DMRG on MPO built using SparseOperatorBuilder

[Somasundaram-Rahul]

Hi all,

I am working through the documentation here:
https://quimb.readthedocs.io/en/latest/operator/operator-basics.html#jordan-wigner-transformation

Once the exact energy is computed from the sparse matrix (as demonstrated in the documentation), I attempted to compute the energy using DMRG, but this doesn't match with the exact one.

I am quite new to quimb and to tensor networks in general, so any help would be greatly appreciated!
Here is the full code for reproducibility:

import quimb as qu
import quimb.tensor as qtn
import quimb.experimental.operatorbuilder as qop

Lx = 4
Ly = 3

sites = [
    (s, i, j)
    for s in ("u", "d")
    for i in range(Lx)
    for j in range(Ly)
]
nsites = len(sites)

symmetry = "U1U1"
sector = (nsites // 2, nsites // 4), (nsites // 2, nsites // 4)

hilbert_space = qop.HilbertSpace(
    sites=sites,
    symmetry=symmetry,
    sector=sector,
)

H = qop.SparseOperatorBuilder(hilbert_space=hilbert_space)

t = 1.0
U = 8.0

for i in range(Lx):
    for j in range(Ly):
        for s in ("u", "d"):
            # hopping
            if i < Lx - 1:
                # the relative ordering within the term now matters too
                H.add_term(-t, ("+", (s, i, j)), ("-", (s, i + 1, j)))
                H.add_term(-t, ("+", (s, i + 1 , j)), ("-", (s, i, j)))
            if j < Ly - 1:
                H.add_term(-t, ("+", (s, i, j)), ("-", (s, i, j + 1)))
                H.add_term(-t, ("+", (s, i, j + 1)), ("-", (s, i, j)))

        # interaction
        H.add_term(U, ("n", ("u", i, j)), ("n", ("d", i, j)))


H.jordan_wigner_transform()

H_sparse = H.build_sparse_matrix()

energy, psi = qu.eigh(H_sparse, k=1)

print ("Exact Energy: ",energy[0])

H_mpo = H.build_mpo()

dmrg = qtn.DMRG2(H_mpo)
dmrg.solve(verbosity=1)

print ("DMRG Energy: ",dmrg.energy)

[jcmgray]

Hi @Somasundaram-Rahul, yes this is because the MPO and DMRG routine doesn't take into account the U1U1 symmetry yet, so it is just finding a groundstate of a different sector. It would be good to add a warning to the build_mpo method about this. Practically, for the fermi-hubbard model you can add a chemical potential of U / 2 to make the half-filling sector the lowest energy, then they should match.