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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 python command lines for running the simulations are found in tutorial5a.py. Alternatively, you can use the Vistrails file:

import pyalps
import numpy as np
import matplotlib.pyplot as plt
import pyalps.pyplot

#prepare the input parameters
parms=[]
coulombparam=[[1.8,0.45],[2.2,0.55],[2.8,0.7]]
for cp in coulombparam: 
    parms.append(
            { 
              'CHECKPOINT'          : 'dump',
              'CONVERGED'           : 0.01,
              'F'                   : 10,
              'FLAVORS'             : 4,
              'H'                   : 0,
              'H_INIT'              : 0.,
              'MAX_IT'              : 20,
              'MAX_TIME'            : 180,
              'MU'                  : 0,
              'N'                   : 1000,
              'NMATSUBARA'          : 1000,
              'N_FLIP'              : 0,
              'N_MEAS'              : 10000,
              'N_MOVE'              : 0,
              'N_ORDER'             : 50,
              'N_SHIFT'             : 0,
              'OMEGA_LOOP'          : 1,
              'OVERLAP'             : 0,
              'SEED'                : 0,
              'SOLVER'              : 'Hybridization',
              'SYMMETRIZATION'      : 1,
              'TOLERANCE'           : 0.3,
              't'                   : 1,
              'SWEEPS'              : 100000000,
              'BETA'                : 30,
              'THERMALIZATION'      : 10,
              'U'                   : cp[0],
              'J'                   : cp[1],
              't0'                  : 0.5,
              't1'                  : 1,
              'G0TAU_INPUT'         :'G0_tau_input_u_'+str(cp[0])+'_j_'+str(cp[1])
            }
        )

#write the input file and run the simulation
for p in parms:
    input_file = pyalps.writeParameterFile('parm_u_'+str(p['U'])+'_j_'+str(p['J']),p)
    res = pyalps.runDMFT(input_file)

The parameter files to run the simulation on the command line 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.

As discussed in the previous tutorial, the (non-)metallicity of the Green's function is best observed by plotting the data on a logarithmic scale:

flavors=parms[0]['FLAVORS']
listobs=[]   
for f in range(0,flavors):
    listobs.append('Green_'+str(f))
  
ll=pyalps.load.Hdf5Loader()
data = ll.ReadMeasurementFromFile(pyalps.getResultFiles(pattern='parm_u_*h5'), respath='/simulation/results/G_tau', measurements=listobs, verbose=True)
for d in pyalps.flatten(data):
    d.x = d.x*d.props["BETA"]/float(d.props["N"])
    d.y = -d.y
    d.props['label'] = r'$U=$'+str(d.props['U'])
plt.figure()
plt.yscale('log')
plt.xlabel(r'$\tau$')
plt.ylabel(r'$G(\tau)$')
plt.title('Hubbard model on the Bethe lattice')
pyalps.pyplot.plot(data)
plt.legend()
plt.show()

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