TCI4Keldysh

This code computes imaginary- and real-frequency 4-point vertices from the multipoint numerical renormalization group (mpNRG) in quantics tensor train (QTT) format. The vertices can be obtained in their 'full' form or decomposed into a 3d core and lower-dimensional asymptotic contributions. Further functionalities include the computation of four-point vertices on dense, possibly nonlinear, grids and the computation of 2-4-point correlators, either in QTT format or on a dense, linear grid.

Getting Started

For user instructions and further details, compile the mini-manual under 'docs/manual.tex'. Note, however, that you will need correctly formatted partial spectral functions (.mat files) to use this code. These are normally provided by the mpNRG code by Lee et. al. [Lee2021].

Conventions

1. Matsubara frequencies with index $n\in -N,...,N-1$
   • bosonic frequencies $\omega_n = 2 n \pi T$
   • fermionic frequencies $\nu_n = (2n+1) \pi T$ in r-channel convention the frequencies read $(\omega, \nu, \nu')$ with \omega=bosonic transferfrequency and $\nu(')=$fermionic frequencies.
2. operators in correlators:
   • 2p/4p: arbitrary order
   • 3p: in $G[\vec{O}]$ we always have $\vec{O} = (Q, F, F)$, i.e., first operator is a (bosonic) auxiliary operator, the other two are regular (fermionic) operators $F\in{c,c^\dagger}$.
3. abbreviations:
   • IE/sIE/aSI: ((a-)symmetric) improved estimator
   • TD: Tucker decomposition
   • MF/KF: Matsubara / Keldysh formalism
   • 2p,3p,4p: p=point
   • GF: correlator/Green's function
   • a,p,t: channels, i.e., different kinds of frequency parametrizations of the vertex
   • K1/2: asymptotic contributions to the vertex
   • $\Gamma$: four-point vertex

References

1. [Frankenbach2025]: "Compressing local vertex functions from the multipoint numerical renormalization group using quantics tensor cross interpolation", https://doi.org/10.1103/jx7h-lsqk
2. [Lihm2024]: "Symmetric improved estimators for multipoint vertex functions", doi:10.1103/PhysRevB.109.125138
3. [Kugler2021]: "Multipoint Correlation Functions: Spectral Representation and Numerical Evaluation", doi: 10.1103/PhysRevX.11.041006
4. [Lee2021]: "Computing Local Multipoint Correlators Using the Numerical Renormalization Group", doi:10.1103/PhysRevX.11.041007
5. [Fernandez2025]: "Learning tensor networks with tensor cross interpolation: New algorithms and libraries", doi:10.21468/SciPostPhys.18.3.104

List of contributors

Markus Frankenbach
Anxiang Ge