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Quantum Mechanics

Matsubara Frequency Table

Matsubara grids are the Fourier modes compatible with a thermal correlator’s imaginary-time boundary condition. The half-step shift of the fermionic grid removes its zero-frequency mode. For the derivation, see Bosonic and Fermionic Matsubara Frequencies.

Bosonic and fermionic thermal grids

Section titled “Bosonic and fermionic thermal grids”

Let β=1/(kBT)\beta=1/(k_{\mathrm B}T) and Lτ=βℏL_\tau=\beta\hbar. Here imaginary-time evolution uses K=H−μN\mathcal K=H-\mu N and the full grand-canonical Fock-space trace. Fixed-NN traces support the usual periodic relation for sector-preserving observables; number-changing fields cannot be cycled through the fixed-sector projector as if it were absent.

Channel Boundary condition Energy grid
Bosonic F(τ+Lτ)=F(τ)F(\tau+L_\tau)=F(\tau) Ωm=2πm/β\Omega_m=2\pi m/\beta
Fermionic F(τ+Lτ)=−F(τ)F(\tau+L_\tau)=-F(\tau) εn=(2n+1)π/β\varepsilon_n=(2n+1)\pi/\beta
Channel Angular-frequency grid Zero mode and reflected index
Bosonic ωm=2πm/(βℏ)\omega_m=2\pi m/(\beta\hbar) m=0m=0; reflection m↦−mm\mapsto-m
Fermionic ωn=(2n+1)π/(βℏ)\omega_n=(2n+1)\pi/(\beta\hbar) no zero mode; reflection n↦−n−1n\mapsto-n-1

The grid indices range over all integers. Energies equal ℏ\hbar times angular frequencies. Setting ℏ=kB=1\hbar=k_{\mathrm B}=1 hides this units distinction without removing it.

Transform and Sum Measures

Section titled “Transform and Sum Measures”

For either energy grid, write its entries as ζn\zeta_n. A consistent transform pair is

FE(iζn)=1ℏ∫0βℏdτ eiζnτ/ℏF(τ),F(τ)=1β∑ne−iζnτ/ℏFE(iζn).\begin{aligned} F_E(i\zeta_n)&=\frac1\hbar\int_0^{\beta\hbar}d\tau\, e^{i\zeta_n\tau/\hbar}F(\tau),\\ F(\tau)&=\frac1\beta\sum_n e^{-i\zeta_n\tau/\hbar}F_E(i\zeta_n). \end{aligned}

If instead Fω(iωn)=∫0βℏdτ eiωnτF(τ)F_\omega(i\omega_n)=\int_0^{\beta\hbar}d\tau\,e^{i\omega_n\tau}F(\tau), the inverse prefactor is 1/(βℏ)1/(\beta\hbar) and Fω=ℏFEF_\omega=\hbar F_E. Changing the grid label alone does not change the transform measure; changing the forward normalization does.

Routing and Parity

Section titled “Routing and Parity”
  • boson ±\pm boson is bosonic;
  • fermion ++ fermion and fermion −- fermion are 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 iεn+μ−ϵki\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.

Zero modes and numerical cutoffs

Section titled “Zero modes and numerical cutoffs”
  • Preserve paired indices: mm with −m-m, and nn with −n−1-n-1.
  • Treat the bosonic zero mode separately when infrared behavior is important.
  • The smallest nonzero bosonic energy is 2π/β2\pi/\beta; the smallest fermionic magnitude is π/β\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 iεn→ωi\varepsilon_n\to\omega without boundary data.

For example, the reflected partner of n=0n=0 is n=−1n=-1, giving energies ±π/β\pm\pi/\beta. A symmetric fermionic cutoff uses n=−M,…,M−1n=-M,\ldots,M-1, not −M,…,M-M,\ldots,M. The bosonic zero-frequency value can also contain a static fluctuation contribution absent from the retarded spectrum; see the spectral representation.

References

Section titled “References”