Cluster-DMFT Lanczos Exact Diagonalization

A Lanczos based solver for the Cluster Dynamical Mean-Field Theory using the N_up:N_dw implementation in the normal phase (plus long-range magnetic or charge order).

The code is based on:

The code structure is as follow:

The set of modules compile into a top layer named CDMFT_ED.f90
The actual implementation of the DMFT equations is case by case performed in a driver program, usually placed in the directory drivers.
In the driver code the user must includes the CDMFT_ED, SCIFOR and DMFT_TOOLS modules and call the necessary procedures to solve the DMFT equations.

The only two bath types implemented to date are the 'replica' and 'general' ones, in which the bath consists of multiple, noninteracting copies of the original interacting cluster, with different hybridization schemes: a unique $V$ for each replica in the former case, meaning that the i-th site in the cluster hybridizes with the j-th site in the replica, with an amplitude $V_{ij} = V\delta_{ij}$; a more general $V_{ij} = V_i \delta_{ij}$ for the latter. For both cases the bath can be initialized in two ways, through the unified ed_set_Hbath interface: one can directly pass the local hamiltonian of the cluster (going with strict semantics for 'replicas') or a symmetry-informed expansion in the form $H^\mathrm{bath} = \sum \lambda^\mathrm{sym}H^\mathrm{sym}$, where the $H^\mathrm{sym}$ terms are arrays isomorphic with $H^\mathrm{loc}$, describing a well defined symmetry of the model, and the $\lambda^\mathrm{sym}$ coefficients are contained in a ([Nineq])x[Nbath]x[Nsym]-sized array, so to allow each replica to have a different expansion on the same unique basis. No control over the initial guess of the $V_{ij}$ values is given to the user, at the moment.

An example, solving the Hubbard model on the square lattice, is contained in the file drivers/cdn_hm_2dsquare.f90.

DEVELOPMENT

MILESTONE 1

Write code for the spin-decoupled case
Code compiles and runs without errors
Check exact diagonalization comparing results with known 2x2 plaquette data (without bath)
Check single-site cluster case against single-site DMFT code
Check 1d Hubbard chain against literature

MILESTONE 2

Include complex routines for diagonalization and bath
Test 2d BHZ model

MILESTONE 3

Add real-space CDMFT case for finite systems
Test Kane-Mele model with real-space CDMFT
Test 3d BHZ model for non spin-coupling choices of cluster
Rewrite the code for the general spin-coupled case

MILESTONE 4

Add cluster density matrix (CDM) computation
Test and benchmark CDM computation
Build smaller-rank reduced DMs from CDM
Check local RDM against semi-analytical build (Norb=1)
Replicate the single-site results in Su, Dai,Tong 2013.
Replicate the (vanishing T) 2x2 results in C. Walsh et al. 2019.