Source code for netket.operator._hamiltonian

# Copyright 2021 The NetKet Authors - All rights reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#    http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

from typing import List, Optional, Sequence, Union
from numba import jit

import numpy as np
import jax.numpy as jnp
import math

from netket.graph import AbstractGraph, Graph
from netket.hilbert import AbstractHilbert, Fock
from netket.utils.types import DType

from . import spin, boson
from ._local_operator import LocalOperator
from ._graph_operator import GraphOperator
from ._discrete_operator import DiscreteOperator
from ._local_operator_helpers import _dtype


class SpecialHamiltonian(DiscreteOperator):
    def to_local_operator(self):
        raise NotImplementedError(
            "Must implemented to_local_operator for {}".format(type(self))
        )

    def conjugate(self, *, concrete: bool = True):
        return self.to_local_operator().conjugate(concrete=concrete)

    def __add__(self, other):
        if type(self) is type(other):
            res = self.copy()
            res += other
            return res

        return self.to_local_operator().__add__(other)

    def __sub__(self, other):
        if type(self) is type(other):
            res = self.copy()
            res -= other
            return res

        return self.to_local_operator().__sub__(other)

    def __radd__(self, other):
        if type(self) is type(other):
            res = self.copy()
            res += other
            return res

        return self.to_local_operator().__radd__(other)

    def __rsub__(self, other):
        if type(self) is type(other):
            res = self.copy()
            res -= other
            return res

        return self.to_local_operator().__rsub__(other)

    def __iadd__(self, other):
        if type(self) is type(other):
            self._iadd_same_hamiltonian(other)
            return self

        return NotImplemented

    def __isub__(self, other):
        if type(self) is type(other):
            self._isub_same_hamiltonian(other)
            return self

        return NotImplemented

    def __mul__(self, other):
        return self.to_local_operator().__mul__(other)

    def __rmul__(self, other):
        return self.to_local_operator().__rmul__(other)

    def _op__matmul__(self, other):
        if hasattr(other, "to_local_operator"):
            other = other.to_local_operator()
        return self.to_local_operator().__matmul__(other)

    def _op__rmatmul__(self, other):
        if hasattr(other, "to_local_operator"):
            other = other.to_local_operator()

        return self.to_local_operator().__matmul__(other)


class Ising(SpecialHamiltonian):
    r"""
    The Transverse-Field Ising Hamiltonian :math:`-h\sum_i \sigma_i^{(x)} +J\sum_{\langle i,j\rangle} \sigma_i^{(z)}\sigma_j^{(z)}`.

    This implementation is considerably faster than the Ising hamiltonian constructed by summing :class:`~netket.operator.LocalOperator` s.
    """

[docs] def __init__( self, hilbert: AbstractHilbert, graph: AbstractGraph, h: float, J: float = 1.0, dtype: DType = None, ): r""" Constructs the Ising Operator from an hilbert space and a graph specifying the connectivity. Args: hilbert: Hilbert space the operator acts on. h: The strength of the transverse field. J: The strength of the coupling. Default is 1.0. dtype: The dtype of the matrix elements. Examples: Constructs an ``Ising`` operator for a 1D system. >>> import netket as nk >>> g = nk.graph.Hypercube(length=20, n_dim=1, pbc=True) >>> hi = nk.hilbert.Spin(s=0.5, N=g.n_nodes) >>> op = nk.operator.Ising(h=1.321, hilbert=hi, J=0.5, graph=g) >>> print(op) Ising(J=0.5, h=1.321; dim=20) """ assert ( graph.n_nodes == hilbert.size ), "The size of the graph must match the hilbert space" super().__init__(hilbert) if dtype is None: dtype = jnp.promote_types(_dtype(h), _dtype(J)) dtype = np.empty((), dtype=dtype).dtype self._dtype = dtype self._h = np.array(h, dtype=dtype) self._J = np.array(J, dtype=dtype) self._edges = np.asarray( [[u, v] for u, v in graph.edges()], dtype=np.intp, )
@property def h(self) -> float: """The magnitude of the transverse field""" return self._h @property def J(self) -> float: """The magnitude of the hopping""" return self._J @property def edges(self) -> np.ndarray: return self._edges @property def is_hermitian(self) -> bool: return True @property def dtype(self) -> DType: return self._dtype
[docs] def conjugate(self, *, concrete=True): # if real if isinstance(self.h, float) and isinstance(self.J, float): return self else: raise NotImplementedError
[docs] @staticmethod @jit(nopython=True) def n_conn(x, out): r"""Return the number of states connected to x. Args: x (matrix): A matrix of shape (batch_size,hilbert.size) containing the batch of quantum numbers x. out (array): If None an output array is allocated. Returns: array: The number of connected states x' for each x[i]. """ if out is None: out = np.empty( x.shape[0], dtype=np.int32, ) out.fill(x.shape[1] + 1) return out
@property def max_conn_size(self) -> int: """The maximum number of non zero ⟨x|O|x'⟩ for every x.""" return self.hilbert.size + 1
[docs] def copy(self, *, dtype: Optional[DType] = None): if dtype is None: dtype = self.dtype graph = Graph(edges=[list(edge) for edge in self.edges]) return Ising(hilbert=self.hilbert, graph=graph, J=self.J, h=self.h, dtype=dtype)
[docs] def to_local_operator(self): # The hamiltonian ha = LocalOperator(self.hilbert, dtype=self.dtype) if self.h != 0: for i in range(self.hilbert.size): ha -= self.h * spin.sigmax(self.hilbert, i) if self.J != 0: for (i, j) in self.edges: ha += self.J * ( spin.sigmaz(self.hilbert, i) * spin.sigmaz(self.hilbert, j) ) return ha
def _iadd_same_hamiltonian(self, other): if self.hilbert != other.hilbert: raise NotImplementedError( "Cannot add hamiltonians on different hilbert spaces" ) self._h += other.h self._J += other.J def _isub_same_hamiltonian(self, other): if self.hilbert != other.hilbert: raise NotImplementedError( "Cannot add hamiltonians on different hilbert spaces" ) self._h -= other.h self._J -= other.J @staticmethod @jit(nopython=True) def _flattened_kernel( x, sections, edges, h, J, ): n_sites = x.shape[1] n_conn = n_sites + 1 x_prime = np.empty( ( x.shape[0] * n_conn, n_sites, ), dtype=x.dtype, ) mels = np.empty(x.shape[0] * n_conn, dtype=h.dtype) diag_ind = 0 for i in range(x.shape[0]): mels[diag_ind] = 0.0 for k in range(edges.shape[0]): mels[diag_ind] += J * x[i, edges[k, 0]] * x[i, edges[k, 1]] odiag_ind = 1 + diag_ind mels[odiag_ind : (odiag_ind + n_sites)].fill(-h) x_prime[diag_ind : (diag_ind + n_conn)] = np.copy(x[i]) for j in range(n_sites): x_prime[j + odiag_ind][j] *= -1.0 diag_ind += n_conn sections[i] = diag_ind return x_prime, mels
[docs] def get_conn_flattened( self, x, sections, pad=False, ): r"""Finds the connected elements of the Operator. Starting from a given quantum number x, it finds all other quantum numbers x' such that the matrix element :math:`O(x,x')` is different from zero. In general there will be several different connected states x' satisfying this condition, and they are denoted here :math:`x'(k)`, for :math:`k=0,1...N_{\mathrm{connected}}`. This is a batched version, where x is a matrix of shape (batch_size,hilbert.size). Args: x (matrix): A matrix of shape (batch_size,hilbert.size) containing the batch of quantum numbers x. sections (array): An array of size (batch_size) useful to unflatten the output of this function. See numpy.split for the meaning of sections. pad (bool): no effect here Returns: matrix: The connected states x', flattened together in a single matrix. array: An array containing the matrix elements :math:`O(x,x')` associated to each x'. """ return self._flattened_kernel( np.asarray(x), sections, self._edges, self._h, self._J, )
def _get_conn_flattened_closure(self): _edges = self._edges _h = self._h _J = self._J fun = self._flattened_kernel def gccf_fun(x, sections): return fun(x, sections, _edges, _h, _J) return jit(nopython=True)(gccf_fun) def __repr__(self): return f"Ising(J={self._J}, h={self._h}; dim={self.hilbert.size})" class Heisenberg(GraphOperator): r""" The Heisenberg hamiltonian on a lattice. """
[docs] def __init__( self, hilbert: AbstractHilbert, graph: AbstractGraph, J: Union[float, Sequence[float]] = 1.0, sign_rule=None, *, acting_on_subspace: Union[List[int], int] = None, ): """ Constructs an Heisenberg operator given a hilbert space and a graph providing the connectivity of the lattice. Args: hilbert: Hilbert space the operator acts on. graph: The graph upon which this hamiltonian is defined. J: The strength of the coupling. Default is 1. Can pass a sequence of coupling strengths with coloured graphs: edges of colour n will have coupling strength J[n] sign_rule: If True, Marshal's sign rule will be used. On a bipartite lattice, this corresponds to a basis change flipping the Sz direction at every odd site of the lattice. For non-bipartite lattices, the sign rule cannot be applied. Defaults to True if the lattice is bipartite, False otherwise. If a sequence of coupling strengths is passed, defaults to False and a matching sequence of sign_rule must be specified to override it acting_on_subspace: Specifies the mapping between nodes of the graph and Hilbert space sites, so that graph node :code:`i ∈ [0, ..., graph.n_nodes - 1]`, corresponds to :code:`acting_on_subspace[i] ∈ [0, ..., hilbert.n_sites]`. Must be a list of length `graph.n_nodes`. Passing a single integer :code:`start` is equivalent to :code:`[start, ..., start + graph.n_nodes - 1]`. Examples: Constructs a ``Heisenberg`` operator for a 1D system. >>> import netket as nk >>> g = nk.graph.Hypercube(length=20, n_dim=1, pbc=True) >>> hi = nk.hilbert.Spin(s=0.5, total_sz=0, N=g.n_nodes) >>> op = nk.operator.Heisenberg(hilbert=hi, graph=g) >>> print(op) Heisenberg(J=1.0, sign_rule=True; dim=20) """ if isinstance(J, Sequence): # check that the number of Js matches the number of colours assert len(J) == max(graph.edge_colors) + 1 if sign_rule is None: sign_rule = [False] * len(J) else: assert len(sign_rule) == len(J) for i in range(len(J)): subgraph = Graph(edges=graph.edges(filter_color=i)) if sign_rule[i] and not subgraph.is_bipartite(): raise ValueError( "sign_rule=True specified for a non-bipartite lattice" ) else: if sign_rule is None: sign_rule = graph.is_bipartite() elif sign_rule and not graph.is_bipartite(): raise ValueError("sign_rule=True specified for a non-bipartite lattice") self._J = J self._sign_rule = sign_rule sz_sz = np.array( [ [1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1], ] ) exchange = np.array( [ [0, 0, 0, 0], [0, 0, 2, 0], [0, 2, 0, 0], [0, 0, 0, 0], ] ) if isinstance(J, Sequence): bond_ops = [ J[i] * (sz_sz - exchange if sign_rule[i] else sz_sz + exchange) for i in range(len(J)) ] bond_ops_colors = list(range(len(J))) else: bond_ops = [J * (sz_sz - exchange if sign_rule else sz_sz + exchange)] bond_ops_colors = [] super().__init__( hilbert, graph, bond_ops=bond_ops, bond_ops_colors=bond_ops_colors, acting_on_subspace=acting_on_subspace, )
@property def J(self) -> float: """The coupling strength.""" return self._J @property def uses_sign_rule(self): return self._sign_rule def __repr__(self): return f"Heisenberg(J={self._J}, sign_rule={self._sign_rule}; dim={self.hilbert.size})" class BoseHubbard(SpecialHamiltonian): r""" An extended Bose Hubbard model Hamiltonian operator, containing both on-site interactions and nearest-neighboring density-density interactions. """
[docs] def __init__( self, hilbert: AbstractHilbert, graph: AbstractGraph, U: float, V: float = 0.0, J: float = 1.0, mu: float = 0.0, dtype: DType = None, ): r""" Constructs a new BoseHubbard operator given a hilbert space, a graph specifying the connectivity and the interaction strength. The chemical potential and the density-density interaction strength can be specified as well. Args: hilbert: Hilbert space the operator acts on. U: The on-site interaction term. V: The strength of density-density interaction term. J: The hopping amplitude. mu: The chemical potential. dtype: The dtype of the matrix elements. Examples: Constructs a BoseHubbard operator for a 2D system. >>> import netket as nk >>> g = nk.graph.Hypercube(length=3, n_dim=2, pbc=True) >>> hi = nk.hilbert.Fock(n_max=3, n_particles=6, N=g.n_nodes) >>> op = nk.operator.BoseHubbard(hi, U=4.0, graph=g) >>> print(op.hilbert.size) 9 """ assert ( graph.n_nodes == hilbert.size ), "The size of the graph must match the hilbert space." assert isinstance(hilbert, Fock) super().__init__(hilbert) if dtype is None: dtype = jnp.promote_types(_dtype(U), _dtype(V)) dtype = jnp.promote_types(dtype, _dtype(J)) dtype = jnp.promote_types(dtype, _dtype(mu)) dtype = np.empty((), dtype=dtype).dtype self._dtype = dtype self._U = np.asarray(U, dtype=dtype) self._V = np.asarray(V, dtype=dtype) self._J = np.asarray(J, dtype=dtype) self._mu = np.asarray(mu, dtype=dtype) self._n_max = hilbert.n_max self._n_sites = hilbert.size self._edges = np.asarray(list(graph.edges())) self._max_conn = 1 + self._edges.shape[0] * 2 self._max_mels = np.empty(self._max_conn, dtype=self.dtype) self._max_xprime = np.empty((self._max_conn, self._n_sites))
@property def is_hermitian(self): return True @property def dtype(self): return self._dtype @property def edges(self) -> np.ndarray: return self._edges @property def U(self): """The strength of on-site interaction term.""" return self._U @property def V(self): """The strength of density-density interaction term.""" return self._V @property def J(self): """The hopping amplitude.""" return self._J @property def mu(self): """The chemical potential.""" return self._mu
[docs] def copy(self): graph = Graph(edges=[list(edge) for edge in self.edges]) return BoseHubbard( hilbert=self.hilbert, graph=graph, J=self.J, U=self.U, V=self.V, mu=self.mu, dtype=self.dtype, )
[docs] def to_local_operator(self): # The hamiltonian ha = LocalOperator(self.hilbert, dtype=self.dtype) if self.U != 0 or self.mu != 0: for i in range(self.hilbert.size): n_i = boson.number(self.hilbert, i) ha += (self.U / 2) * n_i * (n_i - 1) - self.mu * n_i if self.J != 0: for (i, j) in self.edges: ha += self.V * ( boson.number(self.hilbert, i) * boson.number(self.hilbert, j) ) ha -= self.J * ( boson.destroy(self.hilbert, i) * boson.create(self.hilbert, j) + boson.create(self.hilbert, i) * boson.destroy(self.hilbert, j) ) return ha
def _iadd_same_hamiltonian(self, other): if self.hilbert != other.hilbert: raise NotImplementedError( "Cannot add hamiltonians on different hilbert spaces" ) self._mu += other.mu self._U += other.U self._J += other.J self._V += other.V def _isub_same_hamiltonian(self, other): if self.hilbert != other.hilbert: raise NotImplementedError( "Cannot add hamiltonians on different hilbert spaces" ) self._mu -= other.mu self._U -= other.U self._J -= other.J self._V -= other.V @property def max_conn_size(self) -> int: """The maximum number of non zero ⟨x|O|x'⟩ for every x.""" # 1 diagonal element + 2 for every coupling return 1 + 2 * len(self._edges) @staticmethod @jit(nopython=True) def _flattened_kernel( x, sections, edges, U, V, J, mu, n_max, max_conn, mels=None, x_prime=None, pad=False, ): batch_size = x.shape[0] n_sites = x.shape[1] mels_allocated = False x_prime_allocated = False # When executed as a closure those must be allocated inside the numba jitted function if mels is None: mels_allocated = True mels = np.empty(batch_size * max_conn, dtype=U.dtype) if x_prime is None: x_prime_allocated = True x_prime = np.empty((batch_size * max_conn, n_sites), dtype=x.dtype) if pad: x_prime[:, :] = 0 mels[:] = 0 sqrt = math.sqrt Uh = 0.5 * U diag_ind = 0 for b in range(batch_size): mels[diag_ind] = 0.0 x_prime[diag_ind] = np.copy(x[b]) for i in range(n_sites): # chemical potential mels[diag_ind] -= mu * x[b, i] # on-site interaction mels[diag_ind] += Uh * x[b, i] * (x[b, i] - 1.0) odiag_ind = 1 + diag_ind for e in range(edges.shape[0]): i, j = edges[e][0], edges[e][1] n_i = x[b, i] n_j = x[b, j] mels[diag_ind] += V * n_i * n_j # destroy on i create on j if n_i > 0 and n_j < n_max: mels[odiag_ind] = -J * sqrt(n_i) * sqrt(n_j + 1) x_prime[odiag_ind] = np.copy(x[b]) x_prime[odiag_ind, i] -= 1.0 x_prime[odiag_ind, j] += 1.0 odiag_ind += 1 # destroy on j create on i if n_j > 0 and n_i < n_max: mels[odiag_ind] = -J * sqrt(n_j) * sqrt(n_i + 1) x_prime[odiag_ind] = np.copy(x[b]) x_prime[odiag_ind, j] -= 1.0 x_prime[odiag_ind, i] += 1.0 odiag_ind += 1 if pad: odiag_ind = (b + 1) * max_conn diag_ind = odiag_ind sections[b] = odiag_ind x_prime = x_prime[:odiag_ind] mels = mels[:odiag_ind] # if not allocated return copies if not x_prime_allocated: x_prime = np.copy(x_prime) if not mels_allocated: mels = np.copy(mels) return x_prime, mels
[docs] def get_conn_flattened(self, x, sections, pad=False): r"""Finds the connected elements of the Operator. Starting from a given quantum number x, it finds all other quantum numbers x' such that the matrix element :math:`O(x,x')` is different from zero. In general there will be several different connected states x' satisfying this condition, and they are denoted here :math:`x'(k)`, for :math:`k=0,1...N_{\mathrm{connected}}`. This is a batched version, where x is a matrix of shape (batch_size,hilbert.size). Args: x (matrix): A matrix of shape (batch_size,hilbert.size) containing the batch of quantum numbers x. sections (array): An array of size (batch_size) useful to unflatten the output of this function. See numpy.split for the meaning of sections. Returns: matrix: The connected states x', flattened together in a single matrix. array: An array containing the matrix elements :math:`O(x,x')` associated to each x'. """ # try to cache those temporary buffers with their max size total_size = x.shape[0] * self._max_conn if self._max_mels.size < total_size: self._max_mels = np.empty(total_size, dtype=self._max_mels.dtype) self._max_xprime = np.empty((total_size, x.shape[1]), dtype=x.dtype) elif x.dtype != self._max_xprime.dtype: self._max_xprime = self._max_xprime.astype(x.dtype) return self._flattened_kernel( np.asarray(x), sections, self._edges, self._U, self._V, self._J, self._mu, self._n_max, self._max_conn, self._max_mels, self._max_xprime, pad, )
def _get_conn_flattened_closure(self): _edges = self._edges _U = self._U _V = self._V _J = self._J _mu = self._mu _n_max = self._n_max _max_conn = self._max_conn fun = self._flattened_kernel # do not pass the preallocated self._max_mels and self._max_xprime because they are frozen in a closure # and become read only def gccf_fun(x, sections): return fun( x, sections, _edges, _U, _V, _J, _mu, _n_max, _max_conn, ) return jit(nopython=True)(gccf_fun)