mirror of https://github.com/abinit/abinit.git
194 lines
9.7 KiB
Markdown
194 lines
9.7 KiB
Markdown
---
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authors: HeXu
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---
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# Lattice Wannier functions (LWFs).
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!!! tip
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This feature is still under heavy development. Tests and feedbacks are welcome!
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It is not recommended to be used in production at this stage.
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Please contact the developers (x.he@uliege.be) for feedbacks and help.
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This tutorial introduces the methods for constructing lattice Wannier funtions (LWFs)[[cite:Rabe1995]].
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You will learn how to use the SCDM-k (selected columns of the density matrix in k-space -
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[[cite:Damle2015]] [[cite:Damle2017]] and [[cite:Damle2018]]) method for constructing the LWFs,
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and how to plot the phonon band structure from the LWFs and compare it with the full phonon band structures.
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The LWF is a close analogy to the electron Wannier functions, but for atomic displacements.
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Each LWF is formed by a group of atomic displacement within a certain range.
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Together all LWFs form a localized basis set for the atomic distortions, which could span the (sub)space for the atomic vibrations (phonons).
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One typical use case is to build effective Hamiltonian of atomic displacement.
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In many phenomena, only a few distortion modes are important.
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For example, in a structural phase transition the soft modes are often related,
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whereas the phonons with much higher frequency are probably less relevant.
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Thus if the LWFs can well represent the relevant phonon modes, they could reduce the number of degrees of freedom in the study of such phenomena.
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In anaddb, LWFs can be constructed with the SCDM-k algorithm or the projected Wannier function (PWF) algorithm.
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Both methods were initially proposed for electron Wannier functions and were adapted for the LWFs.
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In this tutorial, we will first learn about the algorithms and the practical usage of the methods.
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## SCDM-k method
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### Introduction to the SCDM-k algorithm
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The SCDM-k method is based on two observations: 1. The columns of the density matrix (DM) are often localized,
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given that the DM is expressed in local basis set.
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2. The DM are often low ranked, i.e. they can be spanned by a basis set with size smaller than the original basis set.
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A $m\times n$ matrix $\mathbf{A}$ can be approximately decomposed as another matrix formed by its columns, a $m\times k$ matrix $\mathbf{A[:,c]}$ multiplied by a $k\times n$ matrix, and a permutation matrix $\Pi$ with size $n\times n$.
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The matrix $\mathbf{A[:,c]}$ is selected so that it best represents the $\mathbf{A}$ matrix.
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Thus it can be used as a basis set for describing $\mathbf{A}$. In our case, it can be used as the Wannier function.
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The DM is defined as $\mathbf{\rho}=\sum_i \Psi_if(E_i)\Psi_i^\dagger$,where $H\Psi_i=E_i\Psi_i$,
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where $E_i$ and $\Psi_i$ are the eigen energy and eigen function of the Hamiltonian,
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and $f$ is the occupation function, which is often defined as the Fermi function for the electrons.
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Thus the DM contains information from where the $f$ is large, whereas neglecting the information where the $f$ is close to 0.
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Thus, if the LWF can effectively span the DM, it is good for describing the physics within the range where $f$ is significant.
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This makes $f$ effectively an disentanglement function, which encodes the energy range of interest.
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Instead of using the Fermi function, we can take other functions so that we can select other kinds of energy range,
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thereby giving the "generalized DM".
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In anaddb, three kinds of functions could be used corresponding to the "lwf_disentangle" paramter in anaddb, as shown in the figure below.
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- The unity function, which is 1 everywhere. This means all the energies are treated equally.
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- The erfc function, which is very similar to the Fermi function with two parameters, $\mu$ and $\sigma$.
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It approaches 1 and 0 below and above $\mu$, respectively.
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And the smearing of the function is defined as $\sigma$.
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For example the center panel of the figure is $\mu=0$ and $\sigma=0.2.
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This type of function is often used to build LWFs where low-frequency phonons are of interests.
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- The Gauss function which is highest at the center ($\mu$) and close to 0 away from it.
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The width is defined by $\sigma$. The right panel shows the function with $\mu=0$ and $\sigma=0.2$.
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This type of function is often used if the phonon within some energy range are of interests.
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More generally, the function $f$ does not have to be solely a function of the $E_i$. For example, it can also take the $\Psi_i$ as input.
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We can define a few states as "anchor states", and project the states onto them, and multiply the $f$ by this projection, so that the eigenstates which are most similar to the anchor states are emphasized.
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This method can be very useful when the phonon modes of interests can hardly be distinguished from other modes by energy.
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In anaddb this can be enabled by setting [[lwf_anchor_proj@anaddb]] to 1.
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In the case of a periodic crystal, the columns of the density matrix for all wave vector are localized and are continuous in reciprocal space.
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Therefore by transforming these columns to the real space, one could get localized functions.
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In practice, we start from finding the columns of the DM for an anchor $q$-points.
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Then we take the same columns for each wave vector $q$ in a Monkhorst-Pack grid.
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As these columns are smooth in reciprocal space, the transformations to real space are localized.
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The size of the Monkhorst-Pack grid can be specified by [[lwf_ngqpt@anaddb]].
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Note that it does not have to be the same as [[ngqpt]], as we can use a different $q$-points grid by interpolating the inter-atomic force constant.
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### Steps to construct the LWF using the SCDM method
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Now we take BaTiO$_3$ as an example to learn how to construct the LWFs with the SCDM method.
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You can download the DDB file for the BaTiO$_3$ here:*
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{% dialog tests/tutorial/Input/BTO.DDB %}
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and the anaddb input file:
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{% dialog tests/tutorial/Input/tlwf_1.abi%}
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**Before starting, you might to consider working in a different subdirectory than for the other lessons. Why not create "Work_LWF"?**
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We can walk through the following steps to build the LWF.
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1. Turn on the options required for the construction of LWFs. The interatomic force constants should first be computed, which requires "ifcflag 1" and a few parameters related as in the "Inter-atomic force constant info" section in the input file below. To enable the construction of LWFs with the SCDM-k method, the "lwfflag 1" is used.
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2. Identify the phonon modes of interests.
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In the example of BaTiO$_3$, as we are interested in the ferroelectricity which is related to the three soft modes at $q=\Gamma$.
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We can select $\Gamma$ as the anchor point with "lwf_anchor_qpt 0 0 0" and the number of wannier function to be 3 with the option "lwf_nwann 3"
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3. Identify the disentanglement function.
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The lwf_disentangle parameter is set to 1 to use the "unity" function. We'll tune this to see the effect of using different disentanglement functions.
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We can also enable the projection to the anchor points with "lwf_anchor_proj 1".
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The anchor points are given by the indices of the band at the anchor $q$-point ("lwf_anchor_iband"), which counts from the lowest frequency.
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The example anaddb input looks like follows:
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```
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!File paths
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ddb_filepath = "BTO.DDB"
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output_file = "tlwf_1.abo"
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!Flags
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ifcflag 1
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lwfflag 1
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# Interatomic force constant info
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ngqpt 4 4 4
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q1shft 0 0 0
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nqshft 1
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asr 2
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chneut 1
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dipdip 0
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dipquad 0
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quadquad 0
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# Lattice Wannier function parameters
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lwf_nwann 3
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lwf_ngqpt 4 4 4
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lwf_anchor_qpt 0 0 0
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lwf_anchor_iband 1 2 3
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lwf_disentangle 1
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lwf_anchor_proj=1
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```
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4. Plot the phonon band structure and tune the parameters to get the desired set of LWFs.
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If a q-point path is given in the anaddb input, an lwf_PHFRQ file is generated together with the usual PHFRQ file containing the phonon frequency.
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The format of the two files are the same. The only difference is that, the number of bands equals the number of Wannier functions whereas it is 3 times the number of atoms in the full phonon band structure. Below is an example of the input to specify the q-point path. We can add these lines to the input to get the full and LWF phonon band structures.
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```
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ndivsm 20
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nqpath 8
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qpath
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0.0 0.0 0.0
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0.0 0.5 0.0
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0.5 0.5 0.0
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0.0 0.0 0.0
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0.5 0.5 0.5
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0.0 0.5 0.0
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0.5 0.5 0.0
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0.5 0.5 0.5
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```
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We can use the script provided in the scripts/post_processing directory in the ABINIT package, named compare_phbands.py to plot the two band structure in one figure. The detailed guide for the usage of the script can be found by the command:
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```
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python compare_phbands.py --help
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```
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Here we can use the command to generate the figure, both into the file compare.png and to the screen.
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```
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python compare_phbands.py tlwf_1 -o compare.png -s
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```
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We can see the two band structures. The blue lines form the full phonon band structure and the green dots form the LWF phonon band structure.
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It shows that the phonon band structure from LWFs follows the phonon branches connected with the three soft modes, and goes to high frequencies.
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The LWFs are not unique. They can be adapted to different purposes. For example, we could use a disentanglement function which emphasize the low-frequency part.
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Here we try with a erfc function with $\mu=400 cm^{-1}$ and a smearing of $\sigma=150cm^{-1}$.
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```
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lwf_disentangle 2
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lwf_mu 400.0
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lwf_sigma 150.00
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```
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This give the band structure, which is similar in the low-frequency but are significantly different in the high-frequency range.
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