Variational wave functions for the spin-Peierls transition in the Su-Schrieffer-Heeger model with quantum phonons

We introduce variational wave functions to evaluate the ground-state properties of spin-phonon coupled systems described by the Su-Schrieffer-Heeger model. Quantum spins and phonons are treated on equal footing within a Monte Carlo sampling, and different regimes are investigated. We show that the proposed variational Ansatz yields good agreement with previous density-matrix renormalization group results in one dimension and is able to accurately describe the spin-Peierls transition. This variational approach is constrained neither by the magnetoelastic-coupling strength nor by the dimensionality of the systems considered, thus allowing future investigations in more general cases, which are relevant to spin-liquid and topological phases in two spatial dimensions.

Figure 1

Relative error of the variational energies with respect to Lanczos results $\delta E$ for a chain of $L=8$ sites. The relative error is plotted (as a percentage) as a function of $g/\omega$ for three values of the adiabatic parameter, $\omega /J=0.1$ (top panel), $\omega /J=1$ (middle panel), and $\omega /J=10$ (bottom panel). We note that the scale of the vertical axis is different in the three panels. Two sets of data are shown: blue circles represent the results obtained with the ${\mathcal{J}}_n$ spin-phonon Jastrow factor [Eq. (8)], while red squares correspond to the results obtained with the ${\mathcal{J}}_X$ spin-displacement Jastrow factor [Eq. (10)]. Error bars are smaller than the size of the dots.

Figure 2

Order parameter for the lattice deformation $\mathrm{\Delta }X$ [Eq. (14)] as a function of $g/\omega$. Different lattice sizes $L$ are considered, as well as three different regimes, $\omega /J=0.1$ (left panel), $\omega /J=1$ (middle panel), and $\omega /J=10$ (right panel). Error bars are smaller than the size of the dots. The gray shaded area marks the region in which $\mathrm{\Delta }X$ becomes finite in the thermodynamic limit. The hatched area denotes the position of the critical point (and its uncertainty) according to density-matrix renormalization group calculations [39].

Figure 3

Finite-size scaling of the Fourier-transformed dimer-dimer correlations $D^2$ [Eq. (15)]. Results for $\omega /J=0.1$ (left panel), $\omega /J=1$ (middle panel), and $\omega /J=10$ (right panel) are shown. We note that the scale of the vertical axis is different in the left panel to account for the different orders of magnitude of the correlation functions. The error bars are smaller than the size of the dots.

Figure 4

Energy gain (in units of $J$) due to the SSH coupling of the spins with phonons. The difference between the variational energies of the full SSH system, $E_{\mathrm{var}}(g,\omega ,J)$, and the one of the simple Heisenberg model, $E_{\mathrm{var}}(g=0,\omega =0,J)$, where no phonons are present, is displayed. The results are obtained for a chain with $L=250$ sites as a function of $g/\omega$ for $\omega /J=0.1$ (left panel), $\omega /J=1$ (middle panel), and $\omega /J=10$ (right panel). We note that the scale of the vertical axis is different in the various panels in order to account for the different orders of magnitude of the energy gain. As in Fig. 2, the gray shaded area marks the region in which we observe the onset of Peierls dimerization. The error bars are smaller than the size of the dots.

Figure 5

Average number of phonons per site $\left[\langle n_j\rangle \right]$ as a function of $g/\omega$ for $\omega /J=0.1$ (left panel), $\omega /J=1$ (middle panel), and $\omega /J=10$ (right panel). The results refer to a chain with $L=250$ sites. As in Fig. 2, the gray shaded area marks the region in which we observe the onset of Peierls dimerization. The error bars are smaller than the size of the dots.