Hamiltonian¶

class Hamiltonian(N)[source]¶

Bases: object

Contains methods for initializing, creating and manipulating Hamiltonian matrices.

Parameters:	N (int) – number of nodes to initialize the Hamiltonian object with.
Returns:	: creates an unallocated PETSc matrix on communicator `petsc4py.PETSc.COMM_WORLD`. This is accessed via the attribute `Hamiltonian.mat`.
Return type:	petsc4py.PETSc.Mat
Returns:	: an associated `EigSolver` object is also created at `Hamiltonian.EigSolver`.
Return type:	`EigSolver`

Method Summary¶

`Hamiltonian.Emax`(**kwargs)	Returns the maximum Eigenvalue of the matrix.
`Hamiltonian.Emin`(**kwargs)	Returns the minimum Eigenvalue of the matrix.
`Hamiltonian.createLine`([d, amp])	Create a 1 particle infinite line Hamiltonian.
`Hamiltonian.createLine2P`([d, amp, interaction])	Create a 2 particle infinite line Hamiltonian.
`Hamiltonian.createLine3P`([d, amp, interaction])	Create a 3 particle infinite line Hamiltonian.
`Hamiltonian.destroy`()	Destroys all associated PETSc matrices.
`Hamiltonian.importAdj`(filename, filetype[, ...])	Create a Hamiltonian from an imported adjacency matrix.
`Hamiltonian.importAdjToH`(filename, filetype)	Create a Hamiltonian from an imported adjacency matrix.
`Hamiltonian.reinitialize`()	Destroy and reinitialize the PETSc matrix `Hamiltonian.mat`.

Method Details¶

Hamiltonian.Emax(**kwargs)[source]¶

Returns the maximum Eigenvalue of the matrix.

Example

>>> Emax = Hamiltonian.Emax()

Hamiltonian.Emin(**kwargs)[source]¶

Returns the minimum Eigenvalue of the matrix.

Example

>>> Emin = Hamiltonian.Emin()

Hamiltonian.createLine(d=[0], amp=[0.0])[source]¶

Create a 1 particle infinite line Hamiltonian.

Parameters:	d (array of ints) – an array containing integers indicating the nodes where diagonal defects are to be placed (e.g. `d=[0,1,4]`). amp (array of floats) – an array containing floats indicating the diagonal defect amplitudes corresponding to each element in `d` (e.g. `amp=[0.5,-1,4.2]`). interaction (float) – the amplitude of interaction between the walkers when located on the same vertex.
Returns:	: `Hamiltonian.mat`, the Hamiltonian matrix.
Return type:	`petsc4py.PETSc.Mat()`

Important

The length of amp and d must be identical.

Fortran interface

This function calls the Fortran function hamiltonian_p1_line().

Hamiltonian.createLine2P(d=[0], amp=[0.0], interaction=0.0)[source]¶

Create a 2 particle infinite line Hamiltonian.

Parameters:	d (array of ints) – an array containing integers indicating the nodes where diagonal defects are to be placed (e.g. `d=[0,1,4]`). amp (array of floats) – an array containing floats indicating the diagonal defect amplitudes corresponding to each element in `d` (e.g. `amp=[0.5,-1,4.2]`). interaction (float) – the amplitude of interaction between the walkers when located on the same vertex.
Returns:	: `Hamiltonian.mat`, the Hamiltonian matrix.
Return type:	`petsc4py.PETSc.Mat()`

Important

The length of amp and d must be identical.

Fortran interface

This function calls the Fortran function hamiltonian_p2_line().

Hamiltonian.createLine3P(d=[0], amp=[0.0], interaction=0.0)[source]¶

Create a 3 particle infinite line Hamiltonian.

Parameters:	d (array of ints) – an array containing integers indicating the nodes where diagonal defects are to be placed (e.g. `d=[0,1,4]`). amp (array of floats) – an array containing floats indicating the diagonal defect amplitudes corresponding to each element in `d` (e.g. `amp=[0.5,-1,4.2]`). interaction (float) – the amplitude of interaction between the walkers when located on the same vertex.
Returns:	: `Hamiltonian.mat`, the Hamiltonian matrix.
Return type:	`petsc4py.PETSc.Mat()`

Important

The length of amp and d must be identical.

Fortran interface

This function calls the Fortran function hamiltonian_p3_line().

Hamiltonian.destroy()[source]¶: Destroys all associated PETSc matrices.

Hamiltonian.importAdj(filename, filetype, d=[0], amp=[0.0], layout='spring', delimiter=None)[source]¶

Create a Hamiltonian from an imported adjacency matrix.

Parameters:	filename (str) – path to the file containing the adjacency matrix of the graph filetype (str) – the filetype of the imported adjacency matrix. `'txt'` - an \(N\times N\) dense 2D array in text format. `'bin'` - an \(N\times N\) PETSc binary matrix. d (array of ints) – an array containing integers indicating the nodes where diagonal defects are to be placed (e.g. `d=[0,1,4]`). amp (array of floats) – an array containing floats indicating the diagonal defect amplitudes corresponding to each element in `d` (e.g. `amp=[0.5,-1,4.2]`). layout (str) – the format to store the position of the nodes. `spring` (default) - spring layout. `circle` - nodes are arranged in a circle. `spectral` - nodes are laid out according to the spectrum of the graph. `random` - nodes are arranged in a random pattern. delimiter (str) – this is passed to numpy.genfromtxt in the case of strange delimiters in an imported `txt` file.
Returns:	:this allocates and builds a Hamiltonian matrix at `Hamiltonian.mat`.
Return type:	`petsc4py.PETSc.Mat()`
Returns:	: the original adjacency matrix is also stored as a PETSc matrix at `Hamiltonian.Adj`.
Return type:	`petsc4py.PETSc.Mat()`

Important

The number of nodes in the imported adjacency matrix must match the number of nodes the Hamiltonian object is initialized with.
The length of amp and d must be identical.

Note

This function calls Python functions located at pyCTQW.MPI.io, and is somewhat slower than importAdjToH(), but with the advantage of a more flexible text file importer.
For the ability to generate 2 or 3 particle Hamiltonians with or without interactions, please see importAdjToH().

Hamiltonian.importAdjToH(filename, filetype, d=[0.0], amp=[0.0], p='1', layout='spring', delimiter=None, interaction=0.0, bosonic=False)[source]¶

Create a Hamiltonian from an imported adjacency matrix.

Args:

filename (str): path to the file containing the adjacency matrix of the graph

filetype (str): the filetype of the imported adjacency matrix.

'txt' - an \(N\times N\) dense 2D array in text format.

'bin' - an \(N\times N\) PETSc binary matrix.

d (array of ints): an array containing integers indicating the nodes

where diagonal defects are to be placed (e.g. d=[0,1,4]).

amp (array of floats): an array containing floats indicating the diagonal defect

amplitudes corresponding to each element in d (e.g. amp=[0.5,-1,4.2]).

p (str) : ('1'``|‘2’|‘3’``) - specify whether to create a 1, 2 or 3 particle Hamiltonian.

layout (str): the format to store the position of the nodes.

spring (default) - spring layout.

circle - nodes are arranged in a circle.

spectral - nodes are laid out according to the spectrum of the graph.

random - nodes are arranged in a random pattern.

delimiter (str): this is passed to numpy.genfromtxt

in the case of strange delimiters in an imported txt file.

interaction (float): the amplitude of interaction between the walkers when located on the same vertex.

bosonic (bool): if true, a Bosonic Hamiltonian is generated.

Returns:

: * Hamiltonian.mat (petsc4py.PETSc.Mat) - the Hamiltonian matrix.

Hamiltonian.Adj (petsc4py.PETSc.Mat) - the adjacency matrix stored as a PETSc matrix.

Hamiltonian.nodePos (array) - the \((x,y)\) coordinates of the graph vertices.

Hamiltonian.lineX (array) - the \(x\) coordinates of the edges connecting graph vertices.

Hamiltonian.lineY (array) - the \(y\) coordinates of the edges connecting graph vertices.

Important

The number of nodes in the imported adjacency matrix must match the number of nodes the Hamiltonian object is initialized with.

The length of amp and d must be identical.

Fortran interface

This function calls the Fortran function importadjtoh() and is faster than importAdjToH(), but with a ‘pickier’ text file importer.

Hamiltonian.reinitialize()[source]¶: Destroy and reinitialize the PETSc matrix Hamiltonian.mat.