Random Phase Approximation

The random phase approximation (RPA) describes the collective linear response of an interacting many-body system by letting a simple reference system respond self-consistently to the field induced by its own density fluctuation. In its elementary density form, RPA replaces the exact irreducible polarization by the independent-particle polarization and sums the resulting feedback to all orders.

The central structure is

$$\text{external field} \longrightarrow \text{density fluctuation} \longrightarrow \text{induced field} \longrightarrow \text{density fluctuation}.$$

Although each polarization step is computed from noninteracting particles, the infinite feedback chain can produce behavior absent from every finite perturbative order: screened interactions, collective poles, and plasma oscillations.

RPA is not a claim that phases are literally random, and it is not a generic synonym for “include fluctuations.” It is a specific closure of a response or particle–hole problem. Its reliability depends on the interaction range, density, reference state, channel, dimension, frequency, and observable.

Helpful background. Kubo response supplies the source convention, density operators supply the Fourier normalization, and the ideal Fermi gas supplies the occupations.

Density response and the RPA family

Several related approximations are called RPA:

1. Direct density RPA uses a noninteracting polarization bubble and a direct density interaction.
2. Time-dependent Hartree RPA is the same closure obtained by linearizing a time-dependent Hartree equation.
3. RPA with exchange or time-dependent Hartree–Fock includes exchange in the particle–hole kernel.
4. Finite-basis RPA solves a forward-and-backward particle–hole eigenvalue problem around a mean-field reference.
5. Ring RPA sums ring diagrams in response functions or the correlation energy.
6. Density-functional RPA uses a chosen independent-particle response and the bare Coulomb kernel in an adiabatic-connection energy formula.

These forms share a small-oscillation or repeated particle–hole structure, but they are not numerically identical. Every use of “RPA” should state:

• the reference propagator or orbitals;
• the interaction kernel;
• whether exchange or another vertex kernel is included;
• whether the target is response, excitation energies, or ground-state correlation energy.

This page derives direct density RPA first and then maps the main variants around it.

Source and Sign Conventions

Let a potential energy $U_{\mathrm{ext}}$ couple to the number density:

$$\delta K_{\mathrm{ext}}(t) = \int d^d r\, U_{\mathrm{ext}}(\mathbf r,t) n(\mathbf r).$$

In momentum and frequency space, define the retarded polarization by

$$\delta n(\mathbf q,\omega) = \Pi^R(\mathbf q,\omega) U_{\mathrm{ext}}(\mathbf q,\omega).$$

A positive static potential energy repels particles, so a stable fermion gas has

$$\Pi^R(\mathbf q\to0,0) < 0.$$

The Kubo Formula uses a local chemical-potential source,

$$\delta K_{\mathrm{ext}} = - \int d^d r\, \delta\mu_{\mathrm{ext}}n,$$

and defines

$$\delta n = \chi_{nn}^R \delta\mu_{\mathrm{ext}}.$$

Because

$$U_{\mathrm{ext}} = -\delta\mu_{\mathrm{ext}},$$

the two conventions are related by

$$\Pi^R = -\chi_{nn}^R.$$

This page writes screening formulas with $\Pi$, for which the familiar dielectric denominator is $1-v\Pi$. When translating to the chemical-potential susceptibility, replace $\Pi$ by $-\chi$.

Density–Density Interaction

Consider a translationally invariant reference Hamiltonian

$$K_0 = \sum_{\mathbf k,\sigma} \xi_{\mathbf k} c_{\mathbf k\sigma}^\dagger c_{\mathbf k\sigma}.$$

Define the Fourier density

$$n_{\mathbf q} = \sum_{\mathbf k,\sigma} c_{\mathbf k\sigma}^\dagger c_{\mathbf k+\mathbf q,\sigma}.$$

Thus $n_{\mathbf q}=\int d^dr\,e^{-i\mathbf q\cdot\mathbf r}n(\mathbf r)$; it equals $\rho_{-\mathbf q}$ in the density-operator convention. Here $\mathbf q$ is a wave vector, so the transferred momentum is $\hbar\mathbf q$. Normalize $\delta n(\mathbf q)=\delta\langle n_{\mathbf q}\rangle/\mathcal V$ and take $U_{\mathrm{ext}}(\mathbf q)$ to be an energy amplitude. The retarded kernel is

$$\Pi^R(\mathbf q,t)=-\frac{i}{\hbar\mathcal V}\theta(t) \langle[n_{\mathbf q}(t),n_{-\mathbf q}(0)]\rangle .$$

With time transform $\int dt\,e^{i\omega t}$ this gives the denominator below, without assuming inversion symmetry.

A direct density interaction can be written

$$K_{\mathrm{int}} = \frac{1}{2\mathcal V} \sum_{\mathbf q} v_{\mathbf q} : n_{\mathbf q} n_{-\mathbf q} :.$$

The normal-ordering convention and treatment of $\mathbf q=0$ must be stated. For a Coulomb electron gas, a uniform positive background cancels the divergent uniform Hartree term. One usually excludes $\mathbf q=0$ from the fluctuating interaction and imposes overall neutrality.

The scalar derivation below assumes one density channel and a real even kernel,

$$v_{\mathbf q} = v_{-\mathbf q}.$$

Multicomponent, spin, orbital, and sublattice systems require matrices in the corresponding channel space.

Independent-Particle Polarization

The noninteracting response of $K_0$ is the Lindhard polarization:

$$\begin{aligned} \Pi_0^R(\mathbf q,\omega) ={}& \frac{1}{\mathcal V} \sum_{\mathbf k,\sigma} \frac{ f(\xi_{\mathbf k}) - f(\xi_{\mathbf k+\mathbf q}) }{ \hbar\omega + \xi_{\mathbf k} - \xi_{\mathbf k+\mathbf q} + i0^+ }. \end{aligned}$$

The spin sum is explicit. If a spin degeneracy factor is pulled out, it must not be inserted a second time in the density of states.

Here $f(\xi)=(e^{\beta\xi}+1)^{-1}$ is the grand-canonical Fermi–Dirac occupation, including its zero-temperature limit. This formula is exact for that independent reference system. It is not yet RPA.

Origin of the numerator

The source component proportional to $n_{-\mathbf q}$ moves one fermion from $\mathbf k$ to $\mathbf k+\mathbf q$. Its response is measured by the conjugate density $n_{\mathbf q}$. The occupation difference

$$f(\xi_{\mathbf k}) - f(\xi_{\mathbf k+\mathbf q})$$

implements Pauli blocking. At zero temperature and positive frequency, an absorptive process requires:

$$\xi_{\mathbf k}<0, \qquad \xi_{\mathbf k+\mathbf q}>0.$$

The initial state is occupied and the final state is empty.

Origin of the denominator

The energy transfer for one particle–hole excitation is

$$\Delta\xi_{\mathbf k,\mathbf q} = \xi_{\mathbf k+\mathbf q} - \xi_{\mathbf k}.$$

The retarded denominator has a pole when

$$\hbar\omega = \Delta\xi_{\mathbf k,\mathbf q},$$

with $+i0^+$ selecting causal boundary conditions.

Real and imaginary parts

Using

$$\frac{1}{x+i0^+} = \mathcal P\frac{1}{x} - i\pi\delta(x),$$

one obtains

$$\operatorname{Im} \Pi_0^R(\mathbf q,\omega) ={} - \frac{\pi}{\mathcal V} \sum_{\mathbf k,\sigma} \left[ f(\xi_{\mathbf k}) - f(\xi_{\mathbf k+\mathbf q}) \right] \times \delta \left( \hbar\omega + \xi_{\mathbf k} - \xi_{\mathbf k+\mathbf q} \right).$$

For $\omega>0$, the absorptive part is nonpositive in this convention. The real and imaginary parts are linked by causality through dispersion relations.

Static and Uniform Limits

At exactly $\mathbf q=0$,

$$n_{\mathbf q=0} = \widehat N.$$

If particle number is conserved, then for nonzero frequency

$$\Pi_0^R(\mathbf 0,\omega) = 0.$$

The thermodynamic static limit is different. Taking $\omega=0$ and then $\mathbf q\to0$ gives

$$\lim_{\mathbf q\to0} \Pi_0^R(\mathbf q,0) = - \frac{\partial n}{\partial\mu}.$$

At zero temperature,

$$\frac{\partial n}{\partial\mu} = \nu_{\mathrm{tot}}(E_{\mathrm F}),$$

where $\nu_{\mathrm{tot}}$ includes all internal degeneracies.

Thus

$$\Pi_0^R(\mathbf q\to0,0) = -\nu_{\mathrm{tot}}(E_{\mathrm F}).$$

The static and uniform limits encode different protocols. RPA does not remove this order-of-limits issue.

Particle–Hole Continuum

The set of allowed independent particle–hole energies forms a continuum in the thermodynamic limit. For a three-dimensional parabolic band at zero temperature,

$$\epsilon_{\mathbf k} = \frac{\hbar^2k^2}{2m},$$

the positive-frequency continuum lies within

$$\omega_{\min}(q) \le \omega \le \omega_{\max}(q),$$

where

$$\omega_{\min}(q) = \max \left( 0, \frac{\hbar q^2}{2m} - v_{\mathrm F}q \right),$$

and

$$\omega_{\max}(q) = \frac{\hbar q^2}{2m} + v_{\mathrm F}q.$$

For $q>0$, in the interior of this region,

$$\operatorname{Im}\Pi_0^R \ne 0.$$

An excitation can decay into particle–hole pairs there. Outside the continuum, the independent response is purely reactive in the ideal zero-temperature model.

Self-Consistent-Field Derivation

The total potential energy seen by the particles is

$$U_{\mathrm{tot}} = U_{\mathrm{ext}} + U_{\mathrm{ind}}.$$

In direct Hartree response,

$$U_{\mathrm{ind}}(\mathbf q,\omega) = v_{\mathbf q} \delta n(\mathbf q,\omega).$$

Approximate the response to the total field by the independent-particle polarization:

$$\delta n = \Pi_0^R U_{\mathrm{tot}}.$$

Substitution gives

$$\delta n = \Pi_0^R \left( U_{\mathrm{ext}} + v_{\mathbf q}\delta n \right).$$

Move the feedback term to the left:

$$\left( 1 - v_{\mathbf q}\Pi_0^R \right) \delta n = \Pi_0^R U_{\mathrm{ext}}.$$

Therefore

$$\Pi_{\mathrm{RPA}}^R(\mathbf q,\omega) = \frac{ \Pi_0^R(\mathbf q,\omega) }{ 1 - v_{\mathbf q} \Pi_0^R(\mathbf q,\omega) }.$$

This is the scalar direct-RPA density response.

Physical sign check

For a repulsive interaction,

$$v_{\mathbf q}>0.$$

In the static limit,

$$\Pi_0^R<0.$$

Hence

$$1-v_{\mathbf q}\Pi_0^R > 1.$$

The magnitude of the static density response is reduced: the induced Hartree field opposes the external perturbation. If a convention gives static antiscreening for a simple repulsive Coulomb gas, recheck the source sign and the sign of the induced Hartree field.

Chemical-potential susceptibility form

Using

$$\chi_0^R = -\Pi_0^R,$$

the same result is

$$\chi_{\mathrm{RPA}}^R = \frac{ \chi_0^R }{ 1 + v_{\mathbf q}\chi_0^R }.$$

Both formulas describe the same physics. Mixing the numerator convention from one with the denominator convention from the other produces a false pole.

Dielectric Function

Define the longitudinal RPA dielectric function by

$$\varepsilon_{\mathrm{RPA}}(\mathbf q,\omega) = 1 - v_{\mathbf q} \Pi_0^R(\mathbf q,\omega).$$

The total and external potential energies obey

$$U_{\mathrm{tot}} = \varepsilon_{\mathrm{RPA}}^{-1} U_{\mathrm{ext}}.$$

The screened interaction is

$$W_{\mathrm{RPA}}^R(\mathbf q,\omega) = \frac{ v_{\mathbf q} }{ \varepsilon_{\mathrm{RPA}}(\mathbf q,\omega) } = \frac{ v_{\mathbf q} }{ 1-v_{\mathbf q}\Pi_0^R }.$$

The identities

$$\Pi_{\mathrm{RPA}}^R = \Pi_0^R \varepsilon_{\mathrm{RPA}}^{-1}$$

and

$$W_{\mathrm{RPA}}^R = v + v\Pi_{\mathrm{RPA}}^Rv$$

are useful checks in the scalar case.

The experimentally relevant loss function for longitudinal charge excitations is often

$$- \operatorname{Im} \frac{1}{ \varepsilon_{\mathrm{RPA}}(\mathbf q,\omega) }.$$

It can show both broad particle–hole weight and sharp collective peaks.

Diagrammatic Bubble Chain

The line, vertex, loop-sign, and irreducibility conventions used in this section are introduced in Diagrammatic Methods Preview.

Let $P$ denote the polarization irreducible with respect to cutting one interaction line. The exact reducible polarization satisfies the operator equation

$$\Pi = P + Pv\Pi.$$

Formally,

$$\Pi = \left( 1-Pv \right)^{-1} P.$$

Direct RPA makes the approximation

$$P \approx \Pi_0,$$

where $\Pi_0$ is the single independent-particle bubble. The series is

$$\Pi_{\mathrm{RPA}} ={} \Pi_0 + \Pi_0v\Pi_0 + \Pi_0v\Pi_0v\Pi_0 + \cdots.$$

For commuting scalar quantities,

$$\Pi_{\mathrm{RPA}} = \frac{\Pi_0}{ 1-v\Pi_0 }.$$

Likewise,

$$W_{\mathrm{RPA}} ={} v + v\Pi_0v + v\Pi_0v\Pi_0v + \cdots.$$

The geometric series converges literally only when $|v\Pi_0|<1$ (or the corresponding matrix spectral radius is below one). The self-consistent inverse defines the resummed response beyond that domain by analytic continuation where it exists. Small-$q$ Coulomb screening often lies outside the convergent power series. This inverse can have a collective pole even when a single bubble has only a continuum.

What diagrams are absent

Direct RPA omits:

• exchange vertex corrections between the particle and hole;
• self-energy insertions unless already built into the reference propagator;
• crossed interaction lines;
• short-range local-field corrections;
• multipair irreducible vertices.

Using dressed one-particle energies with a bare vertex is not automatically conserving. Propagators and vertices must be matched when conservation laws are essential.

The historical name

Bohm and Pines separated coherent long-wavelength density motion from microscopic terms whose phases largely cancel in a dense electron gas. Modern RPA is defined by the deterministic response closure above; it introduces neither random noise nor decoherence.

Static Coulomb Screening

For a three-dimensional Coulomb interaction in Gaussian electrostatic units,

$$v_q = \frac{ 4\pi e^2 }{ \epsilon_{\mathrm b}q^2 },$$

where $\epsilon_{\mathrm b}$ is the relative background permittivity and $e^2$ has dimensions of energy times length. In SI formulas replace $4\pi e^2/\epsilon_{\mathrm b}$ by $e_{\mathrm{SI}}^2/(\epsilon_0\epsilon_{\mathrm b})$.

At small $q$ and zero frequency,

$$\Pi_0^R(q,0) \simeq -\nu_{\mathrm{tot}}(E_{\mathrm F}).$$

Define the Thomas–Fermi wave number

$$k_{\mathrm{TF}}^2 = \frac{ 4\pi e^2 }{ \epsilon_{\mathrm b} } \nu_{\mathrm{tot}}(E_{\mathrm F}).$$

Then

$$\varepsilon_{\mathrm{RPA}}(q,0) \simeq 1 + \frac{ k_{\mathrm{TF}}^2 }{ q^2 },$$

and

$$W_{\mathrm{RPA}}(q,0) \simeq \frac{ 4\pi e^2 }{ \epsilon_{\mathrm b} \left( q^2+k_{\mathrm{TF}}^2 \right) }.$$

Fourier transformation gives the Yukawa form

$$W_{\mathrm{TF}}(r) = \frac{ e^2 }{ \epsilon_{\mathrm b}r } e^{-k_{\mathrm{TF}}r}.$$

Long-range Coulomb repulsion is converted into a short-ranged effective interaction on the scale

$$\ell_{\mathrm{TF}} = k_{\mathrm{TF}}^{-1}.$$

In the Thomas–Fermi model, $W/v=\Pi_{\mathrm{RPA}}/\Pi_0=q^2/(q^2+k_{\mathrm{TF}}^2)$. The ratio is one half at $q=k_{\mathrm{TF}}$: long-wavelength fields are strongly screened. Applying this curve across several screening wave numbers requires those wave numbers still to be small compared with $k_{\mathrm F}$.

Thomas–Fermi is only the long-wavelength limit

Replacing $\Pi_0(q,0)$ by its $q\to0$ value discards the nonanalytic structure near $q=2k_{\mathrm F}$. The full static Lindhard function produces Friedel oscillations. In three dimensions their asymptotic form is oscillatory with wave number $2k_{\mathrm F}$ and an algebraic envelope rather than a pure exponential.

Thus:

• Thomas–Fermi screening captures smooth long-wavelength response;
• static RPA with full $\Pi_0(q,0)$ captures the independent-particle $2k_{\mathrm F}$ structure;
• vertex and local-field corrections alter short-range quantitative behavior.

Long-Wavelength Plasmon

Continuity and the longitudinal $f$-sum rule constrain the high-frequency, small-$q$ polarization of a parabolic band:

$$\Pi_0^R(q,\omega) \simeq \frac{ nq^2 }{ m(\omega+i0^+)^2 }$$

when

$$\omega \gg qv_{\mathrm F}.$$

Insert the three-dimensional Coulomb interaction:

$$\varepsilon_{\mathrm{RPA}}(q,\omega) \simeq 1 - \frac{ 4\pi ne^2 }{ m\epsilon_{\mathrm b}\omega^2 }.$$

A collective mode occurs when

$$\varepsilon_{\mathrm{RPA}}(q,\omega) = 0.$$

Therefore

$$\omega(q\to0) = \omega_{\mathrm p},$$

where

$$\omega_{\mathrm p}^2 = \frac{ 4\pi ne^2 }{ m\epsilon_{\mathrm b} }.$$

The three-dimensional bulk plasmon remains at finite frequency as $q\to0$. This gap is a consequence of the long-range Coulomb interaction, not a one-particle band gap.

Dimensional dependence

For a two-dimensional layer with three-dimensional Coulomb fields,

$$v_q = \frac{ 2\pi e^2 }{ \epsilon_{\mathrm b}q }.$$

The same long-wavelength argument gives

$$\omega_{\mathrm p}^2(q) \simeq \frac{ 2\pi ne^2 }{ m\epsilon_{\mathrm b} } q,$$

so

$$\omega_{\mathrm p}(q) \propto \sqrt q.$$

One must not transplant the constant three-dimensional plasma frequency into a two-dimensional electron layer.

Collective Poles and Landau Damping

A real zero of

$$\operatorname{Re} \varepsilon_{\mathrm{RPA}}(q,\omega)$$

is not sufficient for a sharp mode. Its damping depends on

$$\operatorname{Im} \varepsilon_{\mathrm{RPA}}(q,\omega).$$

Outside the particle–hole continuum at zero temperature,

$$\operatorname{Im}\Pi_0^R = 0,$$

and an ideal RPA pole can lie on the real axis.

Inside the continuum,

$$\operatorname{Im}\Pi_0^R \ne 0.$$

The collective density oscillation can decay into particle–hole pairs. The pole moves away from the real axis and the spectral peak broadens. This collisionless decay is Landau damping.

Landau damping does not require impurity scattering or quasiparticle collisions. Conversely, direct RPA omits collisional damping from interactions beyond the bubble chain. A small numerical broadening $i\eta$ should not be interpreted automatically as a physical lifetime.

Operator and Matrix Form

For inhomogeneous or multicomponent systems, response functions are kernels or matrices. The direct RPA equation is

$$\boldsymbol\Pi_{\mathrm{RPA}} = \boldsymbol\Pi_0 + \boldsymbol\Pi_0 \mathbf v \boldsymbol\Pi_{\mathrm{RPA}}.$$

Thus

$$\boldsymbol\Pi_{\mathrm{RPA}} = \left( \mathbf 1 - \boldsymbol\Pi_0\mathbf v \right)^{-1} \boldsymbol\Pi_0.$$

Matrix order matters. In a noncommuting basis, writing a scalar quotient is ambiguous.

The dielectric matrix is

$$\boldsymbol\varepsilon = \mathbf 1 - \mathbf v \boldsymbol\Pi_0,$$

with $\mathbf U_{\mathrm{tot}}=\boldsymbol\varepsilon^{-1}\mathbf U_{\mathrm{ext}}$ and $\boldsymbol\Pi_{\mathrm{RPA}}=\boldsymbol\Pi_0\boldsymbol\varepsilon^{-1}$. Local-field effects in crystals appear as off-diagonal reciprocal-lattice components rather than one scalar $\varepsilon(q,\omega)$.

Finite-Basis Particle–Hole RPA

Let $i,j$ label occupied reference orbitals and $a,b$ unoccupied orbitals. A small-amplitude excitation operator is written

$$Q_\nu^\dagger = \sum_{ai} \left( X_{ai}^{(\nu)} c_a^\dagger c_i - Y_{ai}^{(\nu)} c_i^\dagger c_a \right).$$

Linearized time-dependent mean-field theory gives the non-Hermitian eigenproblem

$$\begin{pmatrix} A & B \\ -B^* & -A^* \end{pmatrix} \begin{pmatrix} X^{(\nu)} \\ Y^{(\nu)} \end{pmatrix} = \hbar\omega_\nu \begin{pmatrix} X^{(\nu)} \\ Y^{(\nu)} \end{pmatrix}.$$

The matrices $A$ and $B$ contain orbital energy differences and the chosen particle–hole interaction kernel.

The metric normalization is

$$\left( X^{(\nu)} \right)^\dagger X^{(\nu)} - \left( Y^{(\nu)} \right)^\dagger Y^{(\nu)} = 1$$

for a positive-frequency, positive-energy mode about an energetically stable reference. Real frequencies alone do not imply this sign: an unstable energy saddle can have a real negative-energy mode.

The backward amplitudes $Y$ couple the excitation to its conjugate particle–hole motion in the linearized equations. Time-dependent Hartree–Fock still evolves a determinant; a correlated quasiboson vacuum is a further interpretation of the RPA amplitudes, not an exact many-fermion state.

Tamm–Dancoff approximation

Setting

$$B = 0$$

removes the backward amplitudes and gives the Tamm–Dancoff approximation. It is Hermitian when $A$ is Hermitian, but it generally loses part of the small-oscillation structure and associated sum-rule properties.

Stability

If the underlying mean-field stationary point is stable in the relevant variation space, the RPA frequencies satisfy the corresponding reality conditions. Imaginary frequencies signal an instability of the reference state rather than a stable excitation with imaginary energy.

Broken continuous symmetries can produce zero-frequency modes. A defective zero mode can produce polynomial growth even with no imaginary eigenvalue; see the explicit example in Bogoliubov theory.

Direct RPA and RPA with Exchange

Direct RPA retains repeated direct interactions between density bubbles. It does not include exchange vertex corrections.

Linearized time-dependent Hartree–Fock includes both direct and exchange contributions in the particle–hole kernel and is often called:

• RPA with exchange;
• RPAE;
• TDHF;
• full particle–hole RPA in some finite-system communities.

These labels are community dependent. A calculation should specify its kernel rather than relying on the acronym.

For a local Hubbard interaction, charge and spin channels can acquire different signs and degeneracy factors after spin decomposition. The Coulomb charge formula

$$1-v_q\Pi_0$$

must not be copied blindly into a spin channel. Derive the channel vertex from the Hamiltonian and source convention.

Relation to Mean Field

Static Hartree theory finds a self-consistent one-point density. Time-dependent Hartree theory lets that density move. Linearizing around a stationary solution produces direct RPA.

Schematically,

$$n = n_0+\delta n,$$

and

$$U_{\mathrm H}[n] \simeq U_{\mathrm H}[n_0] + \frac{\delta U_{\mathrm H}}{\delta n} \delta n.$$

The first term belongs to the stationary reference. The second is the residual kernel that feeds the fluctuation back into itself.

This relation explains why RPA is often described as “mean field plus Gaussian fluctuations” or “small-amplitude time-dependent mean field.” It does not imply that every Gaussian theory is RPA.

Relation to BCS Response

Around a paired saddle, the response space contains normal and anomalous densities. A gauge-consistent treatment must include phase fluctuations and vertex corrections associated with the same pairing interaction that generated the gap.

The static BCS Mean-Field Theory page stops at the saddle and its quasiparticles. RPA-like Gaussian response around that saddle generates collective pair modes and restores conservation constraints absent from a bare quasiparticle bubble.

Using only a gapped bubble while omitting the matching phase vertex can violate gauge invariance. Nambu’s analysis of superconducting response is an important historical example of this general principle.

Ring Correlation Energy

RPA can also approximate a ground-state correlation energy by summing ring diagrams. At zero temperature, a common homogeneous direct-RPA expression is

$$\frac{ E_{\mathrm c}^{\mathrm{RPA}} }{ \mathcal V } ={} \frac{\hbar}{2} \int \frac{d^d q}{ (2\pi)^d } \int_0^\infty \frac{d\omega}{\pi} \times \left[ \ln \left( 1-v_q\Pi_0(q,i\omega) \right) + v_q\Pi_0(q,i\omega) \right].$$

The added $v\Pi_0$ removes the first-order one-bubble contribution, leaving a correlation energy beyond the separately treated Hartree and Fock terms; it is not a subtraction of the Hartree field alone. Expanding the logarithm,

$$\ln(1-x)+x = - \frac{x^2}{2} - \frac{x^3}{3} - \cdots,$$

shows that the expression begins with the second-order ring. This Taylor expansion requires $|x|<1$; the logarithmic expression on the imaginary axis can remain well defined beyond its series radius.

At imaginary frequency, the integrand is smooth and avoids real-axis collective poles. The formula still requires:

• a clearly defined reference response $\Pi_0$;
• consistent treatment of Hartree and exchange;
• ultraviolet and basis convergence;
• care in metals near $q=0$;
• no double counting with other correlation terms.

For the high-density homogeneous electron gas, ring resummation captures the leading singular correlation contribution. That controlled asymptotic result does not make direct RPA quantitatively exact at arbitrary density.

Conservation Laws and Sum Rules

An approximate response should be tested against exact constraints.

Particle-number conservation

At $\mathbf q=0$ and finite $\omega$,

$$\Pi^R(\mathbf 0,\omega) = 0$$

for a closed number-conserving system. A nonzero result often indicates an inconsistent vertex, finite-basis artifact, or mishandled limit.

High-frequency moment

For a Galilean-invariant parabolic system,

$$\Pi^R(q,\omega) \sim \frac{ nq^2 }{ m\omega^2 }$$

at large $\omega$. This coefficient underlies the longitudinal $f$-sum rule.

Compressibility

For a neutral system with short-range forces, the static long-wavelength full response is $-\partial n/\partial\mu=-n^2\kappa_T$, under the appropriate thermodynamic limits. For a Coulomb fluid with a compensating background, this relation instead concerns the proper polarization $P$ when the background varies to maintain neutrality:

$$\lim_{q\to0}P(q,0)=-n^2\kappa_T,\qquad \Pi(q,0)=\frac{P(q,0)}{1-v_qP(q,0)} \underset{q\to0}{\sim}-\frac1{v_q}.$$

The screened response therefore vanishes as $q^2$ in three dimensions even for finite compressibility. Direct RPA sets $P=\Pi_0$, so its proper compressibility is that of the reference gas. Adding a ring correlation energy to the equation of state and differentiating it generally gives a different compressibility unless the response approximation is improved consistently.

Causality

A stable retarded response has no poles in the upper half-plane. Damped poles lie below the real axis; ideal undamped modes appear on its boundary with the retarded $i0^+$ prescription. Real and imaginary parts must satisfy Kramers–Kronig relations. Retarded and Advanced Response gives the canonical adjoint, analyticity, spectral-discontinuity, and subtraction checks.

Positivity of spectral weight

For a physical density response, the dynamic structure factor is nonnegative. In the potential-energy convention,

$$- \operatorname{Im}\Pi^R(q,\omega) \ge 0$$

for positive frequency.

Direct RPA built from a consistent independent-particle response respects important longitudinal constraints, but not every mixture of dressed propagators and bare vertices does.

When RPA Is Controlled

RPA is especially well motivated in:

High-density Coulomb systems

At high density, individual interaction effects are weak while long-range small-$q$ processes remain collectively important. Ring diagrams capture the leading logarithmic correlation energy.

Long wavelengths

Collective charge response at small $q$ is strongly constrained by conservation laws and the long-range Coulomb kernel. RPA often captures the correct plasmon scale even when short-range correlations need improvement.

Large flavor or degeneracy limits

With many fermion components and suitable coupling scaling, bubble chains can dominate over vertex corrections. Large-N and Saddle-Point Methods Preview states the model-scaling and fluctuation-counting conditions behind that claim.

Weak residual interactions

Around a stable mean-field reference, linearized particle–hole motion can provide accurate collective frequencies and transition strengths when omitted multipair configurations are perturbative.

Control in one regime does not transfer automatically to all observables. A good long-wavelength dielectric function can coexist with a poor short-range pair correlation.

What RPA Misses

Exchange vertices

The independent fermion bubble already includes Fermi occupations and Pauli blocking. Direct RPA omits exchange corrections to the interaction vertex between particle and hole. This can matter strongly in spin response and finite systems.

Short-range correlation holes

Repeated long-range screening does not fully enforce the local avoidance structure of strongly repulsive particles.

Self-energy and vertex consistency

RPA often uses bare or mean-field propagators. Quasiparticle mass, lifetime, and spectral-weight changes require self-energy corrections, which should be paired with compatible vertices.

Multiple particle–hole pairs

The basic finite-basis RPA excitation operator contains one particle–hole pair and its backward partner. Two-particle–two-hole configurations produce fragmentation and damping beyond elementary RPA.

Strong coupling and Mott physics

A bubble expansion around itinerant independent particles cannot by itself create the full local Hilbert-space reorganization of a Mott insulator.

Critical fluctuations

Near a strongly fluctuation-dominated transition, Gaussian small oscillations can give incorrect exponents or miss nonperturbative physics.

Collisions

Direct collisionless RPA includes Landau damping but not generic collision integrals, impurity scattering, or multipair decay.

Improvements Beyond Direct RPA

Common extensions include:

• RPA with exchange or time-dependent Hartree–Fock;
• local-field factors;
• self-consistent RPA;
• second RPA with two-particle–two-hole configurations;
• Bethe–Salpeter kernels;
• time-dependent density-functional kernels;
• conserving Baym–Kadanoff approximations;
• $GW$ self-energy calculations using an RPA screened interaction;
• vertex-corrected polarization functions.

For a scalar density channel, define an exchange-correlation kernel $f_{\mathrm{xc}}=-vG$. Then

$$\Pi=\frac{\Pi_0}{1-v(1-G)\Pi_0}.$$

The denominator describes response to the auxiliary field containing both Hartree and exchange-correlation feedback. The physical dielectric function instead compares the external field with the total electrostatic field:

$$\varepsilon=\frac1{1+v\Pi} =\frac{1-v(1-G)\Pi_0}{1+vG\Pi_0}.$$

Only for $G=0$ does it equal the response denominator. A definition of $G$ and its frequency, momentum and sum-rule constraints are needed before using this extension.

Exercises

Two-dimensional plasmon

Use $\Pi_0\simeq nq^2/(m\omega^2)$ to derive the long-wavelength collective frequency of a two-dimensional gas with

$$v_q = \frac{ 2\pi e^2 }{ \epsilon_{\mathrm b}q }.$$

Solution

Use the same long-wavelength polarization:

$$\varepsilon_{\mathrm{RPA}} = 1 - \frac{ 2\pi e^2 }{ \epsilon_{\mathrm b}q } \frac{ nq^2 }{ m\omega^2 }.$$

The pole equation gives

$$\omega^2 = \frac{ 2\pi ne^2 }{ m\epsilon_{\mathrm b} } q.$$

Therefore

$$\omega(q) \propto \sqrt q.$$

The mode approaches zero as $q\to0$, unlike the three-dimensional bulk plasmon.

Particle–hole boundaries

For a parabolic dispersion, show that the energy transfer is

$$\omega = \frac{\hbar q^2}{2m} + \frac{\hbar\mathbf k\cdot\mathbf q}{m}.$$

Use $|\mathbf k|\le k_{\mathrm F}$ to obtain the upper continuum edge and explain the lower edge.

Solution

The one-particle energy difference is

$$\begin{aligned} \epsilon_{\mathbf k+\mathbf q} - \epsilon_{\mathbf k} &= \frac{\hbar^2}{2m} \left[ (\mathbf k+\mathbf q)^2-k^2 \right] \\ &= \frac{\hbar^2q^2}{2m} + \frac{\hbar^2\mathbf k\cdot\mathbf q}{m}. \end{aligned}$$

Divide by $\hbar$:

$$\omega = \frac{\hbar q^2}{2m} + \frac{\hbar\mathbf k\cdot\mathbf q}{m}.$$

The largest value occurs for $\mathbf k$ at the Fermi surface parallel to $\mathbf q$:

$$\omega_{\max} = \frac{\hbar q^2}{2m} + v_{\mathrm F}q.$$

For $q<2k_{\mathrm F}$, occupied and empty states can be chosen arbitrarily close across the Fermi surface, so positive excitation energies extend to zero:

$$\omega_{\min} = 0.$$

For $q>2k_{\mathrm F}$, the smallest allowed transfer occurs with $\mathbf k$ antiparallel to $\mathbf q$:

$$\omega_{\min} = \frac{\hbar q^2}{2m} - v_{\mathrm F}q.$$

Combining both cases,

$$\omega_{\min} = \max \left( 0, \frac{\hbar q^2}{2m} - v_{\mathrm F}q \right).$$

Ring expansion

Expand

$$\ln(1-x)+x$$

through fourth order and identify which interaction orders enter the RPA correlation energy.

Solution

The logarithm has expansion

$$\ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} + O(x^5).$$

Adding $x$ cancels the linear term:

$$\ln(1-x)+x = - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} + O(x^5).$$

With

$$x = v\Pi_0,$$

the correlation energy begins with the second-order ring. The cubic and quartic terms are the three- and four-bubble rings. The subtraction leaves correlation beyond the separately included Hartree and Fock contributions. The displayed power series is valid for $|x|<1$.

A screened response with finite compressibility

Let a static proper polarization be $P=-D$ with $D>0$, and let $v_q=C/q^2$ with $C>0$. Find the full response, its small-$q$ limit and the thermodynamic compressibility. Does a small full response imply a small compressibility?

Solution

The self-consistent result is $\Pi=-Dq^2/(q^2+CD)$. It tends to $-q^2/C$, whereas $\kappa_T=D/n^2$ remains finite when the uniform background follows the density. The vanishing full response comes from the electrostatic cost of a charged fluctuation, not from a thermodynamically incompressible neutralized fluid.

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