Spectral properties

This submodule provides functions that plot spectral properties (mostly the spectral function) as a function of frequency and/or wavevector.

List of functions

pyqcm.spectral.DoS(w, eta=0.1, label=0, sum=False, progress=True, labels=None, colors=None, file=None, data_file='dos.tsv', plt_ax=None, **kwargs)

Plots the density of states (DoS) as a function of frequency

Parameters

  • w (float) – the frequency range is from -w to w if w is a float. If w is a tuple then the range is (wmax[0], wmax[1]). w can also be an explicit list of real frequencies, or of complex frequencies (in which case eta is ignored)

  • eta (float) – Lorentzian broadening, if w is real

  • label (int) – label of the model instance

  • sum (boolean) – if True, the sum of the DoS of all lattice orbitals is plotted in addition to each orbital individually

  • progress (boolean) – if True, prints computation progress

  • labels ([str]) – labels of the different curves

  • colors ([str]) – colors of the different curves

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.spectral.Fermi_surface(nk=64, label=0, orb=None, quadrant=False, plane='xy', k_perp=0.0, file=None, plt_ax=None, **kwargs)

Plots the Fermi surface of the non-interacting model (2D)

Parameters

  • nk (int) – number of wavevectors on each side of the grid

  • label (int) – label of the model instance

  • orb (int) – if None, plots all the orbitals. Otherwise just plots the FS for that orbital (starts at 1)

  • quadrant (boolean) – if True, plots the first quadrant of a square Brillouin zone only

  • plane (str) – momentum plane, ‘xy’=’z’, ‘yz’=’x’=’zy’ or ‘xz’=’zx’=’y’

  • k_perp (float) – momentum component in the third direction (in multiple of \(\pi\))

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.spectral.G_dispersion(nk=64, label=0, orb=None, period='G', contour=False, inv=False, quadrant=False, datafile=None, max=None, k_perp=0.0, plane='xy', file=None, plt_ax=None, **kwargs)

Plots the eigenvalues of the inverse Green function at zero frequency

Parameters

  • nk (int) – number of wavevectors on each side of the grid

  • label (int) – label of the model instance

  • orb (int) – if 0, plots all the orbitals. Otherwise just shows the plot for that orbital (starts at 1)

  • period (str) – periodization scheme (‘G’, ‘M’)

  • contour (boolean) – True for a contour plot; otherwise a 3D plot.

  • inv (boolean) – True if the inverse eigenvalues (inverse energies) are plotted instead

  • quadrant (boolean) – if True, plots the first quadrant of a square Brillouin zone only

  • datafile (str) – if different from None, just writes the data in a file and does not plot

  • max (float) – energy range (from -max to max)

  • k_perp (float) – momentum component in the third direction (in multiple of \(pi\))

  • plane (str) – momentum plane, ‘xy’=’z’, ‘yz’=’x’=’zy’ or ‘xz’=’zx’=’y’

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.spectral.Luttinger_surface(nk=200, label=0, orb=1, quadrant=False, k_perp=0, plane='xy', file=None, plt_ax=None, **kwargs)

Plots the Luttinger surface (zeros of the Green function) in the Brillouin zone (2D)

Parameters

  • nk (int) – number of wavevectors on each side of the grid

  • label (int) – label of the model instance

  • orb (int) – orbital number (starts at 1)

  • quadrant (boolean) – if True, plots the first quadrant of a square Brillouin zone only

  • k_perp (float) – for 3D models, value of the component of k perpendicular to the plane

  • plane (str) – for 3D models, plane of the plot (‘z’=’xy’, ‘y’=’xz’, ‘x=’yz’)

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.spectral.cluster_spectral_function(wmax=6, eta=0.05, imaginary=False, clus=0, label=0, offset=2, full=False, opt=None, spin_down=False, blocks=False, file=None, plt_ax=None, realpart=False, color='b', **kwargs)

Plots the spectral function of the cluster in the site basis

Parameters

  • wmax (float) – the frequency range is from -wmax to wmax if w is a float. If wmax is a tuple then the range is (wmax[0], wmax[1]). wmax can also be an explicit list of real frequencies

  • eta (float) – Lorentzian broadening

  • imaginary (boolean) – If True, the frequency range is along the imaginary frequency axis

  • clus (int) – label of the cluster within the super unit cell (starts at 0)

  • label (int) – label of the model instance

  • offset (float) – vertical offset in the plot between the curves associated to successive wavevectors

  • full (boolean) – if True, plots off-diagonal components as well

  • self (boolean) – if True, plots the self-energy instead of the spectral function

  • opt (str) – if None, G is computed. Other options are ‘self’ (self-energy) and ‘hyb’ (hybridization function)

  • spin_down (boolean) – if True, plots the spin down part, if different

  • blocks (boolean) – if True, gives the GF in the symmetry basis (block diagonal)

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • color – matplotlib color of the curves

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

the array of frequencies, the spectral weight

pyqcm.spectral.gap(k, orb=1, threshold=0.001)

Computes the spectral gap for a series of wavevectors

:param k : set of wavevectors :param int orb : orbital number (starts at 1) :param float threshold : weight below which a Lehmann contribution is deemed zero returns: an array of gap values

pyqcm.spectral.hybridization_function(wmax=6, eta=0.01, imaginary=False, clus=0, realpart=False, label=0, file=None, plt_ax=None, **kwargs)

This function plots the imaginary part of the hybridization function Gamma as a function of frequency. Only the diagonal elements are plotted, but for all clusters if there is more than one. The arguments have the same meaning as in plot_spectrum, except ‘realpart’ which, if True, plots the real part instead of the imaginary part.

Parameters

  • wmax (float) – the frequency range is from -wmax to wmax if w is a float. If wmax is a tuple then the range is (wmax[0], wmax[1]). wmax can also be an explicit list of real frequencies

  • eta (float) – Lorentzian broadening

  • imaginary (boolean) – If True, the frequency range is along the imaginary frequency axis

  • clus (int) – cluster index (starts at 0)

  • realpart (boolean) – if True, the real part of the Green function is shown, not the imaginary part

  • label (int) – label of the model instance

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.spectral.mdc(nk=200, eta=0.1, label=0, orb=None, spin_down=False, quadrant=False, opt='GF', k_perp=0, freq=0.0, max=None, plane='xy', size=1.0, band_basis=False, sym=None, file=None, plt_ax=None, **kwargs)

Plots the spectral weight at zero frequency in the Brillouin zone (2D)

Parameters

  • nk (int) – number of wavevectors on each side of the grid

  • eta (float) – Lorentzian broadening

  • label (int) – label of the model instance

  • orb (int) – if None, sums all the orbitals. Otherwise just shows the weight for that orbital (starts at 1)

  • spin_down (boolean) – true is the spin down sector is to be computed (applies if mixing = 4)

  • quadrant (boolean) – if True, plots the first quadrant of a square Brillouin zone only

  • opt (str) – The quantity to plot. ‘GF’ = Green function, ‘self’ = self-energy, ‘Z’ = quasi-particle weight

  • k_perp (float) – momentum component in the third direction (in multiple of pi)

  • freq (float) – frequency at which the spectral function is computed (0 by default)

  • max (float) – maximum value of the plotting range (if None, maximum of the data)

  • plane (str) – momentum plane, ‘xy’=’z’, ‘yz’=’x’=’zy’ or ‘xz’=’zx’=’y’

  • band_basis (boolean) – uses the band basis instead of the orbital basis (for multiband models)

  • sym (str) – symmetrization option for the mdc

  • size (float) – size of the plot, in multiple of the default (2 pi on the side)

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

the contour plot object

pyqcm.spectral.mdc_anomalous(nk=200, w=0.1j, label=0, orbitals=(1, 1), self=False, im_part=False, quadrant=False, k_perp=0.0, plane='xy', file=None, plt_ax=None, **kwargs)

Plots the anomalous Green function or self-energy (2D)

Parameters

  • nk (int) – number of wavevectors on each side of the grid

  • w (complex) – complex frequency at which the Green function is computed

  • label (int) – label of the model instance

  • orbitals (int) – shows the weight for orbitals (b1,b2) (starts at 1), or numpy array of spin-Nambu projection

  • self (boolean) – if True, plots the anomalous self-energy instead of the spectral function

  • im_part (boolean) – if True, plots the imaginary part instead of the real part

  • quadrant (boolean) – if True, plots the first quadrant of a square Brillouin zone only

  • k_perp (float) – for 3D models, value of the component of k perpendicular to the plane

  • plane (str) – for 3D models, plane of the plot (‘z’=’xy’, ‘y’=’xz’, ‘x=’yz’)

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.spectral.momentum_profile(op, nk=50, label=0, quadrant=False, k_perp=0.0, plane='xy', file=None, plt_ax=None, **kwargs)

Plots the momentum-resolved average of operator op in the Brillouin zone (2D)

Parameters

  • op (str) – name of the lattice operator

  • nk (int) – number of wavevectors on each side of the grid

  • label (int) – label of the model instance

  • quadrant (boolean) – if True, plots the first quadrant of a square Brillouin zone only

  • k_perp (float) – momentum component in the third direction (in multiple of pi)

  • plane (str) – momentum plane, ‘xy’=’z’, ‘yz’=’x’=’zy’ or ‘xz’=’zx’=’y’

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.spectral.plot_dispersion(nk=64, label=0, spin_down=False, orb=None, contour=False, datafile=None, quadrant=False, k_perp=0, plane='xy', file=None, plt_ax=None, view_angle=None, **kwargs)

Plots the dispersion relation in the Brillouin zone (2D)

Parameters

  • nk (int) – number of wavevectors on each side of the grid

  • label (int) – label of the model instance

  • spin_down (boolean) – True is the spin down sector is to be computed (applies if mixing = 4)

  • orb (int) – if None, sums all the orbitals. Otherwise just shows the weight for that orbital (starts at 1)

  • contour (boolean) – True if a contour plot is produced instead of a 3D plot.

  • datafile (str) – if given, name of the data file (no extension please) in which the data is printed, for plotting with an external program. Does not plot. Will produce one file per orbital, with the .tsv extension.

  • quadrant (boolean) – if True, plots the first quadrant of a square Brillouin zone only

  • k_perp (float) – momentum component in the third direction (in multiple of pi)

  • plane (str) – momentum plane, ‘xy’=’z’, ‘yz’=’x’=’zy’ or ‘xz’=’zx’=’y’

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • view_angle (tuple) – optional projection angle to pass to view_init() in the format of (elevation, azimuth) in degrees

  • kwargs – keyword arguments passed to the matplotlib ‘plot_surface’ function

Returns

None

pyqcm.spectral.segment_dispersion(path='triangle', nk=64, label=0, file=None, plt_ax=None, **kwargs)

Plots the dispersion relation in the Brillouin zone along a wavevector path

Parameters

  • path (str) – wavevector path, as used by the function wavevector_path()

  • nk (int) – number of wavevectors on each side of the grid

  • label (int) – label of the model instance

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.spectral.spectral_function(wmax=6.0, eta=0.05, path='triangle', nk=32, label=0, orb=None, offset=2, opt='A', Nambu_redress=True, inverse_path=False, title=None, file=None, plt_ax=None, **kwargs)

Plots the spectral function \(A(\mathbf{k},\omega)\) along a wavevector path in the Brillouin zone. This version plots the spin-down part with the correct sign of the frequency in the Nambu formalism.

Parameters

  • wmax (float) – the frequency range is from -wmax to wmax if w is a float. If wmax is a tuple then the range is (wmax[0], wmax[1]). wmax can also be an explicit list of real frequencies

  • eta (float) – Lorentzian broadening

  • path (str) – a keyword that is passed to pyqcm.wavevector_path() to produce a set of wavevectors along a path, or a tuple

  • nk (int) – the number of wavevectors along each segment of the path (passed to pyqcm.wavevector_grid())

  • label (int) – label of the instance of the model

  • orb (int) – if not None, only plots the spectral function associated with this orbital number (starts at 1). If None, sums over all orbitals.

  • offset (float) – vertical offset in the plot between the curves associated to successive wavevectors

  • opt (str) – ‘A’ : spectral function, ‘self’ : self-energy, ‘Sx’ : spin (x component), ‘Sy’ : spin (y component)

  • Nambu_redress (boolean) – if True, evaluates the Nambu component at the opposite frequency

  • inverse_path (boolean) – if True, inverts the path (k –> -k)

  • title (str) – optional title for the plot. If None, a string with the model parameters will be used.

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.spectral.spectral_function_Lehmann(path='triangle', nk=32, label=0, orb=1, offset=0.1, lims=None, file=None, plt_ax=None, **kwargs)

Plots a Lehmann representation of the spectral function along a wavevector path in the Brillouin zone. Singularities are plotted as impulses with heights proportionnal to the residue.

Parameters

  • path – if a string, keyword passed to pyqcm.wavevector_path() to produce a set of wavevectors; else, explicit list of wavevectors (N x 3 numpy array).

  • nk (int) – the number of wavevectors along each segment of the path (passed to pyqcm.wavevector_path())

  • label (int) – label of the instance of the model

  • orb (int) – only plots the spectral function associated with this orbital number (starts at 1)

  • offset (float) – vertical offset in the plot between the curves associated to successive wavevectors

  • lims ((float,float)) – limits of the plot in frequency (2-tuple)

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

None

pyqcm.spectral.spin_mdc(nk=200, eta=0.1, label=0, orb=None, quadrant=False, opt='spin', freq=0.0, max=None, k_perp=0, plane='xy', band_basis=False, file=None, plt_ax=None, **kwargs)

Plots the spin spectral weight at zero frequency in the Brillouin zone (2D)

Parameters

  • nk (int) – number of wavevectors on each side of the grid

  • eta (float) – Lorentzian broadening

  • label (int) – label of the model instance

  • orb (int) – if None, sums all the orbitals. Otherwise just shows the weight for that orbital (starts at 1)

  • quadrant (boolean) – if True, plots the first quadrant of a square Brillouin zone only

  • opt (str) – The quantity to plot. ‘spin’ = spin texture, ‘spins’ = spin texture (saturated), ‘sz’ = z-component, ‘spinp’ = modulus of xy-component

  • freq (float) – frequency at which the spectral function is computed (0 by default)

  • max (float) – maximum value of the plotting range (if None, maximum of the data)

  • k_perp (float) – momentum component in the third direction (in multiple of pi)

  • plane (str) – momentum plane, ‘xy’=’z’, ‘yz’=’x’=’zy’ or ‘xz’=’zx’=’y’

  • band_basis – uses the band basis instead of the orbital basis (for multiband models)

  • file (str) – if not None, saves the plot in a file with that name

  • plt_ax – optional matplotlib axis set, to be passed when one wants to collect a subplot of a larger set

  • kwargs – keyword arguments passed to the matplotlib ‘plot’ function

Returns

The contour plot object