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Introduction

sparse-ir compresses the imaginary-time Green’s functions of finite-temperature many-body theory. A Green’s function on a dense grid in imaginary time or Matsubara frequency carries far fewer independent numbers than grid points, and the library provides three ways to exploit that:

  • The intermediate representation (IR) basis. An orthonormal basis obtained from a singular value expansion of the analytic-continuation kernel. Its size grows only logarithmically with the cutoff \(\Lambda = \beta\omega_\mathrm{max}\), and a matching set of sparse sampling points in \(\tau\) and in Matsubara frequency is enough to recover the expansion. See Sparse sampling and Transformation from and to IR.
  • The discrete Lehmann representation (DLR). A sum of simple poles on a fixed real-frequency grid, chosen by an interpolative decomposition of the kernel. It needs no IR basis, comes with its own sampling nodes, and can also be built from an existing IR basis. See Discrete Lehmann representation.
  • MiniPole. A data-adapted compression: ESPRIT extracts a handful of complex poles from one particular function, which also gives an analytic continuation. See MiniPole.

The IR basis and the DLR are fixed representations shared by every function with the same \(\beta\), \(\omega_\mathrm{max}\) and accuracy; MiniPole adapts to the data. For how the three are related, where each came from and what each is convenient for, see IR, DLR and MiniPole: history and comparison on the theory site.

How this book is organised

Getting started covers installation and the conventions used throughout, including the Matsubara-frequency index; read Conventions before comparing numbers with another code. Representations introduces the IR basis, the DLR and MiniPole. Analytic continuation goes from imaginary-time data back to real frequency. Applied examples solve complete many-body problems with the IR basis.

Most pages follow a notebook of the Python and Julia tutorials, sparse-ir-tutorial-v2; each page says which one.

How this book is built

The Rust code on these pages is included from the example programs in docs/tutorial-code/ rather than copied by hand, so it is the code that is actually compiled and run, and every figure was drawn from numbers those programs wrote. Most examples are checked against reference values from the Python implementation; the rest, such as the IR-independent DLR and MiniPole, check themselves against closed-form results. Each page ends with the command that reproduces it; run it from docs/tutorial-code.