Electron–Phonon Fields and Retarded Interactions

Integrating a harmonic lattice displacement produces an attractive electron interaction that is nonlocal in imaginary time. Its frequency dependence can enhance pairing, charge order, and mass renormalization, but an attractive kernel alone proves neither superconductivity nor a unique ordered channel.

Required background. Use effective lattice Hamiltonians and many-body coherent-state path integrals. Helpful background. Migdal’s theorem and vertex validity develops the controlled pairing approximation.

Holstein field and Gaussian integration

For a local optical phonon, take

$$H=H_e+\omega_0\sum_i b_i^\dagger b_i +g\sum_i(b_i+b_i^\dagger)(n_i-\bar n).$$

This local molecular-crystal coupling is the Holstein model Holstein 1959, pp. 325–342.

With $X=b+b^\dagger$, the free Matsubara propagator in this normalization is

$$D_0(i\nu_n)=\frac{2\omega_0}{\nu_n^2+\omega_0^2}.$$

The Euclidean phonon action has the Gaussian form $S_X=\frac12XD_0^{-1}X+gX\rho$. Completing the square and integrating $X$ gives

$$S_{\mathrm{int}}^{\mathrm{ret}} =-\frac{g^2}{2}\sum_{i,\nu_n} \rho_i(-i\nu_n)D_0(i\nu_n)\rho_i(i\nu_n), \qquad V_{\mathrm{eff}}(i\nu_n)=-g^2D_0(i\nu_n).$$

The sign is an exact Gaussian result. In the antiadiabatic limit $|\nu_n|\ll\omega_0$, it approaches the instantaneous attraction $-2g^2/\omega_0$ in this $X$ convention. A different normalization of displacement moves factors between $g$ and $D_0$ but leaves $g^2D_0$ invariant.

Competing effects

The same density attraction contributes in particle–particle and particle–hole channels. At or near commensurate filling, a Holstein model may favor a charge-density wave; away from it, pairing may dominate. Dressing by phonons narrows electronic motion and produces a polaronic mass enhancement. The result also competes with screened Coulomb repulsion, whose retardation is not captured by simply adding a static number.

Migdal–Eliashberg theory requires a small adiabatic/vertex parameter and a controlled electronic state. Near a van Hove point, in a narrow band, at strong coupling, or near a Mott transition, vertex corrections and double counting with a downfolded interaction can be large. Anharmonicity and multiple phonon branches require extending the Gaussian kernel.

Exercises

Complete the square in $S_X$ to verify the sign and factor $1/2$ of the effective interaction.

Solution

$S_X=\frac12(X+gD_0\rho)D_0^{-1}(X+gD_0\rho)-\frac{g^2}{2}\rho D_0\rho$. The shifted Gaussian contributes only a determinant, leaving the negative retarded density interaction.

References

• Theodore Holstein, “Studies of Polaron Motion: Part I. The Molecular-Crystal Model,” Annals of Physics 8 (1959) 325–342, doi:10.1016/0003-4916(59)90002-8.