Ising Chains
Purpose and structure
These spin-$\tfrac12$ chains combine Ising $ZZ$ interactions with transverse and optional longitudinal fields. The next-nearest-neighbor variant adds frustration. They are standard small-system benchmarks for phase-transition intuition, quantum simulation, and variational algorithms.
Hamiltonians
$$ H_{\rm TFIM}=-J\sum_i Z_iZ_{i+1}-h\sum_iX_i, $$
$$ H_{\rm long}=-J\sum_iZ_iZ_{i+1}-h_x\sum_iX_i-h_z\sum_iZ_i, $$
$$ H_{\rm NNN}=-J_1\sum_iZ_iZ_{i+1}-J_2\sum_iZ_iZ_{i+2}-h\sum_iX_i. $$
The operators are Pauli matrices, not spin operators divided by two.
Basis and scaling
The computational basis has dimension $2^N$. Builders currently return dense
DenseHamiltonian arrays with Pauli-term metadata.
Package use
from quantum_lattice_models import transverse_field_ising
H = transverse_field_ising(n_sites=6, j=1.0, h=0.7, periodic=False)
quantum-lattice create transverse_field_ising --n-sites 6 --j 1 --h 0.7 --output ising.json
Parameters
| Builder | Parameter | Type | Default | Constraint |
|---|---|---|---|---|
transverse_field_ising |
n_sites |
int |
4 |
>= 1 |
transverse_field_ising |
j |
float |
1.0 |
|
transverse_field_ising |
h |
float |
0.5 |
|
transverse_field_ising |
periodic |
bool |
False |
|
longitudinal_field_ising |
n_sites |
int |
4 |
>= 1 |
longitudinal_field_ising |
j |
float |
1.0 |
|
longitudinal_field_ising |
h_x |
float |
0.5 |
|
longitudinal_field_ising |
h_z |
float |
0.1 |
|
longitudinal_field_ising |
periodic |
bool |
False |
|
next_nearest_neighbor_ising |
n_sites |
int |
5 |
>= 1 |
next_nearest_neighbor_ising |
j1 |
float |
1.0 |
|
next_nearest_neighbor_ising |
j2 |
float |
0.25 |
|
next_nearest_neighbor_ising |
h |
float |
0.5 |
|
next_nearest_neighbor_ising |
periodic |
bool |
False |
Validation and cautions
The zero-field three-site spectrum is checked analytically. Dense memory grows as $4^N$; inspect the estimated dimension and memory before increasing $N$.