EDIpack2.0 Fortran Library
Here we give an overview of the structure of the EDIpack2.0 library, with a detailed description of the relevant modules and procedures.
Library Frontend
The edipack2 represents the user interface or Fortran API
of the library, giving access to all therequried procedures and
variables needed to solve a quantum impurity problem.
Core Solver Routines
The module of the EDIpack2.0 library which wraps up the solver into three functions according to three steps: initialization, solution and finalization.
General Environment
This include a set of modules which contains input variables
ed_input_vars, global ones and shared classes/data structure in
ed_vars_global, global memory (de)allocation and evaluation
of symmetry sectors dimensions ed_setup as well as
generic functions which are used throughout the code ed_aux_funx.
ed_vars_global contains the definition of simple data
structures, such as gfmatrix storing all weights and poles of the Green’s functions or
effective_bath gathering the different bath components
according to value of bath_type and ed_mode.
Sparse Matrix
The module ed_sparse_matrix contains the implementation of a
suitable Compact Row Stored sparse matrix, used to store sparse Hamiltonians in each symmetry
sector. The data structure essentially corresponds to an array of
rows as dynamical vectors, one for the values and one for the columns index of
the non-zero elements of the matrix. This enables \(O(1)\)
access. If MPI parallelization is enabled the structure includes an
automatic rows split balanced among the different threads and a
further subdivision of the columns/values in local and non-local
blocks.
EigenSpace
In ed_eigenspace we implement an ordered
single linked list to efficiently store the lower part of the energy
spectrum, including eigenvectors, eigenvalues, symmetry sector index and
twin states, i.e. states in sectors with exchanged quantum numbers which do
have same energy.
Sectors
The direct diagonalization of the Hamiltonian for a finite system becomes quickly impossible due to the exponential increase of its size \(D^{Ns}\), where \(D\) is the dimension of the local Hilbert space (\(D=4\) for spin-\(1/2\) electrons).
A way to partially soften such severe behavior is to take into account conserved quantities. For any quantity \({\cal Q}\) such that \([H,{\cal Q}]=0\), we can separate the structure of the Hamiltonian matrix into different blocks, each corresponding to a discrete value \(q\) of the spectrum of \(Q\). Analyzing one-by-one such symmetry sector it becomes possible to construct at least part of the energy spectrum.
In ed_sector we implement the construction of the symmetry
sectors for the three cases considered in EDIpack2.0, that we
recall are:
\(\vec{Q}=[\vec{N}_\uparrow,\vec{N}_\downarrow]\) for which either the number of total or orbital spin up and down electrons is conserved: NORMAL
\(\vec{Q}=S_z\) with conserved total magnetization: SUPERConducting
\(\vec{Q}=N_{\rm tot}\) where spin degrees freedom is not fully conserved: NON-SU(2)
Bath
The construction and the handling of the bath is a crucial part of the description of the generic single impurity Anderson problem. The bath is described by two set of parameters: the local hamiltonian \(\hat{h}^p\) and the hybridization \(\hat{V}^p\), for \(p=1,\dots,N_{bath}\). The first describes the local properties of each bath element (be that a single electronic level or a more complex structure made of few levels), the second describes the coupling with the impurity.
In EDIpack2.0 the bath is handled using a Reverse Communication Strategy.
The user accesses the bath as a double precision rank-1 array
containing in a given order all the parameters. This array is passed
as input to EDIpack2.0 procedures and dumped into an internal data
structure, implemented in ed_bath_dmft.
We implemented different bath topologies, which can be
selected using the variable bath_type =
normal, hybrid, replica, general .
For bath_type = normal (hybrid) a number nbath of electronic
levels are coupled to each orbital level (to any orbital level) of the
impurity site. The bath local Hamiltonian is diagonal
\(\hat{h}^p\equiv\epsilon^p_a\delta_{ab}\) while the
hybridizations are: \(\hat{V}^p=V^p_{a}\delta_{ab}\)
(\(\hat{V}^p=V^p_{ab}\)).
If ed_mode = superc the bath includes a set of parameters
\(\Delta_p\) describing the superconductive amplitude on each bath
level.
For bath_type = replica (general) a number nbath of copies of the
impurity structure are coupled to the impurity itself. Each bath
element is made of a number \(N_{orb}\) of electronic levels,
i.e. the number of orbitals in the impurity site.
The hybridization to the impurity site is
\(\hat{V}^p=V^p_{a}\delta_{ab}\) (\(\hat{V}^p=V^p_{ab}\)).
The local bath Hamiltonian is \(\hat{h}^p = \sum_{m=1}^{M} \lambda^p_m O_m\).
The set \(\{O\}_m\) is a user defined matrix basis for the impurity
Hamiltonian or, equally, for the local Hamiltonian of the lattice
problem. The numbers \(\lambda^p_m\in{\mathbb R}\) are
variational parameters.
Note
The enumeration of the total bath electronic levels is different
among the different cases. This number is automatically evaluated
upon calling get_bath_dimension(), see
ed_bath_dim.
Note
The replica, general bath topologies are available also for
the superconductive case ed_mode = superc. In this case the
structure of the matrix basis should be set to the proper
multi-orbital Nambu basis, so that off-diagonal blocks corresponds
to anomalous components.
Hamiltonian
This part of the EDIpack2.0 code implements the setup of the
sector Hamiltonian in each operational modes, corresponding to the
choice of one of the symmetries implemented in the code selected by the variable ed_mode = normal, superc, nosu2. See
ed_sector for more info about the symmetries implemented in
the code.
Any of three different mode is implemented in a distinct class of modules performing all the operations required for the construction of the Hamiltonian and the definition of the corresponding matrix vector product, required by the Arpack/Lanczos.
Exact Diagonalization
This part of the EDIpack2.0 code implements the exact
diagonalization of the general, single-site, multi-orbital quantum
impurity problem in each of the specific symmetry implemented in the
code. The operational modes are selected by the variable
ed_mode = normal, superc, nosu2. See
ed_sector for more info about the symmetries implemented
in the code.
As above, we implemented the three different channels in distinct class of modules.
Green’s Functions
This part of the EDIpack2.0 code implements the calculation of the
impurity interacting Green’s functions, self-energy functions and
impurity susceptibilities. Calculations are performed in each operational mode, corresponding to the
choice of the specific symmetry implemented in the code, i.e. which
quantum numbers are to be conserved. The operational modes are selected
by the variable ed_mode = normal, superc, nosu2. See
ed_sector for more info about the symmetries implemented in
the code.
Observables
This part of the EDIpack2.0 code implements the calculation of the
impurity observables and static correlations, such as density,
internal energy or double occupation. Calculations are performed in
each operational mode, corresponding to the choice of the specific
symmetry implemented in the code, i.e. which quantum numbers are to be
conserved. The operational modes are selected by the variable
ed_mode = normal, superc, nosu2. The observables
are printed in plain text files and are accessible to the user
through the routines lited in ed_io. See ed_sector
for more info about the symmetries implemented in the code.
Input/Output
This module provides access to the results of the exact diagonalization. All quantities such as dynamical response functions, self-energy components or impurity observables can be retrieved using specific functions. Additionally we provide procedure to perform on-the-fly re-calculation of the impurity Green’s functions and self-energy on a given arbitrary set of points in the complex frequency domain.
\(\chi^2\) Fit
In this module we provide to the user a generic function
ed_chi2_fitgf() performing the minimization of a user provided
Weiss field against the corresponding model of non-interacting
Anderson Green’s function with the aim of updating the user bath parameters.