Thermodynamics of a spin-$\frac{1}{2}$ chain coupled to Einstein phonons

A high-order series expansion is employed to study the thermodynamical properties of a $S=1∕2$ chain coupled to dispersionless phonons. The results are obtained without truncating the phonon subspace since the series expansion is performed formally in the overall exchange coupling $J$. The results are used to investigate various parameter regimes, e.g., the adiabatic and antiadiabatic limit as well as the intermediate regime, which is difficult to investigate by other methods. We find that dynamic phonon effects become manifest when more than one thermodynamic quantity is analyzed.

Figure 1

(Color online) Schematic picture of the spin-phonon model as defined by Eq. (1).

Figure 2

(Color online) Dlog-Padé extrapolations of the specific heat. Various extrapolations are compared for three different sets of parameters.

Figure 3

(Color online) Dlog-Padé extrapolations of the specific heat compared to QMC data. The left plot shows data for the values $g=0.66$, $\omega =0.66J$, and the right plot depicts the parameters $g=0.44$, $\omega =0.88J$.

Figure 4

(Color online) Truncated series of the susceptibility. Various sets of parameters are shown. The exactly known result for the Heisenberg model serves as a reference. The left panels illustrate the adiabatic limit whereas the right panels show the results in the antiadiabatic limit.

Figure 5

(Color online) Dlog-Padé extrapolations of the susceptibility. Various extrapolation schemes are compared for three different sets of parameters. The parameters in the first two rows of panels correspond to the gapped regime whereas the lowest row depicts results in the gapless regime.

Figure 6

(Color online) Dlog-Padé extrapolations of the susceptibility compared to QMC data. The left plot shows data for the values $g=0.66$ and $\omega =0.66J$ (gapped phase), and the right plot depicts data for the values $g=0.44$ and $\omega =0.88J$ (gapless phase).

Figure 7

(Color online) “Theoretical” experiment: The data for $g=1$ and ${\omega }_0=J$ (symbols) are taken as (mock) experimental input. It is fitted by the susceptibility of static spin model (solid lines). The temperature $T_{\max }$ denotes at which temperature the susceptibility has its maximum; this value sets the natural energy scale of the problem.

Figure 8

(Color online) Comparison of the specific heat for the parameter sets determined from the susceptibility in Fig. 8. For the upper plot the phonon frequency is assumed to be known beforehand. For the lower plot no a priori knowledge is assumed for the phonon frequency; thus, it is also fitted. The line styles are the same as those used in Fig. 7; that means, the exchange couplings used are those from the corresponding curves in Fig. 7.