Hubbard Model¶

Ground-state DMRG for the 1D Hubbard model with U(1)×SU(2) symmetry.

\[ H = -t \sum_{\langle i,j \rangle, \sigma} (c^\dagger_{i\sigma} c_{j\sigma} + \text{h.c.}) + U \sum_i n_{i\uparrow} n_{i\downarrow} \]

At half-filling (µ = 0) the Hubbard model is a Mott insulator for any U > 0. Its ground-state energy is known exactly via the Bethe ansatz.

1. Setup¶

import alice
alice.configure_logging()

from alice import build_interaction, build_hamiltonian, MPS, dmrg

2. Build the Hamiltonian¶

U(1)×SU(2) symmetry conserves total particle number and total spin simultaneously, achieving the best compression ratio:

config = {
    "geometry": {"lattice": "chain", "lx": 8, "bcx": "OBC", "n2x": True},
    "model": {"category": "conductor", "label": "Hubbard",
              "symmetry": "U1,SU2", "t": 1.0, "U": 4.0, "mu": 0.0},
}
interactions, spc, L = build_interaction(config)
hamiltonian = build_hamiltonian(interactions, L, spc)
print(f"L = {L}, MPO bond dims: {hamiltonian.bond_dims}")

Alternatively, use U(1)×U(1) to conserve spin-up and spin-down counts separately:

config["model"]["symmetry"] = "U1,U1"

3. Build an initial MPS¶

For U(1)×SU(2), the product-state initialization targets the half-filled singlet sector (total particle number L, total spin S=0). Building the initial MPS correctly for non-Abelian symmetry is non-trivial; it is typically done via iterative diagonalization (see examples/examples_dmrg/dmrg_heisenberg.py for a reference implementation).

A simpler approach is to start from a 2-site DMRG sweep with a small random MPS:

# For a demonstration, use the 2-site scheme to grow bonds from a product state.
opts = dmrg.Options(
    scheme   = '2s',
    n_sweeps = 30,
    max_bond = 128,
    e_tol    = 1e-8,
)

4. Run DMRG¶

summary = dmrg.run(mps, hamiltonian, opts)

print(f"Energy:        {summary.energy:.10f}")
print(f"Energy/site:   {summary.energy / L:.10f}")
print(f"Converged:     {summary.converged}")
print(f"Max bond dim:  {max(summary.bond_dims)}")

For L=8, U=4, t=1, OBC, the ground-state energy is approximately E ≈ -3.67.

5. Environment caching for large chains¶

For long chains where all environment blocks do not fit in RAM, enable disk caching:

opts = dmrg.Options(
    scheme        = '2s',
    n_sweeps      = 20,
    max_bond      = 256,
    env_cache_dir = "/tmp/alice_envs",   # write blocks to disk
    env_window    = 4,                    # keep 4 blocks in memory
    env_async_io  = True,                 # overlap I/O with computation
)

The env_cache_dir directory is created automatically. Each block is stored as a .pt file and re-loaded transparently as the sweep progresses.