Heisenberg Chain

Purpose and structure

The anisotropic nearest-neighbor Heisenberg chain supports independent $XX$, $YY$, and $ZZ$ couplings and a uniform longitudinal field. It is useful for spin correlations, symmetry studies, and exact-diagonalization benchmarks.

$$ H=\sum_i(J_xX_iX_{i+1}+J_yY_iY_{i+1}+J_zZ_iZ_{i+1}) +g\sum_iZ_i. $$

The package uses Pauli products directly and a positive sign for field.

Basis and scaling

The full computational basis has dimension $2^N$. Dense and CSR sparse builders are available. When $J_x=J_y$, total Pauli-$Z$ magnetization $M=\sum_i Z_i$ is conserved and the fixed-sector dimension is

$$ \binom{N}{(N-M)/2}. $$

Sector basis states retain their full computational-basis integer indices.

Package use

from quantum_lattice_models import heisenberg_chain, heisenberg_chain_sector

H = heisenberg_chain(n_sites=5, jx=1.0, jy=0.8, jz=1.2, field=0.1)
sector = heisenberg_chain_sector(
    n_sites=6,
    magnetization=0,
    jx=1.0,
    jy=1.0,
    jz=1.2,
)
H_sector = sector.matrix

Parameters

Validation and cautions

Hermiticity and real spectra are tested. The isotropic limit is $J_x=J_y=J_z$. Fixed-magnetization construction rejects $J_x\ne J_y$ because that anisotropy does not conserve total $Z$ magnetization. Reduced matrices are tested against the corresponding full-space blocks.

Related: XXZ chain, Heisenberg ladder.