XXZ

The spin-1/2 XXZ chain is available from the common Polfed.Models namespace through xxz_hamiltonian.

using Polfed.Models: xxz_hamiltonian

H = xxz_hamiltonian(L, Lup, J, Delta, W; kwargs...)

For the full Julia constructor signature, argument types, keyword meanings, and return types, see Models, in particular xxz_hamiltonian.

Model Definition

The Hamiltonian is

\[H = J \sum_{i=1}^{L} \left( S_i^x S_{i+1}^x + S_i^y S_{i+1}^y + \Delta S_i^z S_{i+1}^z \right) + \sum_{i=1}^{L} h_i S_i^z.\]

The constructor is fixed to spin-1/2 chains and works directly in a chosen magnetization sector:

  • L is the chain length,
  • Lup is the number of spin-up sites,
  • the total magnetization is $S_z = Lup - L/2$.

For even L, the zero-magnetization sector is therefore given by Lup = L ÷ 2.

Disorder and boundary conventions follow the common Polfed.Models API:

  • W is the random-field disorder strength,
  • if fields is not provided, then h_i are sampled uniformly from [field - W, field + W],
  • boundary=:periodic wraps bonds around the chain,
  • boundary=:open omits boundary-crossing bonds.

Parameters

  • L: chain length.
  • Lup: number of spin-up sites in the fixed spin-1/2 sector.
  • J: nearest-neighbor exchange scale.
  • Delta: Ising anisotropy $\Delta$.
  • W: disorder width for longitudinal fields.
  • boundary: either :periodic or :open.
  • field: center of the disorder window.
  • fields: explicit field realization. When passed, it is used directly.
  • rng: random-number generator used when disorder is sampled internally.
  • use_sparse: if true, return a sparse matrix; otherwise return a dense matrix.

Basic Example

using Polfed
using Polfed.Models: xxz_hamiltonian
using LinearAlgebra
using Random

rng = MersenneTwister(1234)

# constructing the Hamiltonian
L = 16
Lup = L ÷ 2
J = 1.0
Delta = 1.0
W = 0.5

H = xxz_hamiltonian(
    L,
    Lup,
    J,
    Delta,
    W;
    boundary=:periodic,
    rng=rng,
    use_sparse=true,
)

# generating a random initial state
x0 = rand(rng, size(H, 1))
x0 ./= norm(x0)

# setting POLFED parameters
howmany = 40
target = :middle

vals, vecs = polfed(H, x0, howmany, target)

Relation to Advanced Tutorials

The advanced tutorial pages build on this same model and show how to exploit its structure for faster mappings and GPU workflows:

For the detailed Julia constructor reference, see Models.

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