def test_dmrg_excited(eps=1.e-12):
# checks ground state and 2 excited states (in same symmetry sector) for a small system
# (without truncation)
L, g = 8, 1.3
bc = 'finite'
model_params = dict(L=L,
J=1.,
g=g,
bc_MPS=bc,
conserve='parity',
verbose=0)
M = TFIChain(model_params)
# compare to exact solution
ED = ExactDiag(M)
ED.build_full_H_from_mpo()
ED.full_diagonalization()
# Note: energies sorted by chargesector (first 0), then ascending -> perfect for comparison
print("Exact diag: E[:5] = ", ED.E[:5])
print("Exact diag: (smalles E)[:10] = ", np.sort(ED.E)[:10])
psi_ED = [ED.V.take_slice(i, 'ps*') for i in range(5)]
print("charges : ", [psi.qtotal for psi in psi_ED])
# first DMRG run
psi0 = mps.MPS.from_product_state(M.lat.mps_sites(), [0] * L, bc=bc)
dmrg_pars = {
'verbose': 0,
'N_sweeps_check': 1,
'lanczos_params': {
'reortho': False
},
'diag_method': 'lanczos',
'combine': True
}
eng0 = dmrg.TwoSiteDMRGEngine(psi0, M, dmrg_pars)
E0, psi0 = eng0.run()
assert abs((E0 - ED.E[0]) / ED.E[0]) < eps
ov = npc.inner(psi_ED[0], ED.mps_to_full(psi0), 'range', do_conj=True)
assert abs(abs(ov) - 1.) < eps # unique groundstate: finite size gap!
# second DMRG run for first excited state
dmrg_pars['verbose'] = 1.
dmrg_pars['orthogonal_to'] = [psi0]
psi1 = mps.MPS.from_product_state(M.lat.mps_sites(), [0] * L, bc=bc)
eng1 = dmrg.TwoSiteDMRGEngine(psi1, M, dmrg_pars)
E1, psi1 = eng1.run()
assert abs((E1 - ED.E[1]) / ED.E[1]) < eps
ov = npc.inner(psi_ED[1], ED.mps_to_full(psi1), 'range', do_conj=True)
assert abs(abs(ov) - 1.) < eps # unique groundstate: finite size gap!
# and a third one to check with 2 eigenstates
dmrg_pars['orthogonal_to'] = [psi0, psi1]
# note: different intitial state necessary, otherwise H is 0
psi2 = mps.MPS.from_singlets(psi0.sites[0],
L, [(0, 1), (2, 3), (4, 5), (6, 7)],
bc=bc)
eng2 = dmrg.TwoSiteDMRGEngine(psi2, M, dmrg_pars)
E2, psi2 = eng2.run()
print(E2)
assert abs((E2 - ED.E[2]) / ED.E[2]) < eps
ov = npc.inner(psi_ED[2], ED.mps_to_full(psi2), 'range', do_conj=True)
assert abs(abs(ov) - 1.) < eps # unique groundstate: finite size gap!
def test_dmrg_excited(eps=1.e-12):
# checks ground state and 2 excited states (in same symmetry sector) for a small system
# (without truncation)
L, g = 8, 1.1
bc = 'finite'
model_params = dict(L=L,
J=1.,
g=g,
bc_MPS=bc,
conserve='parity',
verbose=0)
M = TFIChain(model_params)
# compare to exact solution
ED = ExactDiag(M)
ED.build_full_H_from_mpo()
ED.full_diagonalization()
# Note: energies sorted by chargesector (first 0), then ascending -> perfect for comparison
print("Exact diag: E[:5] = ", ED.E[:5])
# first DMRG run
psi0 = mps.MPS.from_product_state(M.lat.mps_sites(), [0] * L, bc=bc)
dmrg_pars = {
'verbose': 1,
'N_sweeps_check': 1,
'lanczos_params': {
'reortho': True
}
}
eng0 = dmrg.EngineCombine(psi0, M, dmrg_pars)
E0, psi0 = eng0.run()
assert abs((E0 - ED.E[0]) / ED.E[0]) < eps
ov = npc.inner(ED.V.take_slice(0, 'ps*'),
ED.mps_to_full(psi0),
do_conj=True)
assert abs(abs(ov) - 1.) < eps # unique groundstate: finite size gap!
# second DMRG run for first excited state
dmrg_pars['orthogonal_to'] = [psi0]
psi1 = mps.MPS.from_product_state(M.lat.mps_sites(), [0] * L, bc=bc)
eng1 = dmrg.EngineCombine(psi1, M, dmrg_pars)
E1, psi1 = eng1.run()
assert abs((E1 - ED.E[1]) / ED.E[1]) < eps
ov = npc.inner(ED.V.take_slice(1, 'ps*'),
ED.mps_to_full(psi1),
do_conj=True)
assert abs(abs(ov) - 1.) < eps # unique groundstate: finite size gap!
# and a third one to check with 2 eigenstates
dmrg_pars['orthogonal_to'] = [psi0, psi1]
# note: different intitial state necessary, otherwise H is 0
psi2 = mps.MPS.from_product_state(M.lat.mps_sites(), [0, 1] * (L // 2),
bc=bc)
eng2 = dmrg.EngineCombine(psi2, M, dmrg_pars)
E2, psi2 = eng2.run()
print(E2)
assert abs((E2 - ED.E[2]) / ED.E[2]) < eps
ov = npc.inner(ED.V.take_slice(2, 'ps*'),
ED.mps_to_full(psi2),
do_conj=True)
assert abs(abs(ov) - 1.) < eps # unique groundstate: finite size gap!
def example_exact_diagonalization(L, Jz):
xxz_pars = dict(L=L, Jxx=1., Jz=Jz, hz=0.0, bc_MPS='finite')
M = XXZChain(xxz_pars)
product_state = ["up", "down"] * (xxz_pars['L'] // 2) # this selects a charge sector!
psi_DMRG = MPS.from_product_state(M.lat.mps_sites(), product_state)
charge_sector = psi_DMRG.get_total_charge(True) # ED charge sector should match
ED = ExactDiag(M, charge_sector=charge_sector, max_size=2.e6)
ED.build_full_H_from_mpo()
# ED.build_full_H_from_bonds() # whatever you prefer
print("start diagonalization")
ED.full_diagonalization() # the expensive part for large L
E0_ED, psi_ED = ED.groundstate() # return the ground state
print("psi_ED =", psi_ED)
print("run DMRG")
dmrg.run(psi_DMRG, M, {'verbose': 0}) # modifies psi_DMRG in place!
# first way to compare ED with DMRG: convert MPS to ED vector
psi_DMRG_full = ED.mps_to_full(psi_DMRG)
print("psi_DMRG_full =", psi_DMRG_full)
ov = npc.inner(psi_ED, psi_DMRG_full, axes='range', do_conj=True)
print("<psi_ED|psi_DMRG_full> =", ov)
assert (abs(abs(ov) - 1.) < 1.e-13)
# second way: convert ED vector to MPS
psi_ED_mps = ED.full_to_mps(psi_ED)
ov2 = psi_ED_mps.overlap(psi_DMRG)
print("<psi_ED_mps|psi_DMRG> =", ov2)
assert (abs(abs(ov2) - 1.) < 1.e-13)
assert (abs(ov - ov2) < 1.e-13)
# -> advantage: expectation_value etc. of MPS are available!
print("<Sz> =", psi_ED_mps.expectation_value('Sz'))
def test_dmrg_diag_method(engine, diag_method, tol=1.e-6):
bc_MPS = 'finite'
model_params = dict(L=6, S=0.5, bc_MPS=bc_MPS, conserve='Sz', verbose=0)
M = SpinChain(model_params)
# chose total Sz= 4, not 3=6/2, i.e. not the sector with lowest energy!
# make sure below that we stay in that sector, if we're supposed to.
init_Sz_4 = ['up', 'down', 'up', 'up', 'up', 'down']
psi_Sz_4 = mps.MPS.from_product_state(M.lat.mps_sites(), init_Sz_4, bc=bc_MPS)
dmrg_pars = {
'verbose': 1,
'N_sweeps_check': 1,
'combine': True,
'max_sweeps': 5,
'diag_method': diag_method,
'mixer': True,
}
ED = ExactDiag(M)
ED.build_full_H_from_mpo()
ED.full_diagonalization()
if diag_method == "ED_all":
charge_sector = None # allow to change the sector
else:
charge_sector = [2] # don't allow to change the sector
E_ED, psi_ED = ED.groundstate(charge_sector=charge_sector)
DMRGEng = dmrg.__dict__.get(engine)
print("DMRGEng = ", DMRGEng)
print("setting diag_method = ", dmrg_pars['diag_method'])
eng = DMRGEng(psi_Sz_4.copy(), M, dmrg_pars)
E0, psi0 = eng.run()
print("E0 = {0:.15f}".format(E0))
assert abs(E_ED - E0) < tol
ov = npc.inner(psi_ED, ED.mps_to_full(psi0), 'range', do_conj=True)
assert abs(abs(ov) - 1) < tol
def test_MPO_var(L=8, tol=1.e-13):
xxz_pars = dict(L=L,
Jx=1.,
Jy=1.,
Jz=1.1,
hz=0.1,
bc_MPS='finite',
conserve=None)
M = SpinChain(xxz_pars)
psi = random_MPS(L, 2, 10)
exp_val = M.H_MPO.expectation_value(psi)
ED = ExactDiag(M)
ED.build_full_H_from_mpo()
psi_full = ED.mps_to_full(psi)
exp_val_full = npc.inner(psi_full,
npc.tensordot(ED.full_H, psi_full, axes=1),
axes='range',
do_conj=True)
assert abs(exp_val - exp_val_full) / abs(exp_val_full) < tol
Hsquared = M.H_MPO.variance(psi, 0.)
Hsquared_full = npc.inner(psi_full,
npc.tensordot(ED.full_H,
npc.tensordot(ED.full_H,
psi_full,
axes=1),
axes=1),
axes='range',
do_conj=True)
assert abs(Hsquared - Hsquared_full) / abs(Hsquared_full) < tol
var = M.H_MPO.variance(psi)
var_full = Hsquared_full - exp_val_full**2
assert abs(var - var_full) / abs(var_full) < tol
def test_tebd(bc_MPS, g=0.5):
L = 2 if bc_MPS == 'infinite' else 6
# xxz_pars = dict(L=L, Jxx=1., Jz=3., hz=0., bc_MPS=bc_MPS)
# M = XXZChain(xxz_pars)
# factor of 4 (2) for J (h) to change spin-1/2 to Pauli matrices
model_pars = dict(L=L, Jx=0., Jy=0., Jz=-4., hx=2. * g, bc_MPS=bc_MPS, conserve=None)
M = SpinChain(model_pars)
state = ([[1, -1.], [1, -1.]] * L)[:L] # pointing in (-x)-direction
psi = MPS.from_product_state(M.lat.mps_sites(), state, bc=bc_MPS)
tebd_param = {
'verbose': 2,
'dt': 0.01,
'order': 2,
'delta_tau_list': [0.1, 1.e-4, 1.e-8],
'max_error_E': 1.e-9,
'trunc_params': {
'chi_max': 50,
'trunc_cut': 1.e-13
}
}
engine = tebd.Engine(psi, M, tebd_param)
engine.run_GS()
print("norm_test", psi.norm_test())
if bc_MPS == 'finite':
psi.canonical_form()
ED = ExactDiag(M)
ED.build_full_H_from_mpo()
ED.full_diagonalization()
psi_ED = ED.groundstate()
Etebd = np.sum(M.bond_energies(psi))
Eexact = np.min(ED.E)
print("E_TEBD={Etebd:.14f} vs E_exact={Eex:.14f}".format(Etebd=Etebd, Eex=Eexact))
assert (abs((Etebd - Eexact) / Eexact) < 1.e-7)
ov = npc.inner(psi_ED, ED.mps_to_full(psi), do_conj=True)
print("compare with ED: overlap = ", abs(ov)**2)
assert (abs(abs(ov) - 1.) < 1.e-7)
# Test real time TEBD: should change on an eigenstate
Sold = np.average(psi.entanglement_entropy())
for i in range(3):
engine.run()
Enew = np.sum(M.bond_energies(psi))
Snew = np.average(psi.entanglement_entropy())
assert (abs(Enew - Etebd) < 1.e-8)
assert (abs(Sold - Snew) < 1.e-5) # somehow we need larger tolerance here....
if bc_MPS == 'infinite':
Etebd = np.average(M.bond_energies(psi))
Eexact = e0_tranverse_ising(g)
print("E_TEBD={Etebd:.14f} vs E_exact={Eex:.14f}".format(Etebd=Etebd, Eex=Eexact))
Sold = np.average(psi.entanglement_entropy())
for i in range(2):
engine.run()
Enew = np.average(M.bond_energies(psi))
Snew = np.average(psi.entanglement_entropy())
assert (abs(Etebd - Enew) < 1.e-7)
assert (abs(Sold - Snew) < 1.e-5) # somehow we need larger tolerance here....
def test_dmrg(bc_MPS, combine, mixer, n, g=1.2):
L = 2 if bc_MPS == 'infinite' else 8
model_params = dict(L=L, J=1., g=g, bc_MPS=bc_MPS, conserve=None, verbose=0)
M = TFIChain(model_params)
state = [0] * L # Ferromagnetic Ising
psi = mps.MPS.from_product_state(M.lat.mps_sites(), state, bc=bc_MPS)
dmrg_pars = {
'verbose': 5,
'combine': combine,
'mixer': mixer,
'chi_list': {
0: 10,
5: 30
},
'max_E_err': 1.e-12,
'max_S_err': 1.e-8,
'N_sweeps_check': 4,
'mixer_params': {
'disable_after': 15,
'amplitude': 1.e-5
},
'trunc_params': {
'svd_min': 1.e-10,
},
'max_N_for_ED': 20, # small enough that we test both diag_method=lanczos and ED_block!
'max_sweeps': 40,
'active_sites': n,
}
if not mixer:
del dmrg_pars['mixer_params'] # avoid warning of unused parameter
if bc_MPS == 'infinite':
# if mixer is not None:
# dmrg_pars['mixer_params']['amplitude'] = 1.e-12 # don't actually contribute...
dmrg_pars['start_env'] = 1
res = dmrg.run(psi, M, dmrg_pars)
if bc_MPS == 'finite':
ED = ExactDiag(M)
ED.build_full_H_from_mpo()
ED.full_diagonalization()
E_ED, psi_ED = ED.groundstate()
ov = npc.inner(psi_ED, ED.mps_to_full(psi), 'range', do_conj=True)
print("E_DMRG={Edmrg:.14f} vs E_exact={Eex:.14f}".format(Edmrg=res['E'], Eex=E_ED))
print("compare with ED: overlap = ", abs(ov)**2)
assert abs(abs(ov) - 1.) < 1.e-10 # unique groundstate: finite size gap!
var = M.H_MPO.variance(psi)
assert var < 1.e-10
else:
# compare exact solution for transverse field Ising model
Edmrg = res['E']
Eexact = e0_tranverse_ising(g)
print("E_DMRG={Edmrg:.12f} vs E_exact={Eex:.12f}".format(Edmrg=Edmrg, Eex=Eexact))
print("relative energy error: {err:.2e}".format(err=abs((Edmrg - Eexact) / Eexact)))
print("norm err:", psi.norm_test())
Edmrg2 = np.mean(psi.expectation_value(M.H_bond))
Edmrg3 = M.H_MPO.expectation_value(psi)
assert abs((Edmrg - Eexact) / Eexact) < 1.e-10
assert abs((Edmrg - Edmrg2) / Edmrg2) < max(1.e-10, np.max(psi.norm_test()))
assert abs((Edmrg - Edmrg3) / Edmrg3) < max(1.e-10, np.max(psi.norm_test()))
def check_dmrg(L=4, bc_MPS='finite', engine='EngineCombine', mixer=None, g=1.5):
model_params = dict(L=L, J=1., g=g, bc_MPS=bc_MPS, conserve=None, verbose=0)
M = TFIChain(model_params)
state = [0] * L # Ferromagnetic Ising
psi = mps.MPS.from_product_state(M.lat.mps_sites(), state, bc=bc_MPS)
dmrg_pars = {
'verbose': 5,
'engine': engine,
'mixer': mixer,
'chi_list': {
0: 10,
5: 30
},
'max_E_err': 1.e-12,
'max_S_err': 1.e-8,
'N_sweeps_check': 4,
'mixer_params': {
'disable_after': 6,
'amplitude': 1.e-5
},
'trunc_params': {
'svd_min': 1.e-10,
},
'lanczos_params': {
'reortho': True,
'N_cache': 20
},
'max_sweeps': 40,
}
if mixer is None:
del dmrg_pars['mixer_params'] # avoid warning of unused parameter
if bc_MPS == 'infinite':
if mixer is not None:
dmrg_pars['mixer_params']['amplitude'] = 1.e-12 # don't actually contribute...
dmrg_pars['start_env'] = 1
res = dmrg.run(psi, M, dmrg_pars)
if bc_MPS == 'finite':
ED = ExactDiag(M)
ED.build_full_H_from_mpo()
ED.full_diagonalization()
psi_ED = ED.groundstate()
ov = npc.inner(psi_ED, ED.mps_to_full(psi), do_conj=True)
print("E_DMRG={Edmrg:.14f} vs E_exact={Eex:.14f}".format(Edmrg=res['E'], Eex=np.min(ED.E)))
print("compare with ED: overlap = ", abs(ov)**2)
assert abs(abs(ov) - 1.) < 1.e-10 # unique groundstate: finite size gap!
else:
# compare exact solution for transverse field Ising model
Edmrg = res['E']
Eexact = e0_tranverse_ising(g)
print("E_DMRG={Edmrg:.12f} vs E_exact={Eex:.12f}".format(Edmrg=Edmrg, Eex=Eexact))
print("relative energy error: {err:.2e}".format(err=abs((Edmrg - Eexact) / Eexact)))
print("norm err:", psi.norm_test())
Edmrg2 = np.mean(psi.expectation_value(M.H_bond))
assert abs((Edmrg - Eexact) / Eexact) < 1.e-10
assert abs((Edmrg - Edmrg2) / Edmrg2) < max(1.e-10, np.max(psi.norm_test()))
xxz_pars = dict(L=4, Jxx=1., Jz=1., hz=0.0, bc_MPS='finite')
M = XXZChain(xxz_pars)
ED = ExactDiag(M, [0])
ED.build_full_H_from_mpo()
# ED.build_full_H_from_bonds() # whatever you prefer
print("start diagonalization")
ED.full_diagonalization()
psi_ED = ED.groundstate()
print("psi_ED =", psi_ED)
print("start DMRG")
product_state = [0, 1] * (xxz_pars['L'] // 2) # this selects a charge sector!
psi_DMRG = MPS.from_product_state(M.lat.mps_sites(), product_state)
res = run_DMRG(psi_DMRG, M, {'verbose': 0})
# first way to compare ED with DMRG: convert MPS to ED vector
psi_DMRG_full = ED.mps_to_full(psi_DMRG)
print("psi_DMRG_full =", psi_DMRG_full)
ov = abs(npc.inner(psi_ED, psi_DMRG_full, do_conj=True))
print("|<psi_ED|psi_DMRG>| =", ov)
assert (abs(ov - 1.) < 1.e-13)
# second way: convert ED vector to MPS
psi_ED_mps = ED.full_to_mps(psi_ED)
ov = psi_ED_mps.overlap(psi_DMRG)
print("|<psi_ED_mps|psi_DMRG>| =", abs(ov))
assert (abs(abs(ov) - 1.) < 1.e-13)
# -> advantange: expectation_value etc. of MPS are available!
print("<Sz> =", psi_ED_mps.expectation_value('Sz'))