Update quickstart. To be finished.

doc/quickstart/01_anderson.rst

@@ -37,7 +37,7 @@ The initialization of the code is:
 
       
 where we load both the `EDIpack2.0` and `SciFortran` libraries through
-their main module :f:mod:`edipac2` and :f:mod:`scifor`. We also define
+their main module :f:mod:`edipack2` and :f:mod:`scifor`. We also define
 some local variables and proceed with reading the input file
 :code:`"inputED.conf"` using the `EDIpack2.0` function
 :f:func:`ed_read_input`.

doc/quickstart/02_dmft.rst

@@ -1,3 +1,160 @@
 ED solver for Dynamical Mean-Field Theory
 #########################################
 
+In this section we take a step further to show how to integrate the
+`EDIpack2.0` as a solver for Dynamical Mean-Field Theory calculations. 
+Specifically, here we discuss the step-by-step implementation of a Fortran program (see
+below) solving this model using DMFT with the `EDIpack2.0` exact
+diagonalization algorithm at :math:`T=0`.
+
+Similar to the previous section, we consider a Fermi-Hubbard
+model defined on a Bethe lattice DOS
+:math:`\rho(x)=\frac{1}{2D}\sqrt{D^2-x^2}`. The preamble of the
+program includes variable definitions (a old-looking Fortran
+specialty) and input file read:
+
+
+.. code-block:: fortran
+
+   program ed_hm_bethe
+      USE EDIPACK2
+      USE SCIFOR
+      USE DMFT_TOOLS
+      implicit none
+
+      !> Number of discretization points for the Bethe DOS 
+      integer,parameter                           :: Le=5000
+      !> Bethe half-bandwidth = energy unit
+      real(8),parameter                           :: D=1d0
+      !> Bethe DOS and linear dispersion
+      real(8),dimension(Le)                       :: DOS
+      real(8),dimension(Le)                       :: ene
+      !>Bath:
+      real(8),allocatable                         :: Bath(:)
+      !> local dynamical functions, rank-5 [Nspin,Nspin,Norb,Norb,L]
+      !   (we could use other format )
+      complex(8),allocatable,dimension(:,:,:,:,:) :: Weiss,Smats,Sreal,Gmats,Greal,Weiss_
+      !> input file
+      character(len=16)                           :: finput
+
+      !> Use SciFortran parser to retrieve the name of input file 
+      call parse_cmd_variable(finput,"FINPUT",default='inputED.conf')
+      
+      !> READ THE input using EDIpack procedure: 
+      call ed_read_input(trim(finput))
+
+
+In this  we load both the `EDIpack2.0` and `SciFortran` libraries through
+their main module :f:mod:`edipack2` and :f:mod:`scifor`. We  define
+some local variables and  read the input file
+(default :code:`"inputED.conf"`) using the `EDIpack2.0` function :f:func:`ed_read_input`.
+
+
+The next step is to construct the Bethe lattice DOS, we use
+`SciFortran` procedure to simplify the task:
+
+.. code-block:: fortran
+
+   Ene = linspace(-D,D,Le,mesh=de)
+   DOS = dens_bethe(Ene,D)
+
+
+Once the local dynamical functions have been allocated, we can proceed
+to allocate the user side bath :f:var:`bath` and initialise the ED
+solver by calling the :f:func:`ed_init_solver` procedure:
+
+
+.. code-block:: fortran
+
+   Nb=ed_get_bath_dimension()
+   allocate(bath(Nb))
+   call ed_init_solver(bath)
+
+
+On input the :f:var:`bath` is guessed from a flat distribution
+centered around zero and with half-width :f:var:`ed_hw_bath`. If a
+file :f:var:`hfile` with suffix `.restart` containing bath parameters is found in the run
+directory then the bath is read from file.
+
+We are now ready to perform a DMFT self-consistency cycle. In the
+present it looks like:
+
+.. code-block:: fortran
+   :linenos:
+   :emphasize-lines: 6, 19
+
+   iloop=0;converged=.false.
+   do while(.not.converged.AND.iloop<nloop)
+     iloop=iloop+1
+     
+     !> Solve the effective impurity problem
+     call ed_solve(bath)
+     
+     !> Impurity Self-energy on Matsubara axis
+     call ed_get_sigma(Smats,'m')
+
+     !> Build a local Green's function using the Impurity Self-energy
+     wfreq = pi/beta*(2*arange(1,Lmats)-1)   !automatic Fortran allocation
+     do i=1,Lmats
+        zeta= xi*wfreq(i)+xmu - Smats(1,1,1,1,i)
+        Gmats(1,1,1,1,i) = sum(DOS(:)/( zeta-Ene(:) ))*de  ! One can do better than this of course 
+     enddo
+
+     !> Self-consistency: get the new Weiss field:
+     Weiss(1,1,1,1,:) = one/(one/Gmats(1,1,1,1,:) + Smats(1,1,1,1,:))
+     !> Mix to avoid trapping:
+     if(iloop>1)Weiss = wmixing*Weiss + (1.d0-wmixing)*Weiss_
+
+     !> Close the self-consistency fitting the new bath:
+     call ed_chi2_fitgf(Weiss,bath,ispin=1)
+     
+     !>Check convergence
+     converged =( sum(abs(Weiss(1,1,1,1,:)-Weiss_(1,1,1,1,:)))/sum(abs(Weiss(1,1,1,1,:))) )<dmft_error
+     Weiss_=Weiss     
+   enddo
+
+
+The first step, line 9, is to call the :f:func:`ed_solve` procedure in
+`EDIpack2` which solve the quantum impurity problem defined by a given
+input bath :f:var:`bath`. On exit, all the ED related quantities are
+stored in the memory, ready to be retrieved upon call.
+For instance we retrieve the Matsubara self-energy
+:math:`\Sigma(i\omega_n)` using the procedure :f:func:`ed_get_sigma`
+and store the result in the array :f:var:`Smats`.
+
+Next, lines 12-16, we evaluate the local Green's function
+:math:`\int^{D}_{-D} d\epsilon \frac{\rho(\epsilon)}{\zeta-\epsilon}`
+where :math:`\zeta=i\omega_n+\mu-\Sigma(i\omega_n)`.
+This function is used to update the Weiss field :math:`{\cal G}_0`
+using the **self-consistency** relation (line 19):
+
+.. math::
+
+   {\cal G}_0(i\omega_n) = \left[ G^{-1}_{loc}(i\omega_n) + \Sigma(i\omega_n)\right]^{-1}
+
+
+The closing step of the DMFT cycle, specific of the Exact
+Diagonalization solver, is to project the obtained Weiss field onto
+the set of Anderson non-interacting Green's function
+:math:`G^{And}_0(i\omega_n;\vec{b})` describing a discretized bath of
+:f:var:`Nb` parameters. This step is performed using the complementary
+method :f:func:`ed_chi2_fitgf`, which optimize the bath parameters
+by minimizing the distance between such two functions. See line 24.
+
+The cycle close with a simple error check on the Weiss field itself. 
+
+
+.. raw:: html
+
+   <hr>
+
+
+In the following we present some results obtained with this simple
+method.
+
+
+
+
+.. image:: 02_dmft_fig.svg
+   :class: with-border
+   :width: 800px