def __call__(self, link, staple, mask):
verbose = g.default.is_verbose(
"metropolis"
) # need verbosity categories [ performance, progress ]
project_method = self.params["project_method"]
step_size = self.params["step_size"]
number_accept = 0
possible_accept = 0
t = g.timer("metropolis")
t("action")
action = g.component.real(g.eval(-g.trace(link * g.adj(staple)) *
mask))
t("lattice")
V = g.lattice(link)
V_eye = g.identity(link)
t("random")
self.rng.element(V, scale=step_size, normal=True)
t("update")
V = g.where(mask, V, V_eye)
link_prime = g.eval(V * link)
action_prime = g.component.real(
g.eval(-g.trace(link_prime * g.adj(staple)) * mask))
dp = g.component.exp(g.eval(action - action_prime))
rn = g.lattice(dp)
t("random")
self.rng.uniform_real(rn)
t("random")
accept = dp > rn
accept *= mask
number_accept += g.norm2(accept)
possible_accept += g.norm2(mask)
link @= g.where(accept, link_prime, link)
t()
g.project(link, project_method)
# g.message(t)
if verbose:
g.message(
f"Metropolis acceptance rate: {number_accept / possible_accept}"
)
def left_increment(dst, src_left, scale):
dst = g.util.to_list(dst)
src_left = g.util.to_list(src_left)
group = dst[0].otype
algebra = group.cartesian()
assert src_left[0].otype.__name__ == algebra.__name__
for src_left_mu, dst_mu in zip(src_left, dst):
if group.__name__ != algebra.__name__:
dst_mu @= group.compose(
g.project(g.convert(g(scale * src_left_mu), group), "defect"),
dst_mu)
else:
dst_mu @= group.compose(g(scale * src_left_mu), dst_mu)
def __call__(self, link, staple, mask):
verbose = g.default.is_verbose(
"su2_heat_bath"
) # need verbosity categories [ performance, progress ]
project_method = self.params["project_method"]
# params
niter = self.params["niter"]
# temporaries
grid = link.grid
u2 = g.lattice(grid, g.ot_matrix_su_n_fundamental_group(2))
u2_eye = g.identity(u2)
one = g.identity(g.complex(grid))
zero = g.complex(grid)
zero[:] = 0
eps = g.complex(grid)
eps[:] = grid.precision.eps * 10.0
xr = [g.complex(grid) for i in range(4)]
a = [g.complex(grid) for i in range(4)]
two_pi = g.complex(grid)
two_pi[:] = 2.0 * np.pi
accepted = g.complex(grid)
d = g.complex(grid)
V_eye = g.identity(link)
# pauli
pauli1, pauli2, pauli3 = tuple([g.lattice(u2) for i in range(3)])
ident = g.identity(u2)
pauli1[:] = 1j * np.array([[0, 1], [1, 0]], dtype=grid.precision.complex_dtype)
pauli2[:] = 1j * np.array([[0, 1j], [-1j, 0]], dtype=grid.precision.complex_dtype)
pauli3[:] = 1j * np.array([[1, 0], [0, -1]], dtype=grid.precision.complex_dtype)
# counter
num_sites = round(g.norm2(g.where(mask, one, zero)))
# shortcuts
inv = g.component.pow(-1.0)
# go through subgroups
for subgroup in link.otype.su2_subgroups():
V = g.eval(link * g.adj(staple))
# extract u2 subgroup following Kennedy/Pendleton
link.otype.block_extract(u2, V, subgroup)
u2 @= u2 - g.adj(u2) + g.identity(u2) * g.trace(g.adj(u2))
udet = g.matrix.det(u2)
adet = g.component.abs(udet)
nzmask = adet > eps
u2 @= g.where(nzmask, u2, u2_eye)
udet = g.where(nzmask, udet, one)
xi = g.eval(0.5 * g.component.sqrt(udet))
u2 @= 0.5 * u2 * inv(xi)
# make sure that su2 subgroup projection worked
assert g.group.defect(u2) < u2.grid.precision.eps * 10.0
xi @= 2.0 * xi
alpha = g.component.real(xi)
# main loop
it = 0
num_accepted = 0
accepted[:] = 0
d[:] = 0
while (num_accepted < num_sites) and (it < niter):
self.rng.uniform_real(xr, min=0.0, max=1.0)
xr[1] @= -g.component.log(xr[1]) * inv(alpha)
xr[2] @= -g.component.log(xr[2]) * inv(alpha)
xr[3] @= g.component.cos(g.eval(xr[3] * two_pi))
xr[3] @= xr[3] * xr[3]
xrsq = g.eval(xr[2] + xr[1] * xr[3])
d = g.where(accepted, d, xrsq)
thresh = g.eval(one - d * 0.5)
xrsq @= xr[0] * xr[0]
newly_accepted = g.where(xrsq < thresh, one, zero)
accepted = g.where(mask, g.where(newly_accepted, newly_accepted, accepted), zero)
num_accepted = round(g.norm2(g.where(accepted, one, zero)))
it += 1
if verbose:
g.message(f"SU(2)-heatbath update needed {it} / {niter} iterations")
# update link
a[0] @= g.where(mask, one - d, zero)
a123mag = g.component.sqrt(g.component.abs(one - a[0] * a[0]))
phi, cos_theta = g.complex(grid), g.complex(grid)
self.rng.uniform_real([phi, cos_theta])
phi @= phi * two_pi
cos_theta @= (cos_theta * 2.0) - one
sin_theta = g.component.sqrt(g.component.abs(one - cos_theta * cos_theta))
a[1] @= a123mag * sin_theta * g.component.cos(phi)
a[2] @= a123mag * sin_theta * g.component.sin(phi)
a[3] @= a123mag * cos_theta
ua = g.eval(a[0] * ident + a[1] * pauli1 + a[2] * pauli2 + a[3] * pauli3)
b = g.where(mask, g.adj(u2) * ua, ident)
link.otype.block_insert(V, b, subgroup)
link @= g.where(accepted, V * link, link)
# check
check = g.where(accepted, ua * g.adj(ua) - ident, 0.0 * ident)
delta = (g.norm2(check) / g.norm2(ident)) ** 0.5
assert delta < grid.precision.eps * 10.0
check = g.where(accepted, b * g.adj(b) - ident, 0.0 * ident)
delta = (g.norm2(check) / g.norm2(ident)) ** 0.5
assert delta < grid.precision.eps * 10.0
check = g.where(accepted, V * g.adj(V) - V_eye, 0.0 * V_eye)
delta = (g.norm2(check) / g.norm2(V_eye)) ** 0.5
assert delta < grid.precision.eps * 10.0
# project
g.project(link, project_method)
################################################################################
# Test projection schemes on promoting SP to DP group membership
################################################################################
V0 = g.convert(rng.element(g.mcolor(grid_sp)), g.double)
for method in ["defect_left", "defect_right"]:
V = g.copy(V0)
I = g.identity(V)
I_s = g.identity(g.complex(grid_dp))
for i in range(3):
eps_uni = (g.norm2(g.adj(V) * V - I) / g.norm2(I))**0.5
eps_det = (g.norm2(g.matrix.det(V) - I_s) / g.norm2(I_s))**0.5
g.message(
f"Before {method} iteration {i}, unitarity defect: {eps_uni}, determinant defect: {eps_det}"
)
g.project(V, method)
assert eps_uni < 1e-14 and eps_det < 1e-14
################################################################################
# Test SU(2) fundamental and conversion to adjoint
################################################################################
rng = g.random("test")
for eps_ref, grid in [(1e-6, grid_sp), (1e-12, grid_dp)]:
g.message(
f"Test SU(2) fundamental and adjoint conversion on grid {grid.precision.__name__}"
)
U = [g.matrix_su2_fundamental(grid) for i in range(2)]
rng.element(
def projected_convert(x, otype):
return g.project(g.convert(x, otype), "defect")
group_defect = g.group.defect(Vt_split[t])
g.message(f"Distance to group manifold: {group_defect}")
if group_defect > 1e-12:
g.message(
f"Time slice {t} on split grid {Vt_split[t].grid.srank} has group_defect = {group_defect}"
)
sys.exit(1)
g.message("Unsplit")
g.unsplit(Vt, Vt_split, cache)
g.message("Project to group (should only remove rounding errors)")
Vt = [g.project(vt, "defect") for vt in Vt]
g.message("Test")
# test results
for t in range(Nt):
f = g.qcd.gauge.fix.landau([Usep[mu][t] for mu in range(3)])
dfv = f.gradient(Vt[t], Vt[t])
theta = g.norm2(dfv).real / Vt[t].grid.gsites / dfv.otype.Nc
g.message(f"theta[{t}] = {theta}")
g.message(f"V[{t}][0,0,0] = ", Vt[t][0, 0, 0])
if theta > p_theta_eps or np.isnan(theta):
g.message(f"Time slice{t} did not converge: {theta} >= {p_theta_eps}")
sys.exit(1)
# merge time slices
def update_field(P, U, ep):
for Pmu, Umu in zip(P, U):
Umu @= g.project(g.matrix.exp(g(ep * Pmu)), "defect") * Umu