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ROADMAP

Milestone 0: rewrite of the original implementation by Zhang, Rozenberg and Nájera

  • Port to MATLAB the modern python implementation by Óscar Nájera and crosscheck the basic output it provides.

  • Check the effect of the self-consistency condition by inspecting the spectral functions at one loop vs at convergence, with both the initial guesses for the bath. [it requires to implement a proper self-consistency control, possibly with linear mixing]

  • Analyze the U-driven MIT by extracting the quasiparticle weight from the self-energy, determining Uc2 and Uc1 points at some relevant temperatures and capturing first-order behaviour in between.

  • Complete the original tutorial by inspecting the T-driven MIT and defining the supercritical behaviour (bad metal and pseudogap phases).

> v0.1

Milestone I: functional phase diagrams, pseudo-order-parameter zoo, performance

  • Compute a mottness-marker based on the divergence of the scattering rate (Im[Sigma(0)]... very sensible, basically inexpensive), so to obtain sharper phase diagrams with respect to the Z-derived ones. It should also give insight about the supercritical phases: bad metal and pseudogap. [this yet to check]

    At the moment we implement the "strenght of correlations", S = norm[Sigma(0)-Sigma(∞)], as defined in PRL 114 185701 (2015), with actual neat results: the marker is almost zero accross the whole FL phase and starts increasing very fast in the Mott insulator.

  • Compute the Luttinger integral, as defined in PRB 90 075150 (2014). Since it appears to be quantized at very low temperatures, it could become the definitive flag for quantum phase diagrams; much better than Z or S for it is an integer. [→ easier phase-boundary recognition!]

    Luttinger Theorem currently works for very low temperatures only. It might well be an inherent limitation, restricting its domain to the quantum Mott transition. Also note that to have a sharp first order step-up at the transition a very highly frequency resolution is needed, so to make the IPT solver performance-critical! [solved brilliantly with fast convolutions, see the solver-optimization section below]

  • SOLVER-OPTIMIZATION: make the SOPT run faster, by optimizing the needed convolutions. [implemented a pow2-optimized FFTW-based custom algorithm that actually greatly improves the cpu-time for the wres=2^15 calculation: almost a x10 overall speedup!]

  • LOOP-OPTIMIZATION: insert a "restarting" protocol for lines and full phase diagram spans. The gloc0=0 condition appers to be too unstable to obtain accurate UC1 values. Furthermore this would most probably speed up a lot the convergence, by lowering the required number of iterations.

  • HPC-OPTIMIZATION: configure an interface to cluster facilities and define the scheduling resources for optimal running [no distributed computing, just built-in handling of shared-memory parallelization]

  • Implement an ergonomic 'full-roundtrip' protocol, so to enable suitable explorations of the coexistence region.

  • Start writing markdown docs, with the aim to document the implementation details (usage instruction should go directly on the README instead...)

Milestone II: more lattices

  • Add a bunch of different particle-hole symmetric lattices, such as finite-coordination Bethe, honeycomb, 2d-square, 1d-chain, 3d-cubic, 3d-bcc... the main inspiration comes from the mighty GFtools by DerWeh.

  • Rewrite the code for the particle-hole broken case so to enable even more lattices, like kagome, fcc, triangular.

> lattices-branch

Milestone III: imaginary-axis formalism and new entries to the zoo

  • Write an efficient analytic continuation to the imaginary-axis, so to retrieve the Matsubara representation of gloc and sloc.

  • Overload the quantities that can be computed with both real- and imaginary-axis formalism. Determine which version is to be preferred and when.

    • Quasiparticle weight: Z = 1/(1-Im∑(iπT)/πT)
    • Luttinger integral: look at arXiv:2202.00426

  • Add relevant quantities that are accessible only via Mastubara summations [e.g. double occupancy, ref. to Phys. Rev. B 93, 155162 (2016)]

  • Compute the Local Entanglement Entropy, as defined in Rev. Mod. Phys. 80, 517 (2008) (section V.F) and used in Mod. Phys. Lett. B 2013 27:05 to characterize the MIT on the Bethe lattice.

> matsubara-branch

Milestone IV: topological madness

  • Try to reproduce the main result of PRB 102 081110(R), namely the Lanczos tridiagonalization of the self-energy leading to the mapping of the quantum MIT to a generalized SSH SP-TQPT. The original result is achieved within DMFT/NRG and an insane bath dimension, so if we succeed this could be even a ReScience submission (given everything is Octave compatible).