hqs_quantum_solver.greens_function

hqs_quantum_solver.greens_function#

Computation of Green’s functions.

This module implements the computation of the Green’s function, as described in Braun and Schmitteckert 2014.

References:#

Braun, A., and P. Schmitteckert. 2014. “Numerical Evaluation of Green’s Functions Based on the Chebyshev Expansion.” Phys. Rev. B 90 (16): 165112. doi:10.1103/PhysRevB.90.165112.

Functions

retarded_greens_function(omegas, space, ...)

Calculate frequency-resolved retarded Green's function via Chebyshev expansion.
retarded_greens_function(omegas: ndarray, space: VectorSpaceT, space_plus: VectorSpaceT, space_minus: VectorSpaceT, hamiltonians: Callable[[VectorSpaceT], OperatorProtocol], right_operators: Callable[[VectorSpaceT, VectorSpaceT], list[OperatorProtocol]], left_operators: Callable[[VectorSpaceT, VectorSpaceT], list[OperatorProtocol]] | None = None, eta: float = 0.1, energy_eigenpair: tuple[float, ndarray] | None = None, chebyshev_degree: int = 30, spectral_buffer: float = 0.02) → ndarray#

Calculate frequency-resolved retarded Green’s function via Chebyshev expansion.

The frequency-resolved retarded Green’s function is given by

\[G^r_{\hat{A}_j, \hat{B}_k}(\omega) = \int_{-\infty}^\infty G^r_{\hat{A}_j, \hat{B}_k}(t) e^{(-\eta + \mathrm{i}\omega)t} \,dt \,,\]

where \(G^r_{\hat{A}_j, \hat{B}_k}(t)\) is the retarded Green’s function,

\[G^r_{\hat{A}_j, \hat{B}_k}(t) = -\mathrm{i} \theta(t) \langle \{ \hat{A}_j(t), \hat{B}_k(0) \} \rangle \,.\]

Inhere, \(\{ \cdot, \cdot \}\) denotes the anti-commutator, \(\theta\) is the Heaviside step function, and \(\hat{A}_j(t)\) and \(\hat{B}_k(t)\) are the operators \(\hat{A}_j\) and \(\hat{B}_k\) in the Heisenberg picture, i.e.,

\[\hat{A}_j(t) = e^{\mathrm{i} tH} \hat{A}_j e^{-\mathrm{i} tH} \quad\text{and}\quad \hat{B}_k(t) = e^{\mathrm{i} tH} \hat{B}_k e^{-\mathrm{i} tH} \,.\]

The operator \(H\) is the Hamiltonian of the system.

The expectation value is computed for a vector \(| \psi \rangle\), which has to be an eigenstate of the Hamiltonian for this function to work and is by default choosen to be the groundstate.

Parameters:

  • omegas (ndarray) – The array of frequencies at which to evaluate the Green’s function.

  • space (Any) – The vector space of the vector \(| \psi \rangle\).

  • space_plus (Any) – The vector space of the vector \(\hat{B} | \psi \rangle\).

  • space_minus (Any) – The vector space of the vector \(\hat{A} | \psi \rangle\).

  • hamiltonians (Callable[[Any], OperatorProtocol]) – Function that returns the Hamiltonian for a given vector space. This function will only be called with the argument space, space_plus, and space_minus.

  • right_operators (Callable[[Any, Any], list[OperatorProtocol]]) – Function that returns the list of operators \(\hat{B}_k\) for a given pair of operator domain and codomain. Will be called with (space, space_plus) and (space_minus, space).

  • left_operators (Callable[[Any, Any], list[OperatorProtocol]] | None) –

    Function that returns the list of operators \(\hat{A}_j\) for a given pair of operator domain and codomain. Will be called with (space_plus, space) and (space, space_minus).

    When not specified, assumes that \(\hat{A}_j = \hat{B}_j^\dagger\). In this case fewer computations need to be performed.

  • eta (float) – The value of \(\eta\) must be strictly positive for the Fourier integral to converge. Increasing \(\eta\) increases the convergence of the method, while smoothing out the resulting function.

  • energy_eigenpair (tuple[float, np.ndarray] | None) – The eigenvalue and corresponding eigenvector \(| \psi \rangle\) for which to compute the expectation value. By default the groundstate is chosen.

  • chebyshev_degree (int) – The degree of the Chebyshev polynomial used for the approximation.

  • spectral_buffer (float) – Value from \([0, 1)\). Security factor for the selection of the Chebyshev approximation interval. The value needs to be in the order of the relative error of the eigenvalue bound estimates.

Returns:

Returns a three-dimensional array G[omega, j, k], where the first index corresponds to the frequency, the second index to the index \(j\), and the third index to the index \(k\).

Return type:

ndarray

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