Thermal Green Functions

A thermal Green function packages equilibrium propagation, occupation, operator statistics, and equal-time algebra into one imaginary-time two-point function.

For a normal fermionic single-particle channel,

$$\mathcal G_{ab}(\tau) = - \left\langle \mathcal T_\tau c_a(\tau)c_b^\dagger(0) \right\rangle_\beta.$$

Its negative-time branch gives the occupied one-body density matrix, its positive-time branch gives the complementary empty weight, and the jump at zero is fixed by the canonical anticommutator. The derivations below use a positive-temperature grand-canonical Fock-space trace for number-changing fields; fixed-number canonical projections need separate treatment.

Helpful background. Imaginary time supplies operator evolution; Matsubara frequencies supplies parity-dependent Fourier grids. The single-particle Green-function discussion explains addition and removal spectra.

Thermal Green-function definitions and conventions

Equilibrium generator

Use a self-adjoint generator with a finite Gibbs trace at $\beta>0$, or a specified limit of local equilibrium correlators. Use that generator consistently in the density operator and imaginary-time evolution:

$$\rho_\beta = \frac{ e^{-\beta\mathcal K} }{ \mathcal Z }, \qquad \mathcal Z = \operatorname{Tr} e^{-\beta\mathcal K}.$$

For a grand-canonical state with conserved particle number, $[H,N]=0$,

$$\mathcal K = H-\mu N.$$

Imaginary-time operators are

$$A(\tau) = e^{\tau\mathcal K/\hbar} A e^{-\tau\mathcal K/\hbar}.$$

Using $H$ in this evolution while retaining $H-\mu N$ in the Gibbs weight changes where $\mu$ appears. Either convention can be translated into the other, but they must not be mixed silently.

Thermal interval

Write

$$L_\tau := \beta\hbar, \qquad 0 \leq \tau \lt L_\tau.$$

The notation keeps $\tau$ in time units and $\beta$ in inverse-energy units.

Matsubara-energy transform

For a periodic or antiperiodic function $\mathcal C(\tau)$, this page uses

$$\mathcal C(i\zeta_n) = \frac1\hbar \int_0^{L_\tau} d\tau\, e^{i\zeta_n\tau/\hbar} \mathcal C(\tau),$$

with inverse

$$\mathcal C(\tau) = \frac1\beta \sum_n e^{-i\zeta_n\tau/\hbar} \mathcal C(i\zeta_n).$$

The symbol $\zeta_n$ is generic. For a bosonic channel,

$$\zeta_n = \Omega_m = \frac{2\pi m}{\beta}.$$

For a fermionic channel,

$$\zeta_n = \nu_n = \frac{(2n+1)\pi}{\beta}.$$

What Is Being Correlated?

Specify the operators, ordering, and overall prefactor before comparing two Green-function formulas. Some useful choices are:

Object	Operators	Common definition	Boundary class
fermion propagator	$c_a$, $c_b^\dagger$	$-\langle\mathcal T_\tau c_a c_b^\dagger\rangle$	antiperiodic
boson propagator	$b_a$, $b_b^\dagger$	$-\langle\mathcal T_\tau b_a b_b^\dagger\rangle$	periodic
coordinate correlator	$q$, $q$	$+\langle\mathcal T_\tau q q\rangle$	periodic
density correlator	$\delta n$, $\delta n$	$+\langle\mathcal T_\tau\delta n\,\delta n\rangle$	periodic
spin correlator	$\delta S^\alpha$, $\delta S^\beta$	convention dependent	periodic
pair correlator	$\Delta$, $\Delta^\dagger$	convention dependent	periodic

The first two rows share a conventional leading minus sign, but their exchange signs differ. The next rows illustrate that a bosonic or even channel need not use that leading minus sign at all.

Graded Imaginary-Time Ordering

Let $A$ and $B$ have definite fermion parities

$$p_A, p_B \in \{0,1\}.$$

Imaginary-time ordering is

$$\mathcal T_\tau A(\tau)B(\tau') ={} \theta(\tau-\tau') \times A(\tau)B(\tau') + (-1)^{p_Ap_B} \theta(\tau'-\tau) \times B(\tau')A(\tau).$$

When both operators are odd, reversing them contributes a minus sign. If either is even, no graded exchange sign appears.

For a channel-dependent overall convention $s_{AB}$, define

$$\mathcal C_{AB}(\tau-\tau') = s_{AB} \left\langle \mathcal T_\tau A(\tau)B(\tau') \right\rangle_\beta.$$

The fixed sign $s_{AB}$ and the exchange sign $(-1)^{p_Ap_B}$ have different origins:

• $s_{AB}$ is chosen by convention;
• $(-1)^{p_Ap_B}$ follows from graded operator algebra.

Changing the first does not change the thermal boundary class. Dropping the second changes the physics.

Piecewise form

Set $\tau'=0$ and suppose the two insertions have the same parity $p$. With

$$\eta_p = (-1)^p,$$

the ordered correlator is

$$\mathcal C_{AB}(\tau) = \begin{cases} s_{AB} \langle A(\tau)B(0)\rangle_\beta, & \tau>0, \\[4pt] s_{AB}\eta_p \langle B(0)A(\tau)\rangle_\beta, & \tau<0. \end{cases}$$

The two branches contain different operator products. At finite temperature both may be nonzero because the initial thermal state need not be the ground state or the vacuum.

Stationarity and Time Differences

Since

$$[\rho_\beta,\mathcal K] = 0,$$

an equilibrium two-point function is invariant under a common imaginary-time translation:

$$\mathcal C_{AB}(\tau,\tau') = \mathcal C_{AB}(\tau-\tau',0).$$

This statement assumes:

• a time-independent equilibrium generator;
• no explicit imaginary-time dependence in the operators beyond Heisenberg evolution;
• a well-defined thermal trace or equilibrium state.

A driven state, a transient preparation, or a source varying around the thermal contour generally depends on both time arguments separately. Matsubara frequency is then not a complete label.

Thermal Boundary Conditions

The KMS condition relates reversed-order correlators through trace cyclicity. Graded ordering supplies the sign needed to restore the chosen order. For a two-point channel generated by parity $p$,

$$\mathcal C_p(\tau+L_\tau) = (-1)^p \mathcal C_p(\tau).$$

Thus:

$$\begin{aligned} \mathcal C_{\mathrm B}(\tau+L_\tau) &= \mathcal C_{\mathrm B}(\tau), \\ \mathcal C_{\mathrm F}(\tau+L_\tau) &= - \mathcal C_{\mathrm F}(\tau). \end{aligned}$$

The sign is assigned by the parity of the complete operator channel:

• $c_a$ is odd, so a normal fermion propagator is antiperiodic;
• $b_a$ is even under fermion parity, so a boson propagator is periodic;
• $c_a^\dagger c_b$ is even, so a density correlator is periodic;
• $c_\downarrow c_\uparrow$ is even, so a pair correlator is periodic.

Antiperiodicity is a property of the ordered thermal correlator. It does not say that a physical fermion state acquires a minus sign after an elapsed laboratory time $\beta\hbar$.

Normal Single-Particle Functions

Fermions

For fermionic operators obeying the canonical anticommutation relations,

$$\left\{ c_a,c_b^\dagger \right\} = \delta_{ab},$$

define

$$\mathcal G_{ab}(\tau) = - \left\langle \mathcal T_\tau c_a(\tau)c_b^\dagger(0) \right\rangle_\beta.$$

Its branches are

$$\mathcal G_{ab}(\tau) = \begin{cases} - \langle c_a(\tau)c_b^\dagger(0)\rangle_\beta, & \tau>0, \\[4pt] + \langle c_b^\dagger(0)c_a(\tau)\rangle_\beta, & \tau<0. \end{cases}$$

The plus sign on the negative-time branch results from multiplying the conventional overall minus by the odd-operator exchange minus.

Bosons

For canonical bosons,

$$\left[ b_a,b_b^\dagger \right] = \delta_{ab},$$

a common nonrelativistic convention is

$$\mathcal D_{ab}(\tau) = - \left\langle \mathcal T_\tau b_a(\tau)b_b^\dagger(0) \right\rangle_\beta.$$

The branches are

$$\mathcal D_{ab}(\tau) = \begin{cases} - \langle b_a(\tau)b_b^\dagger(0)\rangle_\beta, & \tau>0, \\[4pt] - \langle b_b^\dagger(0)b_a(\tau)\rangle_\beta, & \tau<0. \end{cases}$$

There is no exchange minus because bosonic operators are even. The leading minus remains only because this definition chose it.

Some atomic, condensed-matter, and field-theory texts define bosonic propagators with different overall signs or factors of $\hbar$. A formula should be compared only after its definition and transform pair are aligned.

Equal-Time Limits and Contact Terms

The value at exactly $\tau=0$ is less informative than the one-sided limits. Define the fermionic one-body density matrix

$$\rho_{ab}^{(1)} := \left\langle c_b^\dagger c_a \right\rangle_\beta.$$

Then

$$\begin{aligned} \mathcal G_{ab}(0^-) &= \rho_{ab}^{(1)}, \\ \mathcal G_{ab}(0^+) &= - \left( \delta_{ab} - \rho_{ab}^{(1)} \right). \end{aligned}$$

Their difference is the canonical contact term:

$$\mathcal G_{ab}(0^+) - \mathcal G_{ab}(0^-) = - \delta_{ab}.$$

For bosons, let

$$n_{ab}^{(b)} := \left\langle b_b^\dagger b_a \right\rangle_\beta.$$

The same leading-minus convention gives

$$\begin{aligned} \mathcal D_{ab}(0^-) &= - n_{ab}^{(b)}, \\ \mathcal D_{ab}(0^+) &= - \left( \delta_{ab} + n_{ab}^{(b)} \right), \end{aligned}$$

and therefore

$$\mathcal D_{ab}(0^+) - \mathcal D_{ab}(0^-) = - \delta_{ab}.$$

The equal-time jumps look identical in this convention, but their boundary gluing differs:

• the fermion propagator is antiperiodic;
• the boson propagator is periodic.

Exact free-mode examples with $\beta\xi=\beta\epsilon=\ln3$ give $f=1/4$ and $n_{\mathrm B}=1/2$. The curves are $-\tfrac34 e^{-\xi\tau/\hbar}$ and $-\tfrac32e^{-\epsilon\tau/\hbar}$ on $0<\tau<L_\tau=\beta\hbar$. Both contact jumps equal $-1$, shown on the same vertical scale. The endpoint at $L_\tau^-$ equals the negative of the fermion value at $0^-$, and the boson value itself.

Occupations from frequency data

Using the inverse transform, the fermionic density matrix is

$$\rho_{ab}^{(1)} = \lim_{\epsilon\downarrow0} \frac1\beta \sum_n e^{i\nu_n\epsilon/\hbar} \mathcal G_{ab}(i\nu_n).$$

The factor $e^{i\nu_n\epsilon/\hbar}$ specifies the $0^-$ limit. Omitting it from a conditionally convergent sum can return an ambiguous midpoint rather than the occupation.

For the bosonic leading-minus convention,

$$n_{ab}^{(b)} = - \lim_{\epsilon\downarrow0} \frac1\beta \sum_m e^{i\Omega_m\epsilon/\hbar} \mathcal D_{ab}(i\Omega_m).$$

Matsubara Formalism Preview develops these convergence prescriptions in detail.

Free Fermion Mode

Consider one canonical fermion with

$$\mathcal K = \xi c^\dagger c, \qquad \xi = \epsilon-\mu.$$

Its partition function and occupation are

$$\begin{aligned} \mathcal Z_{\mathrm F} &= 1+e^{-\beta\xi}, \\ f(\xi) &= \frac1{ e^{\beta\xi}+1 }. \end{aligned}$$

The imaginary-time equation of motion gives

$$c(\tau) = e^{-\xi\tau/\hbar}c.$$

Positive-time branch

For

$$0 \lt \tau \lt L_\tau,$$

the operator order is already correct:

$$\begin{aligned} \mathcal G_{\mathrm F}(\tau) &= - e^{-\xi\tau/\hbar} \langle cc^\dagger\rangle_\beta \\ &= - \left[ 1-f(\xi) \right] e^{-\xi\tau/\hbar}. \end{aligned}$$

Negative-time branch

For

$$-L_\tau \lt \tau \lt 0,$$

graded ordering reverses the odd operators:

$$\begin{aligned} \mathcal G_{\mathrm F}(\tau) &= + e^{-\xi\tau/\hbar} \langle c^\dagger c\rangle_\beta \\ &= f(\xi) e^{-\xi\tau/\hbar}. \end{aligned}$$

Both branches matter at finite temperature. The positive-time branch carries the probability that the mode is empty, while the negative-time branch carries the probability that it is occupied.

Antiperiodicity check

The Fermi function obeys

$$\left[ 1-f(\xi) \right] e^{-\beta\xi} = f(\xi).$$

Therefore,

$$\mathcal G_{\mathrm F} (\tau+L_\tau) = - \mathcal G_{\mathrm F}(\tau).$$

At the seam,

$$\begin{aligned} \mathcal G_{\mathrm F}(0^-) &= f(\xi), \\ \mathcal G_{\mathrm F}(0^+) &= - \left[ 1-f(\xi) \right], \\ \mathcal G_{\mathrm F}(L_\tau^-) &= - f(\xi). \end{aligned}$$

The first and third values are related by antiperiodicity. The first and second differ by the canonical jump. Here “canonical” describes the operator algebra, not a fixed-number ensemble.

Frequency-domain result

For a fermionic Matsubara energy $\nu_n$,

$$e^{i\beta\nu_n} = -1.$$

Transforming the positive-time branch gives

$$\begin{aligned} \mathcal G_{\mathrm F}(i\nu_n) ={}& - \left[ 1-f(\xi) \right] \times \frac{ e^{\beta(i\nu_n-\xi)}-1 }{ i\nu_n-\xi } \\ ={}& \frac1{ i\nu_n-\xi }. \end{aligned}$$

The explicit occupation factor cancels against the thermal endpoint factor. Temperature remains encoded in the discrete values of $\nu_n$ and reappears when the inverse transform reconstructs the two time-domain branches.

Limiting checks

If $\xi>0$ and $T\to0$, then

$$f(\xi) \longrightarrow 0,$$

so positive imaginary time describes adding a particle to an empty mode.

If $\xi<0$ and $T\to0$, then

$$f(\xi) \longrightarrow 1,$$

so the positive-time branch vanishes and the negative-time branch records removal from an occupied mode.

At $\xi=0$,

$$f(0) = \frac12,$$

and the propagator is constant on the open interval:

$$\mathcal G_{\mathrm F}(\tau) = - \frac12, \qquad 0<\tau<L_\tau.$$

It still has an equal-time jump and an antiperiodic seam. A flat open-interval curve does not make it bosonic.

Free Boson Mode

Consider one canonical boson with

$$\mathcal K = \varepsilon b^\dagger b.$$

For an unconstrained grand-canonical oscillator, normalizability requires

$$\varepsilon > 0.$$

The partition function and Bose occupation are

$$\begin{aligned} \mathcal Z_{\mathrm B} &= \frac1{ 1-e^{-\beta\varepsilon} }, \\ n_{\mathrm B}(\varepsilon) &= \frac1{ e^{\beta\varepsilon}-1 }. \end{aligned}$$

Evolution gives

$$b(\tau) = e^{-\varepsilon\tau/\hbar}b.$$

Piecewise propagator

Using

$$\mathcal D_{\mathrm B}(\tau) = - \left\langle \mathcal T_\tau b(\tau)b^\dagger(0) \right\rangle_\beta,$$

one obtains, for $0<\tau<L_\tau$,

$$\mathcal D_{\mathrm B}(\tau) = - \left[ 1+n_{\mathrm B}(\varepsilon) \right] e^{-\varepsilon\tau/\hbar}.$$

For $-L_\tau<\tau<0$,

$$\mathcal D_{\mathrm B}(\tau) = - n_{\mathrm B}(\varepsilon) e^{-\varepsilon\tau/\hbar}.$$

Unlike the fermionic case, the negative-time branch keeps the conventional leading minus because exchanging bosonic operators contributes no graded sign.

Periodicity check

The Bose function obeys

$$\left[ 1+n_{\mathrm B}(\varepsilon) \right] e^{-\beta\varepsilon} = n_{\mathrm B}(\varepsilon).$$

Consequently,

$$\mathcal D_{\mathrm B} (\tau+L_\tau) = \mathcal D_{\mathrm B}(\tau).$$

At the seam,

$$\begin{aligned} \mathcal D_{\mathrm B}(0^-) &= - n_{\mathrm B}, \\ \mathcal D_{\mathrm B}(0^+) &= - \left( 1+n_{\mathrm B} \right), \\ \mathcal D_{\mathrm B}(L_\tau^-) &= - n_{\mathrm B}. \end{aligned}$$

The first and third agree by periodicity. The first and second differ by the canonical commutator.

Frequency-domain result

For

$$\Omega_m = \frac{2\pi m}{\beta},$$

one has $e^{i\beta\Omega_m}=1$. Therefore,

$$\begin{aligned} \mathcal D_{\mathrm B}(i\Omega_m) ={}& - \left( 1+n_{\mathrm B} \right) \times \frac{ e^{\beta(i\Omega_m-\varepsilon)}-1 }{ i\Omega_m-\varepsilon } \\ ={}& \frac1{ i\Omega_m-\varepsilon }. \end{aligned}$$

The algebra resembles the fermion result, but the grid and thermal identity differ.

The bosonic zero mode

At $m=0$,

$$\mathcal D_{\mathrm B}(i\Omega_0) = - \frac1\varepsilon.$$

This is finite for $\varepsilon>0$. As $\varepsilon\to0^+$, both the zero-mode propagator and $n_{\mathrm B}$ diverge, while the single-mode grand partition function ceases to be normalizable at $\varepsilon=0$.

In an extended interacting Bose system, condensation and critical zero-mode physics require a separate treatment of the thermodynamic limit, symmetry breaking, and infrared fluctuations. The algebra of one stable oscillator does not by itself establish a phase transition.

Harmonic-Oscillator Coordinate Correlator

The same bosonic mode also illustrates why overall sign conventions must be stated. Let

$$q = \sqrt{ \frac{\hbar}{ 2M\omega_0 } } \left( b+b^\dagger \right), \qquad \varepsilon = \hbar\omega_0.$$

Define the coordinate correlator without a leading minus:

$$C_{qq}(\tau) = \left\langle \mathcal T_\tau q(\tau)q(0) \right\rangle_\beta.$$

For

$$0 \leq \tau \leq L_\tau,$$

the exact result is

$$C_{qq}(\tau) ={} \frac{\hbar}{ 2M\omega_0 } \times \left[ \left( 1+n_{\mathrm B} \right) e^{-\omega_0\tau} + n_{\mathrm B} e^{\omega_0\tau} \right].$$

It satisfies

$$C_{qq}(L_\tau-\tau) = C_{qq}(\tau)$$

and is periodic around the thermal circle.

Its Matsubara transform is

$$C_{qq}(i\Omega_m) = \frac{ \hbar^2 }{ M \left[ \Omega_m^2 + (\hbar\omega_0)^2 \right] }.$$

This correlator is even in $\Omega_m$ and positive at $\tau=0$. It is not the same object as the annihilation-operator propagator

$$\mathcal D_{\mathrm B}(i\Omega_m) = \frac1{ i\Omega_m-\hbar\omega_0 }.$$

Both are correct. They answer different operator questions and use different overall conventions.

Quadratic Many-Orbital Benchmark

Let

$$\mathcal K_0 = \sum_{a,b} c_a^\dagger h_{ab} c_b,$$

with $h=h^\dagger$ on a finite orbital space and a trace over its complete fermionic Fock space. Define the matrix Fermi function

$$f(h) = \left( e^{\beta h} + \mathbf 1 \right)^{-1}.$$

For $0<\tau<L_\tau$,

$$\mathcal G_0(\tau) = - e^{-h\tau/\hbar} \left[ \mathbf 1-f(h) \right].$$

The transform is the resolvent

$$\mathcal G_0(i\nu_n) = \left[ i\nu_n\mathbf 1-h \right]^{-1}.$$

Diagonalizing $h$ reduces this expression to independent free modes. Under a unitary basis change

$$c' = Uc,$$

the Green matrix transforms as

$$\mathcal G_0' = U \mathcal G_0 U^\dagger.$$

For number-conserving quadratic bosons on a finite orbital space, the analogous formulas use

$$n_{\mathrm B}(h) = \left( e^{\beta h} - \mathbf 1 \right)^{-1},$$

provided the unconstrained bosonic Gibbs trace is normalizable. This requires all one-particle eigenvalues of $h$ to be strictly positive. Pairing terms require an enlarged bosonic Nambu description with its commutator metric; the simple matrix replacement $f(h)\mapsto n_{\mathrm B}(h)$ is not sufficient for that case.

Equation of Motion

The equal-time jump appears as a delta-function source. For the normal fermion Green function,

$$\hbar \partial_\tau \mathcal G_{ab}(\tau) ={} - \hbar \delta_{\mathrm F}(\tau) \delta_{ab} - \left\langle \mathcal T_\tau \left[ \mathcal K, c_a(\tau) \right] c_b^\dagger(0) \right\rangle_\beta.$$

Here $\delta_{\mathrm F}$ is the delta distribution compatible with antiperiodic boundary conditions.

For the quadratic generator,

$$\left[ \mathcal K_0,c_a \right] = - \sum_c h_{ac}c_c,$$

so

$$\left[ \hbar\partial_\tau\mathbf 1 + h \right] \mathcal G_0(\tau) = - \hbar \delta_{\mathrm F}(\tau) \mathbf 1.$$

Fourier transformation gives

$$\left[ i\nu_n\mathbf 1-h \right] \mathcal G_0(i\nu_n) = \mathbf 1.$$

For an interacting Hamiltonian, the commutator generally creates higher-order operator products. This is the equation-of-motion hierarchy. Closing that hierarchy requires an exact solution, an approximation, or a self-energy construction.

High-Frequency Check

The canonical contact term fixes the leading large-frequency coefficient. Assuming the next spectral moment is finite, the expansion is:

$$\mathcal G(i\nu_n) = \frac{ \mathbf 1 }{ i\nu_n } + \mathcal O \left( \frac1{\nu_n^2} \right).$$

For the bosonic annihilation-operator propagator with the same leading-minus convention,

$$\mathcal D(i\Omega_m) = \frac{ \mathbf 1 }{ i\Omega_m } + \mathcal O \left( \frac1{\Omega_m^2} \right).$$

The coefficient changes when the operator normalization or overall Green-function convention changes. A coordinate correlator, for example, falls as $1/\Omega_m^2$ because its positive- and negative-energy poles cancel the $1/\Omega_m$ term.

High-frequency asymptotics provide a stringent numerical check. They test the equal-time algebra before any analytic continuation is attempted.

Connected Observable Correlators

For an observable $A$ with nonzero thermal expectation value, define its fluctuation

$$\delta A = A-\langle A\rangle_\beta.$$

A connected imaginary-time correlator is

$$C_{AB}^{\mathrm c}(\tau) = \left\langle \mathcal T_\tau \delta A(\tau) \delta B(0) \right\rangle_\beta.$$

Subtracting the disconnected term removes

$$\langle A\rangle_\beta \langle B\rangle_\beta,$$

which otherwise contributes a static component. Connected subtraction does not remove every static fluctuation: a conserved number can fluctuate between members of the ensemble even when its mean is subtracted. Its contribution is confined to the zero bosonic Matsubara component, as the single-mode example shows.

Density, spin, current, and pair observables are usually even channels, so their thermal correlators are periodic and use bosonic Matsubara energies. Their operator definitions, tensor indices, conserved quantities, and contact terms differ from those of a normal single-particle propagator.

Green Function Versus Response

An imaginary-time Green function and a retarded response function are related through equilibrium spectral structure, but they are not the same function.

Question	Imaginary-time object	Retarded object
ordering	$\mathcal T_\tau$ on the thermal circle	causal support with $\theta(t)$
domain	compact imaginary time or discrete $i\zeta_n$	real time or $E+i0^+$
boundary information	periodic or antiperiodic	retarded boundary condition
direct use	equilibrium averages, perturbation theory, Monte Carlo	causal response and spectra
conversion	requires a common analytic or spectral representation	obtained as a real-axis boundary value

A fermionic single-particle Green function is not ordinarily the response to a classical laboratory source, because a single fermion operator is odd. Density and spin susceptibilities are even observables and do describe response to appropriate classical sources.

Retarded and Advanced Response and Kubo Formula develop the causal side of this distinction.

Interacting Systems

Interactions change the Green function without changing its defining algebra. In frequency space one often writes

$$\mathcal G^{-1}(i\nu_n) = \mathcal G_0^{-1}(i\nu_n) - \Sigma(i\nu_n),$$

or

$$\mathcal G^{-1}(i\nu_n) = i\nu_n\mathbf 1 - h - \Sigma(i\nu_n).$$

The self-energy can:

• shift effective single-particle energies;
• redistribute spectral weight;
• produce finite lifetimes after real-frequency continuation;
• couple orbital, spin, sublattice, or Nambu components;
• carry nontrivial temperature dependence.

It does not remove the canonical equal-time jump. A numerical interacting Green function with the wrong $1/(i\nu_n)$ coefficient violates the operator algebra regardless of how plausible its low-frequency structure looks.

Diagrammatic methods develops self-energy expansions. Single-particle Green functions explains poles, residues, continua, and quasiparticles.

Numerical Evaluation

Exact diagonalization, Monte Carlo, impurity solvers, and tensor-network methods can all compute thermal Green functions. Compare results only after their matrix indices, signs, time grids, and uncertainties are aligned. The equal-time contact and large-frequency moments provide independent checks before any continuation is attempted. DMFT illustrates these checks for matrix-valued impurity and lattice propagators.

Endpoint handling

At a jump, $0^+$ and $L_\tau^-$ are distinct one-sided values: the latter is related to $0^-$, not to $0^+$. Preserve that distinction when extracting occupations. A periodic or phase-adjusted FFT usually samples $jL_\tau/N_\tau$ for $j=0,\ldots,N_\tau-1$ without duplicating the endpoint, while treating the known contact jump analytically.

Fourier transforms of discontinuous data

A canonical single-particle propagator has an equal-time jump. A naive discrete transform of a coarse grid can therefore show ringing or incorrect high-frequency tails. Subtracting a known analytic tail before transformation and adding it back afterward often improves convergence.

Imaginary-time smoothing

Imaginary-time kernels suppress fine real-frequency structure. Two distinct spectra can produce very similar noisy $\tau$ data. Stable forward agreement in imaginary time is necessary, but it does not by itself prove a unique real-frequency reconstruction.

Common pitfalls

The two minus signs have different origins. An overall minus in a propagator definition is conventional. Exchanging two odd fields under time ordering introduces a separate minus. Only the latter determines fermionic antiperiodicity.

An imaginary-time decay is not a lifetime. A decay exponent can select an excitation energy or threshold. A real-time linewidth requires information about the distribution of spectral weight; it cannot generally be read from that exponent alone.

Nambu does not always mean an indefinite spectrum. A fermionic Nambu anticommutator spectral matrix is positive semidefinite and integrates to the identity in canonical coordinates. A bosonic Nambu commutator spectrum instead has a signed metric. Check the bracket and field algebra before applying positivity.

Endpoints do not establish the full function. A correct jump and occupation are necessary checks. They do not establish all-time KMS consistency, spectral positivity, or the accuracy of an inferred spectrum.

Related applications

• Spectral representation derives thermal kernels and retarded boundary values.
• Superconducting proximity effect uses matrix and anomalous Green functions in inhomogeneous systems.
• Correlation-function definitions collects ordering and transform conventions.
• Matsubara frequency table is a compact units and indexing lookup.

Exercises

1. Derive the ordered branches

Let $A$ and $B$ have the same fermion parity $p$, and define

$$\mathcal C_{AB}(\tau) = - \left\langle \mathcal T_\tau A(\tau)B(0) \right\rangle_\beta.$$

Write the positive- and negative-time branches explicitly. Evaluate the exchange sign for $p=0$ and $p=1$.

Solution

For $\tau>0$, graded ordering gives

$$\mathcal T_\tau A(\tau)B(0) = A(\tau)B(0).$$

For $\tau<0$,

$$\mathcal T_\tau A(\tau)B(0) = (-1)^p B(0)A(\tau).$$

Multiplying by the fixed leading minus, the positive-time branch is

$$\mathcal C_{AB}(\tau) = - \langle A(\tau)B(0)\rangle_\beta.$$

The negative-time branch is

$$\mathcal C_{AB}(\tau) = - (-1)^p \langle B(0)A(\tau)\rangle_\beta.$$

For $p=0$, the negative-time branch keeps the leading minus. For $p=1$, the exchange minus cancels it and the negative-time branch has a plus sign.

2. Reconstruct the free fermion propagator

For

$$\mathcal K = \xi c^\dagger c,$$

derive $\mathcal G_{\mathrm F}(\tau)$ on both open branches, verify antiperiodicity, and compute the equal-time jump.

Solution

The evolution is

$$c(\tau) = e^{-\xi\tau/\hbar}c.$$

Using

$$\langle c^\dagger c\rangle_\beta = f(\xi), \qquad \langle cc^\dagger\rangle_\beta = 1-f(\xi),$$

gives, for $0<\tau<L_\tau$,

$$\mathcal G_{\mathrm F}(\tau) = - \left[ 1-f(\xi) \right] e^{-\xi\tau/\hbar}.$$

For $-L_\tau<\tau<0$,

$$\mathcal G_{\mathrm F}(\tau) = f(\xi) e^{-\xi\tau/\hbar}.$$

Since

$$\left[ 1-f(\xi) \right] e^{-\beta\xi} = f(\xi),$$

the two branches obey

$$\mathcal G_{\mathrm F}(\tau+L_\tau) = - \mathcal G_{\mathrm F}(\tau).$$

Finally,

$$\begin{aligned} \mathcal G_{\mathrm F}(0^+) &= - \left( 1-f \right), \\ \mathcal G_{\mathrm F}(0^-) &= f, \end{aligned}$$

so

$$\mathcal G_{\mathrm F}(0^+) - \mathcal G_{\mathrm F}(0^-) = -1.$$

3. Check the free boson mode

For

$$\mathcal K = \varepsilon b^\dagger b, \qquad \varepsilon>0,$$

derive the leading-minus boson propagator, verify periodicity, and find its zero-frequency value.

Solution

The occupation and evolution are

$$n_{\mathrm B} = \frac1{ e^{\beta\varepsilon}-1 }, \qquad b(\tau) = e^{-\varepsilon\tau/\hbar}b.$$

Therefore, for $0<\tau<L_\tau$,

$$\mathcal D_{\mathrm B}(\tau) = - \left( 1+n_{\mathrm B} \right) e^{-\varepsilon\tau/\hbar}.$$

For $-L_\tau<\tau<0$,

$$\mathcal D_{\mathrm B}(\tau) = - n_{\mathrm B} e^{-\varepsilon\tau/\hbar}.$$

The identity

$$\left( 1+n_{\mathrm B} \right) e^{-\beta\varepsilon} = n_{\mathrm B}$$

implies periodicity. Fourier transformation gives

$$\mathcal D_{\mathrm B}(i\Omega_m) = \frac1{ i\Omega_m-\varepsilon }.$$

At $m=0$,

$$\mathcal D_{\mathrm B}(i\Omega_0) = - \frac1\varepsilon.$$

The positivity condition on $\varepsilon$ is also the condition that the unconstrained single-mode Gibbs trace converge.

4. Extract a density matrix

For fermionic operators obeying the canonical anticommutation relations, show that the one-body density matrix is $\mathcal G(0^-)$ and that the complementary empty-state matrix is $-\mathcal G(0^+)$.

Solution

On the negative-time branch,

$$\mathcal G_{ab}(0^-) = \left\langle c_b^\dagger c_a \right\rangle_\beta = \rho_{ab}^{(1)}.$$

On the positive-time branch,

$$\begin{aligned} - \mathcal G_{ab}(0^+) &= \left\langle c_a c_b^\dagger \right\rangle_\beta \\ &= \delta_{ab} - \rho_{ab}^{(1)}. \end{aligned}$$

The second equality uses

$$c_a c_b^\dagger = \delta_{ab} - c_b^\dagger c_a.$$

Thus the two one-sided limits encode occupied and empty weight, and their difference enforces the anticommutator.

5. Transform the oscillator coordinate correlator

Starting from

$$C_{qq}(\tau) ={} \frac{\hbar}{ 2M\omega_0 } \times \left[ \left( 1+n_{\mathrm B} \right) e^{-\omega_0\tau} + n_{\mathrm B} e^{\omega_0\tau} \right],$$

for $0\leq\tau\leq L_\tau$, show that its Matsubara transform is even in $\Omega_m$.

Solution

Let

$$E_0 = \hbar\omega_0.$$

Transforming the two exponentials and using $e^{i\beta\Omega_m}=1$ gives

$$C_{qq}(i\Omega_m) ={} \frac{\hbar}{ 2M\omega_0 } \frac1{ E_0-i\Omega_m } + \frac{\hbar}{ 2M\omega_0 } \frac1{ E_0+i\Omega_m }.$$

Combining the two terms,

$$\begin{aligned} C_{qq}(i\Omega_m) &= \frac{\hbar}{ 2M\omega_0 } \frac{ 2E_0 }{ E_0^2+\Omega_m^2 } \\ &= \frac{ \hbar^2 }{ M \left[ \Omega_m^2 + (\hbar\omega_0)^2 \right] }. \end{aligned}$$

Only $\Omega_m^2$ appears, so the result is even.

6. Solve a quadratic matrix problem

Let

$$h = \begin{pmatrix} \xi_1 & t\\ t^* & \xi_2 \end{pmatrix}.$$

State the free fermion Green matrix in Matsubara space and explain how its equal-time limit is obtained without computing each matrix element separately.

Solution

The Matsubara Green matrix is

$$\mathcal G_0(i\nu_n) = \left[ i\nu_n\mathbf 1-h \right]^{-1}.$$

Diagonalize

$$h = U^\dagger \begin{pmatrix} \lambda_+ & 0\\ 0 & \lambda_- \end{pmatrix} U.$$

The one-body density matrix is the matrix Fermi function

$$\rho^{(1)} = f(h) = U^\dagger \begin{pmatrix} f(\lambda_+) & 0\\ 0 & f(\lambda_-) \end{pmatrix} U.$$

Therefore,

$$\mathcal G_0(0^-) = f(h),$$

while

$$\mathcal G_0(0^+) = - \left[ \mathbf 1-f(h) \right].$$

This uses spectral calculus for the matrix $h$ and avoids separate contour sums for every entry.

7. Diagnose endpoint data

A numerical calculation for one fermion orbital reports

$$\mathcal G(0^+) = -0.72, \qquad \mathcal G(L_\tau^-) = -0.28.$$

It also claims that the function is periodic. Which parts of the data are mutually consistent, and which claim is wrong?

Solution

For a canonical fermion,

$$\mathcal G(0^+) = - \left( 1-f \right).$$

The first value gives

$$f = 0.28.$$

Antiperiodicity requires

$$\mathcal G(L_\tau^-) = - \mathcal G(0^-) = -f,$$

which agrees with the reported value $-0.28$.

The canonical jump is also consistent:

$$\mathcal G(0^+) - \mathcal G(0^-) = -0.72-0.28 = -1.$$

The endpoint values are compatible with fermionic occupation and antiperiodicity. They do not by themselves verify the function at intermediate times. The claim of periodicity is incompatible with these endpoint data and the canonical jump.

References

• A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover (1975) – classic thermal diagrammatic methods.
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010) – finite-temperature Green functions and field-theory organization.
• G. Baym and N. D. Mermin, “Determination of Thermodynamic Green’s Functions”, Journal of Mathematical Physics 2, 232–234 (1961) – analytic conditions for thermodynamic Green functions.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003 reprint of the 1971 edition) – operator, spectral, and finite-temperature Green-function conventions.
• R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems”, Journal of the Physical Society of Japan 12, 570–586 (1957) – equilibrium correlations and response.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000) – thermal propagators and condensed-matter applications.
• P. C. Martin and J. Schwinger, “Theory of Many-Particle Systems. I”, Physical Review 115, 1342–1373 (1959) – equilibrium Green functions, hierarchy relations, and thermal identities.
• T. Matsubara, “A New Approach to Quantum-Statistical Mechanics”, Progress of Theoretical Physics 14, 351–378 (1955) – original imaginary-time many-body formalism.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, CRC Press (2018 reissue) – operator, coherent-state, and functional methods.