The two-dimensional optical Su-Schrieffer-Heeger model: ground state and thermodynamic properties

TL;DR

This work investigates the two-dimensional optical-SSH model, where electron hopping is modulated by differences in neighboring phonon coordinates, using sign-problem-free auxiliary-field quantum Monte Carlo and mean-field theory to map ground-state and finite-temperature phases. The study reveals staggered and armchair valence-bond solid orders and an $O(4)$-symmetric AFM/CDW/SC sector, with finite-temperature VBS transitions that occur at relatively high temperatures due to light polarons. In the antiadiabatic limit, the ground state favors AFM/CDW/SC for any finite EPC, while at finite phonon frequencies a staggered-to-armchair VBS transition occurs at a finite critical coupling, highlighting qualitative differences from Holstein and bond-SSH models. The results provide comprehensive ground-state and finite-temperature phase diagrams and benchmarks for electron-phonon systems with dispersive phonons, offering insights for potential experimental realizations.

Abstract

We investigate the two-dimensional optical Su-Schrieffer-Heeger (SSH) model, in which the electron hopping amplitude is modulated by the difference between neighboring phonon coordinates. Using sign-problem-free auxiliary-field quantum Monte Carlo simulations, complemented by mean-field analysis, we determine the long-range ordered phases as a function of the electron-phonon coupling and phonon frequency. By examining both adiabatic and antiadiabatic regimes, we reveal the emergence of staggered and armchair valence bond solid (VBS) phases, as well as the O(4) antiferromagnetic phase. In addition, finite-temperature simulations show that the VBS transition occurs at critical temperatures significantly higher than in models with local electron-phonon coupling, consistent with the presence of lighter polarons in the metallic regime. These findings establish the ground-state and finite-temperature phase diagrams of the optical SSH model, which emphasize its similarities and contrasts with other electron-phonon systems.

Figures (14)

• Figure 1: The ground state phase diagram of the optical-SSH model for the half-filled square lattice. Black symbols indicate QMC results, while lines are guides to the eye. Blue shading indicates the parameter region where the AFM/CDW/SC phase occurs. The orange and green patterns indicate the type of VBS ordering.
• Figure 2: Bond-bond correlation function $C(r)$ as a function of the distance $r=|\mathbf{i}-\mathbf{j}|$ between two given sites along high-symmetry directions of the square lattice, shown for several temperatures. The inset illustrates the first quadrant of the lattice. Here, and in all subsequent figures, when not shown, error bars are smaller than the symbol size.
• Figure 3: Staggered bond structure factor as a function of $\beta t$, for fixed $\omega_{0}=1$, and (a) $\lambda / t = 0.058$, (b) 0.065, and (c) 0.092, and several system sizes. The solid lines are just guides to the eye.
• Figure 4: (a) Finite-size scaling analysis of the bond structure factors (a) $S_{\rm B} (\pi,\pi)$ and (b) $S_{\rm B} (0,\pi)$ at low temperatures, for different values of $\lambda$. (c) Extrapolated VBS order parameter as a function of $\lambda/t$, for both $(\pi,\pi)$ and $(0,\pi)$ modes. The dashed lines are guides to the eye.
• Figure 5: Finite-size scaling analysis of the bond structure factor $S_{\rm B} (\pi,\pi)$ for different values of $\lambda$, while fixing $\beta t = 40$, (a) $\omega_{0}=2$, (b) $\omega_{0}=4$, and (c) $\omega_{0}=8$.