# Ground state of spinless fermions with DMRG

*Posted on 03 May 2014*

We want to find the ground state energy of a free spinless fermionic system, described by the following Hamiltonian (Corboz et al., 2009):

where \(<rs>\) goes through nearest neighbour pairs in a two-dimensional
lattice. The fermionic operators are subject to the following
constraints:

To calculate the ground state energy, we use the density matrix renormalization group technique, which is included in ALPS (Bauer et al., 2011). The documentation of the package is sparse, and it is often hard to figure out how to do even such a simple task. Error messages from the console are not particularly helpful either.

Some templates for Hamiltonians are included in the /usr/xml/models.xml file. There is a model for spinless fermions, but with fourth-order terms:

where \(t\), \(V\), and \(\mu\) are parameters, and
\(n_i\) is the number operator. Setting \(\gamma\) to zero in
the original Hamiltonian, we can calculate the ground state. The
following Python code generates the problem and prints the ground state
energy and the truncation error:

```
import pyalps
parms = [{
'LATTICE' : "open square lattice", # Set up the lattice
'MODEL' : "spinless fermions", # Select the model
'L' : 4, # Lattice dimension
't' : -1 , # This and the following
'mu' : 2, # are parameters to the
'U' : 0 , # Hamiltonian.
'V' : 0,
'Nmax' : 2 , # These parameters are
'SWEEPS' : 4, # specific to the DMRG
'MAXSTATES' : 100, # solver.
'NUMBER_EIGENVALUES' : 1,
'MEASURE_ENERGY' : 1
}]
prefix = '2D_spinless_fermions'
input_file = pyalps.writeInputFiles(prefix,parms)
res = pyalps.runApplication('dmrg',input_file,writexml=True)
data = pyalps.loadEigenstateMeasurements(pyalps.getResultFiles(prefix=prefix))
for s in data[0]:
print s.props['observable'], ' : ', s.y[0]
```

# References

Corboz, P.; Evenbly, G.; Verstraete,
F. & Vidal, G. Simulation of interacting fermions with entanglement renormalization. *Physics Review A*,
2010, 81, pp. 010303.

Bauer, B.; Carr, L.; Evertz, H.; Feiguin,
A.; Freire, J.; Fuchs, S.; Gamper, L.; Gukelberger, J.; Gull, E.;
Guertler, S. & others. The ALPS project release 2.0: open source software for strongly correlated systems. *Journal of Statistical
Mechanics: Theory and Experiment*, IOP Publishing, 2011, 2011, P05001.