iDMRG

Infinite DMRG (iDMRG) finds the ground-state energy per site of a translationally invariant Hamiltonian directly in the thermodynamic limit.

Background

iDMRG works by repeatedly inserting two new sites into the centre of a growing chain and optimising them with a Lanczos eigensolver. After each growth step the centre bond is truncated via SVD to a maximum bond dimension \(\chi\). The energy-per-site estimate converges as the effective environment builds up around the unit cell.

Key properties of the Tenax implementation:

• 2-site growth algorithm: two sites are added per iteration.

• The bulk Hamiltonian is specified as a single repeated MPO tensor \(W[w_l, \text{ket}, \text{bra}, w_r]\).

• Outer growth loop is a Python for-loop; the Lanczos matvec is @jax.jit compiled.

• Built-in MPO constructors for the Heisenberg chain and cylinder.

Configuration

from tenax import iDMRGConfig

config = iDMRGConfig(
    max_bond_dim=100,       # maximum MPS bond dimension (chi)
    max_iterations=200,     # maximum number of growth steps
    convergence_tol=1e-8,   # stop when |dE/site| < tol
    lanczos_max_iter=50,    # Lanczos iteration cap
    lanczos_tol=1e-12,      # Lanczos convergence tolerance
    svd_trunc_err=None,     # SVD truncation error (None = use max_bond_dim)
    verbose=True,
)

Parameter	Default	Description
max_bond_dim	100	Maximum MPS bond dimension \(\chi\)
max_iterations	200	Maximum 2-site growth steps
convergence_tol	1e-8	Energy-per-site convergence threshold
lanczos_max_iter	50	Maximum Lanczos iterations
lanczos_tol	1e-12	Lanczos convergence tolerance
svd_trunc_err	None	Max SVD truncation error (overrides max_bond_dim if set)
verbose	False	Print per-step diagnostics

Example – Heisenberg chain

from tenax import iDMRGConfig, idmrg, build_bulk_mpo_heisenberg

# Build the bulk W-tensor for the spin-1/2 XXZ Heisenberg model
bulk_mpo = build_bulk_mpo_heisenberg(Jz=1.0, Jxy=1.0, hz=0.0)

config = iDMRGConfig(max_bond_dim=64, max_iterations=200, verbose=True)
result = idmrg(bulk_mpo, config)

print(f"Energy/site: {result.energy_per_site:.10f}")
print(f"Converged:   {result.converged}")
print(f"Steps:       {len(result.energies_per_step)}")
# Reference: E/site ≈ 0.25 - ln(2) ≈ −0.4431471805

build_bulk_mpo_heisenberg returns a DenseTensor with legs ("w_l", "mpo_top", "mpo_bot", "w_r") and shape (5, 2, 2, 5).

Example – Infinite cylinder

build_bulk_mpo_heisenberg_cylinder constructs a bulk MPO for a Heisenberg model on an infinite cylinder of circumference \(L_y\). Each “super-site” encodes an entire ring of \(L_y\) spins (physical dimension \(d = 2^{L_y}\)).

from tenax import iDMRGConfig, idmrg, build_bulk_mpo_heisenberg_cylinder

Ly = 4
bulk_mpo = build_bulk_mpo_heisenberg_cylinder(Ly=Ly, J=1.0)

config = iDMRGConfig(max_bond_dim=200, max_iterations=300)
result = idmrg(bulk_mpo, config)

print(f"E/site (Ly={Ly}): {result.energy_per_site / Ly:.8f}")

Note

build_bulk_mpo_heisenberg_cylinder only accepts even \(L_y\). Odd circumference frustrates antiferromagnetic order on the square lattice.

Result object

idmrg() returns an iDMRGResult named tuple:

Field	Type	Description
energy_per_site	float	Converged energy per site
energies_per_step	list[float]	Energy-per-site estimate at each iteration
mps_tensors	list[Tensor]	2-site unit cell [A_L, A_R]
singular_values	jax.Array	Singular values on the centre bond
converged	bool	Whether the run converged within tolerance

MPO construction

build_bulk_mpo_heisenberg builds a single bulk \(W\)-matrix (shape \(5 \times d \times d \times 5\)) for the spin-1/2 XXZ Heisenberg model:

\[H = J_z \sum_i S^z_i S^z_{i+1} + \frac{J_{xy}}{2} \sum_i (S^+_i S^-_{i+1} + S^-_i S^+_{i+1}) + h_z \sum_i S^z_i\]

For finite chains, use build_mpo_heisenberg() instead (see DMRG). For custom Hamiltonians, see AutoMPO.