Solving the Anderson Impurity model with one impurity site and Nb discrete bath sites in a star geometry for a total number of site Ns=1+Nb. See figure below.
The Hamiltonian is:
where
The Hamiltonian conserves the total number of electrons and the total spin in the z-direction. Thus the problem can be split in independent sectors, or, in mathematical term, the Hamiltonian is block diagonal.
The Hilbert space has 4^Ns basis vector. We choose the following convention to represent a basis vector
Therefore, a wavefunction in that Hilbert space is
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More detailed explanations to come on the model (applying creation and destruction operators on states etc), how to use the codes and cleaning the format of the codes that I never bothered to do.
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While Ns=11 and Ns=12 are possible in principle (precompute the dictionary and save it), this pushes the capability very much. Ns=10 works pretty well on a desktop. For larger Ns such as 11, 12, 13, 14 (even the best implementation running on supercomputers cannot really go larger) it does not make sense to use an implementation in Matlab or Python (this implementation could be slightly more efficient in the dictionnary for example, the convention could be simpler, etc but I never bothered pushing the efficiency to the extreme as alternative are available). Using one of the available C, C++ or Fortran implementations is suggested for Ns > 10.
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However, even without explanation on the codes, if one has a value of Ns, values for all parameters of the Hamiltonian i.e. ed,U,e_2,V_2,...,e_Ns,V_Ns, a fictitious temperature T and a number of Matsubara frequencies Nmat, Every possible quantities for the model can be obtained by running:
wn=pi*T*( 2*(0:Nmat-1)+1 ); [C_ind,table,indice_sector,H_non_zero_ele] = ED_Ns_generate_final(Ns); spar=0 ee=[e_2,e_3,...,e_Ns]; VV=[V_2,V_3,...,V_Ns]; [Gcl,E,EGS,Psi,Psi_GS,NSz_GS,Problem_mat,nd,ndup,nddown,nc,ncup,ncdown,D,an_m,bn2_m,dplusd,an_p,bn2_p,ddplus] = ED_Green_final(wn,ed,U,ee,VV,Ns,C_ind,table,indice_sector,H_non_zero_ele,spar) Gcl is the Green's function of the impurity E contains the smallest energies of each sector EGS is the ground state energy Psi contains the wave functions corresponding to E Psi_Gs is the ground state wave function NSz_GS informs about which sector has the ground state in [N,Sz] N electrons with total Sz spin nd is the impurity electronic density ndup is the impurity electronic density with spin up nddown is the impurity electronic density with spin down nc,ncup,ncdounw the same for the bath D is the double occupy of the impurity an_m,bn2_m,dplusd,an_p,bn2_p,ddplus are the continued fraction coefficients of the Green's function
These codes were used in https://arxiv.org/abs/1408.1143 and https://arxiv.org/abs/1506.08858 .