Phase diagram of the two-dimensional Hubbard-Holstein model

Natanael C Costa,Kazuhiro Seki,Seiji Yunoki,Sandro Sorella

doi:10.1038/s42005-020-0342-2

Abstract

The electron–electron and electron–phonon interactions play an important role in correlated materials, being key features for spin, charge and pair correlations. Thus, here we investigate their effects in strongly correlated systems by performing unbiased quantum Monte Carlo simulations in the square lattice Hubbard-Holstein model at half-filling. We study the competition and interplay between antiferromagnetism (AFM) and charge-density wave (CDW), establishing its very rich phase diagram. In the region between AFM and CDW phases, we have found an enhancement of superconducting pairing correlations, favouring (nonlocal) s-wave pairs. Our study sheds light over past inconsistencies in the literature, in particular the emergence of CDW in the pure Holstein model case.

Highlights

The electron–electron and electron–phonon interactions play an important role in correlated materials, being key features for spin, charge and pair correlations
While Bardeen, Cooper and Schrieffer used this interaction in their seminal work to explain pairing[2], Peierls took it into account to provide a mechanism, based on Fermi surface nesting (FSN), that leads to charge-density wave (CDW)[3]
We highlight [i] the absence of a finite critical e-ph coupling for the pure Holstein model, i.e., the CDW phase sets in for any λ > 0; [ii] the existence of a finite AFM critical point on the line U = λ, which is strongly dependent on the phonon frequency; and [iii] an enhancement of nonlocal swave pairing in the region between the AFM and CDW phases

Summary

Results

Since this metallic-like region seems to be in the negative Ueff side of the phase diagram, the s-wave symmetry is naively expected Such an enhancement of nonlocal s-wave response suggests that short-range charge/spin correlations may suppress the formation of local (s-wave) and NN (s*-wave) Cooper pairs, making the NNN ones (s**-wave) the. It is important to notice that, as a consequence of the Kohn-Luttinger weak coupling argument[56], instabilities in the particle-particle channel are expected in this intermediate region without AFM and CDW orders, which could lead to pairing[57,58,59] These instabilities are believed to occur at very low temperatures, usually not accessible for the current QMC methodologies. We warn that a more precise determination of the nature of this region may require the analysis of dynamical quantities, or long-range effective electronic interactions[43,60]

Discussion

Methods

Code availability