Fermi blockade of the strong electron–phonon interaction in modelled optimally doped high temperature superconductors

We study how manifestations of strong electron–phonon interaction depend on the carrier concentration by solving the two-dimensional Holstein model for the spin-polarized fermions using an approximation free bold-line diagrammatic Monte Carlo method. We show that the strong electron–phonon interaction, obviously present at very small Fermion concentration, is masked by the Fermi blockade effects and Migdal’s theorem to the extent that it manifests itself as moderate one at large carriers densities. Suppression of strong electron–phonon interaction fingerprints is in agreement with experimental observations in doped high temperature superconductors.


Size dependence
To check whether the system size N × N = 16 × 16 is sufficient to reproduce properties of the Holstein model for single polarons when the largest finite-size effects are expected, we calculated various characteristics of the polaron by the diagrammatic Monte Carlo 1, 2 and compared them with known infinite lattice results. In the simulations of finite lattice all momenta in the reciprocal space also form a lattice k x,y = (2π/N) j, −N/2 ≤ j < N/2 .
In Fig. 1 we show how the polaron energy, E, and quasiparticle residue, Z, depend on the lattice size for N = 4, 8, 16, 32, 64, 128, ∞, and conclude that N = 16 results reproduce the infinite system limit with accuracy of three to four significant digits.    Table 1. Relations between the chemical potential, µ, fermion density per site, δ , and ratio between the Fermi energy and phonon frequency e F /ω ph . To establish them one needs to account for skeleton diagrams up to order m.
Convergence properties of the skeleton expansion strongly depend on the fermion density δ (or chemical potential, µ, in the grand canonical ensemble). In Fig. 2. we present our BLDMC data for density dependence on the expansion order at low temperature T = t/20 and different values of µ. At low density one needs to account for vertex corrections up to order 16 to obtain converged results. Note that the chemical potential µ is not directly related to the Fermi energy counted counted from the bottom of the dispersion relation which is strongly renormalized by interactions. Table 1 provides final relations between all quantities, including the required expansion order.

Ground state energy, Z-factor, and spectral function of a single polaron
In Fig. 3 we present the spectral function of a single polaron in the infinite in finite N 2 = 16 2 systems. Nearly perfect agreement (well within the analytic continuation procedure uncertainties) proves that finite-size effects in this case are negligible not only for ground state energies but also for excited states.  To determine the quasiparticle residue and interaction induced energy shift, ∆E = E − (−4t), we rely on the standard reliable method: at large imaginary time the asymptotic decay of the Green's function is given by G(τ) → τ→∞ Z exp(−∆Eτ) , see 1, 2 , allowing one to extract Z and ∆E from a simple exponential fit. The leading correction decays with exponent controlled by the lowest excited state (the second polaron state according to the spectral density analysis). In Fig. 4 we show how Z and E 3/5 estimates change when we move the fitting interval [0.95τ max , 1.05τ max ] to larger values of τ max . It is clear from Fig. 3 that the energy dependence on τ max within the range τ max ∈ [14, 80] is very weak (about 2%). This is in sharp contrast, with the quasiparticle residue estimates: Z increases by nearly 40% when τ max decreases from τ max = 80 to τ max = 14. This sensitivity explains the discrepancy between the calculations performed at finite temperature T = t/20 and at T = 0. We attribute it to the presence of the second polaron state with comparable Z factor and relatively small excitation energy E 2 − E G ≈ 0.17t.

Relation of the single polaron parameters and results of extrapolation procedure for BDMC data
The extrapolation procedure is validated by an excellent agreement between the BDMC result for the ground state energy of single-polarons, E(m → ∞) = −4.89 and the DMC benchmark E 1 = −4.891 . In the same limit, the extrapolated result for the QP residue Z(m → ∞) ≈ 0.33 turns out to be larger than that for single polarons, Z 1 = 0.238. The reason for the discrepancy is a combination of the finite temperature effect and self-trapping phenomenon 3,4 , manifesting itself as a second, low energy, E 2 − E 1 ≈ 0.17 < ω ph , excited polaron state with rather large spectral weight, Z 2 ≈ 0.3, clearly observed in the spectrum of single polarons at T = 0, see Fig. 3. Because of this soft excitation, the standard procedure of extracting Z from the large-τ asymptotic behavior of the imaginary time Green function G(τ) 1, 2 turns out to be sensitive to the choice of the large imaginary time used to fit the data (for τ < 40), whereas the estimate for energy remains accurate even for τ < 20, see Fig. 4 . Therefore, at T = t/20 we detect the QP weight that overestimates Z 1 of single polarons in the ground state. Semi-quantitatively, the finite temperature BDMC result can be understood from the relation Z β /2 = Z 1 + Z 2 exp[−(β /2)(E 2 − E 1 )] ≈ 0.31, which accounts for the activated second polaron state contribution at τ max = β /2.

Dependence of the crossover to the strong coupling regime on the phonon frequency
We studied how the ground state energy E, quasiparticle-residue Z, and average number of phonons N ph in the polaron cloud, for a single Holstein polaron depend on the dimesionless coupling constant λ and the phonon frequency, see 6 Filling factor dependence of the quasiparticle resudue Z in the Hubbard-Holstein model The Hubbard-Holstein Hamiltonian on a square lattice reads: where c † is /b † i are standard notations for electron/phonon creation operators on site i with spin s =↑ / ↓, t is the nearest neighbor hopping amplitude, n is = c † is c is , ω ph = 0.5t is the energy of the local optical mode. U is the repulsive on-site repulsion, and g is the EPI coupling. The dimensionless EPI coupling constant λ = g 2 /(4ω ph t) is chosen to be λ = 1.07, i.e. the same as in the main text. We find that the bi-polaron instability is prevented when the on-site repulsion is increased to U = 2t. Concentration dependence of the quasiparticle residue at the Fermi surface for this model is presented in Fig. 6. The DiagMC simulation for model (1) is far more demanding compared to the spin-polarized case discussed in the main text, and without additional algorithmic developments cannot be carried out to expansion orders higher than (m = 7).