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Physical review. B, Condensed matter

Doping dependence of antiferromagnetic correlations in high-temperature superconductors

M. Inui, Sebastian Doniach, M. Gabay

Published Oct 1, 1988• 148 citations
Publisher

Abstract

We have studied the single-band Hubbard model on a two-dimensional square lattice slightly below half filling. Using the canonical transformation and a form of variational wave function that allows an expansion in terms of hole concentration $\ensuremath{\delta}$ with respect to half filling, we find that a nonzero $\ensuremath{\delta}$ results in a frustration of the antiferromagnetic (AF) order favored by the superexchange interaction. Use of a variational Monte Carlo calculation enables us to construct a phase diagram in the $\frac{t}{U\ensuremath{-}\ensuremath{\delta}}$ plane in which the AF order disappears at $\ensuremath{\delta}\ensuremath{\approx}0.06$ for a typical value of $\frac{t}{U}=\frac{1}{4}$.

DOI
10.1103/physrevb.38.6631

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We have studied the single-band Hubbard model on a two-dimensional square lattice slightly below half filling. Using the canonical transformation and a form of variational wave function that allows an expansion in terms of hole concentration \ensuremathδ\ensuremath{\delta}\ensuremathδ with respect to half filling, we find that a nonzero \ensuremathδ\ensuremath{\delta}\ensuremathδ results in a frustration of the antiferromagnetic (AF) order favored by the superexchange interaction. Use of a variational Monte Carlo calculation enables us to construct a phase diagram in the tU\ensuremath−\ensuremathδ\frac{t}{U\ensuremath{-}\ensuremath{\delta}}U\ensuremath−\ensuremathδt​ plane in which the AF order disappears at \ensuremathδ\ensuremath≈0.06\ensuremath{\delta}\ensuremath{\approx}0.06\ensuremathδ\ensuremath≈0.06 for a typical value of tU=14\frac{t}{U}=\frac{1}{4}Ut​=41​.
To study the relationship between antiferromagnetism (AF) and superconductivity (SC) in cuprate superconductors, we have performed a variational Monte Carlo calculation on the two-dimensional Hubbard model with a diagonal transfer t' [1,2]. The following improvements are introduced into the variational function: (1) Coexistence of AF and d-wave singlet gaps that allows a continuous description of their interplay, (2) band renormalization effect owing to the electron correlation up to the fourth-neighbor hoppings, and (3) refined doublon-holon correlation factors. It is found that the AF and SC orders become coexistent or competitive according to the values of model parameters, U/t, t'/t, and δ (doping rate), and the effective band is strongly renormalized for large U/t and low doping rates, so as to recover the nesting condition favorable to AF.
We investigate the phase diagram of two-dimensional (2D) Hubbard model by employing the optimization variational Monte Carlo method. The 2D Hubbard model is the most simple electronic model for cuprate high-temperature superconductors. The phase diagram consists of three regions; they are antiferromagnetic insulator (AFI) region, superconducting (SC) region and the coexistent region of superconductivity and antiferromgantism. The phase diagram obtained by numerical calculations well agrees with the experimental phase diagram for high-temperature cuprates. We mainly focused on the effect of t′t't′ on the antiferromagnetic (AF) correlation and the AFI region. The area of the AF phase increases when we include t′t't′ and thus the pure ddd-wave SC phase decreases. The AFI phase near half filling decreases as ∣t′∣|t'|∣t′∣ increases.
The coexistence or exclusivity of dx2−y2d_{x^{2} - y^{2}}dx2−y2​-wave superconducting (d-SC) and antiferromagnetic (AF) orders and instability toward phase separation (PS) are reconsidered for the square-lattice t–J model with the diagonal transfer (t′) term near half filling. To reliably treat the strong correlation including the local constraint of no double occupancy, we use a variational Monte Carlo method, in which the band-renormalization effect (BRE) is also incorporated. The results are compared with those recently obtained for the corresponding Hubbard model [J. Phys. Soc. Jpn. 85, 074701 (2016)]. It is found that BRE has a very strong effect on the AF order and greatly modifies the ground-state phase diagram in the t′–δ space (δ: doping rate) proposed in previous studies, especially for large |t′/t|. In a wide range of t′/t (|t′/t| ≤ 0.5) and for any underdoped δ (≲ \lesssim ≲ 0.15), the AF order arises for J/t = 0.3, similarly to in the Hubbard model. In contrast with the Hubbard model, however, the d-SC order also arises down to small δ in a wide range of t′/t (≳ \gtrsim ≳ −0.2) and coexists with the AF order even for t′/t < 0 (type-II AF regime). Furthermore, there is no instability toward PS for any t′/t and δ for J/t = 0.3.
The competition between ddd-wave superconductivity (SC) and antiferromagnetism (AF) in the high-Tc{T}_{c}Tc​ cuprates is investigated by studying the hole- and electron-doped two-dimensional Hubbard model with a recently proposed variational quantum-cluster theory. The approach is shown to provide a thermodynamically consistent determination of the particle number, provided that an overall shift of the on-site energies is treated as a variational parameter. The consequences for the single-particle excitation spectra and for the phase diagram are explored. By comparing the single-particle spectra with quantum Monte Carlo and experimental data, we verify that the low-energy excitations in a strongly correlated electronic system are described appropriately. The cluster calculations also reproduce the overall ground-state phase diagram of the high-temperature superconductors. In particular, they include salient features such as the enhanced robustness of the antiferromagnetic state as a function of electron doping and the tendency towards phase separation into a mixed antiferromagnetic-superconducting phase at low doping and a pure superconducting phase at high (both hole and electron) doping.
Having cuprate superconductors in mind, we study a band-renormalization effect on a coexisting state of antiferromagnetism (AF) and superconductivity (SC) in a t-J model on a square lattice with a diagonal transfer t ′ and three-site hopping terms, using a variational Monte Carlo method. It is found that this t-J model with t ′/ t = –0.3 exhibits coexistence of AF and SC orders in lightly doped regime and robust SC for a wide range of doping rate δ . This result contrasts with the previous results of a large- U/t Hubbard model, in which the two orders are mutually exclusive and a coexisting state does not appear; the SC state appears only in heavily doped regime (δ≳ 0.2), after the pure AF state vanishes at δ ∼ 0.2.
We use a variational Monte Carlo technique to study the ground state of the two-dimensional Hubbard and t-J models on a square lattice. We use a trial wave function that allows a continuous description of the paramagnetic, antiferromagnetic, and superconducting phases, as well as the coexistence of these phases, with no a priori constraint on double occupancy. The phase diagram of both models is given for intermediate coupling (U=10t). We show that the apparent discrepancies between the two models, that appeared in previous variational Monte Carlo studies, are not present when this sufficiently good variational wave function is used. The two models are in good qualitative agreement. We find that except at half-filling a pure antiferromagnetic phase does not exist but is always in coexistence with a d-wave superconducting phase, and is followed by a pure d-wave superconducting phase at doping up to \ensuremath{\delta}\ensuremath{\sim}0.3. The staggered magnetization and the superconducting gap are measured, and a comparison with other analytical or numerical results is made.