Spectral function of the Holstein polaron at finite temperature

We compute the Holstein polaron spectral function on a one-dimensional ring using the finite-temperature $T$ Lanczos method. With increasing $T$ additional features in the spectral function emerge even at temperatures below the phonon frequency. We observe a substantial spread of the spectral weight towards lower frequencies and the broadening of the quasiparticle (QP) peak. In the weak-coupling regime the QP peak merges with the continuum in the high-$T$ limit. In the strong-coupling regime the main features of the low-$T$ spectral function remain detectable up to the highest $T$ used in our calculations. The effective polaron mass shows a nonmonotonic behavior as a function of $T$ at small phonon frequency but increases with $T$ at larger frequencies. The self-energy remains $k$ independent even at elevated $T$ in the frequency range corresponding to the polaron band, while at higher frequencies it develops a distinguishable $k$ dependence. Analytical expressions for the first few frequency moments are derived, and they agree well with those extracted from numerical calculations in a wide-$T$ regime.

Figure 1

$A(\omega ,k=0)$ at ${\omega }_0=1.0$ computed for $\lambda =0.5$ through $\lambda =2.0$ from (a) to (c). Temperature is measured in units of the hopping $t_0=1$. In this and all subsequent figures we have used the following parameters when generating the variational Hilbert space: $N_h=22,L=6,\eta =0.05$, and $M=500$. The Hilbert space spans over $N_1\sim 120000$ states in the subspace of one electron and $N_0=23000$ different phonon degrees of freedom, taking into account all translationally invariant states in the zero-electron case. We have sampled over $R=90$ distinct initial random states and all $k$ sectors. Horizontal red lines with arrows on both sides indicate the corresponding ${\omega }_0$. Red vertical dashed lines point towards positions of QP peaks, while the blue dashed line indicates the position of the EQP peak in (c).

Figure 2

$A(\omega ,k)$ for $\lambda =1$ vs different values of $T/{\omega }_0$ and ${\omega }_0=1$ in (a)–(d) and ${\omega }_0=0.5$ in (e)–(h). The same size of the system was used as in Fig. 1. In addition we have used twisted boundary conditions to compute $A(\omega ,k)$ at 25 equally spaced $k$ points in the interval $k\in [0,\pi ]$ with increments of $\mathrm{\Delta }k=\pi /24$. Note that in all plots from (a) to (h) the same color coding was used to enable direct comparison between different cases. In (h) a free-electron band $\epsilon \left(k\right)=-2t_{0}cos\left(k\right)$ is shown using a dashed line as a guide to the eye. Lorentzian broadening $\eta =0.05$ was used in all cases. In (i) we display $m^{*}/m_0$, where $m_0=1/2t_0$ is the free-electron mass, obtained from fits to the polaron band using the analytical form ${\epsilon }_f\left(k\right)={\sum }_n{a}_{n}cos\left(nk\right)$, up to $n=3$, shown as dashed lines in (a)–(c) and (e)–(g).

Figure 3

$A(\omega ,k)$ and $\mathrm{\Sigma }(\omega ,k)$ for $\lambda ={\omega }_0=1$ computed at values of $T/{\omega }_0=0.1$ in (a) and (c) and 0.5 in (b) and (d). The same size of the system was used as in Figs. 1 and 2. In each plot we present 25 curves computed at equally spaced $k$ points in the interval $k\in [0,\pi ]$ with increments of $\mathrm{\Delta }k=\pi /24$. Lorentzian broadening $\eta =0.05$ was used in all cases, which is also responsible for a small deviation from zero in (c) for $\omega \lesssim -2.7$.

Figure 4

$M_2$ and $M_3$ for different values of $\lambda$ vs $T$ for $k=0$ in (a) and $\pi$ in (b). Open symbols represent moments obtained from numerical results of $A(\omega ,k)$, where instead of Lorentzian broadening we have used a Gaussian one with $\sigma =0.05$ to ensure convergence of integrals. The lines represent analytical results given by Eqs. (8). Insets: $I\left(\omega \right)$ for different values of $\lambda$ and $k=0$ in (a) and $\pi$ in (b). There are six curves for each set of $\lambda$ and $k$ computed at values of $T\in [0.2,0.6,\cdots ,1.8,2.0]$. Plots were obtained by straightforward numerical integrations of $A(\omega ,k)$, such as presented in Figs. 1 and 5.

Figure 5

$A(\omega ,k=\pi )$ at ${\omega }_0=1.0$ computed for $\lambda =0.5$ through $\lambda =2.0$ from (a) to (c). The rest is the same as in Fig. 1.

Figure 6

$A(\omega ,k)$ for $\lambda ={\omega }_0=1$ for two different system sizes, $L=6$ in (a), (c), and (e) and $L=12$ in (b), (d), and (f), and three different values of $T/{\omega }_0$ as specified in plots. In all cases $A(\omega ,k)$ was computed in 25 equally spaced nonequivalent $k$ points in the interval $k\in [0,\pi ]$ with increments of $\mathrm{\Delta }k=\pi /24$. Note that in all plots from (a) to (f) the same color coding was used to enable direct comparison between different cases.

Figure 7

$A\left(\omega \right)$ obtained from Eqs. (C4) and (C5) using $g=2$ and ${\omega }_0=1$, where sums were taken up to $n,m=30$. Note that the temperatures are given in units of ${\omega }_0$. Lorentzian broadening was used with $\eta =0.05$.