Quantum-Matter Correlators and Observable Conventions

Different many-body correlators answer different questions even when they contain the same two operators. Time ordering organizes perturbation theory, retarded ordering gives causal response, lesser and greater functions carry occupation and availability, the Matsubara function lives on discrete imaginary frequencies, and the spectral function supplies the common analytic data from which equilibrium translations are made. The equilibrium dictionary and its analytic continuation are developed in Mahan 2000, ch. 3.

Required background. Coherent-State Path Integrals for Many-Body Systems supplies the Euclidean functional; Thermal Density Operators and the KMS Condition, Imaginary Time and Matsubara Frequencies, and Retarded, Advanced, and Spectral Correlators supply the universal analytic framework.

Helpful background. Retarded, Advanced, and Keldysh Bases extends the dictionary away from equilibrium.

Single-particle real-time functions

For a fermionic annihilation operator $\psi(1)=\psi(t_1,\mathbf x_1)$ in an equilibrium state, choose

\begin{aligned} G^>(1,2)&=-i\langle\psi(1)\psi^\dagger(2)\rangle,\\ G^<(1,2)&=+i\langle\psi^\dagger(2)\psi(1)\rangle,\\ G^R(1,2)&=-i\theta(t_1-t_2) \langle\{\psi(1),\psi^\dagger(2)\}\rangle,\\ G^A(1,2)&=+i\theta(t_2-t_1) \langle\{\psi(1),\psi^\dagger(2)\}\rangle. \end{aligned}

The time-ordered function is

G^T(1,2)=\theta(t_1-t_2)G^>(1,2) +\theta(t_2-t_1)G^<(1,2).

For bosonic single-particle fields, replace the anticommutator in $G^{R,A}$ by a commutator and adjust the sign convention for $G^<$ . Rather than memorize a mixed-statistics table, define every function from its operator ordering before using a thermal identity.

For a stationary homogeneous state, Fourier transform differences and define

A(\mathbf k,\omega) =i[G^>(\mathbf k,\omega)-G^<(\mathbf k,\omega)] =-2\operatorname{Im}G^R(\mathbf k,\omega).

With canonical fermion normalization,

\int_{-\infty}^{\infty}\frac{\mathrm d\omega}{2\pi} A(\mathbf k,\omega)=1.

This convention makes $A\ge0$ in the canonical diagonal single-particle channel. Matrix-valued spectral functions are positive semidefinite, while arbitrary composite-operator commutator spectra need not be positive at every frequency.

Equilibrium occupation and KMS

For fermions in equilibrium,

G^<(\mathbf k,\omega)=if(\omega)A(\mathbf k,\omega), \qquad G^>(\mathbf k,\omega)=-i[1-f(\omega)]A(\mathbf k,\omega),

where $f(\omega)=1/(e^{\beta\omega}+1)$ when $\omega$ is measured relative to the chemical potential. Hence

n_{\mathbf k}=\langle c_{\mathbf k}^\dagger c_{\mathbf k}\rangle =\int\frac{\mathrm d\omega}{2\pi}f(\omega)A(\mathbf k,\omega).

If frequency is measured from the vacuum Hamiltonian instead, the thermal factor is $f(\omega-\mu)$ . This translation is a frequent source of apparent disagreement.

The retarded function follows from the spectral representation

G^R(\mathbf k,z)= \int\frac{\mathrm d\omega'}{2\pi} \frac{A(\mathbf k,\omega')}{z-\omega'}, \qquad \operatorname{Im}z>0.

Causality makes it analytic in the upper half-plane; poles of a stable retarded propagator lie on or below the real axis after continuation.

Matsubara translation

The fermionic imaginary-time function is

G(\tau,\mathbf k) =-\langle\mathrm T_\tau c_{\mathbf k}(\tau)c_{\mathbf k}^\dagger(0)\rangle, \qquad G(\tau+\beta)=-G(\tau).

At $\omega_n=(2n+1)\pi T$ ,

G(i\omega_n,\mathbf k) =\int\frac{\mathrm d\omega'}{2\pi} \frac{A(\mathbf k,\omega')}{i\omega_n-\omega'}.

If the exact analytic function is known, $i\omega_n\to\omega+i0$ yields $G^R$ . Reconstructing that function from finitely many noisy Matsubara data is an ill-posed inverse problem, not a mechanical substitution.

Density response and measurable spectra

For a Hermitian density $n_{\mathbf q}$ ,

\chi^R_{nn}(t,\mathbf q) =-i\theta(t)\langle[n_{\mathbf q}(t),n_{-\mathbf q}(0)]\rangle.

The dynamic structure factor

S(\mathbf q,\omega) =\frac1Z\sum_{mn}e^{-\beta K_m} |\langle n|n_{\mathbf q}|m\rangle|^2 2\pi\delta(\omega-K_n+K_m)

is nonnegative. The fluctuation–dissipation relation in this normalization is

-2\operatorname{Im}\chi^R_{nn}(\mathbf q,\omega) =[1-e^{-\beta\omega}]S(\mathbf q,\omega).

Thus the response spectral density changes sign with $\omega$ , while $S$ does not. Confusing these two objects can produce a false positivity violation.

Selection workflow

Use $G^T$ or Matsubara $G$ to organize equilibrium perturbation theory.
Use $G^R$ for causal propagation, poles, widths, and linear response.
Use $G^<$ for occupied spectral weight and $G^>$ for available addition weight.
Use $A$ to translate among equilibrium one-particle functions.
Use a composite-operator retarded susceptibility for a measured response, including its matrix element and contact terms.

No correlator is “the Green function” without its ordering, state, source convention, and frequency origin.

Exercises

One free fermionic level

For $K=\xi c^\dagger c$ , compute $A(\omega)$ , $G^<(\omega)$ , and $n$ .

Solution

The retarded propagator is $G^R(\omega)=1/(\omega-\xi+i0)$ , so $A(\omega)=2\pi\delta(\omega-\xi)$ . Then $G^<(\omega)=2\pi i f(\xi)\delta(\omega-\xi)$ and $n=\int fA/(2\pi)=f(\xi)$ .

Check detailed balance

Use the Lehmann sum for a Hermitian density to show $S(\mathbf q,-\omega)=e^{-\beta\omega}S(-\mathbf q,\omega)$ .