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arXiv:2303.05013 · SciPost Physics

Electron addition spectral functions of low-density polaron liquids

Alberto Nocera, Mona Berciu

Published Sep 22, 2023• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.21468/scipostphys.15.3.110

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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 use the density matrix renormalization group (DMRG) to compute the phonon spectral function of a one-dimensional spinless Holstein model doped with a low but finite carrier concentration, x≤0.15x \leq 0.15x≤0.15, as a function of the electron-phonon coupling λλλ. To the best of our knowledge, these are the first such results in this regime, complementing extensive prior work at the single-polaron level (x→0x\to 0x→0). We find that significant phonon spectral weight is transferred both below, all the way down to ω=0ω=0ω=0, and above the bare phonon energy ΩΩΩ, in stark contrast with the Kohn-anomaly phenomenology expected in the Migdal limit, where weight remains centered near ΩΩΩ with a kink at q=2kFq=2k_Fq=2kF​. No signature of this 2kF2k_F2kF​ kink appears in our results. This behavior is captured qualitatively by the Random Phase Approximation (RPA), and semi-quantitatively, at negligible extra computational cost, by a ``dressed RPA'' scheme in which the electron addition propagator is renormalized using the Momentum Average (MA) approximation for the low-density electron-polaron. By contrast, adding the lowest-order vertex correction to this dressed scheme produces unphysical negative spectral weight, signaling that vertex and propagator dressings must be treated consistently once the propagators are dressed nonperturbatively. Our results provide an efficient approximation for the phonon spectral functions of low-density polaron metals, a regime relevant to weakly doped insulators.
The Momentum Average (MA) approximation is a variational approximation that has been applied to analytically calculate the spectral function of single polarons and bipolarons, and was found to quantitatively match numerical results apart from the strongly adiabatic limit. The MA approximation was recently extended to a low carrier concentration of spinless fermions in 1D by making the additional assumption that the N-fermion ground-state is approximated well by the mean-field ground-state. This allowed the spectral function to be calculated analytically outside the Migdal limit. Good agreement was found with Density Matrix Renormalization Group simulations for fermion concentrations up to 0.15. In this thesis, the same approximation is extended to calculate the spectral function of a Holstein polaron for low carrier concentrations of spinless fermions on square and cubic lattices. Polaron effective mass and quasiparticle weight were found to depend strongly on the concentration x, especially in 3D. The 2D mass renormalizations were found to give a close match (< 5% error) to Bold Diagrammatic Monte Carlo (BDMC) results from the literature at all x < 0.2 when coupling is weak. However, there were significant quantitative differences between MA and BDMC results at large coupling. A possible explanation for this discrepancy is the presence of a soft excitation close to the polaron ground state. In addition, an efficient numerical scheme to calculate the real-space free electron propagator for a noninteracting Fermi sea is presented, which is used to calculate the self-energy for the 3D Green’s function at finite concentrations.
The Momentum Average (MA) approximation is a variational approximation that has been applied to analytically calculate the spectral function of single polarons and bipolarons, and was found to quantitatively match numerical results apart from the strongly adiabatic limit. The MA approximation was recently extended to a low carrier concentration of spinless fermions in 1D by making the additional assumption that the N-fermion ground-state is approximated well by the mean-field ground-state. This allowed the spectral function to be calculated analytically outside the Migdal limit. Good agreement was found with Density Matrix Renormalization Group simulations for fermion concentrations up to 0.15. In this thesis, the same approximation is extended to calculate the spectral function of a Holstein polaron for low carrier concentrations of spinless fermions on square and cubic lattices. Polaron effective mass and quasiparticle weight were found to depend strongly on the concentration x, especially in 3D. The 2D mass renormalizations were found to give a close match (< 5% error) to Bold Diagrammatic Monte Carlo (BDMC) results from the literature at all x < 0.2 when coupling is weak. However, there were significant quantitative differences between MA and BDMC results at large coupling. A possible explanation for this discrepancy is the presence of a soft excitation close to the polaron ground state. In addition, an efficient numerical scheme to calculate the real-space free electron propagator for a noninteracting Fermi sea is presented, which is used to calculate the self-energy for the 3D Green’s function at finite concentrations.
Abstract We propose a simple variational solution for calculating one-particle spectral functions in lattice models of spinless metals with strong electron-phonon coupling. It is based on a generalization of the Momentum Average variational approximation for single polarons, combined with the assumption that the other fermions in the system are locked into an inert Fermi sea. We expect the method to be accurate for fermion addition spectral functions in metals with a small Fermi energy (nearly empty band), and for fermion removal spectral functions in metals with a large Fermi energy (nearly full band), provided that the characteristic phonon frequency is not too small. Both these regions are far from the region where the Migdal theorem holds, thus our results offer new insights into polaronic behavior in a largely unexplored part of the parameter space. Here, we show results for the Holstein coupling in one-dimension and present ways to gauge their accuracy, but ultimately this will need to be verified against numerical calculations. This variational method can be extended straightforwardly to higher dimensions and other forms of electron-phonon coupling.
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.
Many-polaron systems with finite charge-carrier density are often encountered experimentally. However, until recently, no satisfactory theoretical description of these systems was available even in the framework of simple models such as the one-dimensional spinless Holstein model considered here. In this work, previous results obtained using numerical as well as analytical approaches are reviewed from a unified perspective, focussing on spectral properties which reveal the nature of the quasiparticles in the system. In the adiabatic regime and for intermediate electron-phonon coupling, a carrier-density driven crossover from a polaronic to a rather metallic system takes place. Further insight into the effects due to changes in density is gained by calculating the phonon spectral function, and the fermion-fermion and fermion-lattice correlation functions. Finally, we provide strong evidence against the possibility of phase separation.