FLUE_SU2_heatbath

SU(2) heatbath update

Key points:

• No stdlib random routines are used.
• getLambda2() uses philox_uniform_oo on (0,1),
• getXVec() uses philox_uniform_co on [-1,1)
• The update is staged by direction and colour so the inner site loop can be expressed as do concurrent.

Source: src/gauge_SU2/heatbath.f90

Dependencies

Contents

• SU2_updateLinks
• stapleAt
• getLambda2
• getXVec
• constructXMatrix
• su2_updated_link

Subroutines

SU2_updateLinks

Update all SU(2) links on the lattice.

The update is staged by:

• direction mu
• checkerboard parity colour

so that each do concurrent block updates an independent set of links.

Inputs: U : current lattice beta : gauge coupling master_key : user-provided Philox master key sweep_id : identifies the sweep so keys vary deterministically

Output: UUpdated : updated lattice

subroutine SU2_updateLinks(U, beta, UUpdated, master_key, sweep_id)

Arguments

Name	Type	Intent	Attributes	Description
U	complex(kind=WC)	in	dimension(:, :, :, :, :, :, 😃	Gaugefield [2,2,4,nt,nx,ny,nz]
beta	real(kind=WP)	in		beta-value
UUpdated	complex(kind=WC)	inout	dimension(:, :, :, :, :, :, 😃	Output updated gaugefield [2,2,4,nt,nx,ny,nz]
master_key	integer(kind=C64)	in	dimension(2)	Philox Key
sweep_id	integer	in		A counter for Philox

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stapleAt

Compute the standard Wilson-type staple for an SU(2) link.

The staple is evaluated at the link U_mu(coord), and is the sum of the forward and backward staples in all transverse directions nu != mu.

Attributes: pure

subroutine stapleAt(U, V, coord, mu)

Arguments

Name	Type	Intent	Attributes	Description
U	complex(kind=WC)	in	dimension(:, :, :, :, :, :, 😃	Input gaugefield [2,2,4,nt,nx,ny,nz]
V	complex(kind=WC)	out	dimension(2, 2)	Output Wilson-type staple
coord	integer	in	dimension(4)	Staple is at this coordinate
mu	integer	in		Staple is in this direction

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Functions

getLambda2

Sample lambda^2 for the SU(2) heatbath accept/reject step.

uses Philox draws directly.

Random-number conventions:

• ri(1:3) and rr are strictly in (0,1), so we use philox_fill_uniform_oo.

Counter usage: counter(1) = site id (passed in through counter0) counter(2) = subgroup id (set in constructXMatrix) counter(3) = attempt number (incremented here) counter(4) = phase id = 1 (lambda2 phase)

Attributes: pure

Returns: real(kind=WP)

function getLambda2(alpha, beta, key, counter0) result(lambda2)

Arguments

Name	Type	Intent	Attributes	Description
alpha	real(kind=WP)	in		alpha and beta parameters (beta is fixed for given sim.)
beta	real(kind=WP)	in		alpha and beta parameters (beta is fixed for given sim.)
key	integer(kind=C64)	in	dimension(2)	Philox keys
counter0	integer(kind=C64)	in	dimension(4)	Philox keys

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getXVec

Sample a 3-vector uniformly from the unit ball, then rescale it so that ||xvec|| = sqrt(1 - x0^2).

Random-number convention:

• Each component is drawn from [-1,1), so we use philox_fill_uniform_co.

Counter usage: counter(1) = site id counter(2) = subgroup id counter(3) = attempt number counter(4) = phase id = 2 (x-vector phase)

Attributes: pure

Returns: real(kind=WP)

function getXVec(x0, key, counter0) result(xvec)

Arguments

Name	Type	Intent	Attributes	Description
x0	real(kind=WP)	in		The zeroth random number that sets the condition on the 3-vec of random numbers
key	integer(kind=C64)	in	dimension(2)	Philox key
counter0	integer(kind=C64)	in	dimension(4)	Philox keys

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constructXMatrix

Construct the SU(2) heatbath matrix X used in the update.

This wraps together:

1. sampling lambda^2,
2. constructing x0 = 1 - 2 lambda^2,
3. sampling the spatial part x(1:3),
4. building the corresponding SU(2) matrix.

subgroup_id is carried in counter(2) so the three SU(2) sub-updates inside an SU(3) heatbath update use disjoint Philox substreams.

Attributes: pure

Returns: complex(kind=WC)

function constructXMatrix(alpha, beta, key, counter0, subgroup_id) result(X)

Arguments

Name	Type	Intent	Attributes	Description
alpha	real(kind=WP)	in		alpha and beta parameters (beta fixed for given sim.)
beta	real(kind=WP)	in		alpha and beta parameters (beta fixed for given sim.)
key	integer(kind=C64)	in	dimension(2)	Philox keys
counter0	integer(kind=C64)	in	dimension(4)	Philox Keys
subgroup_id	integer	in		Describes the SU2 subgroup

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su2_updated_link

Compute the new SU(2) link at one site and one direction.

Steps:

1. Compute the staple V.
2. Compute alpha from det(V).
3. Build the heatbath matrix X using Philox.
4. Return the updated link: U' = X * (V / alpha)^dagger

If alpha is too small, the code falls back to a mild update with alpha=1.

Attributes: pure

Returns: complex(kind=WC)

function su2_updated_link(U, beta, coord, mu, key, dims) result(Unew)

Arguments

Name	Type	Intent	Attributes	Description
U	complex(kind=WC)	in	dimension(:, :, :, :, :, :, 😃	Input gaugefield [2,2,4,nt,nx,ny,nz]
beta	real(kind=WP)	in		beta-value
coord	integer	in	dimension(4)	Base coordinate of link to update [it,ix,iy,zi]
mu	integer	in		Direction to update
key	integer(kind=C64)	in	dimension(2)	Philox key
dims	integer	in	dimension(4)	Lattice dimensions [nt,nx,ny,nz]

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