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arXiv:2008.08661 · Physical review. B./Physical review. B

Charge- and pair-density-wave orders in the one-band Hubbard model from dynamical mean field theory

Sidhartha Dash, David Sénéchal

Published Jan 29, 2021• 10 citations
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Abstract

We study the charge-density-wave order and its competition with superconductivity in the one-band Hubbard model for high-${T}_{c}$ superconducting cuprates. We use cluster dynamical mean field theory at $T=0$. The one-band Hubbard model contains only copper atoms, and a physical charge-density wave with charge excesses located on the oxygen atoms manifests itself as a bond-density wave (BDW) within this context. It arises purely out of local correlation effects and also leads to an ${s}^{\ensuremath{'}}$-wave pair-density wave in the presence of $d$-wave superconductivity. The $d$-wave BDW is suppressed on increasing $U$ and is favored on increasing the magnitude of the second-neighbor hopping ${t}^{\ensuremath{'}}$. Further, the $d$-wave BDW order is weakened when in competition with superconductivity, as seen in experiments. Additionally, it behaves quite differently on varying $U$ in the superconducting state compared to the normal state.

DOI
10.1103/physrevb.103.045142

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We study the charge-density-wave order and its competition with superconductivity in the one-band Hubbard model for high-Tc{T}_{c}Tc​ superconducting cuprates. We use cluster dynamical mean field theory at T=0T=0T=0. The one-band Hubbard model contains only copper atoms, and a physical charge-density wave with charge excesses located on the oxygen atoms manifests itself as a bond-density wave (BDW) within this context. It arises purely out of local correlation effects and also leads to an s\ensuremath′{s}^{\ensuremath{'}}s\ensuremath′-wave pair-density wave in the presence of ddd-wave superconductivity. The ddd-wave BDW is suppressed on increasing UUU and is favored on increasing the magnitude of the second-neighbor hopping t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′. Further, the ddd-wave BDW order is weakened when in competition with superconductivity, as seen in experiments. Additionally, it behaves quite differently on varying UUU in the superconducting state compared to the normal state.
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 ddd-wave superconductivity. The charge order has a period of 4 in one direction, is centered on bonds, and has a ddd 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=5U=5U=5 to U=8U=8U=8. Its location and intensity depend strongly on the band dispersion. When probed simultaneously with ddd-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 study the coexistence of pair- (PDW) and charge-density-wave (CDW) states within the single-band $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ and Hubbard models of ddd-wave superconductivity and discuss our results in the context of the experimental observations for the copper-based compounds. In order to take the correlation effects into account with proper precision, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), which goes beyond the renormalized mean-field theory in a systematic manner. According to our analysis of the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ model, the transition between the pure ddd-wave superconducting phase and the coexistent CDW+PDW phase takes place at $\ensuremath{\delta}\ensuremath{\approx}0.18$ (close to the optimal doping), with the modulated phase located in the underdoped regime. The situation is different for the case of the Hubbard model, where a narrow stability regime of a precursor nematic phase sets in preceding the formation of the modulated CDW+PDW state, with decreasing hole doping. The results conclude our comprehensive discussion of the standard phase diagram for high-TC{T}_{C}TC​ superconducting compounds within the DE-GWF variational approach in the single narrow-band case [see J. Spa\l{}ek et al., Phys. Rev. B 95, 024506 (2017); M. Zegrodnik and Spa\l{}ek, Phys. Rev. B 95, 024507 (2017); Phys. Rev. B 96, 054511 (2017); New J. Phys. 20, 063015 (2018)].
Proximity to a Mott insulating phase is likely to be an important physical ingredient of a theory that aims to describe high-temperature superconductivity in the cuprates. Quantum cluster methods are well suited to describe the Mott phase. Hence, as a step toward a quantitative theory of the competition between antiferromagnetism and ddd-wave superconductivity in the cuprates, we use cellular dynamical mean-field theory to compute zero-temperature properties of the two-dimensional square lattice Hubbard model. The ddd-wave order parameter is found to scale like the superexchange coupling JJJ for on-site interaction UUU comparable to or larger than the bandwidth. The order parameter also assumes a dome shape as a function of doping, while, by contrast, the gap in the single-particle density of states decreases monotonically with increasing doping. In the presence of a finite second neighbor hopping ${t}^{\ensuremath{'}}$, the zero-temperature phase diagram displays the electron-hole asymmetric competition between antiferromagnetism and superconductivity that is observed experimentally in the cuprates. Adding realistic third neighbor hopping ${t}^{\ensuremath{''}}$ improves the overall agreement with the experimental phase diagram. Since band parameters can vary depending on the specific cuprate considered, the sensitivity of the theoretical phase diagram to band parameters challenges the commonly held assumption that the doping vs Tc∕Tcmax{T}_{c}∕{T}_{c}^{\mathit{max}}Tc​∕Tcmax​ phase diagram of the cuprates is universal. The calculated angle-resolved photoemission spectrum displays the observed electron-hole asymmetry. The tendency to homogeneous coexistence of the superconducting and antiferromagnetic order parameters is stronger than observed in most experiments but consistent with many theoretical results and with experiments in some layered high-temperature superconductors. Clearly, our calculations reproduce important features of ddd-wave superconductivity in the cuprates that would otherwise be considered anomalous from the point of view of the standard Bardeen--Cooper--Schrieffer approach. At strong coupling, ddd-wave superconductivity and antiferromagnetism naturally appear as two equally important competing instabilities of the normal phase of the same underlying Hamiltonian.
In light of the new experimental and theoretical important developments in high-Tc{T}_{c}Tc​ superconductivity, we revisit the fermionic hot-spot model relevant to the phenomenology of the cuprates. We extend previous results by means of a complete two-loop order renormalization group (RG) framework. Here, we explicitly study the effect of the charge-density-wave (CDW) order parameter with a ddd-wave form factor with the experimentally observed modulation $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},0)$ and $(0,\ifmmode\pm\else\textpm\fi{}{Q}_{0})$ at the infrared-stable nontrivial fixed point obtained previously for this model. Additionally, we proceed to investigate also the so-called pair-density-wave (PDW) order that was recently proposed in the literature as a possible candidate for the ``hidden'' order to describe the pseudogap phase observed in underdoped cuprates. We confirm that although the above two ordering tendencies are also found to be nearly degenerate both at one-loop and two-loop RG orders and linked by an emergent SU(2) pseudospin symmetry, they turn out to be subleading for weaker couplings in the present model to antiferromagnetism, ddd-wave bond-density wave (BDW) order with modulation along Brillouin zone diagonals $(\ifmmode\pm\else\textpm\fi{}{Q}_{0},\ifmmode\pm\else\textpm\fi{}{Q}_{0})$, and ddd-wave singlet superconductivity (SSC). However, as we increase the strength of the initial coupling towards moderate values, we do capture a tendency for the entangled PDW/CDW order to become leading compared to BDW/SSC in the model, which suggests that the former composite order might be indeed a viable concept to describe some cuprate superconductors at high temperatures in the underdoped regime, as has been recently alluded to by many authors in the literature.
Abstract Flat-band systems provide an ideal platform for studying strong electron correlations, in which multiple quasi-degenerate charge orders frequently coexist. Motivated by this context, we investigate the pair-density-wave (PDW) superconducting state in a quasi-one-dimensional Creutz lattice featuring two strictly flat bands. Using a renormalized mean-field approach for the strongly correlated t - J model, we obtain self-consistent solutions that demonstrate a PDW phase—characterized by finite-center-of-mass-momentum Cooper pairing—coexisting with charge density wave (CDW) and bond density wave (BDW) orders. Physically, this coexistence phenomenon originates from the perfect band nesting inherent to the flat-band system, which allows for changes in the particle momentum distributions without energy cost. Here, a pure PDW solution without CDW and BDW orders emerges in the underdoped regime when magnetic effects are neglected, while strong Hubbard correlations favor a pure CDW phase. We conclude that a robust PDW phase, accompanied by CDW and BDW orders, occupies the majority of the parameter space. Furthermore, both the PDW order parameter and its critical temperature scale linearly with the superexchange coupling strength—a hallmark of flat-band superconductivity. Overall, our results reveal that a PDW superconducting state can be realized on the Creutz lattice with strong correlations.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.