Skip to content
Snippets Groups Projects
Commit b51b2de4 authored by patavirt's avatar patavirt
Browse files

Add Sphinx

parent b724ce47
No related branches found
No related tags found
No related merge requests found
Hide whitespace changes
Inline Side-by-side
doc/conf.py 0 → 100644
+ 11
− 0
View file @ b51b2de4
project = 'usadelndsoc'
copyright = '2024, Pauli Virtanen'
author = 'Pauli Virtanen'
release = '0.1'
extensions = []
#templates_path = ['_templates']
exclude_patterns = ['_build', 'Thumbs.db', '.DS_Store']
html_theme = 'alabaster'
#html_static_path = ['_static']
doc/discretization.md deleted 100644 → 0
+ 0
− 249
View file @ b724ce47
# Gauge-invariant discretization
We do the discretization of the Usadel equation as follows. We first
discretize the action $`S`$. The discrete saddle-point equation or expressions
for currents are not derived analytically, but instead obtained from the computer
implementation of the action via automatic differentiation (AD).
This simplifies things somewhat, because proper handling of the gauge fields is
simpler to do in the action formulation.
The action is
```math
S[Q] = \frac{i\pi\nu}{8} \mathrm{Tr}[D(\hat{\nabla}Q)^2 + \eta F_{ij}Q\hat{\nabla}_iQ\hat{\nabla}_jQ + 4 i \Omega Q]
\,,
```
where $`\hat{\nabla}_j Q = \partial_j Q - i[\mathcal{A}_j, Q]`$
and $`\Omega = \epsilon\tau_3 + i\Delta`$, $`\Delta^\dagger = \Delta`$.
We'd like the discretized functional to be also gauge invariant.
We subdivide the space to cells, centered on a rectangular lattice
$`\vec{r}_j=(h j_x, h j_y, h j_z)`$ with $`j_{x,y,z}\in\mathbb{Z}`$
and *h* are the lattice spacings. See the figure below:
![FIG 1. Lattice discretization](img/discretization.svg "FIG 1. Lattice discretization.")
We choose $`Q(j) = Q(\vec{r}_j)`$ to be values of $`Q`$ at the lattice
sites. We define the Wilson link matrices $`U_{ij} =U_{ji}^{-1}
=U(\vec{r}_{i},\vec{r}_{j}) =
\mathrm{Pexp}[i\int_{L(\vec{r}_i,\vec{r}_j)}d\vec{r}'\,\cdot\vec{\mathcal{A}}(\vec{r}')]`$
where $`L(\vec{r}_i,\vec{r}_j)`$ is straight line from $`\vec{r}_j`$
to $`\vec{r}_i`$ and $`\mathrm{Pexp}`$ is the path-ordered integral.
Since the gauge field $`\vec{\mathcal{A}}`$ is fixed, the link
matrices can be computed ahead of time.
We want a discretized theory that is invariant under the discrete gauge transformation
```math
Q(j) \mapsto u(j) Q u(j)^{-1} \,,
```
and
```math
U_{ij}
\mapsto
u(i)U_{ij}u(j)^{-1}
\,,
```
and in the continuum limit reduces to the original theory. Such
discretization has the advantage of being insensitive to the gauge
choice, and having exact discrete conservation laws associated with
the symmetry.
## Derivative term
We discretize the derivatives a
```math
D_{ij}
:=
\frac{1}{h}[Q(i) U_{ij} - U_{ij} Q(j)]
\,,
\qquad
\Rightarrow
Q(i) D_{ij} = - D_{ij} Q(j)
\,,
```
which satisfies an anticommutation relation analogous to continuum derivative.
We then discretize
```math
S_2
=
\mathrm{Tr} D(\hat{\nabla} Q)^2
\mapsto
-D \frac{2d}{|\mathrm{neigh}|} h^d \sum_{\mathrm{neigh}(i,j)}\mathrm{Tr} D_{ij} D_{ji}
```
where $`d`$ is the space dimension and `|neigh|` the number of neighbors on the grid.
Expanding in small $`h`$ we have
```math
S_2
\simeq
\frac{h^{d}d}{|\mathrm{neigh}|}
\sum_{i}\sum_{j\in{}\mathrm{neigh}(i)}\mathrm{Tr}\Bigl(\frac{Q(i) - Q(j)}{h} - [i\mathcal{A}(\frac{\vec{r}_i+\vec{r}_j}{2}), Q(j)]\Bigr)^2
\,,
```
which verifies the reduction to the continuum limit.
## Field strength
The link matrices have the usual continuum limit expansion
```math
U_{ji}
=
\sum_{n=0}^\infty i^n |\vec{r}_i-\vec{r}_j|^n \int_{-1/2}^{1/2}ds_1\int_{-1/2}^{s_1}ds_2\ldots\int_{-1/2}^{s_{n-1}}ds_n\, \mathcal{A}(s_1)\cdots\mathcal{A}(s_n)
```
where $`\mathcal{A}(s)=\mathcal{A}[(\frac{1}{2} - s)\vec{r}_i + (\frac{1}{2} + s)\vec{r}_j]`$.
As well known in lattice QCD, the field strength can be expressed in
terms of the link matrices $`U`$ via a plaquette loop. Consider then
the plaquette (see Fig. 1), with edges $`j\leftarrow{}i`$ and $`k\leftarrow{}l`$ in
direction $`\mu`$ and $`l\leftarrow{}i`$, $`k\leftarrow{}j`$ in direction $`\nu`$, and expand around
$`\vec{r}_0=\frac{\vec{r}_i+\vec{r}_j+\vec{r}_k+\vec{r}_l}{4}`$:
```math
P_{lkji}
=
U_{il}U_{lk}U_{kj}U_{ji}
\,,
```
```math
\Rightarrow
P_{lkji}
\simeq
1
- i h^2
\bigl(
\partial_\mu \mathcal{A}_\nu - \partial_\nu \mathcal{A}_\mu - i[\mathcal{A}_\mu,\mathcal{A}_\nu]
\bigr)
+
\mathcal{O}(h^3)
=
1
+ i h^2 F_{\mu \nu}(\vec{r}_0)
+
\mathcal{O}(h^3)
```
which then allows expressing $`F_{ij}`$ in terms of the link matrices.
## Hall term
In the plaquette of Fig. 1 we have
```math
\mathrm{tr}
F_{\mu\nu}Q \hat{\nabla}_\mu Q \hat{\nabla}_\nu Q
\rvert_{\mu=ji, \nu=li}
\simeq
\frac{i}{h^4}
\mathrm{tr}
(1 - P_{lkji}) U_{ij} D_{ji} Q(i) D_{il} U_{li}
```
```math
=
\frac{-i}{h^4}
\mathrm{tr}
(U_{lk} U_{kj} - U_{li} U_{ij}) D_{ji} Q(i) D_{il}
```
So the structure is $`\frac{-i}{h^4}`$ $`\times`$ (gauge factors for
counterclockwise loop $`-`$ return back clockwise) $`\times`$ $`DQD`$ for
three sites in counterclockwise order. Since only $`Q(i)`$, $`Q(j)`$,
$`Q(l)`$ appear here, one can also think of this as a plaquette corner
with $`Q(i)`$ being the corner.
To do the sum over the $`\mu`$, $`\nu`$ indices, one can average over
corners of the plaquettes as follows, which gives the final discretization
of the Hall term:
```math
S_H
=
\eta
\mathrm{Tr} F_{\mu\nu}Q\hat{\nabla}_\mu Q\hat{\nabla}_\nu Q
```
```math
S_H
\mapsto
\eta
\frac{h^d d}{4ih^4}
\sum_{\mathrm{plaqc}(ijkl; i)}
\mathrm{Tr}
\Bigl(
(U_{lk} U_{kj} - U_{li} U_{ij}) D_{ji} Q(i) D_{il}
\Bigr)
```
```math
S_H
=
\frac{\eta h^{d-4} d}{4i}
\sum_{\mathrm{plaqc}(i; jkl)}
\mathrm{Tr}
\Bigl(
(P_{lkji} - 1)(Q(i) - U_{ij} Q(j) U_{ji} Q(i) U_{il} Q(l) U_{li})
\Bigr)
\,,
```
where $`\mathrm{plaqc}(i;jkl)`$ are the plaquette corners. The last line
follows with direct calculation, and considering the cyclic
permutations.
## Matrix current
We can also calculate the matrix current $`j\mapsto{}i`$ between neighboring cells:
```math
J_{ij}^a
=
\frac{\partial}{\partial\xi}
S[
U_{ij}
\mapsto
U_{ij}
+
T^a U_{ij}\xi
\,,
\;
U_{ji}
\mapsto
U_{ji}
-
U_{ji} T^a \xi
]\rvert_{\xi=0}
```
where $`T^a`$ is an appropriate matrix generator, and the transformation
is inserted to only one of the links.
The local gauge invariance now implies that there is an exact discrete
continuity equation
```math
\sum_{j\in{}N_i} J_{ij}^a
=
\frac{\partial}{\partial\xi}
S[Q(i)\mapsto{}e^{\xi T^a}Q(i)e^{-\xi T^a}]\rvert_{\xi=0}
=
\frac{\partial}{\partial\xi}
S[\Omega(i)\mapsto{}e^{-\xi T^a}\Omega(i)e^{\xi T^a}]\rvert_{\xi=0}
=
R_{i}^a
\,.
```
This is one main advantage of the procedure, in addition to its
invariance vs. gauge fixing.
However, the definition of the matrix current is *asymmetric*, so
that $`J_{ij}\ne{}-J_{ji}`$ as they differ by a quantity of order
$`\mathcal{O}(h^2)`$. The asymmetry arises when $`T^a`$ does not commute
with $`U_{ij}`$, so it is an issue only for the SU(2) component. The
problem is only in the interpretation.
The currents measured at sites *i* and *j* are different: since
the parallel transport of spin between these two locations can imply
rotation due to the gauge field, there's no reason why we should have
$`J_{ij}=-J_{ji}`$, except in the continuum limit where the two points
become close to each other.
## Implementation
The above is sufficient for numerical implementation: it is not
necessary to find out the saddle point equation symbolically. It can
be deduced by parametrizing first so that $`Q^2=1`$ and then using
automatic differentiation (AD). Only a routine evaluating the value of
the action is then needed, in terms of the AD dual variables.
When looking for the saddle point in equilibrium, the problem becomes
even simpler; *Q* has then additional restrictions, and we are looking
for the minimum of the free energy *S*.
doc/discretization.rst 0 → 100644
+ 293
− 0
View file @ b51b2de4
Gauge-invariant discretization
==============================
We do the discretization of the Usadel equation as follows. We first
discretize the action :math:`S`. The discrete saddle-point equation or
expressions for currents are not derived analytically, but instead
obtained from the computer implementation of the action via automatic
differentiation (AD). This simplifies things somewhat, because proper
handling of the gauge fields is simpler to do in the action formulation.
The action is
.. math::
S[Q] = \frac{i\pi\nu}{8} \mathrm{Tr}[D(\hat{\nabla}Q)^2 + \eta F_{ij}Q\hat{\nabla}_iQ\hat{\nabla}_jQ + 4 i \Omega Q]
\,,
where :math:`\hat{\nabla}_j Q = \partial_j Q - i[\mathcal{A}_j, Q]` and
:math:`\Omega = \epsilon\tau_3 + i\Delta`,
:math:`\Delta^\dagger = \Delta`. We’d like the discretized functional to
be also gauge invariant.
We subdivide the space to cells, centered on a rectangular lattice
:math:`\vec{r}_j=(h j_x, h j_y, h j_z)` with
:math:`j_{x,y,z}\in\mathbb{Z}` and *h* are the lattice spacings. See the
figure below:
.. figure:: img/discretization.svg
:alt: FIG 1. Lattice discretization.
FIG 1. Lattice discretization
We choose :math:`Q(j) = Q(\vec{r}_j)` to be values of :math:`Q` at the
lattice sites. We define the Wilson link matrices
:math:`U_{ij} =U_{ji}^{-1} =U(\vec{r}_{i},\vec{r}_{j}) = \mathrm{Pexp}[i\int_{L(\vec{r}_i,\vec{r}_j)}d\vec{r}'\,\cdot\vec{\mathcal{A}}(\vec{r}')]`
where :math:`L(\vec{r}_i,\vec{r}_j)` is straight line from
:math:`\vec{r}_j` to :math:`\vec{r}_i` and :math:`\mathrm{Pexp}` is the
path-ordered integral. Since the gauge field :math:`\vec{\mathcal{A}}`
is fixed, the link matrices can be computed ahead of time.
We want a discretized theory that is invariant under the discrete gauge
transformation
.. math:: Q(j) \mapsto u(j) Q u(j)^{-1} \,,
and
.. math::
U_{ij}
\mapsto
u(i)U_{ij}u(j)^{-1}
\,,
and in the continuum limit reduces to the original theory. Such
discretization has the advantage of being insensitive to the gauge
choice, and having exact discrete conservation laws associated with the
symmetry.
Derivative term
---------------
We discretize the derivatives a
.. math::
D_{ij}
:=
\frac{1}{h}[Q(i) U_{ij} - U_{ij} Q(j)]
\,,
\qquad
\Rightarrow
Q(i) D_{ij} = - D_{ij} Q(j)
\,,
which satisfies an anticommutation relation analogous to continuum
derivative.
We then discretize
.. math::
S_2
=
\mathrm{Tr} D(\hat{\nabla} Q)^2
\mapsto
-D \frac{2d}{|\mathrm{neigh}|} h^d \sum_{\mathrm{neigh}(i,j)}\mathrm{Tr} D_{ij} D_{ji}
where :math:`d` is the space dimension and ``|neigh|`` the number of
neighbors on the grid.
Expanding in small :math:`h` we have
.. math::
S_2
\simeq
\frac{h^{d}d}{|\mathrm{neigh}|}
\sum_{i}\sum_{j\in{}\mathrm{neigh}(i)}\mathrm{Tr}\Bigl(\frac{Q(i) - Q(j)}{h} - [i\mathcal{A}(\frac{\vec{r}_i+\vec{r}_j}{2}), Q(j)]\Bigr)^2
\,,
which verifies the reduction to the continuum limit.
Field strength
--------------
The link matrices have the usual continuum limit expansion
.. math::
U_{ji}
=
\sum_{n=0}^\infty i^n |\vec{r}_i-\vec{r}_j|^n \int_{-1/2}^{1/2}ds_1\int_{-1/2}^{s_1}ds_2\ldots\int_{-1/2}^{s_{n-1}}ds_n\, \mathcal{A}(s_1)\cdots\mathcal{A}(s_n)
where
:math:`\mathcal{A}(s)=\mathcal{A}[(\frac{1}{2} - s)\vec{r}_i + (\frac{1}{2} + s)\vec{r}_j]`.
As well known in lattice QCD, the field strength can be expressed in
terms of the link matrices :math:`U` via a plaquette loop. Consider then
the plaquette (see Fig. 1), with edges :math:`j\leftarrow{}i` and
:math:`k\leftarrow{}l` in direction :math:`\mu` and
:math:`l\leftarrow{}i`, :math:`k\leftarrow{}j` in direction :math:`\nu`,
and expand around
:math:`\vec{r}_0=\frac{\vec{r}_i+\vec{r}_j+\vec{r}_k+\vec{r}_l}{4}`:
.. math::
P_{lkji}
=
U_{il}U_{lk}U_{kj}U_{ji}
\,,
.. math::
\Rightarrow
P_{lkji}
\simeq
1
- i h^2
\bigl(
\partial_\mu \mathcal{A}_\nu - \partial_\nu \mathcal{A}_\mu - i[\mathcal{A}_\mu,\mathcal{A}_\nu]
\bigr)
+
\mathcal{O}(h^3)
=
1
+ i h^2 F_{\mu \nu}(\vec{r}_0)
+
\mathcal{O}(h^3)
which then allows expressing :math:`F_{ij}` in terms of the link
matrices.
Hall term
---------
In the plaquette of Fig. 1 we have
.. math::
\mathrm{tr}
F_{\mu\nu}Q \hat{\nabla}_\mu Q \hat{\nabla}_\nu Q
\rvert_{\mu=ji, \nu=li}
\simeq
\frac{i}{h^4}
\mathrm{tr}
(1 - P_{lkji}) U_{ij} D_{ji} Q(i) D_{il} U_{li}
.. math::
=
\frac{-i}{h^4}
\mathrm{tr}
(U_{lk} U_{kj} - U_{li} U_{ij}) D_{ji} Q(i) D_{il}
So the structure is :math:`\frac{-i}{h^4}` :math:`\times` (gauge factors
for counterclockwise loop :math:`-` return back clockwise)
:math:`\times` :math:`DQD` for three sites in counterclockwise order.
Since only :math:`Q(i)`, :math:`Q(j)`, :math:`Q(l)` appear here, one can
also think of this as a plaquette corner with :math:`Q(i)` being the
corner.
To do the sum over the :math:`\mu`, :math:`\nu` indices, one can average
over corners of the plaquettes as follows, which gives the final
discretization of the Hall term:
.. math::
S_H
=
\eta
\mathrm{Tr} F_{\mu\nu}Q\hat{\nabla}_\mu Q\hat{\nabla}_\nu Q
.. math::
S_H
\mapsto
\eta
\frac{h^d d}{4ih^4}
\sum_{\mathrm{plaqc}(ijkl; i)}
\mathrm{Tr}
\Bigl(
(U_{lk} U_{kj} - U_{li} U_{ij}) D_{ji} Q(i) D_{il}
\Bigr)
.. math::
S_H
=
\frac{\eta h^{d-4} d}{4i}
\sum_{\mathrm{plaqc}(i; jkl)}
\mathrm{Tr}
\Bigl(
(P_{lkji} - 1)(Q(i) - U_{ij} Q(j) U_{ji} Q(i) U_{il} Q(l) U_{li})
\Bigr)
\,,
where :math:`\mathrm{plaqc}(i;jkl)` are the plaquette corners. The last
line follows with direct calculation, and considering the cyclic
permutations.
Matrix current
--------------
We can also calculate the matrix current :math:`j\mapsto{}i` between
neighboring cells:
.. math::
J_{ij}^a
=
\frac{\partial}{\partial\xi}
S[
U_{ij}
\mapsto
U_{ij}
+
T^a U_{ij}\xi
\,,
\;
U_{ji}
\mapsto
U_{ji}
-
U_{ji} T^a \xi
]\rvert_{\xi=0}
where :math:`T^a` is an appropriate matrix generator, and the
transformation is inserted to only one of the links.
The local gauge invariance now implies that there is an exact discrete
continuity equation
.. math::
\sum_{j\in{}N_i} J_{ij}^a
=
\frac{\partial}{\partial\xi}
S[Q(i)\mapsto{}e^{\xi T^a}Q(i)e^{-\xi T^a}]\rvert_{\xi=0}
=
\frac{\partial}{\partial\xi}
S[\Omega(i)\mapsto{}e^{-\xi T^a}\Omega(i)e^{\xi T^a}]\rvert_{\xi=0}
=
R_{i}^a
\,.
This is one main advantage of the procedure, in addition to its
invariance vs. gauge fixing.
However, the definition of the matrix current is *asymmetric*, so that
:math:`J_{ij}\ne{}-J_{ji}` as they differ by a quantity of order
:math:`\mathcal{O}(h^2)`. The asymmetry arises when :math:`T^a` does not
commute with :math:`U_{ij}`, so it is an issue only for the SU(2)
component. The problem is only in the interpretation.
The currents measured at sites *i* and *j* are different: since the
parallel transport of spin between these two locations can imply
rotation due to the gauge field, there’s no reason why we should have
:math:`J_{ij}=-J_{ji}`, except in the continuum limit where the two
points become close to each other.
Implementation
--------------
The above is sufficient for numerical implementation: it is not
necessary to find out the saddle point equation symbolically. It can be
deduced by parametrizing first so that :math:`Q^2=1` and then using
automatic differentiation (AD). Only a routine evaluating the value of
the action is then needed, in terms of the AD dual variables.
When looking for the saddle point in equilibrium, the problem becomes
even simpler; *Q* has then additional restrictions, and we are looking
for the minimum of the free energy *S*.
doc/index.rst 0 → 100644
+ 20
− 0
View file @ b51b2de4
.. usadelndsoc documentation master file, created by
sphinx-quickstart on Tue Apr 23 13:16:43 2024.
You can adapt this file completely to your liking, but it should at least
contain the root `toctree` directive.
usadelndsoc
===========
.. toctree::
:maxdepth: 2
:caption: Contents:
discretization
Indices and tables
==================
* :ref:`genindex`
* :ref:`modindex`
* :ref:`search`
doc/pandoc/convert.sh deleted 100755 → 0
+ 0
− 8
View file @ b724ce47
#!/bin/bash
PDIR=$(dirname "$0")
pandoc --from markdown --to html \
--embed-resources --standalone \
--lua-filter $PDIR/gitlab-math.lua \
--lua-filter $PDIR/fix-links.lua \
--katex \
-o "${1/.md/.html}" "$1"
doc/pandoc/fix-links.lua deleted 100644 → 0
+ 0
− 44
View file @ b724ce47
-- see if the file exists
function file_exists(file)
local f = io.open(file, "rb")
if f then f:close() end
return f ~= nil
end
-- 1. Replaces .md with .html
-- 2. Replaces absolute paths with relative ones
-- 3. Replaces folder links with links to their index / readme
function fix_link (url)
-- Replace md with html
fixed_url = url:gsub("%.md", ".html")
-- Replace project-root-relative (i.e. /path/to/thing.md) paths with relative ones
fixed_url = fixed_url:gsub("^/", (
function(s)
basedir = io.popen("realpath --relative-to=\"$(dirname \"" .. s:sub(2) .. "\")\" .", 'r'):read('*a')
return basedir:gsub("%s", "") .. "/" .. s:sub(2)
end
))
-- Allowed readme / index names
dir_name = {} -- values are functions that return the path to the new link
dir_name["index.md"] = (function(s) return s:gsub("/?$", "/index.html") end)
dir_name["INDEX.md"] = (function(s) return s:gsub("/?$", "/INDEX.html") end)
dir_name["readme.md"] = (function(s) return s:gsub("/?$", "/readme.html") end)
dir_name["README.md"] = (function(s) return s:gsub("/?$", "/README.html") end)
-- Is the url pointing to a folder?
file_or_folder = io.popen("[ -d '" .. fixed_url .. "' ] && echo 'folder'", 'r'):read('*a')
-- If it's a folder, find the readme / index and replace the link with that.
if file_or_folder:gsub("%s", "") == "folder" then
for index,mk_path in pairs(dir_name) do
if file_exists(fixed_url .. "/" .. index) then
fixed_url = mk_path(fixed_url)
break
end
end
end
return fixed_url
end
function Link(link) link.target = fix_link(link.target); return link end
function Image(img) img.src = fix_link(img.src); return src end
-- return {{Meta = Meta}, {Link = Link, Image = Image}}
\ No newline at end of file
doc/pandoc/gitlab-math.lua deleted 100644 → 0
+ 0
− 18
View file @ b724ce47
function Math(el)
if el.mathtype == "InlineMath" then
if el.text:sub(1,1) == '`' and el.text:sub(#el.text) == '`' then
local text = el.text:sub(2,#el.text-1)
return pandoc.Math(el.mathtype, text)
else
local cont = pandoc.read(el.text)
return el
--return { pandoc.Str("$") } .. cont.blocks[1].content .. { pandoc.Str("$") }
end
end
end
function CodeBlock(el)
if el.classes[1] == "math" then
return pandoc.Para({ pandoc.Math("DisplayMath", el.text) })
end
end
meson.build
+ 1
− 1
View file @ b51b2de4
......@@ -41,7 +41,6 @@ incdir_numpy = run_command(py3,
],
check: true
).stdout().strip()
cc = meson.get_compiler('cpp')
numpy_dep = declare_dependency(
compile_args : ['-DNPY_NO_DEPRECATED_API=NPY_1_9_API_VERSION'],
include_directories : [include_directories(incdir_numpy)],
......@@ -66,3 +65,4 @@ deps = [cppad_dep, eigen3_dep]
#executable('main2', ['src/xtest.f95', 'src/adjac/adjac.f95'])
subdir('usadelndsoc')
subdir('doc')
usadelndsoc/meson.build
+ 1
− 1
View file @ b51b2de4
py3.extension_module(
usadelndsoc_pymod = py3.extension_module(
'_core',
['../src/core.cpp'],
dependencies : deps + [numpy_dep, py3_dep, pybind11_dep],
......
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment