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118 lines
3.6 KiB
ReStructuredText
118 lines
3.6 KiB
ReStructuredText
High-Frequency Tail (``TailGf``)
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=========================================
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Definition
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----------------------
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The tail of a Green's function is defined as the behavior of the Green's
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function :math:`G` at large Matsubara frequencies, namely
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.. math:: \mathbf{G}(i\omega_n) \stackrel {=}{\infty} \mathbf{a}_{-1}\cdot i\omega_n + \mathbf{a}_{0} +\mathbf{a}_{1}\cdot \frac{1}{ i\omega_n} +\mathbf{a}_{2}\cdot \frac{1}{ (i\omega_n)^2} +\dots
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Generically, the tail is parametrized by matrix-valued coefficients
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:math:`\mathbf{a}_{i}` (of size :math:`N_1\times N_2`\ )
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.. math:: t = \sum_{i=o_{min}}^{o_{max}} \mathbf{a}_i (i\omega_n)^{-i}
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Implementation
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--------------
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In TRIQS, the tail is contained in an Python object ``TailGf`` with the
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following members:
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- ``data`` is a numpy array representing :math:`\mathbf{a}_{i}` :
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``data[i,m,n]`` :math:`= (\mathbf{a}_i)_{m,n}`\ .
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- ``mask`` is the :math:`N_1\times N_2` numpy array of the maximal
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index :math:`i_{nm}` of the known coefficients (``order_max`` may be
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larger than ``mask``, but all coefficients of indices greater than
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``mask`` are irrelevant)
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- ``N1`` and ``N2`` give the size of each tail coefficient
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:math:`\mathbf{a}_{i}` : :math:`N_1\times N_2`
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- ``size`` is the number of coefficients of the tail.
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- ``__getitem__``and ``__setitem__`` operators: access and set the ith
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coefficient :math:`\mathbf{a}_{i}` with the bracket operator
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- ``__call__`` operator: evaluate the tail at a given frequency
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Example
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-------
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Basic ``TailGf`` object
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~~~~~~~~~~~~~~~~~~~~~~~
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.. runblock:: python
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from pytriqs.gf.local import *
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t = TailGf(shape=(1,1))
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print "t = ",t
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print "t.data.shape = ",t.data.shape
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print "t.order_min = ",t.order_min
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print "t.order_max = ",t.order_max
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print "t.mask = ",t.mask
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print "t[1] = ",t[1]
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t[1]=[1]
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print "t = ",t
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t[-1]=.25
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print "t = ",t
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print "t(100) = ",t(100)
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As a member of a Green's function
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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Basic access
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^^^^^^^^^^^^
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.. runblock:: python
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from pytriqs.gf.local import *
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# Create the Matsubara-frequency Green's function and initialize it
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g = GfImFreq(indices = [1], beta = 50, n_points = 1000, name = "imp")
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g <<= inverse( iOmega_n + 0.5 )
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print "g.tail = ", g.tail
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print "g.tail[2] = ",g.tail[2]
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Fitting tails: ``fit_tail``
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^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Green's functions have a method ``fit_tail`` allowing to fit the data
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contained in the Green's function. ``fit_tail`` is called in the
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following way:
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``fit_tail(fixed_coeff, order_max, fit_start, fit_stop)`` where
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- ``fixed_coeff`` is the :math:`n\times N_1 \times N_2` numpy array of
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know coefficients
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(``fixed_coeff[i,n,m]``:math:`\equiv (\mathbf{a}_{-1+i})_{nm}`\ )
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- ``order_max`` is the maximal index of the coefficients to be
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determined
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- ``fit_start`` and ``fit_stop`` are the frequencies between which to
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fit the data
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In the following example, the Green's function ``g`` defined above is
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fitted between :math:`\omega_n=10` and :math:`\omega_n = 20` with fixed
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coefficients :math:`(\mathbf{a}_{-1})_{00} = 0`\ ,
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:math:`(\mathbf{a}_{0})_{00} = 0` and :math:`(\mathbf{a}_{1})_{00} = 1`
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.. runblock:: python
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from pytriqs.gf.local import *
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g = GfImFreq(indices = [1], beta = 50, n_points = 1000, name = "imp")
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g <<= inverse( iOmega_n + 0.5 )
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g.tail.zero()
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fixed_coeff=numpy.zeros([1,1,3],float)
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fixed_coeff[0,0,0]=0.
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fixed_coeff[0,0,1]=0.
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fixed_coeff[0,0,2]=1.
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order_max = 4
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fit_start = 10.
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fit_stop = 20.
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g.fit_tail(fixed_coeff, order_max, fit_start, fit_stop)
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print "g.tail = ", g.tail
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