Running ED simulation
As an example of the sparse exact diagonalization method, we will obtain the triplet gap as a function of system size for the spin chain model.
First step is import required packages.
import pyalps
import matplotlib.pyplot as plt
import pyalps.plotThen we prepare each set of input parameters and write them into the format ALPS expects.
parms = []
for l in [4, 6, 8, 10, 12, 14, 16]:
for sz in [0, 1]:
parms.append(
{.
'LATTICE' : "chain lattice",.
'MODEL' : "spin",
'local_S' : 1,
'J' : 1,
'L' : l,
'CONSERVED_QUANTUMNUMBERS' : 'Sz',
'Sz_total' : sz
}
)
#write the input file and run the simulation
input_file = pyalps.writeInputFiles('parm2a',parms)Then we run sparsediag for each set of parameters:
res = pyalps.runApplication('sparsediag',input_file)And extract the ground state energies over all momenta for every simulation.
for sim in data:
l = int(sim[0].props['L'])
if l not in lengths: lengths.append(l)
sz = int(sim[0].props['Sz_total'])
all_energies = []
for sec in sim:
all_energies += list(sec.y)
min_energies[(l,sz)]= np.min(all_energies)Finally, we make a plot of the triplet gap as a function of system size.
gapplot = pyalps.DataSet()
gapplot.x = 1./np.sort(lengths)
gapplot.y = [min_energies[(l,1)] -min_energies[(l,0)] for l in np.sort(lengths)]..
gapplot.props['xlabel']='$1/L$'
gapplot.props['ylabel']='Triplet gap $\Delta/J$'
gapplot.props['label']='S=1'
gapplot.props['line']='.'
plt.figure()
pyalps.plot.plot(gapplot)
plt.legend()
plt.xlim(0,0.25)
plt.ylim(0,1.0)
pars = [fw.Parameter(0.411), fw.Parameter(1000), fw.Parameter(1)]
f = lambda self, x, p: p[0]()+p[1]()*np.exp(-x/p[2]())
# we fit only a range from 8 to 14
fw.fit(None, f, pars, np.array(gapplot.y)[2:], np.sort(lengths)[2:])
x = np.linspace(0, 1./min(lengths), 100)
plt.plot(x, f(None, 1/x, pars))
plt.show()The final result should look like this:
