Matsubara Frequency Table

Canonical treatment: Bosonic and Fermionic Matsubara Frequencies owns the derivation, transform conventions, routing arguments, numerical discussion, exercises, and references. This page is the quick lookup table.

Let $\beta=1/(k_{\mathrm B}T)$ and let the imaginary-time circumference be $L_\tau=\beta\hbar$.

Master Table

Channel	Boundary condition	Energy grid	Angular-frequency grid	Zero mode	Index reflection
Bosonic	$F(\tau+L_\tau)=F(\tau)$	$\Omega_m=2\pi m/\beta$	$\omega_m=2\pi m/(\beta\hbar)$	$m=0$	$m\mapsto-m$
Fermionic	$F(\tau+L_\tau)=-F(\tau)$	$\varepsilon_n=(2n+1)\pi/\beta$	$\omega_n=(2n+1)\pi/(\beta\hbar)$	none	$n\mapsto-n-1$

The energy variables have units of energy; the angular frequencies have units of inverse time and obey $\Omega_m=\hbar\omega_m$. In natural units $\hbar=k_{\mathrm B}=1$, the two notations look identical, so the convention must still be declared.

Transform and Sum Measures

One consistent energy convention is

$$F(\tau) = \frac1\beta\sum_n e^{-i\varepsilon_n\tau/\hbar}F(i\varepsilon_n), \qquad F(i\varepsilon_n) = \frac1\hbar \int_0^{\beta\hbar}d\tau\, e^{i\varepsilon_n\tau/\hbar}F(\tau).$$

If angular frequencies are used instead, the inverse-transform prefactor is $1/(\beta\hbar)$. Do not combine the exponent from one convention with the sum measure from the other.

Routing and Parity

• boson $\pm$ boson is bosonic;
• fermion $-$ fermion is bosonic;
• fermion $+$ boson is fermionic;
• a composite operator uses the grid fixed by its thermal boundary condition, not simply by the statistics of its microscopic constituents.

Chemical potential normally appears in the propagator denominator, for example $i\varepsilon_n+\mu-\epsilon_{\mathbf k}$, rather than by redefining the standard grid. Twisted boundary conditions are an intentional generalization and must state their phase convention.

Numerical Reminders

• Preserve paired indices: $m$ with $-m$, and $n$ with $-n-1$.
• Treat the bosonic zero mode separately when infrared behavior is important.
• The smallest nonzero bosonic energy is $2\pi/\beta$; the smallest fermionic magnitude is $\pi/\beta$.
• A Matsubara cutoff is a numerical regulator, not a physical bandwidth.
• Analytic continuation relates imaginary-axis data to a specified retarded, advanced, or other real-frequency object; it is not the substitution $i\varepsilon_n\to\omega$ without boundary data.

Cross-Links

• Imaginary Time
• Thermal Green Functions
• Spectral Representation
• Matsubara Formalism: Preview
• Fourier-Transform Conventions