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Hubbard Model

Finite-temperature thermodynamics via XTRG for the 1D Hubbard chain with Z(2)×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 U = 0 the spin-up and spin-down channels decouple, and each channel is identical to a spinless free-fermion chain at the same chemical potential, so

\[ \log Z_{\text{spinful}}(\beta) = 2 \times \log Z_{\text{spinless}}(\beta) \]

giving an exact reference to validate against. For U ≠ 0 there is no closed-form log Z, so only the U=0 point can be checked this way — see examples/examples_xtrg/xtrg_spinful.py for a runnable script that also reports thermodynamics at finite U.

1. Setup

import alice
alice.configure_logging()  # optional: write diagnostics to .logging/

from alice.network import build_hamiltonian, build_interaction
from alice.network.thermal import thermal_mpo
from alice.algorithm import xtrg

2. Build the Hamiltonian

Z(2)×SU(2) conserves fermion parity and total spin, giving the best compression ratio for this model:

config = {
    "geometry": {"lattice": "chain", "lx": 6, "bcx": "OBC", "n2x": True},
    "model": {"category": "conductor", "label": "Hubbard",
              "symmetry": "Z2,SU2", "t": 1.0, "U": 0.0, "mu": 0.0},
}
interactions, spc, geo = build_interaction(config)
hamiltonian = build_hamiltonian(interactions, geo.L, spc)
print(f"L = {geo.L}, MPO bond dims: {hamiltonian.bond_dims}")

3. Run XTRG

opts = xtrg.Options(
    scheme       = '2s',
    tau_0        = 2 ** -12,
    n_steps      = 10,
    taylor_order = 10,
    max_bond     = 32,
    n_sweeps     = 4,
)

rho0 = thermal_mpo(hamiltonian, opts.tau_0, opts.taylor_order, spc)
summary, artifact = xtrg.run(xtrg.Artifact(rho=rho0, beta=opts.tau_0, step=0), opts)

4. Compare with the exact U=0 solution

import math

def exact_log_z_spinless(L, t, mu, beta):
    eps = [-2.0 * t * math.cos(k * math.pi / (L + 1)) for k in range(1, L + 1)]
    return sum(math.log1p(math.exp(-beta * (e - mu))) for e in eps)

def exact_log_z_spinful(L, t, mu, beta):
    return 2.0 * exact_log_z_spinless(L, t, mu, beta)

for beta, lz in zip(summary.betas, summary.log_z):
    lz_exact = exact_log_z_spinful(geo.L, 1.0, 0.0, beta)
    rel_err = abs(lz - lz_exact) / abs(lz_exact)
    print(f"β = {beta:8.4f},  log Z = {lz:12.8f},  exact = {lz_exact:12.8f},  rel err = {rel_err:.2e}")

5. Interacting case (U ≠ 0)

config["model"]["U"] = 4.0
interactions, spc, geo = build_interaction(config)
hamiltonian = build_hamiltonian(interactions, geo.L, spc)

rho0_u4 = thermal_mpo(hamiltonian, opts.tau_0, opts.taylor_order, spc)
summary_u4, artifact_u4 = xtrg.run(xtrg.Artifact(rho=rho0_u4, beta=opts.tau_0, step=0), opts)
# No closed-form log Z reference exists for U != 0; compare thermodynamics
# (free energy, internal energy, entropy) against the U=0 point for context.

See Also