Numerical simulations on the half-filled 3-D Hubbard model clearly show the onset of Néel order. Simulations of the 2-D electron-phonon Holstein model show the competition between the formation of a Peierls-CDW state and a superconducting state. However, the behavior of the partly filled 2-D Hubbard model is more difficult to determine. At half-filling, the anti-ferromagnetic correlations grow as T is reduced. Doping away from half-filling supresses these correlations, and we find that there is a weak attractive pairing interaction in the d -wave channel. However, the strength of the pair field susceptibility is weak at the temperatures and lattice sizes we have simulated, and the nature of the low-temperature state of the nearly half-filled Hubbard model remains open.
Phase diagram of the two-dimensional Hubbard-Holstein model: enhancement of
s-wave pairing between charge and magnetic orders
We investigate the role of electron-electron and electron-phonon interactions 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, favoring (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.
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.
The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite
Tcdw in two dimensions. Quantum Monte Carlo calculations have determined
Tcdw for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in
d=2 is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine
Tsc for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, where
t is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher
T. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.
We use determinant quantum Monte Carlo to study the half-filled `bond-Holstein' model on a square lattice. We find that the model exhibits a charge-density-wave (CDW) phase transition with a critical temperature
Tcdw considerably higher than that of the canonical `site-Holstein' model. Using a finite-size scaling analysis of the charge structure factor
Scdw, we obtain
Tcdw to greater than one percent accuracy. At the same time, local observables also show clear signatures consistent with the transition temperatures inferred from our scaling analysis. We attribute the enhanced CDW tendencies to a phonon-mediated nearest-neighbor electron repulsion that is directly proportional to the dimensionless electron-phonon coupling
λ in the atomic (
t→0) limit. This behavior contrasts with the site-Holstein case, where the same limit yields only an on-site attraction. We supplement our analysis with results from several unsupervised machine learning methods, which not only confirm our estimates of
Tcdw but also provide insight into the high-temperature crossover between a metallic and bipolaron liquid regime.
In this work, we investigate the impact of Rashba spin-orbit coupling (RSOC) on the formation of charge-density wave (CDW) and superconducting (SC) phases in the Holstein model on a half-filled square lattice. Using unbiased finite-temperature Quantum Monte Carlo simulations, we go beyond mean-field approaches to determine the ground state order parameter as a function of RSOC and phonon frequency. Our results reveal that the Rashba metal is unstable due to particle-hole instabilities, favoring the emergence of a CDW phase for any RSOC value. In the limit of a pure Rashba hopping, the model exhibits a distinct behavior with the appearance of four Weyl cones at half-filling, where quantum phase transitions are expected to occur at strong interactions. Indeed, a quantum phase transition, belonging to the Gross-Neveu Ising universality class between a semi-metal and CDW emerges at finite phonon frequency dependent coupling
λc. In the antiadiabatic limit we observe an enhanced symmetry in the infrared that unifies SC and CDW orders. These results advance our understanding of competing CDW and SC phases in systems with spin-orbit coupling, providing insights that may help clarify the behavior of related materials.