ALPS 2 Tutorials:DMFT-07 Hirsch-Fye

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This is a short tutorial for the ALPS DMFT code. It should illustrate the Dynamical Mean Field Theory and in particular showcase some application of the new continuous-time impurity solvers.

As of now, two impurity solver algorithms are discussed: the discrete time Hirsch Fye code that serves mostly as a pedagogic example code; and a state-of-the-art hybridization expansion code.

The first tutorial will introduce the Metal - AFM insulator transition as a function of temperature in infinite dimension, using a HF solver. Tutorial II will repeat the same exercise with a hybridization expansion solver. Tutorial III illustrates the metal to paramagnetic insulator transition as a function of interaction. And tutorial IV will give an introduction to the so-called orbitally selective Mott transition.

Tutorial: Hirsch Fye Code

We start by running a discrete time Monte Carlo code: the Hirsch Fye code. For concreteness, we reproduce Fig. 11 in the DMFT review by Georges it et al.. The series of six curves shows that the system, a Hubbard model on the Bethe lattice with interaction U=3D/\sqrt{2} at half filling, enters an antiferromagnetic phase.

Hirsch Fye is described in here, and this review also provides an open source implementation for the codes. More information can also be found in [Blüumer's PhD]. While many improvements have been developed (see e.g. Alvarez08 or Nukala09, the algorithm has been replaced by continuous-time algorithms.

The Hirsch Fye simulation will run for about a minute per iteration. The parameter files for running this simulation can be found in the \verb#examples/hirschfye# directory. The main parameters are:

SEED = 0;# & Monte Carlo Random Number Seed \\
THERMALIZATION = 10000;# & Thermalization Sweeps \\
SWEEPS = 1000000;# & Total Sweeps to be computed \\
MAX_TIME = 60;# & Maximum time to run the simulation \\
BETA = 12.;# & Inverse temperature \\
SITES = 1;# & This is a single site DMFT simulation, so Sites is 1 \\
N = 16;# & Number of time slices (you will see that this parameter is rather small) \\
NMATSUBARA = 500;# & The number of Matsubara frequencies \\
U = 3;# & Interaction energy \\
t = 1;# & hopping parameter. For the Bethe lattice considered here $W=2D=4t$\\
MU = 0;# & Chemical potential \\
H = 0;# & Magnetic field \\
PARAMAGNET = 0;# & We are not enforcing a paramagnetic self consistency condition \\
SOLVER = /opt/alps/bin/hirschfye;#& The external Hirsch Fye solver

To start a simulation add dmft to your path and type:

dmft hirschfye.param

The code will run for up to 10 iterations. In the directory in which you run the program you will find Green's functions files \verb#G_tau_?# in your output directory. Plot them to obtain Fig.~11 of \onlinecite{Georges96}. You will also find self energies (\verb#selfenergy_?#) and Green's functions in frequency space \verb#G_omega_?#.

As a discrete time method, HF suffers from $\Delta\tau$ - errors. Pick a set of parameters and run it for sucessively larger $N$!

\section{Tutorial: Hybridization Code} We will now reproduce the same result with a continuous-time Quantum Monte Carlo code: the Hybridization code of Ref.~\cite{Werner06}. The parameters are the same, apart from the command for the solver: \verb#SOLVER=Werner#.

After running these simulations compare the output to the Hirsch Fye results. To rerun a simulation, you can specify a starting solution by defining \verb#G0OMEGA_INPUT#, e.g. copy \verb#G0omga_output# to \verb#G0_omega_input#, specify \verb#G0OMEGA_INPUT = G0_omega_input# in the parameter file and rerun the code.

You will notice that the results are relatively noisy. The reason for that is that the expansion order at such high temperatures is very small, and therefore the measurement inefficcient. You can imporove statistics by increasing the total run time (\verb#MAX_TIME#) or by running it on more than one CPU. For running it with MPI, try \verb#mpirun -np procs dmft_mpi parameter_file#.

It is also instructive to run these calculations with a CT-INT code. This code performs an expansion in the interaction (instead of the hybridization).

\section{Tutorial: Mott Transition} It is sometimes instructive to enforce a paramagnetic solution and thereby supress long-ranged order in a DMFT simulation. For this the up and down spin of the Green's functions are symmetrized (parameter \verb#PARAMAGNET = 1;#). We use this to investigate the ``Mott transition in single-site DMFT, as a function of interaction at fixed temperature $\beta t=20$ (see also Fig. $2$ in \cite{Gull07}). Starting from a non-interacting solution we see in the imaginary time Green's function that the solution is metallic for $U/t \leq 4.5$, and insulating for $U/t\geq 5$. A coexistence region could be found by starting from an insulating (or atomic) solution and trying to convert it for smaller $U$. A good overview of this topic is given in \cite{Blumer}.

Imaginary time Green's functions are not easy to interpret, and therefore many authors employ analytic continuation methods. There are however two distinctive features: the value at $\beta$ corresponds to $-n$, the negative value of the density. And the value at $\beta/2$ goes for decreasing temperature to a value characteristic of $A(0)$, the spectrum at the Fermi energy: $\beta G(\beta/2) {T\rightarrow 0 \atop \rightarrow} A(0)$. From a temperature dependence of the imaginary time Green's function we can therefore immediately see if we have a metal or an insulator.

\section{Orbitally Selective Mott Transition} An interesting phenomenon in Multi-Orbital models is the orbitally selective Mott transition\cite{Anisimov02}. A variant of this, a ``momentum-selective Mott transition, has recently been discovered in cluster calculations. The OSMT is usually investigated in a two-band model, with one wide and one narrow band. We choose here a case with two orbitals and density-density like interactions of $U'=U/2$, $J=U/4$, $2 t_1 = t_2 =1$, and $U$ between $1.6$ and $2.8$, where the first case shows a Fermi liquid-like behavior in both orbital, the $U=2.2$ is orbitally selective, and $U=2.8$ is insulating in both orbitals.