SsTC Driver

Solid state task constructor driver

This is a modern Fortran library sister to WannInt. This library is meant to provide a backend for automation of sampling and integration workflows of functions defined on the Brillouin zone (BZ) of a crystal.

Working principle

Many quantities in solid state physics (Berry curvature, optical responses, transport properties...) are expressed as a function, or as the integral of a function, defined in reciprocal space. The most general form of such function is

$$ C^{\alpha}(\textbf{k}; \beta), $$

where $\textbf{k}$ is a vector in the BZ, $\alpha$ denotes a set of "integer indices", which take only integer values and $\beta$ denotes a set of "continuous variables", which take only real values. The idea behind the library is to automate the task of sampling a generic function $C$ in the BZ and provide the necessary utilities for a user to express the particular function $C$ he/she wishes to sample.

To abstract the idea of integer and continuous indices, MAC is used.

API

The derived type fortran type, public :: task_specifier is defined.

type(task_specifier) :: tsk

Constructor.

A task is created by calling

fortran call tsk%construct(name, [& int_ind, & cont_data_start, & cont_data_end, & cont_data_steps, & exponent, ]& calculator) where

• character(len=*), intent(in) :: name is the name of the task.
• integer, optional, intent(in) :: int_ind(:) is the dimension specifier of the integer indices $\alpha$ in $C^{\alpha}(\textbf{k}; \beta)$. That is, the size of the array is the number of integer indices, and the value of each element is the number of values that index can take. If int_ind is not present, $C^{\alpha}(\textbf{k}; \beta)$ does not depend on $\alpha$.
• integer, optional, intent(in) :: cont_data_steps(:) is the dimension specifier of the continuous indices $\beta$ in $C^{\alpha}(\textbf{k}; \beta)$. That is, the size of the array is the number of continuous variables, and the value of each element is the number of steps that the variable has been discretized into. If cont_data_steps is not present, $C^{\alpha}(\textbf{k}; \beta)$ does not depend on $\beta$.
• real(dp), optional, intent(in) :: cont_data_start(:) is a real array where each array element specifies the starting sampling point for each of the continuous variables.
• real(dp), optional, intent(in) :: cont_data_end(:) is a real array where each array element specifies the ending sampling point for each of the continuous variables.
• real(dp), optional, intent(in) :: exponent is a real number $e$, which if not present, will sample the variable $\betai$ according to $\betai(j)=$ cont_data_start(i) + (cont_data_end(i) - cont_data_start(i)) $\times (j-1)/$ (cont_data_steps(i) - 1), and if present as $\betai(j) \rightarrow e^{\betai(j)}$. If cont_data_steps(i) = 1, then $\betai(j)=$ `contdata_start(i)`.
• procedure(cs), pass(self) :: calculator is a function with interface cs which corresponds the implementation of $C^{\alpha}(\textbf{k}; \beta)$. See interface.

Component type(container_specifier) :: tsk%idims

Is a MAC container specifier initialized to the dimension specifier given by int_ind.

Component type(container_specifier) :: tsk%cdims

Is a MAC container specifier initialized to the dimension specifier given by cont_data_steps.

Name handle

Is called as,

fortran name = tsk%name()

where character(len=120) :: name.

Initialization query

Is called as,

fortran initialized = tsk%initialized()

where logical :: initialized is .true. if the task has been initialized and .false. otherwise.

Continuous variable retriever

This utility serves to retrieve the value of $\beta_i(j)$. It is called as,

fortran beta = tsk%cdt(var, step)

where real(dp) :: beta has the value $\beta_i(j)$ when var=$i$, step=$j$.

Interface cs

The interface ```fortran use MAC, only: containerspecifier, container use WannInt, only: crystal abstract interface function cs(self, crys, k, other) import dp, taskspecifier, crystal class(task_specifier), intent(in) :: self class(crystal), intent(in) :: crys real(dp), intent(in) :: k(3) class(*), optional, intent(in) :: other

complex(dp) :: cs(self%idims%size(), self%cdims%size())

end function cs end interface `` describes the shape of any calculator. The input values for these types of functions are always an object of classtask_specifier, a [WannInt](https://github.com/irukoa/WannInt/) crystal class objectcrys, areal(dp), dimension(3)variablekrepresenting a vector in the BZ of the crystal and an optional unlimited polymorphic objectother. The output value is acomplex(dp)` rank-2 array which holds, in each dimension, the memory layout index corresponding to a particular permutation of the integer and continuous variables, respectively.

Best practices

• If you need to pass further external information to the calculator (a value for a Dirac delta function smearing for integration tasks, an exception, or an error case...), it is best to extend either task_specifier or crystal, depending on what information needs to be passed.
• It is a clever idea to traverse the whole output array with the aid of MAC's indexing when defining calculators. fortran do r = 1, tsk%cdims%size() r_arr = tsk%cdims%ind(r) cv1 = tsk%cdt(var = 1, step = r_arr(1)) !external variable 1, cv2 = tsk%cdt(var = 2, step = r_arr(2)) !external variable 2, ![...] do i = 1, tsk%idims%size() i_arr = tsk%idims%ind(i) iv1 = i_arr(1) !integer variable 1, iv2 = i_arr(2) !integer variable 2, cs(i, r) = ... !your implementation in terms of iv1, cv1... ![...] enddo enddo
• Most of the times (implementing optical responses, transport properties...), the number of integer and continuous values is known. In these cases it is a clever idea to loop though the variables in a transparent way and then indexing. fortran cvshape = tsk%cdims%shape() ivshape = tsk%idims%shape() do w1 = 1, cvshape(1) !external variable 1, cv1 = tsk%cdt(var = 1, step = w1) do w2 = 1, cvshape(2) !external variable 2, cv2 = tsk%cdt(var = 2, step = w2) ![...] r_arr = [w1, w2, ...] do i1 = 1, ivshape(1) !integer variable 1, do i2 = 1, ivshape(2) !integer variable 2, ![...] i_arr = [i1, i2, ...] !your implementation in terms of i1, cv1... cs(tsk%idims%ind(i_arr), tsk%rdims%ind(r_arr)) = ... enddo enddo ![...] enddo enddo

Sampler

An initialized task is sampled by calling (1st way), fortran call tsk%sample(crys, kpart, store_at [, parallelization, other]) where

• class(crystal), intent(in) :: crys is a WannInt crystal.
• integer, intent(in) :: kpart(3) is a integer array where each component specifies the number of points to partition each dimension of the BZ into. Each component kpart(i) discretizes the reciprocal lattice vector $\mathbf{b}_i$.
• complex(dp), allocatable, intent(out) :: store_at(:, :, :)/store_at(:, :) is a rank-3 or rank-2 array. If rank-3:
  • The first dimension specifies the memory layout representation of $\textbf{k}$ in the order that will be laid out by a MAC container specifier with dimension_specifier = kpart.
  • The second dimension is the memory layout representation of a particular permutation of integer indices $\alpha$ in the order laid out by tsk%idims.
  • The third dimension is the memory layout representation of a particular permutation of continuous variables $\beta$ in the order laid out by tsk%cdims.
• If rank-2:
  • The first dimension is the memory layout representation of a particular permutation of integer indices $\alpha$ in the order laid out by tsk%idims.
  • The second dimension is the memory layout representation of a particular permutation of continuous variables $\beta$ in the order laid out by tsk%cdims.
  • store_at holds an unnormalized sum over all the product(kpart) points given as input.
• character(len=*), optional, intent(in) :: parallelization is an specification of the parallelization method to employ to distribute the sampling. Options are MPI+OMP, MPI, OMP, none. Default is OMP.
• class(*), optional, intent(in) :: other is an unlimited polymorphic object to be passed to the calculator during sampling. See interface.

Or by calling (2nd way), An initialized task is sampled by calling (1st way), fortran call tsk%sample(crys, klist, store_at [, parallelization, other]) where all present parameters are the same as described in the previous calling way except for - real(dp), intent(in) :: klist(3, nk), which is a list of nk points where each klist(:, j) represents a BZ point $\textbf{k}^j=(k^j1, k^j2, k^j3)$, where each $k^ji$ is given in coordinates relative to the reciprocal lattice vectors $\mathbf{b}_i$ (crystal coordinates).

Utilities

The library defines the utilities kpath, kslice. These utilities are of use to create a list of $\textbf{k}$-points of the form klist(3, nk) which traverse a path and a slice of the BZ, respectively.

Kpath

The utility is used as fortran path = kpath(vecs, nkpts) where,

• real(dp), intent(in) :: vecs(nv, 3) are the coordinates of nk points where each vecs(:, j) represents a point $\textbf{k}^j=(k^j1, k^j2, k^j3)$, where each $k^ji$ is given in coordinates relative to the reciprocal lattice vectors $\mathbf{b}_i$ (crystal coordinates).
• integer, intent(in) :: nkpts(nv - 1) each element nkpts(i) is the number of points in a straight line to consider between points vecs(i, :), vecs(i + 1, :).
• real(dp), allocatable :: path(:, :) is the return value. On output, the shape is [3, nk], where nk = sum(nkpts) - (nvec - 2). Each element path(:, j) represents a point $\textbf{k}^j=(k^j1, k^j2, k^j3)$, along the ordered path, where each $k^ji$ is given in coordinates relative to the reciprocal lattice vectors $\mathbf{b}_i$ (crystal coordinates).

Kslice

The utility is used as fortran slice = kslice(corner, vec_a, vec_b, part) where,

• real(dp), intent(in) :: corner(3) are the coordinates of a point $\textbf{k}^j=(k^j1, k^j2, k^j3)$, where each $k^ji$ is given in coordinates relative to the reciprocal lattice vectors $\mathbf{b}_i$ (crystal coordinates). This parameter represents the bottom-left point of the slice.
• real(dp), intent(in) :: vec_a(3), vec_b(3) are the coordinates of vectors $\textbf{k}^1$ and $\textbf{k}^2$, where each $k^ji$ is given in coordinates relative to the reciprocal lattice vectors $\mathbf{b}i$ (crystal coordinates), representing two vectors defining the slice.
• integer, intent(in) :: part(2) is a integer array where each component specifies the number of points to partition each dimension of the BZ into. Each component part(i) discretizes the reciprocal lattice vector $\mathbf{b}_i$.
• real(dp), allocatable :: slice(:, :) is the return value. On output, the shape is [3, nk], where nk = product(part). Each element slice(:, j) represents a point $\textbf{k}^j=(k^j1, k^j2, k^j3)$, where each $k^ji$ is given in coordinates relative to the reciprocal lattice vectors $\mathbf{b}i$ (crystal coordinates). The ordering if $\textbf{k}$ points is given by the order that will be laid out by a MAC container specifier with `dimensionspecifier = part`.

Crystal to cartesian coordinates in k-space

The utility is used as fortran k_cart = crys_to_cart(k_crys, crys) where,

• real(dp), intent(in) :: k_crys(3) are the coordinates of a point $\textbf{k}^j=(k^j1, k^j2, k^j3)$, where each $k^ji$ is given in coordinates relative to the reciprocal lattice vectors $\mathbf{b}_i$ (crystal coordinates).
• class(crystal), intent(in) :: crys is a WannInt crystal.
• real(dp), intent(in) :: k_cart(3) is the return value. On output, the components are the coordinates of a point $\textbf{k}^j=(k^jx, k^jy, k^jz)$, where each $k^ji$ is the component, in $\text{A}^{-1}$, relative to the cartesian frame used to define the real lattice basis of the crystal crys.

Cartesian to crystal coordinates in k-space

The utility is used as fortran k_crys = cart_to_crys(k_cart, crys) where,

• real(dp), intent(in) :: k_cart(3) are the coordinates of a point $\textbf{k}^j=(k^jx, k^jy, k^jz)$, where each $k^ji$ is the component, in $\text{A}^{-1}$, relative to the cartesian frame used to define the real lattice basis of the crystal crys.
• class(crystal), intent(in) :: crys is a WannInt crystal.
• real(dp), intent(in) :: k_crys(3) is the return value. On output, the components are the coordinates of a point $\textbf{k}^j=(k^j1, k^j2, k^j3)$, where each $k^ji$ is given in coordinates relative to the reciprocal lattice vectors $\mathbf{b}_i$ (crystal coordinates).

Path lenght utility

The utility is used as fortran path_length = kpath_length(vecs, nkpts, crys) where,

• real(dp), intent(in) :: vecs(nv, 3) are the coordinates of nk points where each vecs(:, j) represents a point $\textbf{k}^j=(k^j1, k^j2, k^j3)$, where each $k^ji$ is given in coordinates relative to the reciprocal lattice vectors $\mathbf{b}_i$ (crystal coordinates).
• integer, intent(in) :: nkpts(nv - 1) each element nkpts(i) is the number of points in a straight line to consider between points vecs(i, :), vecs(i + 1, :).
• class(crystal), intent(in) :: crys is a WannInt crystal.
• real(dp), allocatable :: path_length(:) is the return value. If the array path has been constructed by using the kpath utility, the ikth element of the path_length array contains the distance traversed along the kpath, in $\text{A}^{-1}$, up until arriving to the point path(:, ik).

Build

An automated build is available for Fortran Package Manager users. This is the recommended way to build and use WannInt in your projects. You can add WannInt to your project dependencies by including

[dependencies] SsTC_driver = { git="https://github.com/irukoa/SsTC_driver.git" } in the fpm.toml file.

MAC's objects fortran type, public :: container_specifier type, extends(container_specifier), public :: container and WannInt's objects and utilities fortran type, public :: crystal public :: diagonalize public :: dirac_delta public :: deg_list public :: schur public :: SVD public :: expsh public :: logu are made public by SsTC_driver.

Limitations

The mayor limitation lies in memory management when sampling with parallelization. The array store_at needs to be dynamically allocated. Its size is given by $($ nk $) \times$ product(int_ind) $\times$ product(cont_data_steps). Before making any calculation it is suggested to check the value of its size and choose a parallelization scheme accordingly. It is recommended to use none or OMP parallelization in PCs and MPI or MPI+OMP in multinode clusters. If the size of store_at is too big, segmentation faults may occur. We recommend using only store_at in its rank-3 version in sampling calls if keeping the index of $\textbf{k}$ points is important (for plotting or for special integration schemes).

We recommend the compilers in the Intel oneAPI toolkit ifort/mpiifort and ifx/mpiifx or the GNU compilers gfortran/mpifort. If possible, we recommend using the --heap-arrays flag for Intel compilers and the -fmax-stack-var-size=n flag for GNU compilers.