Introduction to Ω maxEnt , a tool for analytic continuation of Matsubara data

Dominic Bergeron Université de Sherbrooke Introduction to ΩmaxEnt, a tool for analytic continuation of Matsubara data

Analytic Continuation: • G(τ) or G(iωn) => A(ω)? => invert or

Conditioning Problem: • If we discretize ω => • error on A: • is large => not bounded => analytical continuation unique in principle (Baym and Mermin, J.Math.Phys.1961), but unstable numerically => need constraints on A(ω)

Maximum entropy • Different strategy: minimize

ΩmaxEnt • Use if A(ω) is a piecewise polynomial ⇒ analytical integration • Replace high Matsubara frequencies with contraints on moments • Use adapted ω grid • Input can also be G(τ) • Treats fermionic (A(ω)>0) and bosonic (A(ω)/ω>0) data • General covariance matrix

ΩmaxEnt: how to choose α? • Compute A(ω) for large range: • Three regimes in vs α ⇒ Optimal α located on vs α in log-log • Additional diagnostic tools to asses quality of the result: • A(ωsample) vs α • ΔG=(Gin-Gout)/σ vs ωn • 〈ΔGmΔGm+n〉

Why three regimes in vs α? • High α: A(ω)≈D(ω) ⇒ ≈ const • Intermediate: α(⇒( • Low α: decreases very slowly asα( (?) ⇒ ⇒αS forces A(ω) to be smooth ⇒ prevents fitting noise

ΩMaxEnt : user guide

Introduction to Ω maxEnt , a tool for analytic continuation of Matsubara data