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Physical review. B./Physical review. B

Spectral function of the Holstein polaron at finite temperature

J. Bonča, S. A. Trugman, Mona Berciu

Published Sep 18, 2019• 31 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.100.094307

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We compute the Holstein polaron spectral function on a one-dimensional ring using the finite-temperature TTT Lanczos method. With increasing TTT 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-TTT limit. In the strong-coupling regime the main features of the low-TTT spectral function remain detectable up to the highest TTT used in our calculations. The effective polaron mass shows a nonmonotonic behavior as a function of TTT at small phonon frequency but increases with TTT at larger frequencies. The self-energy remains kkk independent even at elevated TTT in the frequency range corresponding to the polaron band, while at higher frequencies it develops a distinguishable kkk dependence. Analytical expressions for the first few frequency moments are derived, and they agree well with those extracted from numerical calculations in a wide-TTT regime.
How a polaron loses its quasiparticle coherence with increasing temperature is a long-standing open problem. Holstein addressed it in 1959 only in the extreme antiadiabatic, strong-coupling limit, while more recent numerically exact approaches are largely restricted to one dimension. Here we solve this problem on square and simple cubic lattices across the weak-, intermediate-, and strong-coupling regimes. We show that the polaron effective mass m∗m^*m∗ and inverse lifetime 1/τ1/τ1/τ increase monotonically with temperature until, at T∼0.5 ΩT \sim 0.5\,ΩT∼0.5Ω, the quasiparticle peak dissolves into a broad incoherent thermal continuum. The phonon frequency ΩΩΩ therefore defines a universal coherence scale, independent of dimensionality and coupling strength, validating Holstein's prediction far beyond the regime in which it was derived. These results follow from a finite-temperature generalization of the Momentum Average (MA) approximation, yielding a closed-form, diagrammatically derived self-energy that is asymptotically exact in the strong-coupling limit at all temperatures. Benchmark comparisons demonstrate excellent quantitative agreement of the resulting 1D spectral functions with the numerically exact Variational Exact Diagonalization-Finite-Temperature Lanczos Method (VED-FTLM) and finite-TTT Density Matrix Renormalization Group (DMRG).
We investigate the low-energy properties of the Holstein polaron through calculation of the qqq-dependent phonon spectral function using an improved exact-diagonalization technique, defined over a variational Hilbert space. We perform a comprehensive study of the low-energy excitations of the polaron. Beside the energy range, where the additional phonon excitation is unbound, we observe separate coherent peaks which correspond to bound and antibound states of a polaron and additional phonon quanta. These novel states can be observed for intermediate and strong electron-phonon coupling strengths, as well as below and above the unbound one-phonon excitation spectrum. A detailed investigation of their properties is presented. We find good agreement between the phonon spectral function obtained from the first-order strong-coupling perturbation theory and numerical results.
We have computed static and dynamic properties of an electron coupled to hard-core-boson (HCB) degrees of freedom in one spatial dimension. The polaron, an electron dressed with HCB excitations, remains light even in the strong-coupling limit as its effective mass remains of the order of the free-electron mass. This result is in a sharp contrast to the Holstein model where the electron effective mass increases exponentially with the electron-phonon coupling. HCB degrees of freedom mediate the attractive potential between two electrons that form a bound singlet bipolaron state at any nonzero coupling strength. In the low-frequency regime of the electron spectral function we observe a quasiparticle (QP) band that is separated from the continuum of states only in the central part of the Brillouin zone. The quasiparticle weight approaches zero as the QP band enters the continuum where it obtains a finite lifetime. At finite temperature an electron can annihilate thermally excited HCBs. Such thermally activated processes lead to a buildup of the spectral weight below the QP band. While the investigated model bears a resemblance with the Holstein model, we point out many important differences that originate from the binary HCB excitation spectrum, which in turn mimics spin-12\frac{1}{2}21​ degrees of freedom.
We present a systematic study of a mobile impurity immersed in a three-dimensional Fermi sea of fermions at finite temperature, by using the standard non-self-consistent many-body TTT-matrix theory that is equivalent to a finite-temperature variational approach with the inclusion of one-particle-hole excitation. The impurity spectral function is determined in the real-frequency domain, avoiding any potential errors due to the numerical analytic continuation in previous TTT-matrix calculations and the small spectral broadening parameter used in variational calculations. In the weak-coupling limit, we find that the quasiparticle decay rate of both attractive and repulsive polarons does not increase significantly with increasing temperature, and therefore Fermi polarons may remain well defined far above Fermi degeneracy. In contrast, near the unitary limit with strong coupling, the decay rate of Fermi polarons rapidly increases and the quasiparticle picture breaks down close to the Fermi temperature. We analyze in detail the recent ejection and injection radio-frequency (rf) spectroscopy measurements, performed at Massachusetts Institute of Technology (MIT) and at the European Laboratory for Non-Linear Spectroscopy (LENS), respectively. We show that the momentum average of the spectral function, which is necessary to account for the observed rf spectroscopy, has a sizable contribution to the width of the quasiparticle peak in spectroscopy. As a result, the measured decay rate of Fermi polarons could be significantly larger than the calculated quasiparticle decay rate at zero momentum. By taking this crucial contribution into account, we find that there is a reasonable agreement between theory and experiment for the lifetime of Fermi polarons in the strong-coupling regime, as long as they remain well defined.
We study the impact of the many-body effects on the phonon spectral function of Holstein polaron in one-dimension in the antiadiabatic regime by incorporating the contributions from the electron self-energy and vertex corrections within a weak-coupling approach that respects the charge-conserving Ward identity. We find that while the polaronic spectral weight is suppressed due to contribution from the electron self-energy, on the other hand, the same is enhanced due to contribution from the vertex corrections. While strength of both the contributions increases with increasing the wave vector ($\q$) of phonons, they nearly cancel each other for the small-$\q$ modes so that the polaronic spectral weight is weakly affected due to the many-body effects. For the large-$\q$ modes near the zone boundary, the net many-body correction is dominated by the contribution of the electron self-energy which increases faster in comparison to that of the vertex corrections with increasing the wave vector thereby resulting in a significant suppression of the polaronic spectral weight. We find that while the weak-coupling perturbative approach provides a reliable estimation of the impact of the many-body effects deep inside the antiadibatic regime, the renormalization of quasiparticle spectrum must be taken into account for an accurate estimation when the phonon energy approaches the electronic bandwidth.
We examine the range of validity of the second-order cumulant expansion (CE) for the calculation of spectral functions, quasiparticle properties, and mobility of the Holstein polaron. We devise an efficient numerical implementation that allows us to make comparisons in a broad interval of temperature, electron-phonon coupling, and phonon frequency. For a benchmark, we use the dynamical mean-field theory which gives, as we have recently shown, rather accurate spectral functions in the whole parameter space, even in low dimensions. We find that in one dimension, the CE resolves well both the quasiparticle and the first satellite peak in a regime of intermediate coupling. At high temperatures, the charge mobility assumes a power law $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ in the limit of weak coupling and $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}3/2}$ for stronger coupling. We find that, for stronger coupling, the CE gives slightly better results than the self-consistent Migdal approximation (SCMA), while the one-shot Migdal approximation is appropriate only for a very weak electron-phonon interaction. We also analyze the atomic limit and the spectral sum rules. We derive an analytical expression for the moments in CE and find that they are exact up to the fourth order, as opposed to the SCMA where they are exact to the third order. Finally, we analyze the results in higher dimensions.