def sigmay(hilbert: _AbstractHilbert,
site: int,
dtype: _DType = complex) -> _LocalOperator:
"""
Builds the :math:`\\sigma^y` operator acting on the `site`-th of the Hilbert
space `hilbert`.
If `hilbert` is a non-Spin space of local dimension M, it is considered
as a (M-1)/2 - spin space.
:param hilbert: The hilbert space
:param site: the site on which this operator acts
:return: a nk.operator.LocalOperator
"""
import numpy as np
import netket.jax as nkjax
if not nkjax.is_complex_dtype(dtype):
import jax.numpy as jnp
import warnings
old_dtype = dtype
dtype = jnp.promote_types(complex, old_dtype)
warnings.warn(
np.ComplexWarning(
f"A complex dtype is required (dtype={old_dtype} specified). "
f"Promoting to dtype={dtype}."))
N = hilbert.size_at_index(site)
S = (N - 1) / 2
D = np.array(
[1j * np.sqrt((S + 1) * 2 * a - a * (a + 1)) for a in np.arange(1, N)])
mat = np.diag(D, -1) + np.diag(-D, 1)
return _LocalOperator(hilbert, mat, [site], dtype=dtype)
def proj(hilbert: AbstractHilbert,
site: int,
n: int,
dtype: DType = float) -> _LocalOperator:
"""
Builds the projector operator :math:`|n\\rangle\\langle n |` acting on the `site`-th of the
Hilbert space `hilbert` and collapsing on the state with `n` bosons.
If `hilbert` is a non-Bosonic space of local dimension M, it is considered
as a bosonic space of local dimension M.
Args:
hilbert: The hilbert space
site: the site on which this operator acts
n: the state on which to project
Returns:
the resulting operator
"""
import numpy as np
N = hilbert.size_at_index(site)
if n >= N:
raise ValueError("Cannot project on a state above the cutoff.")
D = np.array([0 for m in np.arange(0, N)])
D[n] = 1
mat = np.diag(D, 0)
return _LocalOperator(hilbert, mat, [site], dtype=dtype)
def sigmaz(hilbert: _AbstractHilbert,
site: int,
dtype: _DType = float) -> _LocalOperator:
"""
Builds the :math:`\\sigma^z` operator acting on the `site`-th of the Hilbert
space `hilbert`.
If `hilbert` is a non-Spin space of local dimension M, it is considered
as a (M-1)/2 - spin space.
:param hilbert: The hilbert space
:param site: the site on which this operator acts
:return: a nk.operator.LocalOperator
"""
import numpy as np
N = hilbert.size_at_index(site)
S = (N - 1) / 2
D = np.array([2 * m for m in np.arange(S, -(S + 1), -1)])
mat = np.diag(D, 0)
return _LocalOperator(hilbert, mat, [site], dtype=dtype)
def sigmax(hilbert: AbstractHilbert,
site: int,
dtype: DType = float) -> _LocalOperator:
"""
Builds the :math:`\\sigma^x` operator acting on the `site`-th of the Hilbert
space `hilbert`.
If `hilbert` is a non-Spin space of local dimension M, it is considered
as a (M-1)/2 - spin space.
:param hilbert: The hilbert space
:param site: the site on which this operator acts
:return: a nk.operator.LocalOperator
"""
import numpy as np
N = hilbert.size_at_index(site)
S = (N - 1) / 2
D = [np.sqrt((S + 1) * 2 * a - a * (a + 1)) for a in np.arange(1, N)]
mat = np.diag(D, 1) + np.diag(D, -1)
return _LocalOperator(hilbert, mat, [site], dtype=dtype)
def number(hilbert: AbstractHilbert, site: int, dtype: DType = float) -> _LocalOperator:
"""
Builds the number operator :math:`\\hat{a}^\\dagger\\hat{a}` acting on the
`site`-th of the Hilbert space `hilbert`.
If `hilbert` is a non-Bosonic space of local dimension M, it is considered
as a bosonic space of local dimension M.
Args:
hilbert: The hilbert space
site: the site on which this operator acts
Returns:
The resulting Local Operator
"""
import numpy as np
N = hilbert.size_at_index(site)
D = np.array([m for m in np.arange(0, N)])
mat = np.diag(D, 0)
return _LocalOperator(hilbert, mat, [site], dtype=dtype)
def sigmam(hilbert: AbstractHilbert,
site: int,
dtype: DType = float) -> _LocalOperator:
"""
Builds the :math:`\\sigma^{-} = \\frac{1}{2}(\\sigma^x - i \\sigma^y)` operator acting on the
`site`-th of the Hilbert space `hilbert`.
If `hilbert` is a non-Spin space of local dimension M, it is considered
as a (M-1)/2 - spin space.
:param hilbert: The hilbert space
:param site: the site on which this operator acts
:return: a nk.operator.LocalOperator
"""
import numpy as np
N = hilbert.size_at_index(site)
S = (N - 1) / 2
S2 = (S + 1) * S
D = np.array([np.sqrt(S2 - m * (m - 1)) for m in np.arange(S, -S, -1)])
mat = np.diag(D, -1)
return _LocalOperator(hilbert, mat, [site], dtype=dtype)
def pack_internals(
hilbert: AbstractHilbert,
operators_dict: dict,
constant,
dtype: DType,
mel_cutoff: float,
):
"""
Take the internal lazy representation of a local operator and returns the arrays
needed for the numba implementation.
This takes as input a dictionary with Tuples as keys, the `acting_on` and matrices as values.
The keys represent the sites upon which the matrix acts.
It is assumed that the integer in the tuples are sorted.
Returns a dictionary with all the data fields
"""
op_acting_on = list(operators_dict.keys())
operators = list(operators_dict.values())
n_operators = len(operators_dict)
"""Analyze the operator strings and precompute arrays for get_conn inference"""
acting_size = np.array([len(aon) for aon in op_acting_on], dtype=np.intp)
max_acting_on_sz = np.max(acting_size)
max_local_hilbert_size = max(
[max(map(hilbert.size_at_index, aon)) for aon in op_acting_on])
max_op_size = max(map(lambda x: x.shape[0], operators))
acting_on = np.full((n_operators, max_acting_on_sz), -1, dtype=np.intp)
for (i, aon) in enumerate(op_acting_on):
acting_on[i][:len(aon)] = aon
local_states = np.full(
(n_operators, max_acting_on_sz, max_local_hilbert_size), np.nan)
basis = np.full((n_operators, max_acting_on_sz), 1e10, dtype=np.int64)
diag_mels = np.full((n_operators, max_op_size), np.nan, dtype=dtype)
mels = np.full(
(n_operators, max_op_size, max_op_size - 1),
np.nan,
dtype=dtype,
)
x_prime = np.full(
(n_operators, max_op_size, max_op_size - 1, max_acting_on_sz),
-1,
dtype=np.float64,
)
n_conns = np.full((n_operators, max_op_size), -1, dtype=np.intp)
for (i, (aon, op)) in enumerate(operators_dict.items()):
aon_size = len(aon)
n_local_states_per_site = np.asarray(
[hilbert.size_at_index(i) for i in aon])
## add an operator to local_states
for (j, site) in enumerate(aon):
local_states[i, j, :hilbert.shape[site]] = np.asarray(
hilbert.states_at_index(site))
ba = 1
for s in range(aon_size):
basis[i, s] = ba
ba *= hilbert.shape[aon_size - s - 1]
# eventually could support sparse matrices
# if isinstance(op, sparse.spmatrix):
# op = op.todense()
_append_matrix(
op,
diag_mels[i],
mels[i],
x_prime[i],
n_conns[i],
aon_size,
local_states[i],
mel_cutoff,
n_local_states_per_site,
)
nonzero_diagonal = (np.any(np.abs(diag_mels) >= mel_cutoff)
or np.abs(constant) >= mel_cutoff)
max_conn_size = 1 if nonzero_diagonal else 0
for op in operators:
nnz_mat = np.abs(op) > mel_cutoff
nnz_mat[np.diag_indices(nnz_mat.shape[0])] = False
nnz_rows = np.sum(nnz_mat, axis=1)
max_conn_size += np.max(nnz_rows)
return {
"acting_on": acting_on,
"acting_size": acting_size,
"diag_mels": diag_mels,
"mels": mels,
"x_prime": x_prime,
"n_conns": n_conns,
"local_states": local_states,
"basis": basis,
"nonzero_diagonal": nonzero_diagonal,
"max_conn_size": max_conn_size,
}