# ALPS 2 Tutorials:DMFT-02 Hybridization

# Tutorial 02: Hybridization Expansion CT-HYB / Tutorial 03: Interaction Expansion CT-INT

We will now reproduce the same result with a continuous-time Quantum Monte Carlo code: the Hybridization or CT-HYB code by Werner *et al.*. The parameters are the same, apart from the command for the solver:

SOLVER=Hybridization

You can find the parameter files (called `*tsqrt2`) in the directory `tutorials/dmft-02-hybridization` in the examples.

After running these simulations compare the output to the Hirsch Fye results. To rerun a simulation, you can specify a starting solution by defining `G0OMEGA_INPUT`, e.g. copy `G0omga_output` to `G0_omega_input`, specify `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, which renders the measurement procedure inefficient. You can improve statistics by increasing the total run time (`MAX_TIME`) or by running it on more than one CPU. For running it with MPI, try `mpirun -np procs dmft parameter_file` or consult the man page of your mpi installation.

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).
The corresponding parameter files are very similar, you can find them in the directory `tutorials/dmft-03-interaction`.

# Tutorial 04: Mott Transition

Mott transitions are metal insulator transitions that occur in many materials, e.g. transition metal compounds, as a function of pressure or doping. The review by Imada *et al.* gives an excellent introduction to the subject and mentions V2O3 and the organics as typical examples.

MIT are easily investigated by DMFT as the relevant physics is essentially local (or k-independent): At half filling the MIT can by modeled by a self energy with a pole at $\omega=0$ which splits the noninteracting band into an upper and a lower Hubbard band. In this context it is instructive to suppress antiferromagnetic long range order and enforce a paramagnetic solution in the DMFT simulation, to mimic the paramagnetic insulating phase. For this the up and down spin of the Green's functions are symmetrized (parameter `SYMMETRIZATION = 1;`).

We investigate the *Mott* transition in single-site DMFT, as a function of interaction at fixed temperature `\beta t=20` (see e.g. Fig. 2 in this paper).
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 $U$.

Imaginary time Green's functions are not easy to interpret, and therefore many authors employ analytic continuation methods. There are however two clear features: the value at `\beta` corresponds to `-n`, the negative value of the density (per spin). And the value at `\beta/2` goes for decreasing temperature to `A(\omega=0)`, the spectral function at the Fermi energy: `\beta G(\beta/2 \rightarrow A(0)`. From a temperature dependence of the imaginary time Green's function we can therefore immediately see if the system is metallic or insulating.

The `beta20_U*` parameter files in the directory `tutorials/dmft-04-mott` should show you how to go from a metallic (at small $U$) to an insulating (at large $U$) solution, at fixed $\beta$. The largest value of $U$ is deep within the insulating phase.

# Tutorial 05: Orbitally Selective Mott Transition

An interesting phenomenon in multi-orbital models is the orbitally selective Mott transition, first examined by Anisimov *et al*. A variant of this, a *momentum-selective* Mott transition, has recently been discussed in cluster calculations as a cluster representation of pseudogap physics.

In an orbitally selective Mott transition some of the orbitals involved become Mott insulating as a function of doping or interactions, while others stay metallic.

As a minimal model we consider two bands: a wide band and a narrow band. In addition to the intra-orbital Coulomb repulsion $U$ we consider interactions $U'$, and $J$, with $U' = U-2J$. We limit ourselves to Ising-like interactions - a simplification that is often problematic for real compounds.

We choose here a case with two bandwidth $t1=0.5$ and $t2=1$ and density-density like interactions of $U'=U/2$, $J=U/4$, and $U$ between $1.6$ and $2.8$, where the first case shows a Fermi liquid-like behavior in both orbitals, the $U=2.2$ is orbitally selective, and $U=2.8$ is insulating in both orbitals.

The parameter files can be found in `tutorials/dmft-05-osmt` in the directories `beta30_U1.8_2orbital`, `beta30_U2.2_2orbital` and `beta30_U2.8_2orbital`. A paper using the same sample parameters can be found here.

Tutorial by Emanuel - Please don't hesitate to ask!