r"""2D square lattice.
The dispersion of the 2D square lattice is given by
.. math:: ϵ_{k_x, k_y} = 2t [\cos(k_x) + \cos(k_y)]
which takes values in :math:`ϵ_{k_x, k_y} ∈ [-4t, +4t] = [-D, +D]`.
:half_bandwidth: The half_bandwidth corresponds to a nearest neighbor hopping
of `t=D/4`
"""
import numpy as np
from mpmath import mp
from scipy.special import ellipkm1
from gftool._util import _u_ellipk
[docs]def gf_z(z, half_bandwidth):
r"""Local Green's function of the 2D square lattice.
.. math::
G(z) = \frac{2}{πz} ∫^{π/2}_{0} \frac{dϕ}{\sqrt{1 - (D/z)^2 \cos^2ϕ}}
where :math:`D` is the half bandwidth and the integral is the complete
elliptic integral of first kind. See [economou2006]_.
Parameters
----------
z : complex np.ndarray or complex
Green's function is evaluated at complex frequency `z`.
half_bandwidth : float
Half-bandwidth of the DOS of the square lattice.
The `half_bandwidth` corresponds to the nearest neighbor hopping `t=D/4`
Returns
-------
gf_z : complex np.ndarray or complex
Value of the square lattice Green's function
References
----------
.. [economou2006] Economou, E. N. Green's Functions in Quantum Physics.
Springer, 2006.
Examples
--------
>>> ww = np.linspace(-1.5, 1.5, num=500)
>>> gf_ww = gt.lattice.square.gf_z(ww, half_bandwidth=1)
>>> import matplotlib.pyplot as plt
>>> _ = plt.axhline(0, color='black', linewidth=0.8)
>>> _ = plt.plot(ww, gf_ww.real, label=r"$\Re G$")
>>> _ = plt.plot(ww, gf_ww.imag, '--', label=r"$\Im G$")
>>> _ = plt.ylabel(r"$G*D$")
>>> _ = plt.xlabel(r"$\omega/D$")
>>> _ = plt.xlim(left=ww.min(), right=ww.max())
>>> _ = plt.legend()
>>> plt.show()
"""
z_rel_inv = half_bandwidth/z
elliptic = _u_ellipk(z_rel_inv**2)
gf_z = 2./np.pi/half_bandwidth*z_rel_inv*elliptic
return gf_z
[docs]def dos(eps, half_bandwidth):
r"""DOS of non-interacting 2D square lattice.
Has a van Hove singularity (logarithmic divergence) at `eps = 0`.
Parameters
----------
eps : float np.ndarray 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/4`
Returns
-------
dos : float np.ndarray or float
The value of the DOS.
See Also
--------
gftool.lattice.square.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.square.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
elliptic = ellipkm1(eps_rel[nonzero]**2) # on real axis we can use fast scipy Implementation
dos[nonzero] = 2 / np.pi**2 / half_bandwidth * elliptic
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: 9/64,
6: 25/256,
8: (35/128)**2,
10: (63/256)**2,
12: 0.0508890151977539,
14: 0.0438787937164307,
16: 0.0385653460398316,
18: 0.0343993364367634,
20: 0.031045401134179,
}
[docs]def dos_moment(m, half_bandwidth):
"""Calculate the `m` th moment of the square 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 2D square lattice.
Returns
-------
dos_moment : float
The `m` th moment of the 2D square DOS.
Raises
------
NotImplementedError
Currently only implemented for a few specific moments `m`.
See Also
--------
gftool.lattice.square.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 2D square lattice.
Has a van Hove singularity (logarithmic divergence) at `eps = 0`.
This function is particularity suited to calculate integrals of the form
:math:`∫dϵ DOS(ϵ)f(ϵ)`. If you have problems with the convergence,
consider using :math:`∫dϵ DOS(ϵ)[f(ϵ)-f(0)] + f(0)` to avoid the singularity.
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/4`
Returns
-------
dos_mp : mpmath.mpf
The value of the DOS.
See Also
--------
gftool.lattice.square.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.square.dos_mp, [-1, 0, 1])
mpf('1.0')
>>> eps = np.linspace(-1.1, 1.1, num=500)
>>> dos_mp = [gt.lattice.square.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')
# around 0 we have to double precision for `1 - eps**2` to resolve around singularity
with mp.workdps(mp.dps*2, normalize_output=True):
mm = -mp.powm1(eps / half_bandwidth, mp.mpf('2'))
return 2 / (mp.pi**2 * half_bandwidth) * mp.ellipk(mm)
[docs]def stress_trafo(xi, half_bandwidth):
r"""Single pole integration over the stress tensor function.
In analogy to the Hilbert transformation, we define the stress
tensor transformation as
.. math:: T(ξ) = ∫dϵ \tilde{Φ}_{xx}(ϵ)/(ξ - ϵ)
with the stress tensor function
.. math:: \tilde{Φ}_{xx}(ϵ) ≔ ∑_k 𝜕^2/𝜕k_x^2 δ(ϵ - ϵ_k) = -0.5 * ϵ * DOS(ϵ)
Parameters
----------
xi : complex or complex array_like
Point of evaluation of the transformation
half_bandwidth : float
Half-bandwidth of the square lattice.
References
----------
.. [arsenault2013] Arsenault, L.-F., Tremblay, A.-M.S., 2013. Transport
functions for hypercubic and Bethe lattices. Phys. Rev. B 88, 205109.
https://doi.org/10.1103/PhysRevB.88.205109
"""
return -0.5 * (xi*gf_z(xi, half_bandwidth=half_bandwidth) - 1)