XXZ Chain

Purpose and Hamiltonian

The XXZ chain is the $J_x=J_y$ specialization of the anisotropic Heisenberg chain:

$$ H=J\sum_i(X_iX_{i+1}+Y_iY_{i+1}+\Delta Z_iZ_{i+1}) +g\sum_iZ_i. $$

It is useful for anisotropy, magnetization, gap, and conserved-$S^z$ benchmarks.

Basis and use

The full computational-basis matrix has dimension $2^N$. Dense and CSR sparse builders are available. A fixed total Pauli-$Z$ magnetization sector $M=\sum_i Z_i$ has dimension $\binom{N}{(N-M)/2}$.

from quantum_lattice_models import xxz_chain, xxz_chain_sector

H = xxz_chain(n_sites=6, coupling=1.0, anisotropy=0.7)
sector = xxz_chain_sector(n_sites=10, magnetization=0, anisotropy=0.7)
H_sector = sector.matrix

Parameters

Builder	Parameter	Type	Default	Constraint
`xxz_chain`	`n_sites`	`int`	`4`	>= 1
`xxz_chain`	`coupling`	`float`	`1.0`
`xxz_chain`	`anisotropy`	`float`	`0.7`
`xxz_chain`	`field`	`float`	`0.0`
`xxz_chain`	`periodic`	`bool`	`False`
`xxz_chain_sparse`	`n_sites`	`int`	`4`	>= 1
`xxz_chain_sparse`	`coupling`	`float`	`1.0`
`xxz_chain_sparse`	`anisotropy`	`float`	`0.7`
`xxz_chain_sparse`	`field`	`float`	`0.0`
`xxz_chain_sparse`	`periodic`	`bool`	`False`
`xxz_chain_sector_sparse`	`n_sites`	`int`	`6`	>= 1
`xxz_chain_sector_sparse`	`magnetization`	`int`	`0`
`xxz_chain_sector_sparse`	`coupling`	`float`	`1.0`
`xxz_chain_sector_sparse`	`anisotropy`	`float`	`1.0`
`xxz_chain_sector_sparse`	`field`	`float`	`0.0`
`xxz_chain_sector_sparse`	`periodic`	`bool`	`False`

User notes

xxz_chain delegates to heisenberg_chain; its field term therefore has a positive sign. magnetization is the total Pauli-$Z$ eigenvalue, so it must have the same parity as n_sites and satisfy $|M|\le N$. Sector basis mappings support projection from and embedding into the full computational basis.

Related: Heisenberg chain, XY chain.