"""Bethe lattice with infinite coordination number.
This is in fact no real lattice, but a tree. It corresponds to a semi-circular
DOS.
:half_bandwidth: The half_bandwidth corresponds to a scaled nearest neighbor
hopping of `t=D/2`
"""
import numpy as np
from mpmath import mp
from scipy.special import jv
from gftool.precision import PRECISE_TYPES as _PRECISE_TYPES
_SMALL = np.finfo(np.float64).eps**0.25
[docs]def gf_z(z, half_bandwidth):
r"""
Local Green's function of Bethe lattice for infinite coordination number.
.. math:: G(z) = 2(z - s\sqrt{z^2 - D^2})/D^2
where :math:`D` is the half bandwidth and :math:`s=sgn[ℑ{ξ}]`. See
[georges1996]_.
Parameters
----------
z : complex array_like or complex
Green's function is evaluated at complex frequency `z`.
half_bandwidth : float
Half-bandwidth of the DOS of the Bethe lattice.
The `half_bandwidth` corresponds to the nearest neighbor hopping `t=D/2`.
Returns
-------
complex np.ndarray or complex
Value of the Bethe Green's function.
References
----------
.. [georges1996] Georges et al., Rev. Mod. Phys. 68, 13 (1996)
https://doi.org/10.1103/RevModPhys.68.13
Examples
--------
>>> ww = np.linspace(-1.5, 1.5, num=500)
>>> gf_ww = gt.lattice.bethe.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 = np.array(z / half_bandwidth, dtype=np.complex256)
try:
complex_pres = np.complex256 if z.dtype in _PRECISE_TYPES else complex
except AttributeError:
complex_pres = complex
gf_z = 2./half_bandwidth*z_rel*(1 - np.sqrt(1 - z_rel**-2))
return gf_z.astype(dtype=complex_pres, copy=False)
[docs]def gf_d1_z(z, half_bandwidth):
"""
First derivative of local Green's function of Bethe lattice for infinite coordination number.
Parameters
----------
z : complex array_like or complex
Green's function is evaluated at complex frequency `z`.
half_bandwidth : float
Half-bandwidth of the DOS of the Bethe lattice.
The `half_bandwidth` corresponds to the nearest neighbor hopping `t=D/2`.
Returns
-------
complex np.ndarray or complex
Value of the derivative of the Green's function.
See Also
--------
gftool.lattice.bethe.gf_z
"""
z_rel_inv = np.array(half_bandwidth / z, dtype=np.complex256)
try:
complex_pres = np.complex256 if z.dtype in _PRECISE_TYPES else complex
except AttributeError:
complex_pres = complex
sqrt = np.sqrt(1 - z_rel_inv**2)
gf_d1 = 2. / half_bandwidth**2 * (1 - 1/sqrt)
return gf_d1.astype(dtype=complex_pres, copy=False)
[docs]def gf_d2_z(z, half_bandwidth):
"""
Second derivative of local Green's function of Bethe lattice for infinite coordination number.
Parameters
----------
z : complex array_like or complex
Green's function is evaluated at complex frequency `z`.
half_bandwidth : float
Half-bandwidth of the DOS of the Bethe lattice.
The `half_bandwidth` corresponds to the nearest neighbor hopping `t=D/2`.
Returns
-------
complex np.ndarray or complex
Value of the Green's function.
See Also
--------
gftool.lattice.bethe.gf_z
"""
z_rel = np.array(z / half_bandwidth, dtype=np.complex256)
try:
complex_pres = np.complex256 if z.dtype in _PRECISE_TYPES else complex
except AttributeError:
complex_pres = complex
sqrt = np.sqrt(1 - z_rel**-2)
gf_d2 = 2. / half_bandwidth**3 * z_rel * sqrt / (1 - z_rel**2)**2
return gf_d2.astype(dtype=complex_pres, copy=False)
[docs]def gf_z_inv(gf, half_bandwidth):
r"""
Inverse of local Green's function of Bethe lattice for infinite coordination number.
.. math:: R(G) = (D/2)^2 G + 1/G
where :math:`R(z) = G^{-1}(z)` is the inverse of the Green's function.
Parameters
----------
gf : complex array_like or complex
Value of the local Green's function.
half_bandwidth : float
Half-bandwidth of the DOS of the Bethe lattice.
The `half_bandwidth` corresponds to the nearest neighbor hopping `t=D/2`.
Returns
-------
complex np.ndarray or complex
The inverse of the Bethe Green's function `gf_z(gf_z_inv(g, D), D)=g`.
See Also
--------
gftool.lattice.bethe.gf_z
References
----------
.. [georges1996] Georges et al., Rev. Mod. Phys. 68, 13 (1996)
https://doi.org/10.1103/RevModPhys.68.13
Examples
--------
>>> ww = np.linspace(-1.5, 1.5, num=500) + 1e-4j
>>> gf_ww = gt.lattice.bethe.gf_z(ww, half_bandwidth=1)
>>> np.allclose(ww, gt.lattice.bethe.gf_z_inv(gf_ww, half_bandwidth=1))
True
"""
return (0.5 * half_bandwidth)**2 * gf + 1./gf
[docs]def dos(eps, half_bandwidth):
r"""
DOS of non-interacting Bethe lattice for infinite coordination number.
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.
The `half_bandwidth` corresponds to the nearest neighbor hopping `t=D/2`.
Returns
-------
float np.ndarray or float
The value of the DOS.
See Also
--------
gftool.lattice.bethe.dos_mp : Multi-precision version suitable for integration.
References
----------
.. [economou2006] Economou, E. N. Green's Functions in Quantum Physics.
Springer, 2006.
Examples
--------
>>> eps = np.linspace(-1.1, 1.1, num=500)
>>> dos = gt.lattice.bethe.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()
"""
eps_rel = np.asarray(eps / half_bandwidth)
dos = np.zeros_like(eps_rel)
nonzero = (abs(eps_rel) < 1) | np.iscomplex(eps)
dos[nonzero] = 2. / (np.pi*half_bandwidth) * np.sqrt(1 - eps_rel[nonzero]**2)
return dos
# ∫dϵ ϵ^m DOS(ϵ) for half-bandwidth D=1
# from: integral of dos_mp with mp.workdps(100)
# for m in range(0, 22, 2):
# with mp.workdps(100):
# print(mp.quad(lambda eps: 2 * eps**m * dos_mp(eps), [0, 1]))
# rational numbers obtained by mp.identify
dos_moment_coefficients = {
2: 0.25,
4: 0.125,
6: 5/64,
8: 7/128,
10: 21/512,
12: 0.0322265625,
14: 0.02618408203125,
16: 0.021820068359375,
18: 0.01854705810546875,
20: 0.016017913818359375,
}
[docs]def dos_moment(m, half_bandwidth):
"""
Calculate the `m` th moment of the Bethe 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 Bethe lattice.
Returns
-------
float
The `m` th moment of the Bethe DOS.
Raises
------
NotImplementedError
Currently only implemented for a few specific moments `m`.
See Also
--------
gftool.lattice.bethe.dos
"""
if m % 2: # odd moments vanish due to symmetry
return 0
try:
return dos_moment_coefficients[m] * half_bandwidth**m
except KeyError as keyerr:
raise NotImplementedError('Calculation of arbitrary moments not implemented.') from keyerr
[docs]def dos_mp(eps, half_bandwidth=1):
r"""
Multi-precision DOS of non-interacting Bethe lattice for infinite coordination number.
This function is particularly suited to calculate integrals of the form
:math:`∫dϵ DOS(ϵ)f(ϵ)`.
Parameters
----------
eps : mpmath.mpf or mpf_like
DOS is evaluated at points `eps`.
half_bandwidth : mpmath.mpf or mpf_like
Half-bandwidth of the DOS, DOS(| `eps` | > `half_bandwidth`) = 0.
The `half_bandwidth` corresponds to the nearest neighbor hopping `t=D/2`.
Returns
-------
mpmath.mpf
The value of the DOS.
See Also
--------
gftool.lattice.bethe.dos : Vectorized version suitable for array evaluations.
References
----------
.. [economou2006] Economou, E. N. Green's Functions in Quantum Physics.
Springer, 2006.
Examples
--------
Calculate integrals:
>>> from mpmath import mp
>>> mp.quad(gt.lattice.bethe.dos_mp, [-1, 1])
mpf('1.0')
>>> eps = np.linspace(-1.1, 1.1, num=500)
>>> dos_mp = [gt.lattice.bethe.dos_mp(ee, half_bandwidth=1) for ee in eps]
>>> dos_mp = np.array(dos_mp, dtype=np.float64)
>>> import matplotlib.pyplot as plt
>>> _ = plt.plot(eps, dos_mp)
>>> _ = 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()
"""
eps, half_bandwidth = mp.mpf(eps), mp.mpf(half_bandwidth)
if mp.fabs(eps) > half_bandwidth:
return mp.mpf('0')
return 2 / (mp.pi * half_bandwidth) * mp.sqrt(-mp.powm1(eps / half_bandwidth, mp.mpf('2')))
[docs]def gf_ret_t(tt, half_bandwidth, center=0):
r"""
Retarded-time local Green's function of Bethe lattice with Z=∞.
.. math:: G(t) = -2j * Θ(t) * J_1(Dt)/(Dt)
where :math:`D` is the half bandwidth and :math:`J_1(t)` is the Bessel
function of first kind.
Parameters
----------
tt : float array_like or float
Green's function is evaluated at time `tt`.
half_bandwidth : float
Half-bandwidth of the DOS of the Bethe lattice.
The `half_bandwidth` corresponds to the nearest neighbor hopping `t=D/2`.
center : float
Position of the center of the Bethe DOS.
This parameter is **not** given in units of `half_bandwidth`.
Returns
-------
complex np.ndarray or complex
Value of the retarded-time Bethe Green's function.
Examples
--------
>>> tt = np.linspace(0, 50, 1500)
>>> gf_tt = gt.lattice.bethe.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.real, label=r"$\Re G$")
>>> _ = plt.plot(tt, gf_tt.imag, '--', label=r"$\Im G$")
>>> _ = 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 - 1/8*tt2 + 1/192*tt2**2)
big = retard & ~small
gf[big] = -2j * jv(1, tt[big]) / tt[big]
if np.any(center != 0):
return gf*np.exp(-1j*center*tt)
return gf