qlat.topology — Topology Charge and Plaquette Action Density

Source: qlat/qlat/topology.pyx

Note: Update this document when updating the source file.

Outline

  1. Overview

  2. Plaquette Action Density

  3. Topological Charge

  4. Examples

Overview

topology provides measurements of the plaquette action density and topological charge on gauge field configurations. Two definitions of the topological charge are available:

  • Clover-leaf (_clf variants): uses the basic clover discretization of the field strength tensor.

  • 5-loop improved (default variants): uses the 5-loop improved definition from arXiv:hep-lat/9701012, which reduces lattice artifacts.

The plaquette action density functions compute the sum over all six plaquettes at each site of (1 - 1/3 * Re Tr U_P), which is the standard Wilson action density. The total action is beta * total_volume * action_density.

For a single instanton the action is 8 * pi^2 / g^2, and beta = 6 / g^2.

Plaquette Action Density

gf_plaq_action_density_field

gf_plaq_action_density_field(gf: GaugeField) -> FieldRealD

Return a FieldRealD with multiplicity == 1 containing the plaquette action density at each site:

paf(x) = sum_P (1 - 1/3 * Re Tr U_P)

where the sum runs over the 6 plaquettes originating at site x. The total action is:

Action = beta * total_volume * paf_glb_sum / total_volume

gf_plaq_action_density

gf_plaq_action_density(gf: GaugeField) -> float

Return the spatially averaged plaquette action density as a scalar. Equivalent to:

gf_plaq_action_density_field(gf).glb_sum()[:].item() / total_volume

gf_spatial_plaq_action_density_field

gf_spatial_plaq_action_density_field(gf: GaugeField) -> FieldRealD

Return a FieldRealD with multiplicity == 1 containing the spatial plaquette action density at each site. Only spatial plaquettes (those not involving the time direction) are included in the sum.

gf_spatial_plaq_action_density

gf_spatial_plaq_action_density(gf: GaugeField) -> float

Return the spatially averaged spatial-plaquette action density as a scalar.

Topological Charge

gf_topology_field_clf

gf_topology_field_clf(gf: GaugeField) -> FieldRealD

Return a FieldRealD with multiplicity == 1 containing the topological charge density at each site using the basic clover-leaf discretization of the field strength tensor. This is not the improved definition.

gf_topology_clf

gf_topology_clf(gf: GaugeField) -> float

Return the total topological charge as a scalar, using the basic clover-leaf definition. Equivalent to:

gf_topology_field_clf(gf).glb_sum()[:].item()

gf_topology_field

gf_topology_field(gf: GaugeField) -> FieldRealD

Return a FieldRealD with multiplicity == 1 containing the topological charge density at each site using the 5-loop improved definition from arXiv:hep-lat/9701012 Eq. (2-7). This definition reduces discretization errors compared to the basic clover-leaf method.

gf_topology

gf_topology(gf: GaugeField) -> float

Return the total topological charge as a scalar, using the 5-loop improved definition. Equivalent to:

gf_topology_field(gf).glb_sum()[:].item()

gf_topology_terms_field

gf_topology_terms_field(gf: GaugeField) -> FieldRealD

Return a FieldRealD with multiplicity == 5 containing the five individual loop terms that sum to give the 5-loop improved topological charge density. Useful for analyzing the contribution of each term. The sum over the last axis equals gf_topology_field:

gf_topology_terms_field(gf)[:].sum(-1) == gf_topology_field(gf)[:].ravel()

gf_topology_terms

gf_topology_terms(gf: GaugeField) -> np.ndarray

Return a NumPy array of shape (5,) and dtype float64 containing the globally summed values of each of the five loop terms. The sum of the array equals gf_topology(gf).

Examples

Plaquette Action Density

import qlat as q

size_node_list = [
    [1, 1, 1, 1],
    [1, 1, 1, 2],
    [1, 1, 1, 4],
    [1, 1, 1, 8],
]

q.begin_with_mpi(size_node_list)

total_site = q.Coordinate([4, 4, 4, 8])
geo = q.Geometry(total_site)
gf = q.GaugeField(geo)
rng = q.RngState("seed")
gf.set_rand(rng)

pa = q.gf_plaq_action_density(gf)
print(f"Plaquette action density: {pa}")

pa_field = q.gf_plaq_action_density_field(gf)
print(f"Per-site multiplicity: {pa_field.multiplicity}")

q.end_with_mpi()

Topological Charge (5-loop improved)

import qlat as q

size_node_list = [
    [1, 1, 1, 1],
    [1, 1, 1, 2],
    [1, 1, 1, 4],
    [1, 1, 1, 8],
]

q.begin_with_mpi(size_node_list)

total_site = q.Coordinate([4, 4, 4, 8])
geo = q.Geometry(total_site)
gf = q.GaugeField(geo)
rng = q.RngState("seed")
gf.set_rand(rng)

top = q.gf_topology(gf)
print(f"Topological charge (5-loop): {top}")

q.end_with_mpi()

Comparing Clover-Leaf and 5-loop Topology

import qlat as q

size_node_list = [
    [1, 1, 1, 1],
    [1, 1, 1, 2],
    [1, 1, 1, 4],
    [1, 1, 1, 8],
]

q.begin_with_mpi(size_node_list)

total_site = q.Coordinate([4, 4, 4, 8])
geo = q.Geometry(total_site)
gf = q.GaugeField(geo)
rng = q.RngState("seed")
gf.set_rand(rng)

top_clf = q.gf_topology_clf(gf)
top_5loop = q.gf_topology(gf)

print(f"Clover-leaf topology: {top_clf}")
print(f"5-loop improved:      {top_5loop}")

q.end_with_mpi()

Decomposing the 5-loop Terms

import qlat as q

size_node_list = [
    [1, 1, 1, 1],
    [1, 1, 1, 2],
    [1, 1, 1, 4],
    [1, 1, 1, 8],
]

q.begin_with_mpi(size_node_list)

total_site = q.Coordinate([4, 4, 4, 8])
geo = q.Geometry(total_site)
gf = q.GaugeField(geo)
rng = q.RngState("seed")
gf.set_rand(rng)

terms = q.gf_topology_terms(gf)
print(f"5-loop terms: {terms}")
print(f"Sum of terms: {terms.sum()}")
print(f"Direct 5-loop: {q.gf_topology(gf)}")

q.end_with_mpi()

Spatial Plaquette Action Density

import qlat as q

size_node_list = [
    [1, 1, 1, 1],
    [1, 1, 1, 2],
    [1, 1, 1, 4],
    [1, 1, 1, 8],
]

q.begin_with_mpi(size_node_list)

total_site = q.Coordinate([4, 4, 4, 8])
geo = q.Geometry(total_site)
gf = q.GaugeField(geo)
rng = q.RngState("seed")
gf.set_rand(rng)

pa_total = q.gf_plaq_action_density(gf)
pa_spatial = q.gf_spatial_plaq_action_density(gf)

print(f"Total action density:   {pa_total}")
print(f"Spatial action density: {pa_spatial}")

q.end_with_mpi()