"""
Green's function corresponding to a box DOS.
This doesn't correspond to any real lattice. It is mostly meant as very simple
test case, for which we have analytic expressions.
"""
import numpy as np
_SMALL = np.finfo(np.float64).eps**0.25
[docs]
def gf_z(z, half_bandwidth):
r"""
Local Green's function corresponding to a box DOS.
.. math:: G(z) = \ln(\frac{z + D}{z - D}) / 2D
Parameters
----------
z : complex array_like or complex
Green's function is evaluated at complex frequency `z`.
half_bandwidth : float
Half-bandwidth of the box DOS.
Returns
-------
complex np.ndarray or complex
Value of the Green's function corresponding to a box DOS.
Examples
--------
>>> ww = np.linspace(-1.5, 1.5, num=500)
>>> gf_ww = gt.lattice.box.gf_z(ww, half_bandwidth=1)
>>> import matplotlib.pyplot as plt
>>> _ = plt.plot(ww, gf_ww.real, label=r"$\Re G$")
>>> _ = plt.plot(ww, gf_ww.imag, '--', label=r"$\Im G$")
>>> _ = plt.xlabel(r"$\omega/D$")
>>> _ = plt.ylabel(r"$G*D$")
>>> _ = plt.axhline(0, color='black', linewidth=0.8)
>>> _ = plt.xlim(left=ww.min(), right=ww.max())
>>> _ = plt.legend()
>>> plt.show()
"""
z_rel = z / half_bandwidth
return 0.5 / half_bandwidth * np.emath.log((z_rel + 1) / (z_rel - 1))
[docs]
def dos(eps, half_bandwidth):
r"""
Box-shaped DOS.
Parameters
----------
eps : float array_like or float
DOS is evaluated at points `eps`.
half_bandwidth : float
Half-bandwidth of the DOS, DOS(| `eps` | > `half_bandwidth`) = 0.
Returns
-------
float np.ndarray or float
The value of the DOS.
Examples
--------
>>> eps = np.linspace(-1.1, 1.1, num=500)
>>> dos = gt.lattice.box.dos(eps, half_bandwidth=1)
>>> import matplotlib.pyplot as plt
>>> _ = plt.plot(eps, dos)
>>> _ = plt.xlabel(r"$\epsilon/D$")
>>> _ = plt.ylabel(r"DOS * $D$")
>>> _ = plt.axvline(0, color='black', linewidth=0.8)
>>> _ = plt.ylim(bottom=0)
>>> _ = plt.xlim(left=eps.min(), right=eps.max())
>>> plt.show()
"""
return np.where(abs(eps) < half_bandwidth, 0.5/half_bandwidth, 0)
[docs]
def dos_moment(m, half_bandwidth):
"""
Calculate the `m` th moment of the box DOS.
The moments are defined as :math:`∫dϵ ϵ^m DOS(ϵ)`.
Parameters
----------
m : int
The order of the moment.
half_bandwidth : float
Half-bandwidth of the DOS of the box lattice.
Returns
-------
float
The `m` th moment of the box DOS.
See Also
--------
gftool.lattice.box.dos
"""
if m % 2: # odd moments vanish due to symmetry
return 0
return half_bandwidth**m / (m + 1)
[docs]
def gf_ret_t(tt, half_bandwidth, center=0):
r"""
Local retarded-time local Green's function corresponding to a box DOS.
.. math:: G(t) = -1j Θ(t) \sin(Dt)/Dt
where :math:`D` is the half bandwidth.
Parameters
----------
tt : float array_like or float
Green's function is evaluated at time `tt`.
half_bandwidth : float
Half-bandwidth of the box DOS.
center : float
Position of the center of the box DOS.
This parameter is **not** given in units of `half_bandwidth`.
Returns
-------
complex np.ndarray or complex
Value of the retarded-time Green's function corresponding to a box DOS.
Examples
--------
>>> tt = np.linspace(0, 50, 1500)
>>> gf_tt = gt.lattice.box.gf_ret_t(tt, half_bandwidth=1)
>>> import matplotlib.pyplot as plt
>>> _ = plt.axhline(0, color='black', linewidth=0.8)
>>> _ = plt.plot(tt, gf_tt.imag, label=r"$\Im G$")
>>> _ = plt.plot(tt, gf_tt.real, '--', label=r"$\Re G$")
>>> _ = plt.xlabel(r"$t*D$")
>>> _ = plt.ylabel(r"$G$")
>>> _ = plt.xlim(left=tt.min(), right=tt.max())
>>> _ = plt.legend()
>>> plt.show()
"""
tt = np.asarray(half_bandwidth*tt)
gf = np.zeros_like(tt, dtype=complex)
retard = tt.real >= 0
small = retard & (abs(tt) < _SMALL)
# Taylor expansion for small tt, to avoid 1/tt
tt2 = tt[small]**2
gf[small] = -1j*(1 - 1/6*tt2 + 1/120*tt2**2)
big = retard & ~small
gf[big] = -1j * np.sin(tt)[big] / tt[big]
if np.any(center != 0):
return gf*np.exp(-1j*center*tt)
return gf