantum Monte Carlo simulations, we study the superconducting behavior of a 1/8 doped two-dimensional Hubbard model with imposed unidirectional stripelike charge-density-wave modulation. We find a significant increase of the pairing correlations and critical temperature relative to the homogeneous system when the modulation length scale is sufficiently large. With a separable form of the irreducible particle-particle vertex, we show that optimized superconductivity is obtained for a moderate modulation strength due to a delicate balance between the modulation enhanced pairing interaction, and a concomitant suppression of the bare particle-particle excitations by a modulation reduction of the quasiparticle weight.
We study the competition between stripe states with different periods and a uniform ddd-wave superconducting state in the extended two-dimensional Hubbard model at 1/8 hole doping using infinite projected entangled-pair states (iPEPS). With increasing strength of negative next-nearest neighbor hopping ${t}^{\ensuremath{'}}$, the preferred period of the stripe decreases. For the values of ${t}^{\ensuremath{'}}$ predicted for cuprate high-Tc{T}_{c}Tc​ superconductors, we find stripes with a period 4 in the charge order, in agreement with experiments. Superconductivity in the period 4 stripe is suppressed at 1/81/81/8 doping. Only at larger doping, $0.18\ensuremath{\lesssim}\ensuremath{\delta}<0.25$, the period 4 stripe exhibits coexisting ddd-wave superconducting order. The uniform ddd-wave state is only favored for sufficiently large positive ${t}^{\ensuremath{'}}$.