Antiferromagnetism and Kekulé valence bond order in the honeycomb optical Su-Schrieffer-Heeger-Hubbard model

Abstract

The precise role of e-ph coupling in graphene and related materials on a honeycomb lattice is not yet fully understood, despite extensive research on these systems. Here, we perform sign-problem-free determinant quantum Monte Carlo (DQMC) simulations of the optical Su-Schrieffer-Heeger (oSSH)-Hubbard model on the honeycomb lattice, focusing on the parameters relevant to graphene. Performing finite-size scaling analyzes, we obtain the model's ground state phase diagram, which includes the semi-metal (SM), Kekulé Valence Bond Solid (KVBS), and anti-ferromagnetic (AFM) phases, as well as indications of a small KVBS/AFM coexistence region. We find that a weak to moderate Hubbard repulsion, tuned toward the SM-AFM critical value in the pure honeycomb Hubbard model, enhances KVBS correlations and can even stabilize the KVBS phase. Estimating the effective parameters for graphene places it in the SM region of the phase diagram, but near the SM-KVBS phase boundary. Notably, we predict that increasing either the on-site Hubbard repulsion or the e-ph coupling strength drives graphene toward the KVBS phase rather than the AFM phase, highlighting a synergistic effect that can be exploited to further control the remarkable properties of graphene and related materials.

Figures (4)

• Figure 1: Ground state $U-\lambda$ phase diagram for the Hubbard-oSSH model on a half-filled ($\langle n \rangle = 1$) honeycomb lattice, obtained for $\Omega=0.1t$. Blue and black markers indicate a QCP for Kekulé valence bond solid (KVBS) and antiferromagnetic (AFM) phase transitions, respectively. Filled (empty) markers indicate the transition point found for fixed $U \ (\lambda)$ to determine $\lambda_\mathrm{c} \ (U_\mathrm{c})$ for the relevant phase transition. Refer to Fig. \ref{['fig:corr_ratio_vs_couplings']} for examples of this analysis. The red star mark indicates the approximate location of graphene in the phase diagram. The blue and black lines are guides to the eye.
• Figure 2: The evolution of the KVBS $S_\mathrm{VBS}(\boldsymbol{K})$ and antiferromagnetic $S_\mathrm{AFM}(\boldsymbol{\Gamma})$ structure factors for $\Omega=0.1t$. Results are plotted as a function of (a) $U$ for constant $\lambda=0.13$ and (b) $\lambda$ for constant $U=4.5t$. The left [right] $y$-axis in both panels shows the $S_\mathrm{VBS}(\boldsymbol{ K})$ [$S_\mathrm{AFM}(\boldsymbol{\Gamma})$] values, as indicated by the arrows in the top panel. The colored regions indicate the four distinct regions in the phase diagram, including the coexistence region, consistent with Fig. \ref{['fig:Phase_diagram']}. The blue and red lines are guides for the eye.
• Figure 3: A crossing analysis using the correlation ratios $R_\text{VBS}(\boldsymbol{\boldsymbol{K}})$ and $R_\text{AFM}(\boldsymbol{\boldsymbol{\Gamma}})$ to determine the QCP for the KVBS and AFM phases at $\Omega=0.1t$. Here as we vary the lattice size, we set $\beta t =L$ to determine the location of the ground-sate phase boundaries. (a) $R_\mathrm{VBS}(\boldsymbol{K})$ vs. $\lambda$ for fixed $U=0.2t$. The QCP for the SM-KVBS phase transition is observed at $\lambda_\mathrm{c} \approx 0.175$. (b) $R_\mathrm{VBS}(\boldsymbol{K})$ vs $U$ at $\lambda=0.15$. Two crossing points are observed at $U_\mathrm{c}\approx2.5t$ and $5.3t$, indicating two QCPs for this value of the $e$-ph coupling strength. (c) $R_\mathrm{AFM}(\boldsymbol{\Gamma})$ vs $U$ for a constant $\lambda=0.11$, with a QCP observed at $U \approx 4.05t$. Dashed lines in all panels indicate the estimated location of QCPs while the solid lines serve are guides to the eye.
• Figure 4: AFM $S_\mathrm{AFM}(\boldsymbol{\Gamma})$ (red $\Box$) and KVBS $S_\mathrm{VBS}(\boldsymbol{K})$ (blue $\triangledown$) structure factors as a function of inverse temperature $\beta t$ for an $L=12$ lattice within the coexistence region of the phase diagram ($\Omega=0.1t$, $U=4.5t$, and $\lambda=0.14$).