Abstract An analytic theory for the spectral function for electrons coupled with phonons is formulated in the adiabatic limit. In the case when the chemical potential is large and negative μ → − ∞ the ground state does not have the adiabatic deformation and the spectral function is defined by the standard perturbation theory. In this limit we use the diagram technique in order to formulate an integral equation for the renormalized vertex. The spectral function was evaluated by solving the Dyson’s equation for the self-energy with the renormalized vertex. The moments of the spectral function satisfy the exact sum rules up to the 7th moment. In the case when the chemical potential is pinned at the polaron binding energy the spectral function is defined by the ground state with a nonzero adiabatic deformation. We calculate the spectral function with the finite polaron density in the adiabatic limit. We also demonstrate how the sum rules for higher moments may be evaluated in the adiabatic limit. Contrary to the case of zero polaron density the spectral function with the finite polaron concentration has some contributions which are characteristic for polarons.
Abstract An analytic theory for the spectral function for electrons coupled with phonons is formulated in the adiabatic limit. In the case when the chemical potential is large and negative μ → − ∞ the ground state does not have the adiabatic deformation and the spectral function is defined by the standard perturbation theory. In this limit we use the diagram technique in order to formulate an integral equation for the renormalized vertex. The spectral function was evaluated by solving the Dyson’s equation for the self-energy with the renormalized vertex. The moments of the spectral function satisfy the exact sum rules up to the 7th moment. In the case when the chemical potential is pinned at the polaron binding energy the spectral function is defined by the ground state with a nonzero adiabatic deformation. We calculate the spectral function with the finite polaron density in the adiabatic limit. We also demonstrate how the sum rules for higher moments may be evaluated in the adiabatic limit. Contrary to the case of zero polaron density the spectral function with the finite polaron concentration has some contributions which are characteristic for polarons.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations. The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion. Particular emphasis is laid on the different physics occurring in the adiabatic and anti-adiabatic regimes, respectively. Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit. This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
The phonon spectral function of the one-dimensional Holstein model is obtained within weak and strong-coupling approximations based on analytical self-energy calculations.The characteristic excitations found in the limit of small charge-carrier density are related to the known (electronic) spectral properties of Holstein polarons such as the polaron band dispersion.Particular emphasis is laid on the different physics occurring in the adiabatic and antiadiabatic regimes, respectively.Comparison is made with a cluster approach exploiting exact numerical results on small systems to yield an approximation for the thermodynamic limit.This method, similar to cluster perturbation theory, confirms the analytical findings, and yields accurate results also in the intermediate-coupling regime.
A dynamical mean-field theory of the small polaron problem is presented, which becomes exact in the limit of infinite dimensions. The ground-state properties and the one-electron spectral function are obtained for a single electron interacting with Einstein phonons by a mapping of the lattice problem onto a polaronic impurity model. The one-electron propagator of the impurity model is calculated through a continued fraction expansion, at both zero and finite temperature, for any electron-phonon coupling and phonon energy. In contrast to the ground-state properties, such as the effective polaron mass, which show a continuous behavior as the coupling is increased, spectral properties exhibit a sharp qualitative change at low enough phonon frequency: beyond a critical coupling, one energy gap and then more open in the density of states at low energy, while the high-energy part of the spectrum is broad and can be qualitatively explained by a strong coupling adiabatic approximation. As a consequence, narrow and coherent low-energy subbands coexist with an incoherent featureless structure at high energy. The subbands denote the formation of quasiparticle polaron states. Also, divergencies of the self-energy may occur in the gaps. At finite temperature such an effect triggers an important damping and broadening of the polaron subbands. On the other hand, in the large phonon frequency regime such a separation of energy scales does not exist and the spectrum always has a multipeaked structure.
Spectral functions are important quantities that contain a wealth of information about the quasiparticles of a system, and that can also be measured experimentally. For systems with electron-phonon coupling, good approximations for the spectral function are available only in the Migdal limit (at Fermi energies much larger than the typical phonon frequency, E_F\gg \Omega E F ≫ Ω , requiring a large carrier concentration x x ) and in the single polaron limit (at x=0 x = 0 ). Here we show that the region with x\ll 1 x ≪ 1 ( E_F<\Omega E F < Ω ) can also be reliably investigated with the Momentum Average (MA) variational approximation, which essentially describes the formation of a polaron above an inert Fermi sea. Specifically, we show that for the one-dimensional spinless Holstein model, the MA spectral functions compare favorably with those calculated using variationally exact density matrix renormalization group simulations (DMRG) evaluated directly in frequency-space, so long as x<0.1 x < 0.1 and the adiabaticity ratio \Omega/t>0.5 Ω / t > 0.5 . Unlike in the Migdal limit, here 'polaronic physics’ emerges already at moderate couplings. The relevance of these results for a spinful low- x x metal is also discussed.
We present a highly efficient yet accurate analytical approximation for the Green's function of a Holstein polaron. It is obtained by summing all the self-energy diagrams, but with each self-energy diagram averaged over the momenta of its free propagators. The result becomes exact for both zero bandwidth and for zero electron-phonon coupling and is accurate everywhere in the parameter space. The resulting Green's function satisfies exactly the first six spectral weight sum rules. All higher sum rules are satisfied with great accuracy, becoming asymptotically exact for coupling both much larger and much smaller than the free particle bandwidth. Comparison with existing numerical data also confirms this accuracy. We use this approximation to analyze in detail the redistribution of the spectral weight as the coupling strength varies.
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.