Electron–Phonon Fields and Retarded Interactions

Integrating a harmonic lattice displacement produces an electron interaction that is nonlocal in imaginary time. The Gaussian sign is exactly attractive in the density coordinate coupled to the phonon, but the momentum structure, retardation, Coulomb terms, and electronic state decide whether the observable consequence is mass renormalization, charge order, pairing, a polaron, or no ordered phase at all.

Required background. Use effective lattice Hamiltonians to define the electronic subspace and many-body coherent-state path integrals for the imaginary-time integral.

Phonon modes and the general interaction kernel

For harmonic branches $s$, define the dimensionless displacement

$$X_{\mathbf q s}=b_{\mathbf q s}+b_{-\mathbf q s}^{\dagger}$$

and write

$$\begin{aligned} H_{\mathrm{ph}}&=\sum_{\mathbf q s}\omega_{\mathbf q s} \left(b_{\mathbf q s}^{\dagger}b_{\mathbf q s}+\frac12\right),\\ H_{\mathrm{e-ph}}&=\frac{1}{\sqrt N} \sum_{\mathbf k\mathbf q}\sum_{ab\sigma s} g^s_{ab}(\mathbf k,\mathbf q)\, c_{\mathbf k+\mathbf q,a\sigma}^{\dagger} c_{\mathbf k,b\sigma}X_{\mathbf q s}. \end{aligned}$$

The branch, polarization, orbital labels, and momentum transfer are physical data. Hermiticity requires

$$g^s_{ab}(\mathbf k,\mathbf q) =\bigl[g^s_{ba}(\mathbf k+\mathbf q,-\mathbf q)\bigr]^*.$$

With $D_s^0=\langle X_sX_s\rangle_0$ denoting the positive Euclidean covariance,

$$D_s^0(\mathbf q,i\nu_n) =\frac{2\omega_{\mathbf q s}} {\nu_n^2+\omega_{\mathbf q s}^2}.$$

Some Green-function conventions include a minus sign in the definition of a phonon propagator; stating the covariance convention prevents that sign from migrating into the effective interaction.

Collect the electron bilinear coupled to a branch into the phonon source

$$\mathcal J_s(\mathbf q,i\nu_n)=\frac{1}{\sqrt N} \sum_{\mathbf k ab\sigma} g^s_{ab}(\mathbf k,\mathbf q)\, \bigl[c_{\mathbf k+\mathbf q,a\sigma}^{\dagger} c_{\mathbf k,b\sigma}\bigr]_{i\nu_n}.$$

The phonon action is a Gaussian of the form

$$S_{\mathrm{ph}}+S_{\mathrm{e-ph}} =\frac12XD_0^{-1}X+X\mathcal J.$$

Completing the square and integrating all $X_{\mathbf q s}$ gives

$$S_{\mathrm{med}} =-\frac12\sum_{\mathbf q,\nu_n,s} \mathcal J_s(-\mathbf q,-i\nu_n) D_s^0(\mathbf q,i\nu_n) \mathcal J_s(\mathbf q,i\nu_n).$$

This result is exact for harmonic phonons linearly coupled to the chosen electron bilinears. It is negative semidefinite as a quadratic form in those bilinears. It does not say that every Cooper-pair partial wave is attractive: the matrix elements $g^s_{ab}(\mathbf k,\mathbf q)$ and the fermionic exchange structure still have to be projected into the channel of interest.

Holstein model as a worked example

For one local optical mode per site, 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 is the local molecular-crystal model introduced by Holstein 1959, pp. 325–342. The subtraction $\bar n$ fixes the reference displacement and shifts one-body terms; it does not change the retarded two-body kernel.

Use the Matsubara normalization

$$X_i(\tau)=\frac{1}{\sqrt\beta}\sum_{\nu_n} e^{-i\nu_n\tau}X_i(i\nu_n), \qquad \rho_i(\tau)=\frac{1}{\sqrt\beta}\sum_{\nu_n} e^{-i\nu_n\tau}\rho_i(i\nu_n).$$

Then

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

and the two-particle interaction is

$$V_{\mathrm{ph}}(i\nu_n) =-g^2D_0(i\nu_n) =-\frac{2g^2\omega_0}{\nu_n^2+\omega_0^2}.$$

If an instantaneous onsite repulsion $U$ is also retained, the frequency-dependent local vertex is

$$U_{\mathrm{eff}}(i\nu_n) =U-\frac{2g^2\omega_0}{\nu_n^2+\omega_0^2}.$$

All factors in these expressions depend on the convention $X=b+b^\dagger$. Rescaling to a dimensional displacement changes $g$ and $D_0$ separately but leaves the product $g^2D_0$ unchanged.

Low-frequency, adiabatic, and antiadiabatic limits

Three statements that sound similar are not equivalent.

Statement	Required hierarchy	What follows
Low external frequency	$\lvert\nu_n\rvert\ll\omega_0$ for the process being evaluated	$V_{\mathrm{ph}}(i\nu_n)\simeq-2g^2/\omega_0$ for that process only
Antiadiabatic phonon	$\omega_0$ exceeds every relevant electronic transfer, formally $\omega_0\to\infty$ with $U_{\mathrm{ph}}=2g^2/\omega_0$ fixed	The interaction becomes instantaneous throughout the retained electronic window
Adiabatic phonon	$\omega_0/E_{\mathrm{el}}\ll1$	The lattice is slow and retardation remains important; this alone does not prove a controlled vertex expansion

Thus $|\nu_n|\ll\omega_0$ is a low-frequency approximation, not by itself the antiadiabatic limit. In the genuine antiadiabatic limit,

$$V_{\mathrm{ph}}(i\nu_n)\longrightarrow-U_{\mathrm{ph}}, \qquad U_{\mathrm{eff}}\longrightarrow U-U_{\mathrm{ph}},$$

for every frequency retained by the electronic model. Explicit Hubbard–Holstein calculations use precisely this limiting prescription and show that finite $\omega_0$ can retain important dynamical differences from the instantaneous model Johnston et al. 2013, § II A.

The adiabatic ratio and the conventional metallic vertex estimate are distinct quantities:

$$\eta=\frac{\omega_{\mathrm{ph}}}{E_{\mathrm{el}}}, \qquad \gamma_{\mathrm{vertex}} \sim\lambda_{\mathrm{ME}}\frac{\omega_{\mathrm{ph}}}{E_F}.$$

The electronic scale must be chosen for the states and momentum transfers actually involved. The second estimate applies in a regular broad-band Fermi liquid Migdal 1958, pp. 996–1001; it is not a universal theorem for forward-focused coupling, a flat band, a van Hove point, strong coupling, or a Mott-adjacent metal. Here $\lambda_{\mathrm{ME}}$ is the Fermi-surface Eliashberg coupling, not automatically $\lambda_{\mathrm H}$. At low density, $E_F$ is measured from the relevant band edge and can be far smaller than the full bandwidth.

The isotope dependence also tests conventions. If a dimensional normal coordinate $Q$ couples as $I Q\rho$, then

$$Q=\frac{b+b^\dagger}{\sqrt{2M\omega_0}}, \qquad g=\frac{I}{\sqrt{2M\omega_0}}.$$

Within the harmonic Born–Oppenheimer approximation, take both the force constant $K=M\omega_0^2$ and the electronic derivative $I$ to be isotope independent. Then

$$U_{\mathrm{ph}}=\frac{2g^2}{\omega_0} =\frac{I^2}{M\omega_0^2} =\frac{I^2}{K}$$

is mass independent even though $\omega_0\propto M^{-1/2}$. Isotope substitution changes retardation and dynamical observables; it does not automatically change the static attraction.

Polarons, charge order, and pairing

A unitary Lang–Firsov transformation makes the strong-coupling content visible Lang and Firsov 1963, pp. 1301–1312. With $\rho_i=n_i-\bar n$, use

$$\mathcal U=\exp\!\left[ \frac{g}{\omega_0}\sum_i\rho_i(b_i^\dagger-b_i) \right].$$

It removes the linear local coupling and produces

$$\mathcal U H\mathcal U^\dagger \supset-\frac{g^2}{\omega_0}\sum_i\rho_i^2.$$

For spinful electrons, $\rho_i^2$ contains $2n_{i\uparrow}n_{i\downarrow}$, so the induced onsite attraction is again $U_{\mathrm{ph}}=2g^2/\omega_0$, up to one-body and constant terms fixed by $\bar n$. The hopping acquires displacement operators. In the zero-phonon strong-coupling estimate,

$$t_{\mathrm{eff}}\simeq t\, \exp\!\left[-\left(\frac{g}{\omega_0}\right)^2\right],$$

showing how the same coupling that attracts two electrons can narrow their coherent motion and create heavy small polarons or bipolarons. This estimate is not a weak-coupling mass formula.

The density interaction contributes to both particle–particle and particle–hole correlations. At commensurate filling it can favor a charge-density wave; in another filling or band structure it can enhance pairing. Adding a Hubbard repulsion can instead produce antiferromagnetic, metallic, charge-ordered, or paired regimes. For example, determinant quantum Monte Carlo finds direct antiferromagnetic–charge-density-wave competition in the half-filled two-dimensional Hubbard–Holstein model, with conclusions that depend on phonon frequency and finite-temperature diagnostics Johnston et al. 2013, §§ V–VI. An attractive kernel is therefore an input to a phase calculation, not proof of superconductivity.

The local Holstein coupling should not be used as a synonym for every Peierls mechanism. A bond-coupled Su–Schrieffer–Heeger chain has, schematically,

$$H_{\mathrm{SSH}} =-\sum_{i\sigma} \bigl[t-\alpha(u_{i+1}-u_i)\bigr] \left(c_{i+1,\sigma}^{\dagger}c_{i\sigma}+\mathrm{h.c.}\right) +H_{\mathrm{ph}}.$$

A displacement modulates hopping rather than onsite density. In one dimension the electronic response is enhanced near $q=2k_F$, and a commensurate bond dimerization can open a Peierls gap; the SSH model also supports domain-wall solitons Su, Schrieffer, and Heeger 1979, pp. 1698–1701. In higher dimensions, nesting, phonon polarization, commensurability, and fluctuations decide whether a Peierls or charge-density-wave transition occurs. A local Holstein CDW and a bond-order Peierls state can break related translations while having different form factors and probes.

A common dimensionless convention is

$$\lambda_{\mathrm H}=\frac{U_{\mathrm{ph}}}{W} =\frac{2g^2}{\omega_0 W},$$

where $W$ is a declared bare bandwidth. Other communities define $\lambda$ from a Fermi-surface average of the Eliashberg spectrum. Never compare numerical values without matching definitions.

Coulomb retardation and double counting

The attractive phonon kernel and repulsive Coulomb interaction act over different frequency ranges; subtracting two static numbers generally loses that distinction. In the Morel–Anderson reduction of a regular weak-coupling Cooper problem, a dimensionless repulsion $\mu=N(0)V_C$ is integrated from an electronic cutoff $E_{\mathrm{el}}$ down to a phonon cutoff $\omega_c$. The result is

$$\mu^* =\frac{\mu} {1+\mu\ln(E_{\mathrm{el}}/\omega_c)}.$$

The logarithm reduces the repulsion seen by slow phonon-mediated pairing Morel and Anderson 1962, pp. 1263–1271. This formula assumes a separated cutoff hierarchy, a sufficiently regular density of states and interaction across the eliminated window, and a controlled normal electronic state. It is not a universal replacement for a multiorbital $U(\omega)$ near a Mott transition, in a narrow or flat band, at low density, near a van Hove singularity, or when low-energy plasmons and momentum dependence matter. Do not combine $\mu^*$ with an independently retained microscopic $U$ unless a matching prescription shows which screening and frequency intervals each term represents.

The phonon propagator also has to be counted once. Starting with a bare quadratic $D_0$ and retaining the coupled electrons is consistent: electron polarization generated by the solution dresses the phonon. Inserting an electron-dressed experimental or calculated $D$ and then allowing the same retained electrons to dress it again double counts that self-energy unless the overlapping polarization is subtracted. Quadratic mixing among harmonic branches remains Gaussian and can be diagonalized. Intrinsic anharmonicity or nonlinear electron–phonon coupling, by contrast, makes the bosonic integral non-Gaussian and generates interactions beyond the pairwise kernel.

Validity checks and claim ceiling

The chapter validity map and claim table place retarded interactions after the electronic reduction and before any phase attribution. A usable calculation should answer all of the following.

Question	Required check	If it fails
Which phonons couple?	State branches, eigenvector normalization, momentum dependence, and the electronic orbital basis	A local Einstein mode cannot represent the claimed material interaction
Is the bosonic action Gaussian?	Bound intrinsic anharmonicity and nonlinear coupling; diagonalize quadratic branch mixing	Retain the non-Gaussian phonon action and its induced higher interactions
Is phonon dressing counted once?	Declare whether $D$ is bare or dressed and subtract polarization from electrons retained in the solver	A dressed phonon self-energy is generated twice
Is the static reduction valid?	Compare $\omega_{\mathrm{ph}}$ with the full electronic window and vary the frequency treatment	Do not replace the kernel by $U-U_{\mathrm{ph}}$
Are Coulomb terms consistent?	Match the electronic screening window and expose any double counting or pseudopotential reduction	Do not add an unrelated static attraction and repulsion
Are vertex corrections controlled?	Check coupling, Fermi energy, momentum structure, bandwidth, van Hove points, and proximity to a Mott regime	A bubble or Migdal–Eliashberg result is not controlled
Is an ordered phase established?	Compare pairing, charge, magnetic, and polaronic observables with size, temperature, and covariance control	Report a tendency or crossover, not a phase

The harmonic integration derived here remains exact even when Migdal’s approximation fails; what fails is the subsequent truncation of electronic diagrams or vertices. Continue with Migdal’s theorem and vertex-correction validity before using a small-vertex approximation. That page tests explicitly which scale or kinematic hypothesis fails near a van Hove point, in a flat or narrow band, at low density, at strong coupling, or near a Mott regime.

Common pitfalls

Calling the static value the antiadiabatic limit. A single transfer with $|\nu|\ll\omega_0$ samples the static part of the kernel. Antiadiabaticity requires the phonon to be fast relative to the entire retained electronic problem.

Using “attractive” as a phase label. The same density kernel feeds pairing and charge channels and also dresses the carrier mass. Establish the phase with channel-resolved susceptibilities, stiffness or long-range order, and controlled limits.

Mixing displacement conventions. A dimensional coordinate, a mass-weighted normal mode, and $b+b^\dagger$ have different couplings and propagators. Compare only invariant products such as $g^2D_0$ after translating conventions.

Exercises

Complete the square in the Holstein phonon action and verify both the sign and the factor $1/2$ of the mediated action.

Solution

For each site and frequency,

$$\frac12XD_0^{-1}X+gX\rho =\frac12(X+gD_0\rho)D_0^{-1}(X+gD_0\rho) -\frac{g^2}{2}\rho D_0\rho.$$

The shifted Gaussian contributes a determinant independent of $\rho$. The remaining term is negative and contains $1/2$ because the real displacement modes at opposite frequencies are not independent copies.

Compute $V_{\mathrm{ph}}(i\nu_n)/V_{\mathrm{ph}}(0)$. Explain why its small-$\nu_n$ expansion does not establish antiadiabaticity.

Solution

$$\frac{V_{\mathrm{ph}}(i\nu_n)}{V_{\mathrm{ph}}(0)} =\frac{\omega_0^2}{\nu_n^2+\omega_0^2} =1-\frac{\nu_n^2}{\omega_0^2}+O(\nu_n^4/\omega_0^4).$$

This controls one frequency transfer. To obtain an instantaneous electronic model, the approximation must hold for every transfer that materially contributes, so $\omega_0$ must exceed the retained electronic scales while $2g^2/\omega_0$ is held fixed.

Apply the Lang–Firsov transformation to show why the induced attraction is $2g^2/\omega_0$ for two spin species.

Solution

The transformation shifts $b_i\mapsto b_i-(g/\omega_0)\rho_i$. The oscillator plus coupling becomes $\omega_0b_i^\dagger b_i-(g^2/\omega_0)\rho_i^2$. With $\bar n=0$ for simplicity,

$$n_i^2=(n_{i\uparrow}+n_{i\downarrow})^2 =n_i+2n_{i\uparrow}n_{i\downarrow}.$$

The first term shifts the chemical potential; the second is $-(2g^2/\omega_0)n_{i\uparrow}n_{i\downarrow}$. Hence an existing Hubbard interaction becomes $U-2g^2/\omega_0$ in the instantaneous limit.

For the dimensional coupling $I Q\rho$, show which part of the harmonic attraction changes under isotope substitution when $K$ and $I$ are fixed.

Solution

Since $\omega_0=\sqrt{K/M}$ and $g=I/\sqrt{2M\omega_0}$, the static strength $2g^2/\omega_0=I^2/K$ is unchanged. The characteristic frequency scales as $M^{-1/2}$, so the frequency profile of the kernel, the adiabatic ratio, and dynamical or transition observables can change even though its zero-frequency strength does not.

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.
• Steven Johnston, E. A. Nowadnick, Y. F. Kung, B. Moritz, R. T. Scalettar, and T. P. Devereaux, “Determinant Quantum Monte Carlo Study of the Two-Dimensional Single-Band Hubbard–Holstein Model,” Physical Review B 87 (2013) 235133, doi:10.1103/PhysRevB.87.235133, Open accepted manuscript.
• I. G. Lang and Yu. A. Firsov, “Kinetic Theory of Semiconductors with Low Mobility,” Soviet Physics JETP 16 (1963) 1301–1312, Open PDF.
• A. B. Migdal, “Interaction between Electrons and Lattice Vibrations in a Normal Metal,” Soviet Physics JETP 7 (1958) 996–1001, Open PDF.
• P. Morel and P. W. Anderson, “Calculation of the Superconducting State Parameters with Retarded Electron–Phonon Interaction,” Physical Review 125 (1962) 1263–1271, doi:10.1103/PhysRev.125.1263.
• W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in Polyacetylene,” Physical Review Letters 42 (1979) 1698–1701, doi:10.1103/PhysRevLett.42.1698.