vumps_and_plane_wave.py

```"""Excitations of the transverse field Ising model.

This example uses VUMPS to find the ground state of the transverse field Ising model and uses the plane wave excitation ansatz to compute the first excited states.
"""
# Copyright (C) TeNPy Developers, GNU GPLv3

from tenpy.models.tf_ising import TFIChain
from tenpy.algorithms import vumps, plane_wave_excitation
from tenpy.networks import mps
import numpy as np

def tfi_vumps(g=1.5):
model_params = dict(L=2, J=1., g=g, bc_MPS='infinite', conserve='parity')
M = TFIChain(model_params)
psi = mps.MPS.from_product_state(M.lat.mps_sites(), [0, 0], bc='infinite')
vumps_pars = {
'combine': False,
'N_sweeps_check': 1,
'mixer': False,
'N_sweeps_check': 1,
'trunc_params': {
'chi_max': 32,
'svd_min': 1.e-14,
},
'min_sweeps': 30,
'max_sweeps': 50,
'max_split_err': 1e-8,  # different criteria than DMRG
'max_E_err': 1.e-12,
'max_S_err': 1.e-8,
}

eng = vumps.TwoSiteVUMPSEngine(psi, M, vumps_pars)
E, psi = eng.run()
print(f"ground state energy: {E:.5f}")

# to directly continue with plane wave excitations, return the uniform MPS
uniform_psi = eng.psi

return E, uniform_psi, M

def tfi_excitations(psi_gs, M):
pw_params = {
'lanczos_params': {
'N_max': 50,
},
}
eng_pw = plane_wave_excitation.PlaneWaveExcitationEngine(psi_gs, M, pw_params)

momenta = np.arange(0, np.pi, np.pi/8)  # compute for some momenta
qtotal_change = [1]  # look for excitations in other parity sector
num_ev = 1  # we only compute the lowest dispersion mode

dispersions = []
for p in momenta:
Es, _, _ = eng_pw.run(p, qtotal_change, num_ev=num_ev)
dispersions.append(Es)
print(f"excitation energy for momentum {p/np.pi:.2f} Pi: {Es[0]:.5f}")
return momenta, np.array(dispersions)

def tfi_dispersion(k, g):
# exact dispersion for two site unit cell
return np.min([2*np.sqrt(g**2-2*g*np.cos(k)+1), 2*np.sqrt(g**2-2*g*np.cos(k+np.pi)+1)], axis=0)

if __name__ == "__main__":
E, psi, M = tfi_vumps()
momenta, dispersions = tfi_excitations(psi, M)

# plot and compare to exact results
import matplotlib.pyplot as plt

plt.plot(momenta, dispersions, 'x', label='plane wave ansatz')
plt.plot(np.arange(0, np.pi, 0.1), tfi_dispersion(np.arange(0, np.pi, 0.1), 1.5), ':', label='exact', c='black')
plt.legend()
plt.xlabel('momentum')
plt.ylabel('excitation energy')
plt.show()
```