# Bulk-preventing actions for SU(N) gauge theories

Tobias Rindlisbacher<EMAIL_ADDRESS>Albert Einstein Center for
Fundamental Physics & Institute for Theoretical Physics, University of Bern,
Sidlerstrasse 5, CH-3012 Bern, Switzerland  Kari Rummukainen
<EMAIL_ADDRESS>Ahmed Salami<EMAIL_ADDRESS>Department
of Physics & Helsinki Institute of Physics, P.O. Box 64, FI-00014 University
of Helsinki, Finland

###### Abstract

Lattice gauge field theories may suffer from unphysical ”bulk” phase
transitions at strong lattice gauge coupling. We introduce a one-parameter
family of lattice $\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
gauge actions which, when used in combination with an HMC update algorithm,
prevents the appearance of the bulk phase transition. We briefly discuss the
(presumed) mechanism behind the prevention of the bulk transition and present
test results for different
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ gauge groups.

##  I Introduction

In asymptotically free gauge theories on the lattice the continuum limit is
obtained when the bare lattice gauge coupling vanishes. In practice lattice
simulations are always done at finite lattice spacing, and as long as the
coupling constant is sufficiently small we can analytically extrapolate the
results to the continuum limit. The range of lattice spacings (or coupling
constants) is often limited by the emergence of an unphysical “bulk phase” at
strong lattice gauge coupling, which prevents the analytical connection to the
continuum phase from this region. The value of the lattice coupling where the
transition to the bulk phase happens depends on the lattice gauge group, the
choice of the gauge action and the matter content.

The problem of the bulk phase transition becomes acute in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ gauge theories with
large number of colors and in models with large number of fermion degrees of
freedom. In pure gauge
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ theories with the
standard Wilson plaquette action the bulk transition is a rapid cross-over if
$N\leq 4$ but becomes an increasingly strong first order transition for $N\geq
5$ [1]. Adding fermionic degrees of freedom slows down the evolution of the
coupling constant (i.e. the magnitude of the $\beta$-function is smaller) and
also increase the effective lattice gauge coupling [2]. Depending on the
physical case of interest, these effects require one to use strong bare
coupling. This happens especially in infrared (near-)conformal models, where
the coupling runs very slowly, for example in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$ with large numbers
of fundamental fermions [3, 4, 5, 6, 7],
$\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$ with adjoint
fermions [8, 9, 10],
$\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$ with large Nf [11,
12, 13], and $\operatorname{SU}\mathopen{}\mathclose{{}\left(4}\right)$ with
fermions in the antisymmetric representation [14].

In this work we present a local lattice gauge action, defined on elementary
plaquettes, which efficiently removes the transition to the bulk phase. Our
approach is related to the ”dislocation prevention” method by DeGrand et al.
[15]. For $\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ and
$\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$ gauge groups also
the topological gauge actions that restict the plaquette magnitude [16, 17,
18] are to some extent related. The latter stops being the case for
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ gauge groups with
$N>2$, as the plaquette eigenvalues are then no longer determined by the trace
of the plaquette. We note that our approach removes merely the bulk phase but
does not restrict in any way gauge-topology fluctuations.

Following Wilson’s prescription, the lattice discretization of a
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ gauge theory is
obtained by promoting the Lie algebra valued continuum gauge field,

$A_{\mu}\mathopen{}\mathclose{{}\left(x^{\prime}}\right)=\sum_{a}A^{a}_{\mu}\mathopen{}\mathclose{{}\left(x^{\prime}}\right)\,T^{a}\,\in\,\mathfrak{su}\mathopen{}\mathclose{{}\left(N}\right)\
,$ (1)

with
$\mathopen{}\mathclose{{}\left\\{}\right.T^{a}\mathopen{}\mathclose{{}\left.}\right\\}_{a=1,\ldots,N^{2}-1}$
being a basis of $\mathfrak{su}\mathopen{}\mathclose{{}\left(N}\right)$, to
Lie group valued link variables,

$U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)\,=\,\mathcal{P}\operatorname{e}^{\mathrm{i}\,\int_{ax}^{a\,(x+\widehat{\mu})}\mathrm{d}{x^{\prime}}\,A_{\mu}(x^{\prime})}\,\in\,\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)\
,$ (2)

which can be interpreted as the gauge-parallel transporters along the link
between a site $x$ and a neighboring site $x+\widehat{\mu}$. The leading
$\mathcal{P}$ on the right-hand side of (2) indicate that path-ordering should
be applied when evaluating the exponential of the line-integral. The relation
between the coordinate $x^{\prime}\in\mathbb{R}^{4}$ in (1) and the coordinate
$x\in\mathbb{Z}^{4}$ in (2) is given by $x^{\prime}=a\,x$, where $a$ is the
lattice spacing, and $\widehat{\mu}$ refers to the unit-vector in
$\mu$-direction. Parallel transporters over longer distances are then
expressed as products of consecutive link variables and a lattice gauge action
can be defined in terms of link variables by requiring that in the limit
$\mathopen{}\mathclose{{}\left(a\to 0}\right)$ the lattice gauge action
converges to the continuum gauge action,

$S_{G}=\frac{1}{2\,g_{0}^{2}}\int\mathrm{d}^{4}x^{\prime}\operatorname{tr}\mathopen{}\mathclose{{}\left(F_{\mu\nu}\mathopen{}\mathclose{{}\left(x^{\prime}}\right)F_{\mu\nu}\mathopen{}\mathclose{{}\left(x^{\prime}}\right)}\right)\
.$ (3)

Wilson proposed the gauge action [19]

$S_{G,W}=\frac{\beta}{N}\sum\limits_{x}\sum\limits_{\mu<\nu}\operatorname{Re}\operatorname{tr}\bigl{(}\big{.}\mathbbm{1}-U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}\
,$ (4)

which, as is well known, satisfies the above condition and is here written in
terms of the inverse bare gauge coupling $\beta=2\,N/g_{0}^{2}$ and the
plaquette variables

$U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)=U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)\,U_{\nu}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\,U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x+\widehat{\nu}}\right)\,U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x}\right)\
.$ (5)

The gauge action (4) and improved versions of it [20] are the most commonly
used in Monte Carlo studies of
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ lattice gauge
theories. They are, however, not unique and might not be the best choice for
the study of lattice gauge theories at strong coupling, as they allow the
gauge system to enter the bulk phase. Despite its name, the bulk phase is not
necessarily a proper phase, but simply a region in parameter space of the
lattice theory where lattice artifacts dominate in ensemble averages. As a
consequence the relation between lattice and continuum results becomes very
complicated or can even be lost completely if bulk and continuum phase are
separated by a first order transition.

##  II Avoiding the lattice bulk phase

In this section we propose a characterization of ”bulk configurations” in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ lattice gauge
systems, which allows for the definition of a family of lattice gauge actions,
that separate such bulk configurations from regular ones by an infinite
potential barrier, while still yielding the same naive continuum limit as
Wilson’s plaquette gauge action. In combination with a hybrid Monte Carlo
(HMC) update algorithm, the new gauge actions prevent the gauge system from
entering a bulk phase.

###  II.1 Motivation in
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$

In the $\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ case, the
Wilson gauge action in (4) reduces to

$S_{G,W}=\beta\,\sum\limits_{x}\sum\limits_{\mu<\nu}\operatorname{Re}\bigl{(}\big{.}1-U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}\
,$ (6)

and the Abelian link variables can be written as

$U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)=\operatorname{e}^{\mathrm{i}\,\theta_{x,\mu}}\quad\text{with}\quad\theta_{x,\mu}=a\,A_{\mu}\mathopen{}\mathclose{{}\left(x}\right)\,\in\,\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]\
.$ (7)

Let us now define,

$\Theta_{x,\mu\nu}=\theta_{x,\mu}+\theta_{x+\widehat{\mu},\nu}-\theta_{x+\widehat{\nu},\mu}-\theta_{x,\nu}\,\in\,\mathopen{}\mathclose{{}\left(-4\pi,4\pi}\right]\
,$ (8)

and note that while $\Theta_{x,\mu\nu}$ in (8) can vary in the interval
$\mathopen{}\mathclose{{}\left(-4\pi,4\pi}\right]$, the gauge action (6)
depends only on

$\arg\mathopen{}\mathclose{{}\left(U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)\,\in\,\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]\
.$ (9)

As illustrated in Fig. 1, the gauge action (6) produces a bulk-transition at
$\beta=\beta_{b}\approx 1$. For $\beta<\beta_{b}$, the system is in the bulk
phase, where the lattice spacing, $a$, can be considered large and
$\Theta_{x,\mu\nu}$ from (8) explores the full
$\mathopen{}\mathclose{{}\left(-4\pi,4\pi}\right]$-interval. For
$\beta>\beta_{b}$, the system is in the continuum phase, where the lattice
spacing tends to zero if
$\mathopen{}\mathclose{{}\left(\beta\to\infty}\right)$. In this phase,
$\Theta_{x,\mu\nu}$ can still be outside the
$\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval, but the fraction of
such plaquettes quickly drops as
$\mathopen{}\mathclose{{}\left(\beta\to\infty}\right)$ and most of the time,
one has that
$\Theta_{x,\mu\nu}=\arg\mathopen{}\mathclose{{}\left(U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)$
.

Figure 1: The $\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$
lattice gauge theory with Wilson gauge action (6) undergoes a bulk-transition
at $\beta=\beta_{b}\approx 1$. For $\beta<\beta_{b}$, the system is in the
bulk phase, where the lattice spacing, $a$, can be considered large and
$\Theta_{x,\mu\nu}$ from (8) explores the full
$\mathopen{}\mathclose{{}\left(-4\pi,4\pi}\right]$-interval. For
$\beta>\beta_{b}$, the system is in the continuum phase, where the lattice
spacing tends to zero if $\beta\to\infty$. In this phase, $\Theta_{x,\mu\nu}$
can still be outside the
$\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval, but the fraction of
such plaquettes quickly drops as $\beta$ is increased and one mostly has
$\Theta_{x,\mu\nu}=\arg\mathopen{}\mathclose{{}\left(U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)$
.

  1. (a)

link wraps around $\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval:

$x\phantom{\widehat{\mu}}$$x+\widehat{\mu}$$x+\widehat{\mu}+\widehat{\nu}$$x+\widehat{\nu}\vphantom{\widehat{\mu}}$$\Longrightarrow$$x\phantom{\widehat{\mu}}$$x+\widehat{\mu}$$x+\widehat{\mu}+\widehat{\nu}$$x+\widehat{\nu}\vphantom{\widehat{\mu}}$

  2. (b)

$\Theta_{x,\mu\nu}$ grows continuously:

$x\phantom{\widehat{\mu}}$$x+\widehat{\mu}$$x+\widehat{\mu}+\widehat{\nu}$$x+\widehat{\nu}\vphantom{\widehat{\mu}}$$\Longrightarrow$$x\phantom{\widehat{\mu}}$$x+\widehat{\mu}$$x+\widehat{\mu}+\widehat{\nu}$$x+\widehat{\nu}\vphantom{\widehat{\mu}}$

Figure 2: The figure illustrates the two different ways in which
$\Theta_{x,\mu\nu}$ form (8) can grow beyond the
$\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval: a) a single link
(between $x$ and $x+\widehat{\mu}$) winds around
$\mathopen{}\mathclose{{}\left(\pi,\pi}\right]$ which adds almost $2\,\pi$ to
$\Theta_{x,\mu\nu}$; b) no link wraps around the
$\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval, $\Theta_{x,\mu\nu}$
grows continuously bigger than $\pi$.

The fact that plaquettes with
$\Theta_{x,\mu\nu}\notin\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$ also
appear in the continuum phase indicates that the value of $\Theta_{x,\mu\nu}$
by itself cannot be used to distinguish bulk from continuum configurations.
However, as illustrated in Fig. 2, for smoothly varying link variables,
$U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)$, one can identify two
qualitatively different ways in which plaquettes with
$\Theta_{x,\mu\nu}\notin\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$ can be
produced from a configuration in which initially
$\Theta_{x,\mu\nu}=\arg\mathopen{}\mathclose{{}\left(U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)$
is satisfied for all plaquettes:

  1. (a)

the phase of one of the links of a given plaquette can move across the
boundary of the $\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval and
wrap around, which adds roughly $\pm 2\,\pi$ to the plaquette’s phase angle
$\Theta_{x,\mu\nu}$. As indicated in part (a) of Fig. 2, this kind of
plaquette wrapping can happen also if
$\arg\mathopen{}\mathclose{{}\left(U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)$
is close to $0$. We note that the wrapping link will produce a shift of
roughly $\pm 2\,\pi$ in the phase angle $\Theta_{x,\mu\nu}$ of all plaquettes
that contain this link and not just in the phase angle of the plaquette under
consideration;

  2. (b)

none of the links of a given plaquette wraps around the
$\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval. Instead, as
indicated in part (b) of Fig. 2, the plaquette’s phase angle
$\Theta_{x,\mu\nu}$ has grown close to the
$\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval boundary and finally
crosses it continuously as one of the link variables undergoes a change that
causes $\Theta_{x,\mu\nu}$ to grow further. Unlike in case (a), this
continuous plaquette wrapping can take place for single plaquettes. This can
produce metastabilities as wrapped and unwrapped plaquettes pull in opposite
directions on shared links when the action is minimized. In order to approach
the continuum limit, the wrapping needs to be undone.

As case (b) can occur only if individual plaquette angles are allowed to grow
close to $\pm\pi$, this case is likely to occur only in the bulk phase, where
$\beta$ is so small that the plaquette action (6) cannot grow sufficiently
large as to oppose the entropy-driven randomization of the plaquette variables
in the system. We will therefore introduce in Sec. II.3 a family of actions
that will prevent plaquette wrappings of type (b). As it turns out, this is
sufficient to get rid of the bulk transition. First, however, we discuss in
Sec. II.2 how the concept of continuous type (b) plaquette wrappings can be
generalized from $\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ to
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ plaquettes. The
relation between plaquette wrappings and the production of Dirac monopoles is
discussed in Appendix A.1.

###  II.2 Situation in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$

Compact Lie groups like
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ carry a Riemannian
metric $g$ which is invariant under left and right translation. As a
consequence, the well known bi-invariant Haar measure for group integration
exists, which is given by the volume form of the invariant metric, normalized
so that that the group volume is $1$. The fact that
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ has a Riemannian
metric also means that we can at each point
$U\in\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ define a
tangent space
$\mathrm{T}_{U}\\!\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
and the so called exponential map
$\exp_{U}\,:\,\mathrm{T}_{U}\\!\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)\to\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
that maps vectors
$\omega\in\mathrm{T}_{U}\\!\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
from the tangent space at point $U$ to points
$V=\exp_{U}\mathopen{}\mathclose{{}\left(\omega}\right)$ in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$. The graph of the
function
$\gamma_{U,\omega}\mathopen{}\mathclose{{}\left(t}\right)=\exp_{U}\mathopen{}\mathclose{{}\left(\omega\,t}\right)$,
$t\in\mathopen{}\mathclose{{}\left[}\right.0,1\mathopen{}\mathclose{{}\left.}\right]$
then describes a geodesic segment in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ that start in the
point $U$ in the direction specified by $\omega$ and has geodesic length
$|\omega|_{g\mathopen{}\mathclose{{}\left(U}\right)}$.

The exponential map
$\exp_{U}\,:\,\mathrm{T}_{U}\\!\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)\to\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
is bijective only in a neighborhood
$\Omega_{U}\subset\mathrm{T}_{U}\\!\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
around the origin of
$\mathrm{T}_{U}\\!\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$.
The boundary $\partial\Omega_{U}$ of the region
$\Omega_{U}\subset\mathrm{T}_{U}\\!\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
over which the exponential map $\exp_{U}$ is bijective is called the cut locus
of $U$ in
$\mathrm{T}_{U}\\!\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
and its image
$\exp_{U}\mathopen{}\mathclose{{}\left(\partial\Omega_{U}}\right)$ is,
correspondingly, the cut locus of $U$ in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$.

At the point
$U=\mathbbm{1}\in\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ the
tangent space to $\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
corresponds to the usual Lie algebra
$\mathfrak{su}\mathopen{}\mathclose{{}\left(N}\right)=\mathrm{T}_{\mathbbm{1}}\\!\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
and the exponential map applied to some
$\omega\in\mathfrak{su}\mathopen{}\mathclose{{}\left(N}\right)$ can be
identified with the usual matrix exponential,

$\exp_{\mathbbm{1}}\mathopen{}\mathclose{{}\left(\omega}\right)=\exp\mathopen{}\mathclose{{}\left(\mathrm{i}\,\omega^{a}\,T^{a}}\right)\
,$ (10)

with
$\mathopen{}\mathclose{{}\left\\{}\right.T_{a}\mathopen{}\mathclose{{}\left.}\right\\}_{a=1,\ldots,N^{2}-1}$
being a hermitian basis of
$\mathfrak{su}\mathopen{}\mathclose{{}\left(N}\right)$.

The domain
$\Omega_{\mathbbm{1}}\subset\mathfrak{su}\mathopen{}\mathclose{{}\left(N}\right)=\mathrm{T}_{\mathbbm{1}}\\!\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
over which $\exp_{\mathbbm{1}}$ is bijective is given by the set of hermitian
matrices $\omega\in\mathfrak{su}\mathopen{}\mathclose{{}\left(N}\right)$ whose
eigenvalues
$\mathopen{}\mathclose{{}\left\\{}\right.\phi_{i}\mathopen{}\mathclose{{}\left.}\right\\}_{i=1,\ldots,N}$
satisfy [21]:

$\mathopen{}\mathclose{{}\left|\phi_{i}}\right|<\pi\quad\forall\,i\in\mathopen{}\mathclose{{}\left\\{}\right.1,\ldots,N\mathopen{}\mathclose{{}\left.}\right\\}\
.$ (11)

The image
$\exp_{\mathbbm{1}}\mathopen{}\mathclose{{}\left(\partial\Omega_{\mathbbm{1}}}\right)\subset\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$,
i.e. the cut locus of $\mathbbm{1}$ in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$, is therefore given
by the set of matrices
$V\in\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ for which at
least one eigenvalue $\lambda_{i}=\operatorname{e}^{\mathrm{i}\,\phi_{i}}$,
$i=1,\ldots,N$ is equal to $-1$.

To illustrate this, we pick a
$\omega\in\mathfrak{su}\mathopen{}\mathclose{{}\left(N}\right)$ and write it
as

$\omega=U_{\omega}D_{\omega}U_{\omega}^{\dagger}\ ,$ (12)

with $U_{\omega}\in\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
and
$D_{\omega}=\operatorname{diag}\mathopen{}\mathclose{{}\left(\phi_{1},\ldots,\phi_{N}}\right)$
being a diagonal matrix containing the eigenvalues of $\omega$. The image
$V=\exp_{\mathbbm{1}}\mathopen{}\mathclose{{}\left(\omega}\right)$ of $\omega$
under the exponential map $\exp_{\mathbbm{1}}$ is then,

$V=U_{\omega}\exp\mathopen{}\mathclose{{}\left(\mathrm{i}\,D_{\omega}}\right)U_{\omega}^{\dagger}=U_{\omega}D_{V}U_{\omega}^{\dagger}\
,$ (13)

with
$D_{V}=\operatorname{diag}\mathopen{}\mathclose{{}\left(\lambda_{1},\ldots,\lambda_{N}}\right)$
being the diagonal matrix of eigenvalues of $V$, given by
$\lambda_{i}=\operatorname{e}^{\mathrm{i}\,\phi_{i}}$,
$i\in\mathopen{}\mathclose{{}\left\\{}\right.1,\ldots,N\mathopen{}\mathclose{{}\left.}\right\\}$.
As long as $-\pi<\phi_{i}<\pi$ $\forall
i\in\mathopen{}\mathclose{{}\left\\{}\right.1,\ldots,N\mathopen{}\mathclose{{}\left.}\right\\}$,
the exponential map is a bijection as the inverse map can be defined using,

$\phi_{i}=-\mathrm{i}\,\log\mathopen{}\mathclose{{}\left(\lambda_{i}}\right)\quad,\quad
i=1,\ldots,N\ ,$ (14)

with $\log$ being the principal value complex logarithm. But, if
$\mathopen{}\mathclose{{}\left|\phi_{i}}\right|\geq\pi$ for any
$i\in{1,\ldots,N}$, then (14) breaks down.

We can now generalize the criterion we found in the previous section for the
production of bulk configurations in
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ lattice gauge theory
to the $\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ case. In the
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ case we argued, that
if the phase (8) of any plaquette variable
$U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)$ in the system crosses
continuously the $\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval
boundary (plaquette wrappping of type (b)), then a bulk configuration is
produced. Such a continuous crossing of the
$\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval boundary by the
plaquette phase implies of course that the corresponding plaquette variable
$U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)$ passes through the value
$-1$, which is the cut locus for the exponential map from the Lie algebra
$\mathrm{u}\mathopen{}\mathclose{{}\left(1}\right)=\mathrm{T}_{1}\\!\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)=\mathbb{R}$
to the group manifold
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$. If a plaquette
variable $U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)$ approaches and
smoothly crosses the cut locus value $-1$ the resulting lattice gauge field
described by the link variables can no longer be properly mapped on a
corresponding Lie algebra valued gauge field that is relevant for continuum
physics. The same applies in the case of
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ lattice gauge
theory if a $\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
plaquette variable approaches and smoothly cross the
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ cut locus. The
resulting configuration cannot be mapped on a corresponding Lie algebra valued
gauge configuration that is relevant for continuum physics, which qualifies it
as a bulk configuration.

In order to prevent a
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ gauge system form
entering a bulk phase at strong coupling, the plaquette variables must not
continuously cross the cut locus. As discussed below (11), this means that the
eigenvalues of the plaquette variables must not continuously cross the value
$-1$.

The question of how such a restriction affects the creation of magnetic
monopoles in $\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ gauge
theory is addressed in Appendix A.2.

###  II.3 Bulk-preventing action

To prevent plaquettes from having eigenvalues close to $-1$, we introduce the
following family of gauge actions:

$S_{G,b}\,=\,\frac{2\,\gamma}{n\,N}\sum\limits_{x}\sum\limits_{\mu<\nu}\operatorname{tr}\bigl{(}\big{.}\bigl{(}\big{.}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{-n}-\mathbbm{1}\big{.}\bigr{)}\
,$ (15)

with $n\geq 1$ and

$\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)=\bigl{(}\big{.}\mathbbm{1}+\underbrace{U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}_{\mathclap{\text{plaquette}}}\big{.}\bigr{)}/2.$
(16)

The form of the actions (15) was inspired by the dislocation-prevention action
introduced in [15]. The naive continuum limit of (15) is the same as for the
Wilson gauge action $S_{G,W}$. This can be seen by writing
$U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)=\exp\bigl{(}\big{.}\mathrm{i}\,{s}\,F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}$,
with
${s}\,F^{\prime}_{\mu\nu}={a^{2}}\,F_{\mu\nu}+\mathcal{O}\big{(}a^{3}\big{)}$
and expanding in a power series in $s$. For the local plaquette action
contributing to $S_{G,W}$ from (4) one then finds:

$\operatorname{Re}\operatorname{tr}\mathopen{}\mathclose{{}\left(\mathbbm{1}-U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)={s^{2}}\operatorname{tr}\mathopen{}\mathclose{{}\left(F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)/2\\\
-{s^{4}}\frac{1}{12}\begin{cases}\bigl{(}\big{.}\operatorname{tr}\bigl{(}\big{.}F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}/2\big{.}\bigr{)}^{2}&\text{if}\
N<4\\\ \
\,\operatorname{tr}\bigl{(}\big{.}\bigl{(}\big{.}F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{2}\big{.}\bigr{)}/2&\text{if}\
N\geq 4\end{cases}\\\ +\mathcal{O}\big{(}s^{6}\big{)}\ ,$ (17)

and, correspondingly, for the local plaquette action contributing to $S_{G,b}$
from (15):

$\frac{2}{n}\operatorname{tr}\bigl{(}\big{.}\bigl{(}\big{.}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{-n}-\mathbbm{1}\big{.}\bigr{)}={s^{2}}\operatorname{tr}\mathopen{}\mathclose{{}\left(F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)/2\\\
+{s^{4}}\frac{1+3\,n}{24}\begin{cases}\bigl{(}\big{.}\operatorname{tr}\bigl{(}\big{.}F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}/2\big{.}\bigr{)}^{2}&\text{if}\
N<4\\\ \
\,\operatorname{tr}\bigl{(}\big{.}\bigl{(}\big{.}F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{2}\big{.}\bigr{)}/2&\text{if}\
N\geq 4\end{cases}\\\ +\mathcal{O}\big{(}{s^{6}}\big{)}\ .$ (18)

In the limit $\mathopen{}\mathclose{{}\left(a\to 0}\right)$, one has $s\sim
a^{2}$, $F^{\prime}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\sim
F_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)$, and we see that the two
local actions have the same leading term $\sim a^{4}$, namely:

$a^{4}\,\operatorname{tr}\mathopen{}\mathclose{{}\left(F_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)F_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)/2\
.$ (19)

The actions (15) introduce an infinite potential barrier between bulk and
continuum configurations. This is sufficient to ensure that, if we start a
simulation from a cold configuration (all link variables equal to the
identity) and use a hybrid Monte Carlo (HMC) algorithm to update the gauge
system, no bulk-configurations will be produced.

One could infer that our bulk-preventing setup results in a non-ergodic update
algorithm. However, one should keep in mind that the part of the configuration
space that is not sampled is irrelevant for the continuum limit of the theory.
The algorithm prevents ensemble averages of the lattice system from being
contaminated (or even dominated) by bulk-configurations, which should allow
one to extract continuum physics also at stronger coupling. The same effect
could be achieved by defining a modified measure, which gives zero weight to
bulk configurations. However, this would be difficult to implement as the
latter are hard to identify once they are created. The use of an action (15)
in combination with an HMC algorithm is a proxy to achieve the same effect but
in a simpler and more economic way. An expression for the gauge force,
required for the HMC, is presented in Appendix B.

Let us note that for $\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$
and $\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$, the actions in
(15) alone would have a similar effect as the topological actions discussed in
[22, 16, 17, 18]: the larger the inverse bare coupling $\gamma$, the stronger
the plaquette values are repelled from $-1$ resp. $-\mathbbm{1}$. For
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ with $N>2$ the
effect of (15) is, however, different from the one of the topological action
discussed in [18], as for $N>2$ the trace of the plaquette does no-longer
completely determine the plaquette eigenvalues. An action which can have a
similar effect as our actions (15) has been given in [23]. We also note that
based on the observation that the bulk transition of the Wilson gauge action
(4) is sensitive to the addition of an adjoint plaquette term [24], it has in
[25] been demonstrated that for $N=3$ the bulk phase can be avoided by adding
an appropriately scaled negative adjoint plaquette term to the standard Wilson
gauge action [26, 25].

While for us it is desirable that the actions (15) do not prevent topology
from fluctuating, there have also been attempts to find actions which keep the
topology fixed [27].

##  III Results

Figure 3: Comparison of pure
$\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$ gauge theory
results obtained with the Wilson gauge action (WG), Eq. (4) (orange circles
and brown triangles) and the bulk-preventing (BP) action, Eq. (15), for $n=2$
(black diamonds and blue squares). To guide the eye, the data points are
connected by straight lines. The first row shows the real part of the average
traced plaquette (top left), the temporal Polyakov loop (top center), and the
temporal Polyakov loop variance (top right). The second row shows the
topological susceptibility (bottom left) the gradient flow coupling at
$c=\lambda/N_{s}=0.3$ (bottom center), with flow scale $\lambda=\sqrt{8\,t}$,
corresponding to flow time $t$, and $\lambda_{0}/a$ (bottom right), which is
the flow scale $\lambda$ (in lattice units) at which
$t^{2}\,\mathopen{}\mathclose{{}\left\langle
E\mathopen{}\mathclose{{}\left(t}\right)}\right\rangle=0.26$ with
$\mathopen{}\mathclose{{}\left\langle
E\mathopen{}\mathclose{{}\left(t}\right)}\right\rangle$ being the clover
action density of the flowed gauge field at flow time $t$. On our finite
lattices with spatial size $V_{s}=12^{3}$, the temporal size is set to
$N_{t}=6$ for finite temperature, and to $N_{t}=24$ for zero-temperature. The
results are shown as functions of $\beta-1.65$ (WG) resp. $\gamma$ (BP).

To test whether the bulk-preventing actions (15) deserve their name and
whether they are able to reproduce the same weak coupling results as the
Wilson gauge action, we carried out simulations with pure gauge
$\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$, pure gauge
$\operatorname{SU}\mathopen{}\mathclose{{}\left(5}\right)$, and
$\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$ with $N_{f}=4$
Wilson fermion flavours. With the Wilson gauge action, all three of these
theories enter a bulk phase for sufficiently small values of the inverse gauge
coupling $\beta$. For pure gauge
$\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$, the transition is
a smooth cross-over, while for pure gauge
$\operatorname{SU}\mathopen{}\mathclose{{}\left(5}\right)$ and for the
fermionic $\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$ theory
with sufficiently large fermion hopping parameter $\kappa$ the transition is
of first order. In the following we will discuss the three cases separately.
We use the version of the action (15) with $n=2$. The choice of $n$ should not
affect physical results, but it turns out that a too small value of $n$ will
require also a smaller step size in the HMC trajectories to achieve similar
acceptance rates, which can become computationally more expensive than using
$n=2$.

According to the expansions (17) and (18), the Wilson gauge (WG) action (4)
and bulk-preventing (BP) action (15) agree only to order $s^{2}\sim a^{4}$.
Thus, the inverse bare couplings $\beta$ and $\gamma$ will not be equal in the
weak coupling limit. However, it turns out that locally the two couplings can
be related quite accurately by a constant shift, $\gamma=\beta-\Delta\beta$,
which we will use to directly compare bare lattice results obtained with the
two different actions. Of course, there is in general no need for bare lattice
results, obtained with different actions, to agree. However, it seems that in
the present case, the systems controlled by the WG and the BP action behave in
the weak coupling regime sufficiently equally, so that a direct comparison of
bare lattice results is reasonable.

###  III.1 $\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$ pure
gauge

$\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$ pure gauge theory
with the WG action (4) is known to have a smooth cross-over between the bulk
phase and the continuum-like phase. Thus, for this theory the BP action (15)
is not expected to provide any significant advantage over the WG action and
both actions should give rise to the same results not just at weak, but also
all the way down to strong coupling.

In Fig. 3 we compare results obtained with the WG and the BP action. The WG
data is plotted as function of $\beta-\Delta\beta$ with $\Delta\beta=1.65$ and
the BP data is plotted as function of $\gamma$. The shift $\Delta\beta=1.65$
has been determined by requiring that the ”spatial deconfinement” transition,
at which the spatial Polyakov loop develops a non-zero expectation value
(indicating that the physical spatial volume becomes too small to fit a
meson), occurs for the two actions at the same value of $(\beta-\Delta\beta)$
resp. $\gamma$.

The top-left panel in Fig. 3 shows the average of the traced plaquette and we
note that when plotted against $\beta-1.65$ resp. $\gamma$, the plaquette
values for the two different actions agree remarkable well at sufficiently
weak coupling. Only below $\gamma=\beta-1.65\approx 0.65$, where the WG action
enters the bulk-phase, the plaquette value for the WG action starts to deviate
from the BP one as function of $\gamma=\beta-1.65$.

The strong coupling limit is for both actions obtained by sending their
respective inverse coupling to zero, i.e.
$\mathopen{}\mathclose{{}\left(\gamma\to 0}\right)$ and
$\mathopen{}\mathclose{{}\left(\beta\to 0}\right)$. With the WG action the
strong coupling phase extends over the interval
$\beta\in\mathopen{}\mathclose{{}\left[}\right.0,2.3\mathopen{}\mathclose{{}\left.}\right]$,
while with the BP action, the strong coupling phase extends over the
significantly smaller interval
$\gamma\in\mathopen{}\mathclose{{}\left[}\right.0,0.65\mathopen{}\mathclose{{}\left.}\right]$.
In the strong coupling phase, the system should therefore with the WG action
change more slowly as function of $\beta$ than with the BP action as function
of $\gamma$.

This is indeed what can be observed in the remaining panels of Fig. 3: the
results obtained with the two actions for temporal Polyakov loop (top-center),
Polyakov loop variance (top-right), and topological susceptibility (bottom-
left) are consistent and match for $\gamma=\beta-1.65>0.65$ very nicely as
functions of $\gamma$ resp. $\beta-1.65$, while for $\gamma=\beta-1.65<0.65$,
the WG results change more slowly as function of $\beta$ than the
corresponding BP results do as function of $\gamma$.

The last two panels on the second row of Fig. 3 show gradient flow quantities:
the bottom-center panel shows the gradient flow coupling,
$N\,g_{\mathrm{GF}}^{2}\mathopen{}\mathclose{{}\left(c}\right)$ at
$c=\lambda/N_{s}=0.3$, with $\lambda=\sqrt{8\,t}$ being the flow scale
corresponding to flow time $t$; and the bottom-right panel shows
$\lambda_{0}/a$, which is the flow scale $\lambda$ (in lattice units) at which
$t^{2}\,\mathopen{}\mathclose{{}\left\langle
E\mathopen{}\mathclose{{}\left(t}\right)}\right\rangle=0.26$ [28] with
$E\mathopen{}\mathclose{{}\left(t}\right)$ being the clover action density of
the flowed gauge field at flow time $t$. Both gradient flow quantities have
been corrected for leading finite volume and lattice spacing effects [7] by
replacing in the computation of the continuum finite volume correction given
in [29] the continuum momenta by lattice momenta and adapting the formula to
the case $N_{t}\neq N_{s}$. A refined correction formula that takes into
account the details of the utilized lattice actions for sampling and flowing
of the gauge fields, as well as for measuring
$E\mathopen{}\mathclose{{}\left(t}\right)$, is presented in [30]. Also for the
gradient flow quantities the data obtained with the two actions agrees as
function of $\gamma=\beta-1.65$ if $\gamma=\beta-1.65>0.65$, whereas for
$\gamma=\beta-1.65<0.65$, the WG data changes more slowly as function of
$\beta$ than the BP data does as function of $\gamma$.

As mentioned at the beginning of this section, the shift $\Delta\beta=1.65$
has been determined by requiring that the spatial deconfinement happens at the
same value of $\gamma$ and $\beta-\Delta\beta$. Because of our small spatial
volumes of linear size $N_{s}=12$, this happens already at
$\gamma=\beta-1.65>0.8$.

###  III.2 $\operatorname{SU}\mathopen{}\mathclose{{}\left(5}\right)$ pure
gauge

Figure 4: Same as Fig. 3, but for pure gauge
$\operatorname{SU}\mathopen{}\mathclose{{}\left(5}\right)$ and including the
case $N_{t}=4$ for which the WG action can no longer properly resolve the
finite temperature transition as the latter is forced to occur on top of the
bulk transition. The dashed vertical red line indicates the approximate
location of the bulk transition of the WG action. Note that the shown data was
obtained on small lattices with $N_{s}=12$ and we did not attempt to perform
simulations directly at the pseudo critical points. The peaks visible in the
data for the Polyakov loop variance (top-right panel) do therefore not reflect
the true pseudocritcal behavior; the lines simply connect the available data
points to guide the eye.

Fig. 4 provides the same information as Fig. 3 but for
$\operatorname{SU}\mathopen{}\mathclose{{}\left(5}\right)$ instead of
$\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$. The shift in
$\beta$ required to match $\gamma$ at weak coupling has been set to
$\Delta\beta=14.35$, which, as in the
$\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$-case, is determined
by matching the values of $\gamma$ and $\beta-\Delta\beta$ at which spatial
deconfinement occurs.

For $\operatorname{SU}\mathopen{}\mathclose{{}\left(5}\right)$ the bulk
transition of the WG action is of 1st order [1], which is clearly visible from
the sharp discontinuity in the WG data for the average plaquette at
$\beta-14.35\approx 2.3$, shown in the top-left panel of Fig. 4. In contrast,
with the BP action the plaquette is completely continuous as a function of the
inverse gauge coupling $\gamma$. In the continuum phase, i.e. for
$\gamma=\beta-14.35>2.3$, the average plaquette values obtained with the two
different actions converge only slowly with increasing inverse coupling, while
the average temporal Polyakov loop (top-center), temporal Polyakov loop
variance (top-right) and topological susceptibility (bottom-left), as well as
the gradient flow quantities,
$N\,g_{\mathrm{GF}}^{2}\mathopen{}\mathclose{{}\left(c=0.3}\right)$ (bottom-
center) and $\lambda_{0}/a$ (bottom-right), agree for the two actions almost
immediately when $\gamma=\beta-14.35>2.3$.

The finite temperature transition for $N_{t}=6$ occurs at around
$\gamma=\beta-14.35\approx 2.9$. This is above the value at which the WG
action undergoes the bulk transition, and the properties of the finite
temperature transition should therefore be described equally well with the WG
and with the BP action. The temporal Polyakov loop (top-center) looks indeed
the same for the two actions at $N_{t}=6$, and also the temporal Polyakov loop
variance (top-right) agrees very well for $\gamma=\beta-14.35>2.3$ at
$N_{t}=6$; at $\gamma=\beta-14.35\approx 2.3$ (red vertical dashed line) one
can, however, notice a small jump in the Polyakov loop variance for the WG
action, while the Polyakov loop variance obtained with the BP action behaves
completely regular across this point.

A similar behavior can be observed in the $N_{t}=6$ data for the topological
susceptibility (bottom-left), where the results obtained with the WG and BP
action agree for $\gamma=\beta-14.35>2.3$, but as $\gamma=\beta-14.35$
decreases below the bulk transition point, $\gamma=\beta-14.35\approx 2.3$,
the topological susceptibility obtained with the WG action jumps and
approaches almost immediately its strong-coupling plateau value, while with
the BP action, the topological susceptibility approaches its strong coupling
value much more smoothly.

With $N_{t}=4$, the WG action is no longer able to properly resolve the finite
temperature transition. The data obtained with the BP action suggests that the
finite temperature transition should for $N_{t}=4$ occur at
$\gamma=\beta-14.35\approx 2.1$. With the WG action, the system is in the
bulk-phase at this value of the bare gauge coupling [1]. It appears that the
finite temperature transition cannot take place inside the bulk phase and
occurs therefore on top of the bulk transition. Also the topological
susceptibility obtained with the WG action for $N_{t}=4$ appears to be unable
to decrease as long as the system is in the bulk phase. As in the case of
$N_{t}=6$, the topological susceptibility obtained with the WG action appears
also for $N_{t}=4$ to be stuck at the strong coupling plateau value for
$\gamma=\beta-14.35<2.3$ and to decrease abruptly at
$\gamma=\beta-14.35\approx 2.3$ when the inverse coupling is increased beyond
this point. In contrast, with the BP action the asymptotic strong coupling
value of the topological susceptibilities is also for $N_{t}=4$ approached
smoothly.

The measurements of the gradient flow coupling
$N\,g_{\mathrm{GF}}^{2}\mathopen{}\mathclose{{}\left(c}\right)$ at
$c=\lambda/N_{s}=0.3$ are shown in the bottom-center panel of Fig.4. From this
we conclude that the WG action is not capable of reaching gradient flow
couplings larger than $g_{\mathrm{GF}}^{2}\approx 11$ before hitting the bulk
transition. In the bottom-right panel we show the flow scale $\lambda_{0}/a$
at which $t^{2}\,\mathopen{}\mathclose{{}\left\langle
E\mathopen{}\mathclose{{}\left(t}\right)}\right\rangle=0.47$ [28]. For the WG
action there is a discontinuity in $\lambda_{0}/a$ at the bulk transition
point, indicating that there is a largest reachable lattice spacing. For the
BP action these problems disappear and the gradient flow quantities behave
smoothly. As in the $\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$
case, both gradient flow quantities have been corrected for finite volume and
finite lattice spacing effects.

###  III.3 $\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$ with
$N_{f}=4$ Wilson fermions

For a $\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$ lattice gauge
theory with the Wilson gauge action, the transition between continuum- and
bulk-phase is normally a cross-over. However, if the theory is coupled to
fermions, the transition can turn 1st order.

Figure 5: Sketch of the finite temperature phase diagram for
$\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$ lattice gauge
theory with Wilson gauge action and $N_{f}=4$ Wilson clover fermion flavors
(coupling to the gauge field via two-step stout smeared gauge links). The
parameters $\beta$ and $\kappa$ are, respectively, the inverse gauge coupling
and fermion hopping parameter. The red, dashed line marks the location of bulk
transition (resp. crossover if $\kappa$ is sufficiently small) of the Wilson
gauge action, the blue line shows where the PCAC quark mass, $m_{q}$,
vanishes, and the dashed lines in different shades of gray mark the location
of the finite temperature ”deconfinement” transition lines for different
values of $N_{t}$. The locations of the bulk transition and the $m_{q}=0$ line
were estimated with simulations on a $12^{3}\times 24$ lattice, and the
$N_{t}=6$ ”deconfinement” transition line from simulations on a $12^{3}\times
6$ lattice. Note that the $N_{f}=4$ fermions cause the pseudo-critical $\beta$
to be lower than it would be in the pure-gauge case.

Figure 6: Simulation results for
$\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$ lattice gauge
theory, coupled via 2-step stout smeared links to $N_{f}=4$ degenerate Wilson
clover fermion flavors with hopping parameter $\kappa=0.1358$. The simulations
were carried out on lattices of size $12^{3}\times 6$ (finite temp.) resp.
$12^{3}\times 24$ (zero temp.). As in the previous figures, the orange circles
and brown triangles correspond, respectively, to zero and finite temperature
results obtained with the Wilson gauge action (WG), Eq. (4), and the black
diamonds and blue squares to corresponding results obtained with the bulk-
preventing action (BP), Eq. (15). Data points are connected by straight lines
to guide the eye. The first row shows the PCAC quark mass (top left) and
average unsmeared (top center) and smeared (top right) plaquette as functions
of $\gamma$ (BP) resp. $\beta-4.167$ (WG). The remaining rows show the
quantities as functions of the PCAC quark mass (obtained from the $N_{t}=24$
simulations). The second row shows the chiral condensate (middle left), the
temporal Polyakov loop (middle center), and the topological susceptibility
(middle right), and the third row shows the disconnected piece of the chiral
susceptibility (bottom left), variance of the temporal Polyakov loop (bottom
center), and the integrated auto-correlation time of the topological charge
(bottom right). The shaded areas in the different panels mark the PCAC quark
mass range that cannot be resolved with the WG action for the given simulation
parameters, due to the bulk transition.

As a concrete example for a system where this is the case, we consider
$\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$ lattice gauge
theory with the WG action and with $N_{f}=4$ mass-degenerate, dynamical
Wilson-clover fermion flavors, that couple to the gauge field via 2-step stout
smeared links. Fig. 5 shows a schematic
$\mathopen{}\mathclose{{}\left(\beta,\kappa}\right)$-phase diagram for this
system. The dashed red line marks the location of the bulk transition, the
blue line indicates where the PCAC quark mass, $m_{q}$, crosses zero, and the
dashed black line marks the location of the thermal resp.
”confinement/deconfinement” transition if the temporal lattice size is set to
$N_{t}=6$. The label ”confinement/deconfinement” is put in quotation marks,
because we use the temporal Polyakov loop as approximate order parameter for
deconfinement, despite the presence of dynamical fermions [31, 32]. In QCD
deconfinement is observed to be accompanied by a chrial transition that occurs
at the same temperature; for light fermions, this chiral transition dominates,
whereas in the heavy fermion limit, the ”confinement/deconfinement” transition
of pure gauge theory is approached.

The curves in Fig. 5 are based on parameter scans performed with simulations
on lattices of spatial size $V_{s}=N_{s}^{3}$ with $N_{s}=12$. For the
$m_{q}=0$ and the bulk transition lines, the temporal lattice extent was set
to $N_{t}=24$ (approximating zero-temperature), while for the
”confinement/deconfinement” transition, the indicated $N_{t}=6$ was used. The
additional dashed lines in different shades of gray are not based on actual
simulations; they merely illustrate how the ”confinement/deconfinement”
transition line is expected to change if $N_{t}$ is increased (assuming that
also $N_{s}$ is increased accordingly).

For values of $\kappa$ above $\sim 0.13$ the line where the PCAC quark mass
crosses zero coincides with the bulk transition line. Across these coinciding
lines the system undergoes a first order transition and PCAC quark mass never
passes through the value $m_{q}=0$ but jumps discontinuously from positive to
negative values across the transition line. This is shown in the top left
panel of Fig. 6 where the PCAC quark mass for the WG action (brown circles) is
shown as function of $\beta-4.167$ at $\kappa=0.1358$. To the left of the bulk
transition line, the system is in the unphysical bulk phase, while to the
right of the line the system has unphysical negative PCAC quark mass. Thus,
the lattice does not describe any continuum-related physics for $\kappa>0.13$.
Only for $\kappa<0.13$ there is a range in $\beta$ for which the system is in
the continuum phase and the PCAC quark mass is non-negative.

With $N_{t}=6$, also the ”confinement/deconfinement” line in Fig. 5 is for
$\kappa<0.142$ on top of the bulk transition line. In the displayed range of
$\kappa$ the ”confinement/deconfinement” transition separates from the bulk
transition line only for $\kappa>0.142$, but is then located in the negative
PCAC mass region and hence unphysical. To extract information about the
continuum theory form this lattice system, one would have to increase $N_{t}$
(and, correspondingly, also $N_{s}$ to avoid dominance of finite volume
effects) so that the ”confinement/deconfinement” line fully separates from the
bulk transition line. Thus, we can conclude that it is not possible to reach
the light quark confinement (chiral) phase transition with the WG action using
$N_{t}=6$ lattices. Of course, in the limit
$\mathopen{}\mathclose{{}\left(\kappa\to 0}\right)$, where the quark mass
grows much larger than the deconfinement energy scale, the
”confinement/deconfinement” line is expected to separate from the bulk
transition line also for $N_{t}=6$, as the fermions decouple and the system
reduces to pure gauge
$\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$.

Fig. 6 contains also results obtained with the bulk-preventing action (15). In
this case the bulk transition is absent and the PCAC quark mass approaches
$m_{q}=0$ continuously. The small gap in the data around $m_{q}=0$ is due to
the slowing down caused by the appearance of zero eigenmodes of the Wilson-
Dirac operator when $m_{q}\to 0$. This could be avoided by e.g. using
Schrödinger functional boundary conditions, which remove zero modes.

In the first two panels of the second and third row of Fig. 6 the chiral
condensate (2nd row, left) and disconnected chiral susceptibility (last row,
left), as well as the temporal Polyakov loop (2nd row, center) and
corresponding variance (last row, center) are plotted as functions of the PCAC
quark mass, $m_{q}$. While deep in the strong-coupling and deep in the
negative mass phase the results obtained with the WG action agree with those
obtained using the BP action, the discontinuity in $m_{q}$ with the WG action
(marked by the shaded areas) implies that the WG action cannot be used to
study the transition region. On the other hand, with the BP action there is no
discontinuity in $m_{q}$ and no bulk transition, and the behavior of the
chiral condensate and the Polyakov loop at the finite-temperature phase
transition are resolved.

The remaining two panels of Fig. 6, which show the topological susceptibility
(2nd row, right) and integrated auto-correlation time for the topological
charge itself (last row, right), indicate that also when coupled to fermions,
fluctuations of the gauge-topology are not hindered by the use of the BP gauge
action from (15) and HMC updates. For the values of $m_{q}$ which are
accessible with both actions, both actions yield the same results for the
topological susceptibility. The slightly higher integrated auto-correlation
time for the topological charge with the BP action at $N_{t}=6$ is mostly due
to a different tuning of the acceptance rates for the HMC trajectories.

##  IV Conclusions

We have identified a mechanism which appears to be relevant for the formation
of unphysical ”bulk” configurations and the corresponding occurrence of a
”bulk transition” in simulations of lattice
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ gauge theories
using Wilson’s plaquette gauge action. We proposed a one-parameter family of
alternative gauge actions, which possess the same continuum limit as the
Wilson plaquette gauge action but which, when used in combination with an HMC
update algorithm, prevent bulk-configurations from being created.

We tested our bulk-preventing simulation framework for pure gauge
$\operatorname{SU}\mathopen{}\mathclose{{}\left(2}\right)$, pure gauge
$\operatorname{SU}\mathopen{}\mathclose{{}\left(5}\right)$, and for
$\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$ with $N_{f}=4$
mass-degenerate Wilson-clover fermion flavors with hopping parameter
$\kappa=0.1358$, and which are coupled to the gauge field via 2-step stout
smeared link variables. We found that in all three cases, the bulk-preventing
action (15) with $n=2$ removes the bulk transition and reproduces at
sufficiently weak coupling the same results as the Wilson plaquette action.

In the case of the fermionic
$\operatorname{SU}\mathopen{}\mathclose{{}\left(3}\right)$ theory, the Wilson
gauge action could not be used to study the physical finite temperature phase
transition on $N_{t}=6$ lattices at small quark masses. This is due to the
fact that the bulk transition prevents the system from simultaneously reaching
the physical transition region and small quark masses. On the other hand, with
the bulk-preventing action (15) the bulk transition is absent and $m_{q}$ can
be made arbitrarily small. It is also worth noting that the bulk-preventing
actions do not seem to hinder any processes required for topology
fluctuations.

###### Acknowledgements.

The authors acknowledge support from the Academy of Finland grants 308791,
319066, and 345070. T. R. is supported by the Swiss National Science
Foundation (SNSF) through the grant no. TMPFP2_210064. The authors wish to
acknowledge CSC - IT Center for Science, Finland, and the Finnish Computing
Competence Infrastructure (FCCI) for computational resources.

##  Appendix A Plaquette wrappings and magnetic monopole creation

In this appendix we discuss how in
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ lattice gauge theory
the plaquette wrapping types (a) and (b), introduced in Sec. II.1, are related
to the creation of magnetic Dirac monopoles as defined in [33]. The situation
with monopoles in $\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
lattice gauge theory will be addressed in the second part of this appendix.

###  A.1 Situation in
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$

Dirac monopoles in $\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$
lattice gauge theory, as described in [33], live on the dual lattice and are
defined by a non-zero net physical magnetic flux through the boundary of a
spatial cube. If this flux is non-zero for a given cube, then this cube
contains a Dirac monopole. With our conventions, the corresponding operator
can be written as:

$M\mathopen{}\mathclose{{}\left(x}\right)=\frac{1}{2\pi}\epsilon_{4\sigma\mu\nu}\mathopen{}\mathclose{{}\left(\arg\mathopen{}\mathclose{{}\left(U_{\mu\nu}\mathopen{}\mathclose{{}\left(x+\widehat{\sigma}}\right)}\right)-\arg\mathopen{}\mathclose{{}\left(U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)}\right)\
,$ (20)

which is the lattice analogue of the continuum expression,
$M\mathopen{}\mathclose{{}\left(x}\right)=J_{\mathopen{}\mathclose{{}\left(m}\right)}^{0}$,
with

$J_{\mathopen{}\mathclose{{}\left(m}\right)}^{\rho}\mathopen{}\mathclose{{}\left(x}\right)=\epsilon^{\rho\sigma\mu\nu}\partial_{\sigma}F_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\
,$ (21)

i.e.
$M\mathopen{}\mathclose{{}\left(x}\right)=\operatorname{div}\mathopen{}\mathclose{{}\left(B\mathopen{}\mathclose{{}\left(x}\right)}\right)$.
If one defines integer plaquette variables
$n_{x,\mu\nu}\,\in\,\mathopen{}\mathclose{{}\left\\{}\right.-2,-1,0,1,2\mathopen{}\mathclose{{}\left.}\right\\}$,
so that

$\arg\mathopen{}\mathclose{{}\left(U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)}\right)=\Theta_{x,\mu\nu}-2\,\pi\,n_{x,\mu\nu}\
,$ (22)

with $\Theta_{x,\mu\nu}$ from (8), then $n_{x,\mu\nu}$ counts the amount of
Dirac flux passing through the (oriented) plaquette on which
$U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)$ is defined. The magnetic
monopole operator (20) can then also be written as [33]

$M\mathopen{}\mathclose{{}\left(x}\right)=\epsilon_{4\sigma\mu\nu}\mathopen{}\mathclose{{}\left(n_{x+\widehat{\sigma},\mu\nu}-n_{x,\mu\nu}}\right)\
.$ (23)

From (20) or (23) we see, that in order to produce or destroy a magnetic
monopole, one has to change the net Dirac flux through the boundary of a
spatial cube. Considering the plaquette wrapping types (a) and (b) introduced
in Sec. II.1, a change of the net Dirac flux through the boundary of a spatial
cube can only be achieved with plaquette wrappings of type (b). Fig. 7
illustrates how a type (b) plaquette wrapping of a spatial plaquette produces
in the two adjacent spatial cubes a pair of magnetic monopoles of opposite
charge. The monopoles serve, respectively, as source and sink for the Dirac
flux passing through the wrapped plaquette.

Figure 7: If a single spatial plaquette (blue) picks up a non-zero Dirac flux
(red dashed arrow), the two spatial cubes adjacent to the plaquette get,
respectively, occupied by a monopole (black sphere) and an anti-monopole
(white sphere).

Figure 8: If the phase of the displayed spatial link (long black arrow) wraps
around the $\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval, this
implies that the phases of all plaquettes that contain this link also wrap
around the $\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval. For the
spatial plaquettes (left-hand panel) this means that they pick up a Dirac flux
passing through them, which is such that a closed magnetic flux loop (red
dotted arrow loop) is formed that winds around the phase-wrapped link (long
black arrow). For the temporal plaquettes (right-hand panel), the phase-
wrapping does not correspond to the production of Dirac flux as temporal
plaquettes are associated with the electric and not the magnetic field.

With plaquette wrappings of type (a) on the other hand, only closed Dirac
strings can be produced and therefore no monopole/anti-monopole pairs are
created. The reason for this is, that plaquette wrappings of type (a) are
caused by a link variable wrapping around the
$\mathopen{}\mathclose{{}\left(-\pi,\pi}\right]$-interval. Such a link
wrapping will, however, never affect just a single plaquette but all
plaquettes that contain the given link. If this link is a spatial one, it is
contained in spatial as well as in temporal plaquettes. For the spatial
plaquettes the wrapping creates the aforementioned Dirac flux that forms a
closed loop winding around the shared link, illustrated in the left-hand panel
of Fig. 8. For the temporal plaquettes, sketched in the right-hand panel of
Fig. 8, the wrapping does not give rise to the production of Dirac flux as
temporal plaquettes are associated to the electric and not the magnetic field.
For the same reason, no Dirac flux is produced if the wrapping link is a
temporal one, as this would affect only temporal plaquettes.

###  A.2 Situation in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$

In non-abelian $\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
gauge theory magnetic monopoles can be defined in different ways [34, 35, 36,
37] and can be divided into two categories: singular monopoles which require a
Dirac string, and non-singular monopoles which possess a Dirac string only in
certain gauges. On the lattice singular monopoles can be implemented in terms
of net $\mathrm{Z}_{N}$ Dirac flux through the boundary of spatial cubes [38,
39, 40], with the Dirac flux,
$\sim\exp\mathopen{}\mathclose{{}\left(2\,\pi\,\mathrm{i}\,n_{x,\mu\nu}/N}\right)$,
$n_{x,\mu\nu}\in\mathopen{}\mathclose{{}\left\\{}\right.0,\ldots,N-1\mathopen{}\mathclose{{}\left.}\right\\}$,
associated with a plaquette variable
$U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)$, being defined so that in
the factorization

$U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)=\operatorname{e}^{2\,\pi\,\mathrm{i}\,n_{x,\mu\nu}/N}\,\tilde{U}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)$
(24)

one has

$\arg(\operatorname{tr}(\tilde{U}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)))\in\mathopen{}\mathclose{{}\left(-\pi/N,\pi/N}\right]\
.$ (25)

The possible values for the first factor on the right-hand side of (24)
correspond to the center of
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$:

$\mathrm{Z}\mathopen{}\mathclose{{}\left(\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)}\right)=\mathopen{}\mathclose{{}\left\\{}\right.\mathbbm{1}\,\operatorname{e}^{2\,\pi\,\mathrm{i}\,n/N}\mathopen{}\mathclose{{}\left.}\right\\}_{n\in\mathopen{}\mathclose{{}\left\\{}\right.0,\ldots,N-1\mathopen{}\mathclose{{}\left.}\right\\}}\cong\mathrm{Z}_{N}\
.$ (26)

As the above discussed Dirac monopoles in
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ lattice gauge
theory, also the so defined $\mathrm{Z}_{N}$ monopoles decouple in the (naive)
continuum limit $\mathopen{}\mathclose{{}\left(\beta\to\infty}\right)$ of the
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ lattice gauge
theory. Preventing the formation of $\mathrm{Z}_{N}$ monopoles in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ lattice gauge
theory (with WG action) has for $N=2$ been reported to push the bulk phase to
stronger coupling [39]. Preventing not only $\mathrm{Z}_{N}$ monopoles but
also the formation of $\mathrm{Z}_{N}$ Dirac strings from being formed removes
the bulk phase completely [39, 22, 16]. The continuum limit of the model
remains, however, unaffected by these modifications [41].

To make contact with our discussion in Sec. II.2 about the relation between
the production of bulk configurations in
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ lattice gauge
theory and having $\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
plaquettes continuously crossing the cut locus of $\mathbbm{1}$ on
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$, we note that the
non-trivial $\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ center
elements listed in (26) are all located beyond this cut locus for $N>2$ (resp.
on the cut locus for $N=2$). Therefore, by preventing the continuous cut locus
crossing of $\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$
plaquettes, the production of $\mathrm{Z}_{N}$ Dirac flux though individual
plaquettes is not possible, which should also prevent the production of
$\mathrm{Z}_{N}$ monopole/anti-monopole pairs. This is the analogue of the
type (b) plaquette wrapping from Sec. II.1, which is responsible for the
monopole/anti-monopole pair production in
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ lattice gauge
theory, as described in Fig. 7.

We note that this does not imply that the use of our bulk-preventing actions
(15) in combination with HMC updates would prevent the production of
$\mathrm{Z}_{N}$ monopoles in general; only the production of cutoff scale
monopoles is exlucded. Extended $\mathrm{Z}_{N}$ monpoles with ”thick” strings
[42, 43, 44] attached should still be possible.

Also, the production of non-singular
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$ monopoles [45, 46]
is not prevented by our bulk-preventing setup. Non-singular monopoles could on
the lattice be located by gauge fixing the system to maximal abelian gauge
(MAG) [47] and then performing abelian projection (AP) to the Cartan subgroup
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)^{\mathopen{}\mathclose{{}\left(N-1}\right)}$
[48]. In each of the $\mathopen{}\mathclose{{}\left(N-1}\right)$
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ factors of
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)^{\mathopen{}\mathclose{{}\left(N-1}\right)}$
one can then look for Dirac monopoles with the method from [33], discussed in
the first part of this appendix. It is only in this maximum abelian gauge that
non-singular monopoles show up as Dirac monopoles of the projected
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)^{\mathopen{}\mathclose{{}\left(N-1}\right)}$
subgroup; fixing to another gauge before projecting to
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)^{\mathopen{}\mathclose{{}\left(N-1}\right)}$
results in fewer monopoles being found [49] in the
$\operatorname{U}\mathopen{}\mathclose{{}\left(1}\right)$ subgroup factors. We
have not tried to use the MAG-AP method to monitor the monopole densities
directly in our HMC simulations with the BP actions from (15). However, there
is no reason to expect that the BP setup would prevent continuum-relevant
monopoles from being produced. Since finite temperature instantons (calorons)
can be described as composite objects, consisting of multiple monopoles [50],
the bottom-left panels of Figs. 3,4, which illustrate that topology
fluctuations are unhindered by the bulk-prevention, indicate that also the
caloron constituent monopoles can fluctuate.

##  Appendix B Computing the gauge force

We note that the plaquette variables satisfy
$U^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)=U_{\nu\mu}\mathopen{}\mathclose{{}\left(x}\right)$,
so that also the $\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)$ from
Eq. (16) satisfy
$\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)=\Omega_{\nu\mu}\mathopen{}\mathclose{{}\left(x}\right)$.
We can therefore write the bulk-preventing action from Eq. (15) as

$S_{G,b}\,=\,\frac{\gamma}{n\,N}\sum\limits_{x}\sum\limits_{\mu\neq\nu}\operatorname{tr}\bigl{(}\big{.}\bigl{(}\big{.}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{-n}-\mathbbm{1}\big{.}\bigr{)}\
.$ (27)

Let us now denote by $\delta^{a}_{y,\rho}$ the variation with respect to the
link-variable that lives on the link that points from site $y$ in
$\rho$-direction. We then have:

$\delta^{a}_{y,\rho}\,S_{G,b}\,=\,-\frac{\gamma}{n\,N}\sum\limits_{x}\sum\limits_{\mu\neq\nu}\operatorname{tr}\bigl{\\{}\big{.}\bigl{(}\big{.}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{-n}\\\
\cdot\bigl{(}\big{.}\delta^{a}_{y,\rho}\bigl{(}\big{.}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{n}\big{.}\bigr{)}\\\
\cdot\bigl{(}\big{.}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{-n}\big{.}\bigr{\\}}\\\
=\,-\frac{\gamma}{N}\sum\limits_{x}\sum\limits_{\mu\neq\nu}\operatorname{tr}\bigl{\\{}\big{.}\bigl{(}\big{.}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{-\mathopen{}\mathclose{{}\left(n+1}\right)}\\\
\cdot\underbrace{\delta^{a}_{y,\rho}\bigl{(}\big{.}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}}_{\mathclap{\bigl{(}\big{.}\frac{1}{2}\delta^{a}_{y,\rho}U^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)+\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\bigl{(}\big{.}\frac{1}{2}\delta^{a}_{y,\rho}U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}}}\big{.}\bigr{\\}}\\\
=-\frac{\gamma}{2\,N}\sum\limits_{x}\sum\limits_{\mu\neq\nu}\operatorname{tr}\bigl{\\{}\big{.}\bigl{(}\big{.}\delta^{a}_{y,\rho}U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}A_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\\\
+A^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\bigl{(}\big{.}\delta^{a}_{y,\rho}U^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}\big{.}\bigr{\\}}\\\
=-\frac{\gamma}{N}\sum\limits_{x}\sum\limits_{\mu\neq\nu}\operatorname{tr}\bigl{\\{}\big{.}\bigl{(}\big{.}\delta^{a}_{y,\rho}U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}A_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{\\}}\
,$ (28)

where

$A_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)=\bigl{(}\big{.}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{-\mathopen{}\mathclose{{}\left(n+1}\right)}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\\\
=\bigl{(}\big{.}\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}^{-n}\Omega^{-1}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)\
,$ (29)

and we have used that
$A^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)=A_{\nu\mu}\mathopen{}\mathclose{{}\left(x}\right)$
as $\Omega_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)$ and
$\Omega^{\dagger}_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)$ commute
with each other and with their inverses. If we now carry out the variation of
the plaquette explicitly, we find:

$\delta^{a}_{y,\rho}U_{\mu\nu}\mathopen{}\mathclose{{}\left(x}\right)=\\\
\delta_{x,y}\delta_{\mu\rho}\bigl{(}\big{.}\delta^{a}U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}U_{\nu}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x+\widehat{\nu}}\right)U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x}\right)\\\
+\delta_{x+\widehat{\mu},y}\delta_{\nu\rho}U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)\bigl{(}\big{.}\delta^{a}U_{\nu}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\big{.}\bigr{)}U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x+\widehat{\nu}}\right)U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x}\right)\\\
+\delta_{x+\widehat{\nu},y}\delta_{\mu\rho}U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)U_{\nu}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)\bigl{(}\big{.}\delta^{a}U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x+\widehat{\nu}}\right)\big{.}\bigr{)}U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x}\right)\\\
+\delta_{x,y}\delta_{\nu\rho}U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)U_{\nu}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x+\widehat{\nu}}\right)\bigl{(}\big{.}\delta^{a}U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x}\right)\big{.}\bigr{)}\\\
=\mathrm{i}\,\delta_{x,y}\delta_{\mu\rho}T^{a}U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)U_{\nu}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x+\widehat{\nu}}\right)U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x}\right)\\\
+\mathrm{i}\,\delta_{x+\widehat{\mu},y}\delta_{\nu\rho}U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)T^{a}U_{\nu}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x+\widehat{\nu}}\right)U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x}\right)\\\
-\mathrm{i}\,\delta_{x+\widehat{\nu},y}\delta_{\mu\rho}U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)U_{\nu}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x+\widehat{\nu}}\right)T^{a}U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x}\right)\\\
-\mathrm{i}\,\delta_{x,y}\delta_{\nu\rho}U_{\mu}\mathopen{}\mathclose{{}\left(x}\right)U_{\nu}\mathopen{}\mathclose{{}\left(x+\widehat{\mu}}\right)U^{\dagger}_{\mu}\mathopen{}\mathclose{{}\left(x+\widehat{\nu}}\right)U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x}\right)T^{a}\
,$ (30)

where
$\mathopen{}\mathclose{{}\left\\{}\right.T^{a}\mathopen{}\mathclose{{}\left.}\right\\}_{a=1,\ldots,N^{2}-1}$
are the hermitian generators of
$\operatorname{SU}\mathopen{}\mathclose{{}\left(N}\right)$, normalized so that
$\operatorname{tr}\bigl{(}\big{.}T^{a}T^{b}\big{.}\bigr{)}=\delta^{ab}/2$.
Plugging this into Eq. (28), we obtain after some manipulations:

$\delta^{a}_{y,\rho}\,S_{G,b}\,=\,-\frac{2\,\gamma}{N}\sum\limits_{\nu\neq\rho}\operatorname{Re}\bigl{[}\big{.}\operatorname{tr}\bigl{(}\big{.}\mathrm{i}\,T^{a}U_{\rho\nu}\mathopen{}\mathclose{{}\left(y}\right)A_{\rho\nu}\mathopen{}\mathclose{{}\left(y}\right)\big{.}\bigr{)}\\\
+\operatorname{tr}\bigl{(}\big{.}\mathrm{i}\,T^{a}U_{\rho\mathopen{}\mathclose{{}\left(-\nu}\right)}\mathopen{}\mathclose{{}\left(y}\right)A_{\rho\mathopen{}\mathclose{{}\left(-\nu}\right)}\mathopen{}\mathclose{{}\left(y}\right)\big{.}\bigr{)}\big{.}\bigr{]}\
,$ (31)

with

$U_{\mu\mathopen{}\mathclose{{}\left(-\nu}\right)}\mathopen{}\mathclose{{}\left(x}\right)=U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x-\widehat{\nu}}\right)U_{\nu\mu}\mathopen{}\mathclose{{}\left(x-\widehat{\nu}}\right)U_{\nu}\mathopen{}\mathclose{{}\left(x-\widehat{\nu}}\right)\
$ (32)

being the plaquette that starts and ends at site $x$ and is spanned by the
$\mu$ and the negative $\nu$ direction, and the corresponding A-matrix,

$A_{\mu\mathopen{}\mathclose{{}\left(-\nu}\right)}\mathopen{}\mathclose{{}\left(x}\right)=U^{\dagger}_{\nu}\mathopen{}\mathclose{{}\left(x-\widehat{\nu}}\right)A_{\nu\mu}\mathopen{}\mathclose{{}\left(x-\widehat{\nu}}\right)U_{\nu}\mathopen{}\mathclose{{}\left(x-\widehat{\nu}}\right)\
.$ (33)

## References

  * [1] B. Lucini, M. Teper and U. Wenger, “Properties of the deconfining phase transition in SU(N) gauge theories,” JHEP 02 (2005), 033 doi:10.1088/1126-6708/2005/02/033 [arXiv:hep-lat/0502003 [hep-lat]].
  * [2] A. Hasenfratz and T. A. DeGrand, “Heavy dynamical fermions in lattice QCD,” Phys. Rev. D 49 (1994), 466-473 doi:10.1103/PhysRevD.49.466 [arXiv:hep-lat/9304001 [hep-lat]].
  * [3] F. Bursa, L. Del Debbio, L. Keegan, C. Pica and T. Pickup, “Mass anomalous dimension in SU(2) with six fundamental fermions,” Phys. Lett. B 696 (2011), 374-379 doi:10.1016/j.physletb.2010.12.050 [arXiv:1007.3067 [hep-ph]].
  * [4] T. Karavirta, J. Rantaharju, K. Rummukainen and K. Tuominen, “Determining the conformal window: SU(2) gauge theory with $N_{f}$ = 4, 6 and 10 fermion flavours,” JHEP 05 (2012), 003 doi:10.1007/JHEP05(2012)003 [arXiv:1111.4104 [hep-lat]].
  * [5] V. Leino, T. Rindlisbacher, K. Rummukainen, F. Sannino and K. Tuominen, “Safety versus triviality on the lattice,” Phys. Rev. D 101 (2020) no.7, 074508 doi:10.1103/PhysRevD.101.074508 [arXiv:1908.04605 [hep-lat]].
  * [6] J. Rantaharju, T. Rindlisbacher, K. Rummukainen, A. Salami and K. Tuominen, “Spectrum of SU(2) gauge theory at large number of flavors,” Phys. Rev. D 104 (2021) no.11, 114504 doi:10.1103/PhysRevD.104.114504 [arXiv:2108.10630 [hep-lat]].
  * [7] T. Rindlisbacher, K. Rummukainen, A. Salami and K. Tuominen, “Nonperturbative Decoupling of Massive Fermions,” Phys. Rev. Lett. 129 (2022) no.13, 131601 doi:10.1103/PhysRevLett.129.131601 [arXiv:2110.13882 [hep-lat]].
  * [8] A. J. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuominen, “Spectrum of SU(2) lattice gauge theory with two adjoint Dirac flavours,” JHEP 05 (2009), 025 doi:10.1088/1126-6708/2009/05/025 [arXiv:0812.1467 [hep-lat]].
  * [9] F. Bursa, L. Del Debbio, L. Keegan, C. Pica and T. Pickup, “Mass anomalous dimension in SU(2) with two adjoint fermions,” Phys. Rev. D 81 (2010), 014505 doi:10.1103/PhysRevD.81.014505 [arXiv:0910.4535 [hep-ph]].
  * [10] J. Rantaharju, “Gradient Flow Coupling in the SU(2) gauge theory with two adjoint fermions,” Phys. Rev. D 93 (2016) no.9, 094516 doi:10.1103/PhysRevD.93.094516 [arXiv:1512.02793 [hep-lat]].
  * [11] T. Appelquist et al. [LSD], “Lattice simulations with eight flavors of domain wall fermions in SU(3) gauge theory,” Phys. Rev. D 90 (2014) no.11, 114502 doi:10.1103/PhysRevD.90.114502 [arXiv:1405.4752 [hep-lat]].
  * [12] A. Hasenfratz, C. Rebbi and O. Witzel, “Gradient flow step-scaling function for SU(3) with Nf=8 fundamental flavors,” Phys. Rev. D 107 (2023) no.11, 114508 doi:10.1103/PhysRevD.107.114508 [arXiv:2210.16760 [hep-lat]].
  * [13] T. Appelquist et al. [Lattice Strong Dynamics], “Nonperturbative investigations of SU(3) gauge theory with eight dynamical flavors,” Phys. Rev. D 99 (2019) no.1, 014509 doi:10.1103/PhysRevD.99.014509 [arXiv:1807.08411 [hep-lat]].
  * [14] T. DeGrand, Y. Liu, E. T. Neil, Y. Shamir and B. Svetitsky, “Spectroscopy of SU(4) lattice gauge theory with fermions in the two index anti-symmetric representation,” PoS LATTICE2014 (2014), 275 doi:10.22323/1.214.0275 [arXiv:1412.4851 [hep-lat]].
  * [15] T. DeGrand, Y. Shamir and B. Svetitsky, “Suppressing dislocations in normalized hypercubic smearing,” Phys. Rev. D 90 (2014) no.5, 054501 doi:10.1103/PhysRevD.90.054501 [arXiv:1407.4201 [hep-lat]].
  * [16] V. G. Bornyakov, M. Creutz and V. K. Mitrjushkin, “Modified Wilson action and Z(2) artifacts in SU(2) lattice gauge theory,” Phys. Rev. D 44 (1991), 3918-3923 doi:10.1103/PhysRevD.44.3918
  * [17] O. Akerlund and P. de Forcrand, “U(1) lattice gauge theory with a topological action,” JHEP 06 (2015), 183 doi:10.1007/JHEP06(2015)183 [arXiv:1505.02666 [hep-lat]].
  * [18] D. Nogradi, L. Szikszai and Z. Varga, “Non-abelian lattice gauge theory with a topological action,” JHEP 08 (2018), 032 doi:10.1007/JHEP08(2018)032 [arXiv:1807.05295 [hep-lat]].
  * [19] K. G. Wilson, “Confinement of Quarks,” Phys. Rev. D 10 (1974), 2445-2459 doi:10.1103/PhysRevD.10.2445
  * [20] M. Luscher and P. Weisz, “On-Shell Improved Lattice Gauge Theories,” Commun. Math. Phys. 97 (1985), 59 [erratum: Commun. Math. Phys. 98 (1985), 433] doi:10.1007/BF01206178
  * [21] W. Rossmann, “Lie Groups: An Introduction Through Linear Groups,” Oxford graduate texts in mathematics, 2002, Oxford University Press, isbn: 9780198596837.
  * [22] R. C. Brower, D. A. Kessler and H. Levine, “The Onset of Asymptotically Free Scaling,” Phys. Rev. D 26 (1982), 959 doi:10.1103/PhysRevD.26.959
  * [23] B. B. Brandt, R. Lohmayer and T. Wettig, “Induced QCD II: Numerical results,” JHEP 07 (2019), 043 doi:10.1007/JHEP07(2019)043 [arXiv:1904.04351 [hep-lat]].
  * [24] G. Bhanot and M. Creutz, “Variant Actions and Phase Structure in Lattice Gauge Theory,” Phys. Rev. D 24 (1981), 3212 doi:10.1103/PhysRevD.24.3212
  * [25] A. Hasenfratz, “Infrared fixed point of the 12-fermion SU(3) gauge model based on 2-lattice MCRG matching,” Phys. Rev. Lett. 108 (2012), 061601 doi:10.1103/PhysRevLett.108.061601 [arXiv:1106.5293 [hep-lat]].
  * [26] M. Hasenbusch and S. Necco, “SU(3) lattice gauge theory with a mixed fundamental and adjoint plaquette action: Lattice artifacts,” JHEP 08 (2004), 005 doi:10.1088/1126-6708/2004/08/005 [arXiv:hep-lat/0405012 [hep-lat]].
  * [27] W. Bietenholz, K. Jansen, K. I. Nagai, S. Necco, L. Scorzato and S. Shcheredin, “Exploring topology conserving gauge actions for lattice QCD,” JHEP 03 (2006), 017 doi:10.1088/1126-6708/2006/03/017 [arXiv:hep-lat/0511016 [hep-lat]].
  * [28] T. DeGrand, “Simple chromatic properties of gradient flow,” Phys. Rev. D 95 (2017) no.11, 114512 doi:10.1103/PhysRevD.95.114512 [arXiv:1701.00793 [hep-lat]].
  * [29] Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. H. Wong, “The Yang-Mills gradient flow in finite volume,” JHEP 11 (2012), 007 doi:10.1007/JHEP11(2012)007 [arXiv:1208.1051 [hep-lat]].
  * [30] Z. Fodor, K. Holland, J. Kuti, S. Mondal, D. Nogradi and C. H. Wong, “The lattice gradient flow at tree-level and its improvement,” JHEP 09 (2014), 018 doi:10.1007/JHEP09(2014)018 [arXiv:1406.0827 [hep-lat]].
  * [31] C. E. Detar, O. Kaczmarek, F. Karsch and E. Laermann, “String breaking in lattice quantum chromodynamics,” Phys. Rev. D 59 (1999), 031501 doi:10.1103/PhysRevD.59.031501 [arXiv:hep-lat/9808028 [hep-lat]].
  * [32] F. Karsch, “Deconfinement and chiral symmetry restoration in QCD with fundamental and adjoint fermions,” PoS corfu98 (1998), 008 doi:10.22323/1.001.0008
  * [33] T. A. DeGrand and D. Toussaint, “Topological Excitations and Monte Carlo Simulation of Abelian Gauge Theory,” Phys. Rev. D 22 (1980), 2478 doi:10.1103/PhysRevD.22.2478
  * [34] P. Goddard and D. I. Olive, “New Developments in the Theory of Magnetic Monopoles,” Rept. Prog. Phys. 41 (1978), 1357 doi:10.1088/0034-4885/41/9/001
  * [35] R. A. Brandt and F. Neri, “Magnetic Monopoles in SU(n) Gauge Theories,” Nucl. Phys. B 186 (1981), 84-108 doi:10.1016/0550-3213(81)90094-8
  * [36] E. J. Weinberg, D. London and J. L. Rosner, “Magnetic Monopoles With $Z(n$) Charges,” Nucl. Phys. B 236 (1984), 90-108 doi:10.1016/0550-3213(84)90526-1
  * [37] S. Hollands and M. Muller-Preussker, “Definition of magnetic monopole numbers for SU(N) lattice gauge Higgs models,” Phys. Rev. D 63 (2001), 094503 doi:10.1103/PhysRevD.63.094503 [arXiv:hep-th/9901114 [hep-th]].
  * [38] A. Ukawa, P. Windey and A. H. Guth, “Dual Variables for Lattice Gauge Theories and the Phase Structure of Z(N) Systems,” Phys. Rev. D 21 (1980), 1013 doi:10.1103/PhysRevD.21.1013
  * [39] R. C. Brower, D. A. Kessler and H. Levine, “Monopole Condensation and the Lattice QCD Crossover,” Phys. Rev. Lett. 47 (1981), 621 doi:10.1103/PhysRevLett.47.621
  * [40] P. de Forcrand, C. Korthals-Altes and O. Philipsen, “Screening of Z(N) monopole pairs in gauge theories,” Nucl. Phys. B 742 (2006), 124-141 doi:10.1016/j.nuclphysb.2006.02.035 [arXiv:hep-ph/0510140 [hep-ph]].
  * [41] J. Fingberg, U. M. Heller and V. K. Mitrjushkin, “Scaling in the positive plaquette model and universality in SU(2) lattice gauge theory,” Nucl. Phys. B 435 (1995), 311-338 doi:10.1016/0550-3213(94)00492-W [arXiv:hep-lat/9407011 [hep-lat]].
  * [42] G. Mack and V. B. Petkova, “Comparison of Lattice Gauge Theories with Gauge Groups Z(2) and SU(2),” Annals Phys. 123 (1979), 442 doi:10.1016/0003-4916(79)90346-4
  * [43] G. Mack and V. B. Petkova, “Sufficient Condition for Confinement of Static Quarks by a Vortex Condensation Mechanism,” Annals Phys. 125 (1980), 117 doi:10.1016/0003-4916(80)90121-9
  * [44] R. C. Brower, D. A. Kessler and H. Levine, “Dynamics of SU(2) Lattice Gauge Theories,” Nucl. Phys. B 205 (1982), 77 doi:10.1016/0550-3213(82)90467-9
  * [45] G. ’t Hooft, “Magnetic Monopoles in Unified Gauge Theories,” Nucl. Phys. B 79 (1974), 276-284 doi:10.1016/0550-3213(74)90486-6
  * [46] A. M. Polyakov, “Particle Spectrum in Quantum Field Theory,” JETP Lett. 20 (1974), 194-195 PRINT-74-1566 (LANDAU-INST).
  * [47] C. Bonati and M. D’Elia, “The Maximal Abelian Gauge in SU(N) gauge theories and thermal monopoles for N = 3,” Nucl. Phys. B 877 (2013), 233-259 doi:10.1016/j.nuclphysb.2013.10.004 [arXiv:1308.0302 [hep-lat]].
  * [48] G. ’t Hooft, “Topology of the Gauge Condition and New Confinement Phases in Nonabelian Gauge Theories,” Nucl. Phys. B 190 (1981), 455-478 doi:10.1016/0550-3213(81)90442-9
  * [49] C. Bonati, A. Di Giacomo and M. D’Elia, “Detecting monopoles on the lattice,” Phys. Rev. D 82 (2010), 094509 doi:10.1103/PhysRevD.82.094509 [arXiv:1009.2425 [hep-lat]].
  * [50] K. M. Lee and C. h. Lu, “SU(2) calorons and magnetic monopoles,” Phys. Rev. D 58 (1998), 025011 doi:10.1103/PhysRevD.58.025011 [arXiv:hep-th/9802108 [hep-th]].