Variational tests are performed for current order parameters as probable sources of the pseudogap normal state of cuprates. The calculations are carried out based on the states with correlations of the valence bond type whose formation can induce in principle both the superconducting order of the d symmetry and current phases. It is shown for the t-t′ -U Hubbard models with a large value of U (∼8 t ) and the Hubbard splitting of the conduction band that (1) phases of alternating charge and longitudinal spin currents cannot be realized and (2) transverse spin currents are not compatible with the superconducting order and they could exist against the normal-state background only within a very narrow doping region near the optimal one. This region does not correspond to the region of existence of a pseudogap in cuprates, which refutes the above-mentioned hypothesis of the pseudogap origin. The requirements to the parameters of models for which the consideration of correlations of the valence bond type yields a reasonable phase curve. The existence of current phases in the t-t′-U-V Hubbard models with a strong interaction ( V >0.25t) of particles in neighboring sites is predicted when the d -superconductivity is completely suppressed.
Variational tests are performed for current order parameters as probable sources of the pseudogap normal state of cuprates. The calculations are carried out based on the states with correlations of the valence bond type whose formation can induce in principle both the superconducting order of the d symmetry and current phases. It is shown for the t-t′ -U Hubbard models with a large value of U (∼8 t ) and the Hubbard splitting of the conduction band that (1) phases of alternating charge and longitudinal spin currents cannot be realized and (2) transverse spin currents are not compatible with the superconducting order and they could exist against the normal-state background only within a very narrow doping region near the optimal one. This region does not correspond to the region of existence of a pseudogap in cuprates, which refutes the above-mentioned hypothesis of the pseudogap origin. The requirements to the parameters of models for which the consideration of correlations of the valence bond type yields a reasonable phase curve. The existence of current phases in the t-t′-U-V Hubbard models with a strong interaction ( V >0.25t) of particles in neighboring sites is predicted when the d -superconductivity is completely suppressed.
It is shown that d -symmetry superconductivity due to valence bond correlations is possible. Valence bond correlations are compatible with antiferromagnetic spin order. In order to explictly construct a homogeneous state with the valence bond structure in the two-dimensional Hubbard model for an arbitrary doping, we have used the variational method based on unitary local transformation. Attraction between holes in the d -channel is due to modulation of hopping by the site population in course of the valence bond formation, and corresponding parameters have been calculated variationally. An important factor for the gap width is the increase in the density of states on the Fermi level due to antiferromagnetic splitting of the band. The gap width and its ratio to the T c are 2Δ≃0.1 t and 2Δ/ kT c ≃4.5−4 for U/t ≃8. The correspondence between the theoretical phase diagram and experimental data is discussed. The dependence of T c on the doping δ =| n −1| and the Fermi surface shape are highly sensitive to the weak interaction t ′ leading to diagonal hoppings. In the case of t ′>0 and p -doping, the peak on the curve of T c ( δ ) occurs at the doping δ opt , when the energy of the flattest part of the lower Hubbard subband crosses the Fermi level at k ∼( π ,0). In underdoped samples with δ < δ opt , the anisotropic pseudogap in the normal state corresponds to the energy difference | E ( π ,0)− μ | between this part of the spectrum and the Fermi level.
It is shown that d x 2−y 2 symmetry of superconducting order due to valence bond (VB) type correlations is possible. The VB correlations are compatible with antiferromagnetic (AF) spin order. For the two-dimensional Hubbard model with arbitrary doping, the variational method of local unitary transformations is used to construct explicitly a uniform state with VB structure. The d-channel attraction of holes is a consequence of the modulation of hops by the populations of centers accompanying VB formation, and the parameters of the modulation are determined variationally. The increase in the density of states at the Fermi level accompanying AF splitting of the band, which is absent in the paramagnetic state, is important for the gap width. The gap width and its ratio to T c are of the order of 2Δ≃0.1t and 2Δ/kT c≃4–4.5 with U/t≃8. The agreement between the phase diagram found and experiment is discussed.
') neighbor hoppings. Our work investigates the consequences of increasing hole doping on the instauration of stripes and the behavior of the superconducting order parameter, with a discussion on how the two phenomena affect each other. We consider two different values of the next-nearest neighbor hopping parameter, that are appropriate for describing cuprate superconductors. We observe that stripes are the optimal state in a wide doping range; the stripe wavelength reduces at increasing doping, until stripes melt into a uniform state for large values of doping. Superconducting pair-pair correlations, indicating the presence of superconductivity, are always suppressed in the presence of stripes. Our results suggest that the phase diagram for the single-band Hubbard model is dominated by stripes, with superconductivity being possible only in a narrow doping range between striped states and a nonsuperconducting metal.
The doped Hubbard model is a simple model for high-Tc cuprate superconductors, while its ground state remains a challenge. Here, by performing state-of-the-art variational Monte Carlo calculations for the strong-coupling Hubbard model, we find evidence that the d-wave superconducting phase emerges always near the phase separation region and the superconducting order has one-to-one correspondence with the enhancement of charge compressibility. The order as well as the phase separation are vulnerable to realistic intersite Coulomb interaction, while the superexchange interaction enhances both. An appropriate combination of these two widens the stable superconducting phase.
It is now well established that superconducting cuprates support a charge-density-wave state in the so-called underdoped region of their phase diagram. We investigate the possibility of charge order in the square-lattice Hubbard model, both alone and in coexistence with d-wave superconductivity. The charge order has a period of 4 in one direction, is centered on bonds, and has a d form factor. We use the variational cluster approximation, an approach based on a rigorous variational principle that treats short-range correlations exactly, with two clusters of size $2\ifmmode\times\else\texttimes\fi{}6$ that together tile the infinite lattice and provide a nonbiased unit for a period-4 bond-density wave (BDW). We find that the BDW exists in a finite range of hole doping and increases in strength from U=5 to U=8. Its location and intensity depend strongly on the band dispersion. When probed simultaneously with d-wave superconductivity, the energy is sometimes lowered by the presence of both phases, depending on the interaction strength. Whenever they coexist, a pair-density wave (a modulation of superconducting pairing with the same period and form factor as the BDW) also exists.
We calculate the properties of the two-band Hubbard model using the dynamical cluster approximation. The phase diagram resembles the generic phase diagram of the cuprates, showing a strong asymmetry with respect to electron- and hole-doped regimes, in agreement with experiment. Asymmetric features are also seen in one-particle spectral functions and in the charge, spin, and d-wave pairing-susceptibility functions. We address the possible reduction of the two-band model to a low-energy single-band one, as it was suggested by Zhang and Rice. Comparing the two-band Hubbard model properties with the single-band Hubbard model ones, we have found similar low-energy physics provided that the next-nearest-neighbor hopping term ${t}^{\ensuremath{'}}$ has a significant value $({t}^{\ensuremath{'}}∕t\ensuremath{\approx}0.3)$. The parameter ${t}^{\ensuremath{'}}$ is the main culprit for the electron-hole asymmetry. However, a significant value of ${t}^{\ensuremath{'}}$ cannot be provided in a strict Zhang and Rice [Phys. Rev. B 37, R3759 (1988); 41, 7243 (1990)] picture where the extra holes added into the system bind to the existing Cu holes forming local singlets. We note that by considering approximate singlet states, such as plaquette states, reasonable values of ${t}^{\ensuremath{'}}$, which capture qualitatively the physics of the two-band model, can be obtained. We conclude that a single-band $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U$ Hubbard model captures the basic physics of the cuprates concerning superconductivity, antiferromagnetism, pseudogap, and electron-hole asymmetry, but is not suitable for a quantitative analysis or to describe physical properties involving energy scales larger than about 0.5 eV.