Diagrammatic Monte Carlo study of Fröhlich polaron dispersion in two and three dimensions

We present results for the solution of the large polaron Fröhlich Hamiltonian in 3 dimensions (3D) and 2 dimensions (2D) obtained via the diagrammatic Monte Carlo (DMC) method. Our implementation is based on the approach by Mishchenko [A. S. Mishchenko et al., Phys. Rev. B 62, 6317 (2000)]. Polaron ground state energies and effective polaron masses are successfully benchmarked with data obtained using Feynman's path integral formalism. By comparing 3D and 2D data, we verify the analytically exact scaling relations for energies and effective masses from $3\mathrm{D}\to 2\mathrm{D}$, which provides a stringent test for the quality of DMC predictions. The accuracy of our results is further proven by providing values for the exactly known coefficients in weak- and strong-coupling expansions. Moreover, we compute polaron dispersion curves which are validated with analytically known lower and upper limits in the small-coupling regime and verify the first-order expansion results for larger couplings, thus disproving previous critiques on the apparent incompatibility of DMC with analytical results and furnishing useful reference for a wide range of coupling strengths.

Figure 1

Eighth-order diagram for $G^{\left(2\right)}(\mathbf{k},\tau ,{\tilde{\mathbf{q}}}_1,{\tilde{\mathbf{q}}}_2)$. Note that diagrams in the expansion of $G^{\left(2\right)}$ have two phonon propagators attached to the diagram end. The total polaron wave vector $\mathbf{k}={\mathbf{k}}_1+{\tilde{\mathbf{q}}}_1+{\tilde{\mathbf{q}}}_2$ is conserved at the vertices.

Figure 2

General workflow of the DMC algorithm. The algorithm returns the histogram of the function $Q\left(\right\{y\left\}\right)$.

Figure 3

Detailed workflow of the DMC algorithm as it was used in this paper. The algorithm returns estimates for the lowest eigenenergy $E_0(k,\alpha )$ and the inverse of the effective polaron mass $1/m_{*}\left(\alpha \right)$ for given $\mathbf{k}$ and $\alpha$ values.

Figure 4

Comparison of our results (circles) with previous DMC results by Mishchenko [22] (continuous lines) and with results obtained with Feynman's approach [15] (dashed lines). The top graph shows the polaron ground state energy $E_0(0,\alpha )$ and the bottom graph the logarithm of the polaron effective mass $log{m}_{*}\left(\alpha \right)$ as a function of $\alpha$.

Figure 5

Polaron energy $E_0(0,\alpha )$ as a function of the coupling constant $\alpha$. The modulus of the total wave vector is $k=0$. Results from the Feynman approach are shown as dashed lines. DMC results for 3D systems are depicted as squares and for 2D as circles. $\mathrm{\Delta }{E}_0$ is the difference between Feynman and DMC results. The inset shows the scaling ratio $R_E\left(\alpha \right)=E_0^{2D}(0,\alpha )/E_0^{3D}(0,3\pi \alpha /4)$ between our 2D and 3D DMC results.

Figure 6

Logarithm of the polaron effective mass $m_{*}\left(\alpha \right)$ as a function of the coupling constant $\alpha$. Results from the Feynman approach are shown as dashed lines. DMC results for 3D systems are depicted as squares and for 2D as circles. The inset shows the scaling ratio $R_{m_{*}}\left(\alpha \right)=m_{*}^{2D}\left(\alpha \right)/m_{*}^{3D}\left(3\pi \alpha /4\right)$ between our 2D and 3D DMC results.

Figure 7

Polaron energy $E_0(k,\alpha )-E_0(0,\alpha )$ as a function of the modulus of the total wave vector $k$ in 3D (left, for coupling constants $\alpha =1.0,2.0$, and 3.0) and 2D (right, $\alpha =0.5,1.0$, and 1.5). The continuum edge is shown at $E_c\left(k\right)=1$.

Figure 8

Polaron energy $E_0(k,\alpha )$ in 3D (left) and 2D (right) as a function of the modulus of the total wave vector $k$ for coupling constant $\alpha =0.068$ (top row) and $\alpha =0.5$ (bottom row). Lower and upper bounds and a fitting function to the dispersion are taken from Ref. [20].