Free Fermion¶
Ground-state DMRG for a spinless tight-binding chain.
\[
H = -t \sum_{\langle i,j \rangle} (c^\dagger_i c_j + \text{h.c.}) - \mu \sum_i n_i, \quad t = 1, \; \mu = 0
\]
At half-filling (ยต = 0), the exact ground-state energy for a chain of length L with OBC is
\(E_0 = -t \sum_{k=1}^{L/2} 2\cos\bigl(\pi k / (L+1)\bigr)\).
1. Setup¶
import alice
alice.configure_logging()
from alice import build_interaction, build_hamiltonian, MPS, dmrg
2. Build the Hamiltonian¶
config = {
"geometry": {"lattice": "chain", "lx": 20, "bcx": "OBC", "n2x": True},
"model": {"category": "fermionic", "label": "FreeFermion",
"symmetry": "U1", "t": 1.0, "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}")
3. Build an initial MPS¶
For a half-filled chain with U(1) symmetry, alternate occupied (charge 1) and empty (charge 0) sites:
from nicole import isometry, Direction
from nicole.index import Index, Sector
ndigits = max(2, len(str(L)))
bond_0 = Index([Sector(0, 1)], Direction.OUT)
tensors = []
for i in range(L):
charge = 1 if i % 2 == 0 else 0 # half-filling product state
phys = Index([Sector(charge, 1)], Direction.IN)
t = isometry(bond_0.flip(), phys)
t.insert_index(1, direction=Direction.OUT, itag=f'A{i+1:0{ndigits}d}')
t.retag(0, f'A{i:0{ndigits}d}')
t.retag(2, f's{i:02d}')
tensors.append(t)
mps = MPS(tensors)
4. Run DMRG¶
opts = dmrg.Options(
scheme = '2s',
n_sweeps = 10,
max_bond = 32,
e_tol = 1e-10,
)
summary = dmrg.run(mps, hamiltonian, opts)
import numpy as np
exact = -2 * sum(np.cos(np.pi * k / (L + 1)) for k in range(1, L // 2 + 1))
print(f"DMRG energy: {summary.energy:.10f}")
print(f"Exact energy: {exact:.10f}")
print(f"Error: {abs(summary.energy - exact):.2e}")
The error should be below 1e-10 for max_bond = 32 on a free-fermion chain of length 20.