QED Operators

ed_lgt.operators.QED_operators.QED_Hamiltonian_couplings(lattice_dim, pure_theory, g, m=None)[source]

This function provides the QED Hamiltonian coefficients starting from the gauge coupling g and the bare mass parameter m

Parameters:

  • pure_theory (bool) – True if the theory does not include matter

  • g (scalar) – gauge coupling

  • m (scalar, optional) – bare mass parameter

Returns:

dictionary of Hamiltonian coefficients

Return type:

dict

ed_lgt.operators.QED_operators.QED_check_gauss_law(spin, pure_theory, lattice_dim, gauss_law_ops, threshold=1e-15)[source]

This function perform a series of checks to the gauge invariant dressed-site local basis of the QED Hamiltonian, in order to verify that Gauss Law is effectively satified.

Parameters:

  • spin (scalar, int) – spin representation of the U(1) Gauge field, corresponding to a gauge Hilbert space of dimension (2 spin +1)

  • pure_theory (bool) – If True, the local basis describes gauge invariant states in absence of matter. Defaults to False.

  • lattice_dim (int) – number of lattice spatial dimensions

  • gauss_law_ops (dict) – It contains the Gauss Law operators (for each type of lattice site)

  • threshold (scalar & real, optional) – numerical threshold for checks. Defaults to 1e-15.

Raises:

  • TypeError – If the input arguments are of incorrect types or formats.

  • ValueError – if the gauge basis M does not behave as an Isometry: M^T*M=1

  • ValueError – if the gauge basis does not generate a Projector P=M*M^T

  • ValueError – if the QED gauss law is not satisfied

ed_lgt.operators.QED_operators.QED_dressed_site_operators(spin, pure_theory, U, lattice_dim)[source]

This function generates the dressed-site operators of the QED Hamiltonian in d spatial dimensions for d=1,2,3 (pure or with matter fields) for any possible trunctation of the spin representation of the gauge fields.

Parameters:

  • spin (scalar, int) – spin representation of the U(1) Gauge field, corresponding to a gauge Hilbert space of dimension (2 spin +1)

  • pure_theory (bool) – If true, the dressed site includes matter fields

  • U (str) – which version of U you want to use to obtain rishons: ‘ladder’, ‘spin’, ‘cyclic’

  • lattice_dim (int) – number of lattice spatial dimensions

Returns:

dictionary with all the operators of the QED (pure or full) Hamiltonian

Return type:

dict

ed_lgt.operators.QED_operators.QED_gauge_invariant_states(spin, pure_theory, lattice_dim)[source]

This function generates the gauge invariant basis of a QED LGT in a d-dimensional lattice where gauge (and matter) degrees of freedom are merged in a compact-site notation by exploiting a rishon-based quantum link model.

NOTE: In presence of matter, the gague invariant basis is different for even and odd sites due to the staggered fermion solution

NOTE: The function provides also a restricted basis for sites on the borderd of the lattice where not all the configurations are allowed (the external rishons/gauge fields do not contribute)

Parameters:

  • spin (scalar, int) – spin representation of the U(1) Gauge field, corresponding to a gauge Hilbert space of dimension (2 spin +1)

  • pure_theory (bool,optional) – if True, the theory does not involve matter fields

  • lattice_dim (int, optional) – number of spatial dimensions. Defaults to 2.

Returns:

dictionaries with the basis and the states

Return type:

(dict, dict)

ed_lgt.operators.QED_operators.QED_rishon_operators(spin, pure_theory, U)[source]

This function computes the QED Rishon modes adopted for the U(1) Lattice Gauge Theory for the chosen spin representation of the Gauge field.

Parameters:

  • spin (scalar, int) – spin representation of the U(1) Gauge field, corresponding to a gauge Hilbert space of dimension (2 spin +1)

  • pure_theory (bool) – If true, the dressed site includes matter fields

  • U (str) – which version of U you want to use to obtain rishons: ‘ladder’, ‘spin’

Returns:

dictionary with the rishon operators and the parity

Return type:

dict