At the cake meeting this week we discussed

Density Matrix Embedding: A Simple Alternative to Dynamical Mean-Field Theory

Gerald Knizia and Garnet Kin-Lic Chan

This is an original and promising approach. It is computationally much "cheaper" than DMFT.

There are follow up papers that apply the method to quantum chemistry, the problem of defining the QM/MM boundary, the honeycomb lattice Hubbard model, and calculation of spectral functions. In the latter the bath is frequency dependent.

The system is divided into an "impurity" and a "bath".

The starting point is the Schmidt decomposition of the system quantum state. If M is the dimension of the impurity Hilbert space then there are at most M terms in the decomposition. This limits the amount of entanglement between the impurity and the bath.

Consider a Hubbard model where the impurity is a single site, then M=4.

The reduced Hamiltonian acting on the Schmidt basis states looks like a two-site Hubbard model, which is easy to solve.

[It is not clear why nearest neighbour Coulomb repulsion or more complicated 4-body terms are not included].

Expectation values of this reduced Hamiltonian are exactly the same as for the full model.

But, the problem is one can only exactly construct the Schmidt basis if one already knows the true ground state.

So instead one uses a mean field [Hartree-Fock] Hamiltonian and wave function for the bath. It looks like there is one free parameter, an effective chemical potential. That parameter is determined by a self-consistency condition that makes sure the one-electron density matrix calculated with the bath Hamiltonian is as close as possible to the impurity Hamiltonian. This is the analogue of the self-consistency condition of DMFT.

Some of the results look promising with favourable comparison with exact results for one dimensional Hubbard model obtained using the Bethe ansatz.

Given the low computational cost it is not clear why results were not obtained for larger clusters beyond 2 x 2.

I would be nice to see a longer paper, e.g. a PRB, with more details including a few worked examples, such as those I heard Garnett Chan mention briefly in a talk at Telluride.

All calculations presented are for zero temperature.

Cursory comments are made about it being straight-forward to extend to non-zero temperature. It is very important that this is done. One of the great achievements of DMFT is that it describes the temperature dependence of transport properties, including the crossover with increasing temperature from a Fermi liquid to a bad metal.

I am not sure about the results presented for spectral functions. It is claimed that they capture the "three peak structure" of DMFT but I fail to see that. In DMFT in the metallic phase near the Mott transition one sees a central peak [Kondo resonance] that is clearly separated from the upper and lower Hubbard bands. A sample is below, which also highlights the strong temperature dependence, taken from this preprint.

## Friday, October 3, 2014

### A promising alternative to Dynamical Mean-Field Theory?

Labels:
bad metals,
DMFT,
entanglement,
quantum chemistry,
strong correlations

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