Slave–Boson Mean Field Method

A fast approach to solve a general impurity problem is the slave–boson method [1,2,3]. At the mean field level, it gives the results similar to the famous Gutzwiller approximation [4]. However, it is improvable by performing fluctuations around the saddle point. This approach is accurate as it has been shown recently to give the exact critical value of  in the large degeneracy limit at half–filling [5].

The main idea is to rewrite atomic states consisting of  electrons ,  with help of a set of slave–bosons . In the following, we assume  symmetric case, i.e., equivalence between different states  for fixed . Formulae corresponding to a more general crystal–field case are given in Appendix B. The creation operator of a physical electron is expressed via slave particles in the standard manner [2]. In order to recover the correct non–interacting limit at the mean–field level, the Bose fields  can be considered as classical values found from minimizing a Lagrangian  corresponding to the Anderson impurity Hamiltonian. Two Lagrange multipliers  and  should be introduced in this way, which correspond to the following two constrains:

                                                                              (1)

                                                        (2)

The numbers  are similar to the probabilities  to find an atom in configuration with  electrons in the definition of the atomic Green function:

                                                             (3)

where energy levels are  and combinatorial factor  arrives due to equivalence of all states with  electrons in .

We thus see the physical meaning of the first constrain which is the sum of probabilities to find atom in any state is equal to one, and the second constrain gives the mean number of electrons coinciding with that found from . A combinatorial factor  arrives due to assumed equivalence of all states with  electrons.

Minimization of  with respect to  leads us to the following set of equations to determine the quantities :

                                          (4)

where , determines the mass renormalization, and the coefficients ,  are normalization constants as in Refs. [1,2].  is the total energy of the atom with  electrons in  approximation.

Eq. (4), along with the constrains (1), (2) constitute a set of non–linear equations which have to be solved iteratively. In practice, we consider Eq. (4) as an eigenvalue problem with  being the eigenvalue and  being the eigenvectors of the matrix. The physical root corresponds to the lowest eigenvalue of  which gives a set of  determining the mass renormalization  Since the matrix to be diagonalized depends non–linearly on  via the parameters  and  and also on , the solution of the whole problem assumes the self–consistency: i) we build an initial approximation to  (for example the Hartree–Fock solution) and fix some , ii) we solve eigenvalue problem and find new normalized , iii) we mix new  with the old ones using the Broyden method [6] and build new  and . Steps ii) and iii) are repeated until the self–consistency with respect to  is reached. During the iterations we also vary  to obey the constrains. The described procedure provides a stable computational algorithm for solving AIM and gives us an access to the low–frequency Green’s function and the self–energy of the problem via knowledge of the slope of  and the value  at zero frequency.

The described slave–boson method gives the following expression for the self–energy:

                                                              (5)

The impurity Green function  in this limit is given by the expression

                                                                          (6)

Density–density correlation function  for local states with  electrons is proportional to the number of pairs formed by  particles . Since the probability for  electrons to be occupied is given by: , the physical density–density correlator can be deduced from: . Similarly, the triple occupancy can be calculated from .

If we consider the two–band Hubbard model in  orbitally degenerate case. Hybridization  satisfies the DMFT self–consistency condition of the Hubbard model on a Bethe lattice

                                                                        (7)

The Coulomb interaction is chosen to be  which is sufficiently large to open the Mott gap at integer fillings.

References

[1]  G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. 57, 1362 (1986).

[2]  R. Fresard and G. Kotliar, Phys. Rev. B. 56, 12909 (1997).

[3]  H. Hasegawa, Phys. Rev. B 56, 1196 (1997).

[4]  M. Gutzwiller, Phys. Rev. 134, A923 (1964).

[5]  S. Florens, A. Georges, G. Kotliar, and O. Parcollet, Phys. Rev. B 66, 205102 (2002).

[6]  See, e.g., D. D. Johnson, Phys. Rev. B 38, 12807 (1988).

Running program

· compile program typing "make" at the source files location. Makefile was tested on Linux OS using PGI compiler. Please adjust it depending on operating system and compiler used.

· Edit “input" file which has the following structure:

1	IMOD
0.0	EF
2.0	U
0.016	TEMP
4	NDEG
500	NMSB
500	OMAX
500	NOMG
50	WEND
F	COMPUTE_REAL

where IMOD = 1 is the Hubbard Model, IMOD=3 the Anderson impurity model, EF is the impurity level, U is value of Coulomb repulsion, TEMP is temperature to be used used to create for Matsubara points grid, NDEG is the degeneracy, NMSB is the number of Matsubara point, OMAX is the imaginary frequency cutoff, NOMG is the number is real frequency points, WEND is the real frequency cutoff, COMPUTE_REAL is flag once is “True" tell program to produce the self-consistent solution (last iteration) on real axis.

1. Program’s input consists from one more file (provided the Anderson impurity model item is chosen in “input" file):

“delta.dat" containing the hybridization function.

The structure is the following:

Column #	Value
1	Frequency
2	Re Delta
3	Im Delta

2. Run the program executable (“main" is the default name).

3. Program’s output consists from the following files:

1) “gfsig_iw.dat" containing Green’s function (GF) and the self-energy (SE) on Matsubara axis

2) “gfsig_re.dat" containing Green’s function (GF) and the self-energy (SE) on real axis provided flag "COMPUTE_REAL" is “TRUE".

They have the same structure:

Column #	Value
1	Frequency
2	Re GF
3	Im GF
4	Re SE
5	Im SE

3) “grid_re.dat" containing real frequency grid for the hybridization function (delta).

4) “grid_im.dat" containing Matsubara frequency grid for the hybridization function (delta).

Both files have the same structure:

Column #	Value
1	Frequency

Program content

Makefile	main make file
dmf_cmpdiag.f	The solution of the generalized eigenvalue problem
imp_sunhub.f	Solves Anderson impurity model, returns GF and SE.
imp_sunmas.f	Computes effective mass renormalization.
lib_broy6.f	Broyden mixing.
lib_cinv.f	Finds inverse of square matrix.
lib_csplines.f	Splines complex function from one to another grid.
lib_deriv1.f	Calculates radial derivative.
lib_morefun.f	Calculations of a few auxiliary functions
lib_pade.f	Realization of Páde procedure.
lib_splin3.f	Computed a natural spline approximation of third order.
mod_common.f	File containing common modules used across the program.
mod_dimart.f	File containing common modules used across the program.
mod_init.f	File containing common modules used across the program.
sun.f	Main program.