hamil

Base classes and functions for generating and interacting with Hamiltonian matrices.

class hamil.Hamiltonian(sys, basis, tau=0.1)

Base Hamiltonian class.

The relevant subclass which provides the appropriate matrix elements should be used.

Parameters:
  • sys – object describing system to be studied; used only in the virtual matrix element functions
  • basis (iterable of iterables of single-particle basis functions) – set of many-particle basis functions
  • tau (float) – timestep by which to propogate psip distributions in imaginary time

Note

This is a base class; basis and sys must be appropriate to the actual subclass used and are specific to the required system and underlying many-particle basis set.

sys = None

system to be studied

basis = None

set of many-particle basis functions

nbasis = None

number of many-particle basis functions (i.e. length basis)

hamil = None

Hamiltonian matrix in the basis set of the many-particle basis functions

tau = None

timestep by which to propogate psip distributions in imaginary time

pos_propogator = None

positive propogator matrix, \(T^+\) where \(T = 1 - \tau H = T^+ - T^-\) and \(T^+\) elements are non-negative.

neg_propogator = None

negative propogator matrix, \(T^-\) where \(T = 1 - \tau H = T^+ - T^-\) and \(T^-\) elements are non-negative.

mat_fn_diag(b)

Calculate a diagonal Hamiltonian matrix element.

Warning

Virtual member. Must be appropriately implemented in a subclass.

Parameters:b (iterable of single-particle basis functions) – a many-particle basis function, \(|b\rangle\)
Return type:float
Returns:\(\langle b|H|b \rangle\).
mat_fn_offdiag(b1, b2)

Calculate an off-diagonal Hamiltonian matrix element.

Warning

Virtual member. Must be appropriately implemented in a subclass.

Parameters:
  • b1 (iterable of single-particle basis functions) – a many-particle basis function, \(|b_1\rangle\)
  • b2 (iterable of single-particle basis functions) – a many-particle basis function, \(|b_2\rangle\)
Return type:

float
Returns:

\(\langle b_1|H|b_2 \rangle\).
eigh()
Returns:(eigenvalues, eigenvectors) of the Hamiltonian matrix.
eigvalsh()
Returns:eigenvalues of the Hamiltonian matrix.
negabs_off_diagonal_elements()

Set off-diagonal elements of the Hamiltonian matrix to be negative.

This converts the Hamiltonian into the lesser sign-problem matrix discussed by Spencer, Blunt and Foulkes.

negabs_diagonal_elements()

Set diagonal elements of the Hamiltonian matrix to be negative.

This, when called after negabs_offdiagonal_elements, converts the Hamiltonian into the greater sign-problem matrix discussed by Spencer, Blunt and Foulkes.

propogate(pos_psips, neg_psips)

Propogates a psip (psi-particle) distribution for a single timestep.

Parameters:
  • pos_psips (1D array or list (length nbasis)) – distribution of positive psips at time \(t = n\tau\) on the many-fermion basis set.
  • neg_psips (1D array or list (length nbasis)) – distribution of negative psips at time \(t = n\tau\) on the many-fermion basis set.
Returns:

(next_pos_psips, next_neg_psips) — positive and negative psip distributions at time \(t=(n+1)\tau\).
hamil.hartree_excitation(p1, p2)

Find the excitation connecting two Hartree products.

Parameters:
  • p1 (iterable of single-particle basis functions) – a Hartree product basis function
  • p2 (iterable of single-particle basis functions) – a many-particle basis function
Returns:

(from_1, to_1) where:

from_1

list of single-particle basis functions excited from p1

to_2

list of single-particle basis functions excited into p2
hamil.determinant_excitation(d1, d2)

Find the excitation connecting two Slater determinants.

Parameters:
  • p1 (iterable of single-particle basis functions) – a Slater determinant basis function
  • p2 (iterable of single-particle basis functions) – a Slater determinant basis function
Returns:

(from_1, to_1, nperm) where:

from_1

list of single-particle basis functions excited from d1

to_2

list of single-particle basis functions excited into d2

nperm

number of permutations required to align the two determinants such that the orders of single-particle basis functions are in maximum agreement. This, in general, is not the minimal possible number of permutations but the parity of the permutations, which is all that is required for calculating matrix elements, is correct.
hamil.permanent_excitation(p1, p2)

Find the excitation connecting two permanents.

Parameters:
  • p1 (iterable of single-particle basis functions) – a permanent basis function
  • p2 (iterable of single-particle basis functions) – a permanent basis function
Returns:

(from_1, to_1) where:

from_1

list of single-particle basis functions excited from p1

to_2

list of single-particle basis functions excited into p2