# The DMFT framework

### This page gives hints on how to perform a DMFT calculation with the ABINIT package.

##### Copyright (C) 2016-2017 ABINIT group (BAmadon)

Mentioned in help_features#5.

### Table of content:

- 1. Introduction.
- 2. Related lesson(s) of the tutorial.
- 3. Related input variables.
- 4. Selected input files.
- 5. References.

### 1. Introduction.

DFT fails to describe the ground state and/or the excited states such as many lanthanides, actinides or transition metals. Indeed, exchange correlation functionals are not (yet) able to describe the strong repulsive Coulomb interactions occurring among electrons in partly filled localized d or f orbitals.

A way to improve the description of strongly correlated systems is to explicitly include the strong repulsive Coulomb interactions in the Hamiltonian. Solving it in the static mean field approximation, gives the DFT+U method ([Anisimov1991],[Liechtenstein1995]), implemented in ABINIT [Amadon2008a]. The Dynamical Mean Field Theory [Georges1996] (DMFT), goes beyond, by solving exactly the local correlations for an atom in an effective field (i.e., an Anderson model). The effective field reproduces the effect of the surrounding correlated atoms and is thus self-consistently related to the solution of the Anderson model [Georges1996].

The combination of DFT with DMFT [Georges2004],[Kotliar2006] (usedmft= 1) relies on :

- The definition of correlated orbitals. In ABINIT, we use Wannier functions built using projected local orbitals [Amadon2008]. Wannier functions are unitarily related to a selected set of Kohn Sham (KS) wavefunctions, specified in ABINIT by band indices dmftbandi and dmftbandf. As empty bands are necessary to build Wannier functions, it is required in DMFT calculations that the KS Hamiltonian is correctly diagonalized: use high values for nnsclo and nline. In order to make a first rough estimation of the orbital character of KS bands and choose the band index, the band structure with highlighted atomic orbital character (so called fatbands) can be plotted, using the pawfatbnd variable. Band structures obtained from projected orbitals Wannier functions can also be plotted using plowan_compute and related variables.
- The choice of the screened Coulomb interaction U (upawu) and J (jpawu). Note that up to version 7.10.5 (but not in later versions) jpawu= 0 is required if the density matrix in the correlated subspace is not diagonal.
- The choice of the double counting correction [Amadon2012]. The current default choice in ABINIT is dmft_dc= 1 which corresponds to the full localized limit.
- The method of resolution of the Anderson model. In ABINIT, it can be the Hubbard I method [Amadon2012] (dmft_solv= 2), the Continuous time Quantum Monte Carlo (CTQMC) method [Gull2011],[Bieder2014] (dmft_solv= 5) or the static mean field method (dmft_solv= 1), equivalent to usual DFT+U [Amadon2012]).

The practical solution of the DFT+DMFT scheme is usually presented as a double loop over, first, the local Green's function, and second the electronic local density [Amadon2012]. The number of iterations of the two loops are determined by dmft_iter and nstep. However, in the general case, the most efficient way to carry out fully consistent DFT+DMFT calculations is to keep only the loop governed by nstep, while dmft_iter=1 [Bieder2014], dmft_rslf= 1 (to read the self-energy file at each step of the DFT loop) and prtden= -1 (to be able to restart the calculation of each step of the DFT loop from the density file). Lastly, one linear and one logarithmic grid are used for Matsubara frequencies [Kotliar2006] determined by dmft_nwli and dmft_nwlo (Typical values are 10000 and 100, but convergence should be studied). More information can be obtained in the log file by setting pawprtvol=3.

The main output of the calculations are the imaginary time Green's function , from which spectral functions can be obtained using an external maximum entropy code [Bergeron2016], self-energies, from which quasiparticle renormalization weight can be extracted, the density matrix of correlated orbitals, and the internal energies [Amadon2006]. The electronic entropic contribution to the free energy can also be obtained using dmft_entropy and dmft_nlambda.

The efficient CTQMC code in ABINIT, which is the most time consuming part of DMFT, uses the hybridization expansion [Werner2006],[Gull2011] with a density-density multiorbital interaction [Gull2011]. Moreover, the hybridization function [Gull2011] is assumed to be diagonal in the orbital (or flavor) index. This is exact for cubic symmetry without spin orbit coupling but, in general, one should always check that the off-diagonal terms are much smaller than the diagonal ones. A link to the exact rotationally invariant interaction CTQMC code of the TRIQS library is also available using dmft_solv=7.

As the CTQMC solver uses a Fourier transform, the time grid dmftqmc_l in imaginary space must be chosen so that the Nyquist frequency, defined by π*dmftqmc_l*tsmear, is around 2 or 3 Ha. A convergence study should be performed on this variable. Moreover, the number of imaginary frequencies (dmft_nwlo) has to be set to at least twice the value of dmftqmc_l. Typical numbers of steps for the thermalization (dmftqmc_therm) and for the Monte carlo runs (dmftqmc_n) are 106 and 109 respectively. The random number generator can be initialized with the variable dmftqmc_seed. Several other variables are available. dmftctqmc_order gives a histogram of the perturbation orders during the simulation, dmftctqmc_gmove customizes the global move tries (mainly useful for systems with high/low spin configurations), and dmftctqmc_meas sets the frequency of measurement of quantities.
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### 2. Related lesson(s) of the tutorial.

- The lesson on DFT+DMFT shows how to perform a DFT+DMFT calculation on SrVO3 using projected Wannier functions. Prerequisite : DFT+U.
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### 3. Related input variables.

Compulsory input variables:

... dmft_iter [Dynamical Mean Fied Theory: number of ITERation]
... dmft_nwli [Dynamical Mean Fied Theory: Number of frequency omega (W) in the LInear mesh]
... dmft_nwlo [Dynamical Mean Fied Theory: Number of frequency omega (W) in the LOg mesh]
... dmftbandf [Dynamical Mean Field Theory: BAND: Final]
... dmftbandi [Dynamical Mean Field Theory: BAND: Initial]
... dmftqmc_l [Dynamical Mean Fied Theory: Quantum Monte Carlo time sLices]
... dmftqmc_n [Dynamical Mean Fied Theory: Quantum Monte Carlo Number of sweeps]
... dmftqmc_therm [Dynamical Mean Fied Theory: Quantum Monte Carlo THERMalization]
... usedmft [USE Dynamical Mean Field Theory]

Basic input variables:

... dmft_rslf [Dynamical Mean Fied Theory: Read SeLF energy]
... dmft_solv [Dynamical Mean Fied Theory: choice of SOLVer]

Useful input variables:

... dmft_dc [Dynamical Mean Fied Theory: Double Counting]
... dmft_mxsf [Dynamical Mean Fied Theory: MiXing parameter for the SelF energy]
... dmft_tollc [Dynamical Mean Fied Theory: TOLerance on Local Charge for convergence of the DMFT loop]
... dmftcheck [Dynamical Mean Fied Theory: CHECKs]
... dmftctqmc_check [Dynamical Mean Fied Theory: Continuous Time Quantum Monte Carlo CHECK]
... dmftctqmc_gmove [Dynamical Mean Fied Theory: Continuous Time Quantum Monte Carlo Global MOVEs]
... dmftctqmc_order [Dynamical Mean Fied Theory: Continuous Time Quantum Monte Carlo perturbation ORDER]
... dmftqmc_seed [Dynamical Mean Fied Theory: Quantum Monte Carlo SEED]

Input variables for experts:

... dmft_entropy [Dynamical Mean Fied Theory: ENTROPY]
... dmft_nlambda [Dynamical Mean Fied Theory: Number of LAMBDA points]
... dmft_read_occnd [Dynamical Mean Fied Theory: READ OCCupations (Non Diagonal)]
... dmft_t2g [Dynamical Mean Fied Theory: t2g orbitals]
... dmft_tolfreq [Dynamical Mean Fied Theory: TOLerance on DFT correlated electron occupation matrix for the definition of the FREQuency grid]
... dmftctqmc_basis [Dynamical Mean Fied Theory: Continuous Time Quantum Monte Carlo BASIS]
... dmftctqmc_correl [Dynamical Mean Fied Theory: Continuous Time Quantum Monte Carlo CORRELations]
... dmftctqmc_grnns [Dynamical Mean Fied Theory: Continuous Time Quantum Monte Carlo GReeNs NoiSe]
... dmftctqmc_meas [Dynamical Mean Fied Theory: Continuous Time Quantum Monte Carlo MEASurements]
... dmftctqmc_mov [Dynamical Mean Fied Theory: Continuous Time Quantum Monte Carlo MOVie]
... dmftctqmc_mrka [Dynamical Mean Fied Theory: Continuous Time Quantum Monte Carlo MARKov Analysis]
... dmftctqmc_triqs_nleg [Dynamical Mean Fied Theory: Continuous Time Quantum Monte Carlo perturbation of TRIQS, Number of LEGendre polynomials]
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### 4. Selected input files.

WARNING : as of ABINITv8.6.x, the list of input files provided in the specific section of the topics Web pages is still to be reviewed/tuned. In some cases, it will be adequate, and in other cases, it might be incomplete, or perhaps even useless.
The user can find some related example input files in the ABINIT package in the directory /tests, or on the Web:

tests/v6/Input: t47.in

tests/v7/Input: t21.in t27.in t28.in
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### 5. References.

[Amadon2006] B. Amadon, S. Biermann, A. Georges and F. Aryasetiawan, "The α-γ Transition of Cerium Is Entropy Driven", Phys. Rev. Lett. 96, 066402 (2006).
DOI: 10.1103/PhysRevLett.96.066402.
[Amadon2008] B. Amadon, F. Lechermann, A. Georges, F. Jollet, T. O. Wehling and A. I. Liechtenstein, "Plane-wave based electronic structure calculations for correlated materials using dynamical mean-field theory and projected local orbitals", Phys. Rev. B 77, 205112 (2008).
DOI: 10.1103/PhysRevB.77.205112.
[Amadon2008a] B. Amadon, F. Jollet and M. Torrent, "γ and β cerium: LDA+U calculations of ground-state parameters", Phys. Rev. B 77, 155104 (2008).
DOI: 10.1103/PhysRevB.77.155104.
[Amadon2012] B. Amadon, "A self-consistent DFT+DMFT scheme in the projector augmented wave method: applications to cerium, Ce2O3 and Pu2O3 with the Hubbard I solver and comparison to DFT+U", J. Phys.: Cond. Matt. 24, 075604 (2012).
URL: http://iopscience.iop.org/article/10.1088/0953-8984/24/7/075604.
[Anisimov1991] V. I. Anisimov and O. Gunnarsson, "Density-functional calculation of effective Coulomb interactions in metals", Phys. Rev. B 43, 7570–7574 (1991).
DOI: 10.1103/PhysRevB.43.7570.
[Bergeron2016] Dominic Bergeron and A.-M. S. Tremblay, "Algorithms for optimized maximum entropy and diagnostic tools for analytic continuation", Phys. Rev. E 94, 023303 (2016).
DOI: 10.1103/PhysRevE.94.023303.
[Bieder2014] J. Bieder and B. Amadon, "Thermodynamics of the α-γ transition in cerium from first principles", Phys. Rev. B 89, 195132 (2014).
DOI: 10.1103/PhysRevB.89.195132.
[Georges1996] A. Georges, G. Kotliar, W. Krauth and M.J. Rozenberg, "Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions", Rev. Mod. Phys. 68, 13–125 (1996).
DOI: 10.1103/RevModPhys.68.13.
[Georges2004] A. Georges, "Strongly Correlated Electron Materials: Dynamical Mean-Field Theory and Electronic Structure", AIP Conf. Proc. 715, 3–74 (2004).
DOI: 10.1063/1.1800733.
[Gull2011] E. Gull, A.J. Millis, A. I. Liechtenstein, A.N. Rubtsov, M. Troyer and P. Werner, "Continuous-time Monte Carlo methods for quantum impurity models", Rev. Mod. Phys. 83, 349–404 (2011).
DOI: 10.1103/RevModPhys.83.349.
[Kotliar2006] G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet and C. A. Marianetti, "Electronic structure calculations with dynamical mean-field theory", Rev. Mod. Phys. 78, 865–951 (2006).
DOI: 10.1103/RevModPhys.78.865.
[Liechtenstein1995] A. I. Liechtenstein, V. I. Anisimov and J. Zaanen, "Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators", Phys. Rev. B 52, R5467–R5470 (1995).
DOI: 10.1103/PhysRevB.52.R5467.
[Werner2006] P. Werner, A. Comanac, L. de Medici, M. Troyer and A.J. Millis, "Continuous-Time Solver for Quantum Impurity Models", Phys. Rev. Lett. 97, 076405 (2006).
DOI: 10.1103/PhysRevLett.97.076405.

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