[a]Nicholas Sale

# Persistent homology as a probe for center vortices and deconfinement in
SU(2) lattice gauge theory

Biagio Lucini  Jeffrey Giansiracusa

###### Abstract

Topological Data Analysis (TDA) is a field that leverages tools and ideas from
algebraic topology to provide robust methods for analysing geometric and
topological aspects of data. One of the principal tools of TDA, persistent
homology, produces a quantitative description of how the connectivity and
structure of data changes when viewed over a sequence of scales. We propose
that this presents a means to directly probe topological objects in gauge
theories. We present recent work on using persistent homology to detect center
vortices in SU(2) lattice gauge theory configurations in a gauge-invariant
manner. We introduce the basics of persistence, describe our construction, and
demonstrate that the result is sensitive to vortices. Moreover we discuss how,
with simple machine learning, one can use the resulting persistence to
quantitatively analyse the deconfinement transition via finite-size scaling,
providing evidence on the role of vortices in relation to confinement in Yang-
Mills theories.

##  1 Introduction

Center vortices are topological defects that are observed in lattice quantum
chromodynamics (QCD) simulations [2]. They provide a potential explanation for
the mechanism of confinement in QCD and the deconfinement phase transition
[17, 4, 19], but existing methods to study them in configurations rely on
gauge fixing and projection [9] which suffers from the Gribov copies problem
[10, 16]. In this project we leverage persistent homology [6], a tool from
topological data analysis (TDA), to study center vortices in a gauge invariant
manner. Rather than working with lattice QCD, we consider the $\mathrm{SU}(2)$
lattice gauge theory as a toy model since this also exhibits center vortices
and a deconfinement phase transition.

##  2 Lattice Model and Center Vortices

The 4D $\mathrm{SU}(2)$ lattice gauge theory is specified by
$\mathrm{SU}(2)$-valued variables $U_{\mu}(x)$, taking the form of a $2\times
2$ complex matrix, located on each link $(x,\mu)$ of an $N_{t}\times
N_{s}^{3}$ lattice $\Lambda$ with periodic boundary conditions, where
$\mu\in\\{0,1,2,3\\}$ describes the direction in which the link emanates from
the lattice site $x\in\Lambda$. Gauge invariant observables are obtained as
traces of products of the link variables along closed paths $C$, also known as
Wilson loops $W(C)$. The simplest non-trivial example is the Wilson loop
around a $1\times 1$ plaquette $(x,\mu,\nu)$ of the lattice:

$W_{\mu,\nu}(x)=\frac{1}{2}\,tr\Big{[}U_{\mu}(x)\,U_{\nu}(x+\hat{\mu})\,U_{\mu}^{\dagger}(x+\hat{\nu})\,U_{\nu}^{\dagger}(x)\Big{]}.$

We use this to define the Wilson action given a configuration
$\mathbf{U}=\\{U_{\mu}(x)\\}_{(x,\mu)}$ as

$S(\mathbf{U})=-\frac{\beta}{4}\sum_{x,\mu<\nu}W_{\mu,\nu}(x)$ (1)

where $\beta=4/g^{2}$ and $g$ is the gauge coupling parameter. This in turn
allows us to define the vacuum expectation value of any given observable
$A(\mathbf{U})$ as

$\langle A\rangle=\frac{\int
d\mathbf{U}\,A(\mathbf{U})\,e^{-S(\mathbf{U})}}{\int
d\mathbf{U}\,e^{-S(\mathbf{U})}}$ (2)

where $d\mathbf{U}=\prod_{x,\mu}dU_{\mu}(x)$ is a product of Haar measures
over $\mathrm{SU}(2)$ for each link variable. In practice we estimate
expectations using Monte Carlo methods, where Eq. (2) becomes a simple mean of
the observed values.

We introduce center vortices following [17]. Fix a time slice at time $t$.
Given two closed oriented curves $C$ and $C^{\prime}$ in that 3-dimensional
slice with linking number $m$, a loop operator $B(C^{\prime},t)$ can be
defined that has the following commutation algebra with the Wilson loop
$W(C,t)$:

$W({C},t)B({C^{\prime}},t)-(-1)^{m}B({C^{\prime}},t)W({C},t)=0.$ (3)

The operator $B(C^{\prime},t)$ is called the ’t Hooft loop which, when acting
on a gauge configuration, creates a magnetic flux with the resulting
observable effect of multiplication of all Wilson loops around curves ${C}$
with linking number 1 with ${C^{\prime}}$ by $-1$. The ’t Hooft loop is
therefore said to be a vortex creation operator. Since the center of the group
$Z(\mathrm{SU}(2))=\\{I,-I\\}\cong\mathbb{Z}_{2}$ plays a role (as exposed by
the factor $(-1)^{m}$), the vortices are called center vortices. Allowing the
curve $C^{\prime}$ to vary continuously over time slices, we see that a vortex
traces out a surface in 4-space, closed by the periodic boundary conditions.

The above describes a ’thin’ vortex. In practice center vortices have some
finite thickness, so that only larger Wilson loops may fully link with them
and obtain the full center charge. Loops that partially link may still obtain
a partial charge, being multiplied by some matrix lying between $I$ and $-I$
in $\mathrm{SU}(2)$.

To explicitly insert a thin vortex into the system to study we will make use
of the trick of imposing twisted boundary conditions [18]. The idea is that we
negate the contribution to the action of the co-closed collection of
plaquettes

$T=\\{((0,0,y,z),0,1)\,\,|\,\,0\leq y,z<N_{s}\\}$

corresponding to a surface wrapping round the latter two spatial dimensions of
the lattice. The action with twisted boundary conditions becomes

$S_{T}(\mathbf{U})=-\frac{\beta}{4}\Bigg{[}\sum_{\begin{subarray}{c}x,\mu<\nu\\\
(x,\mu,\nu)\not\in
T\end{subarray}}W_{x,\mu,\nu}-\sum_{\begin{subarray}{c}x,\mu<\nu\\\
(x,\mu,\nu)\in T\end{subarray}}W_{x,\mu,\nu}\Bigg{]}$ (4)

which we refer to as the twisted action. This modification of the action
allows the lattice to support an odd number of center vortices wrapping in the
$yz$ plane, which is not allowed by the usual periodic boundary conditions of
the Wilson action. We can think of this twisted action as explicitly inserting
a thin vortex into the system on the surface defined by $T$, so that the
system is forced to generate a (thick) vortex to cancel it out. We shall
denote expectations calculated with respect to this twisted action by $\langle
A\rangle_{\text{twist}}$, where $A$ is a generic observable.

##  3 Persistent Homology

Persistent homology takes a nested sequence of topological spaces and produces
a topological summary called a barcode or persistence diagram [6] (see [3, 7,
13, 8] for useful references). In this case we assign a filtered cubical
complex $F_{\boldsymbol{U}}:\mathbb{R}\rightarrow\text{CubicalComplex}$ (such
that $s\leq t$ implies $F(s)\subseteq F(t)$) to each configuration
$\boldsymbol{U}=\\{U_{\sigma}(x)\\}$ and compute the persistent homology of
this filtered cubical complex. Our construction is based on Wilson loops and
therefore yields gauge-invariant persistence diagrams.

The idea is to explicitly construct a cubical model of vortex surfaces, under
the assumption that vortices are thin. Vortex sheets live on the dual lattice
$\Lambda^{*}$. We therefore consider a decomposition of the spacetime manifold
into a cubical complex $Y$ given by $\Lambda^{*}$ in which there is a vertex
for each dual lattice site, an edge for each link in the dual lattice, a
2-cube for each dual plaquette, etc. Note that there is a bijection between
the 2-cubes of this cubical complex (i.e. dual plaquettes) with the plaquettes
of the original lattice $\Lambda$, pairing a plaquette with the dual plaquette
that intersects it at a single point. We will construct a filtration of $Y$ by
letting each 2-cube enter at a filtration index given by the value of the
Wilson loop around the plaquette it is paired with in the bijection.

To define the filtered complex we give a filtration index $f(c)\in\mathbb{R}$
for each cube $c$ in $Y$ specifying when it appears. Then
$F_{\boldsymbol{U}}(s)$ is the subcomplex of $Y$ consisting of all cubes $c$
for which $f(c)\leq s$. That is,

$F_{\boldsymbol{U}}(s)=f^{-1}(-\infty,s].$

Since we are attempting to model vortex surfaces, we will initially specify
when the 2-cubes are to enter the filtered complex and then introduce the
cubes of other dimensions based on these.

Our construction of the function $f$ is the following:

  1. 1.

For each 2-cube $c$, i.e. dual plaquette, we set $f(c)$ equal to the value of
the Wilson loop around the plaquette paired with it by the bijection.

  2. 2.

Since a 2-cube is not allowed to be included before its constituent 1-cubes
and 0-cubes in a cubical complex, setting $f(c)$ for these to be the smallest
value of $f$ of any of the 2-cubes they are incident to.

  3. 3.

For the 3-cubes and 4-cubes we follow a clique-like rule, setting $f(c)$ for
these to be the largest value of $f$ of any of the 2-cubes contained in their
boundary.

Thus for $s<-1$, $F_{\boldsymbol{U}}(s)$ is the empty complex and for $s\geq
1$, $F_{\boldsymbol{U}}(s)$ is the filled in tiling of spacetime, homeomorphic
to a 4-torus due to the periodic boundary conditions. Going between these
values, the first cubes to enter $F_{\boldsymbol{U}}$ are surfaces made up of
plaquettes in bijection with Wilson loops that are close to $-1$. The idea
therefore is that thin vortex surfaces will enter the filtered complex early.
Moreover, since small Wilson loops like those considered here still pick up a
partial charge from thick vortices, surfaces representing those thick vortices
ought to enter the filtered complex earlier than they otherwise would have. We
expect to detect these closed surfaces in persistent $H_{2}$. We may also see
other topological features such as the presence of handles or holes in
$H_{1}$, as well as the transient low-persistence points in persistent $H_{0}$
and $H_{1}$ that arise as the surface forms near the start of the filtration.

##  4 Detecting Vortices with Twisted Boundary Conditions

We first test if the persistent homology can distinguish between
configurations generated using the Wilson action and configurations generated
using the twisted action. That is, if it detects an inserted vortex.

For $N_{s}\in\\{12,16,20\\}$, fixing $N_{t}=4$, we generate $200$
configurations using the Wilson action (1) and $200$ configurations using the
twisted action (4) for each $\beta\in\\{1.5,1.6,\dots 2.9\\}$. Configurations
are generated using the HiRep software [5] with 1 heatbath step and 4
overelaxation steps for each Monte Carlo step and a sample taken every 100
Monte Carlo steps.

Figure 1 shows example persistence diagrams obtained using the two different
actions and in the two phases of the model.

Figure 1: Sample persistence diagrams of individual configurations obtained
using the following actions and values of $\beta$: (a) Wilson, $\beta=1.5$ (b)
twisted, $\beta=1.5$ (c) Wilson, $\beta=2.9$ (d) twisted, $\beta=2.9$. The
arrow in (d) indicates the point $(b,\infty)\in PH_{2}$ with the smallest
birth index $b$. Note the distance between it and the others.

In the deconfined phase, one of the infinite death points in $H_{2}$ is born
much earlier for the twisted action. This represents a surface which wraps the
periodic boundary conditions of the lattice entering our filtration early:
i.e. the inserted vortex. We therefore define the following observable based
on the persistence diagram of a configuration

$m_{2}=\min\big{\\{}\,b\,\,\big{|}\,\,(b,\infty)\in PH_{2}\,\big{\\}}.$

The expected value of $m_{2}$ for different lattice sizes with the Wilson
action and twisted action are shown in Figure 2(a). Note that there is no
difference between the expectations estimated using the different actions well
into the confined phase, but in the deconfined phase the curves separate. As
the lattice size increases, the point at which the curves diverge approaches
the critical $\beta$ of the phase transition from below. These observations
motivate measuring the difference between the expected values using different
actions

$O_{m_{2}}=\langle m_{2}\rangle-\langle m_{2}\rangle_{\text{twist}}$

as a phase indicator which will be zero in the confined phase and non-zero in
the deconfined phase, similar to the definition of an order parameter but
without the requirement to detect any symmetry breaking. A finite-size scaling
analysis of this quantity yields the curve collapse in Figure 2(b), computed
numerically using the Nelder-Mead method following [1]. The resulting
estimates of $\beta_{c}$ and $\nu$,

$\displaystyle\beta_{c}$ $\displaystyle=2.291\pm 0.019$ $\displaystyle\nu$
$\displaystyle=0.614\pm 0.079,$

are in agreement with our reference estimate $\beta_{c}=2.2986(6)$ from [12]
and $\nu=0.629971(4)$ from [11]. Error estimates are obtained by performing
$2000$ bootstraps.

(a)

(b)

Figure 2: (a) The expected value of the observable $m_{2}$ as a function of
$\beta$ plotted for different values of $N_{s}$ and with the Wilson and
twisted actions. (b) The curve collapse of $O_{m_{2}}$ using $\beta_{c}=2.291$
and $\nu=0.614$. Error bars are not shown for clarity but are comparable to
those in (a).

##  5 Detecting the Deconfinement Transition Without Twisted Boundary
Conditions

Using a machine learning framework inspired by that in [14], we investigate if
it is possible to extract the critical $\beta$ and critical exponent $\nu$ of
the deconfinement transition for $N_{t}=4$ using configurations sampled using
the Wilson action alone.

For lattices of size $4\times N_{s}^{3}$ with $N_{s}\in\\{12,16,20,24\\}$, we
train a $k$-nearest neighbours classifier ($k=30$) on the concatenated
$PH_{0}$, $PH_{1}$, $PH_{2}$ and $PH_{3}$ persistence images of $200$
configurations sampled at each $\beta$ in the confined and deconfined regions
given in Table 1. The classifier is then used to produce a predicted
classification $O_{k\mathrm{NN}}$ for $200$ configurations sampled for each
value of $\beta$ in the critical region. The resulting curve is shown in
Figure 3.

Region |  $\beta$
---|---
Confined |  2.2 , 2.21, 2.22, 2.23, 2.24
Deconfined |  2.36, 2.37, 2.38, 2.39, 2.4
Critical |  2.25, 2.26, 2.27, 2.275, 2.28, 2.285, 2.29, 2.295, 2.298, 2.299, 2.3, 2.301, 2.302, 2.305, 2.31, 2.315, 2.32, 2.325, 2.33, 2.34, 2.35
Table 1: Values of $\beta$ sampled at for the $N_{t}=4$ phase transition.

Figure 3: Plot showing our phase indicator $\langle O_{k\mathrm{NN}}\rangle$
as a function of $\beta$ for $N_{t}=4$. The points show the measured
expectations and the curve is the output of histogram reweighting these
measurements.

Assuming a known value of $\nu=0.629971$, we can do a finite-size scaling
analysis by extracting the pseudo-critical point for each $N_{s}$ via the
implicit formula $\langle O_{k\mathrm{NN}}\rangle(\beta_{c}(N_{s}))=0.5$ then
fitting these to the straight line ansatz
$\beta_{c}(N_{s})-\beta_{c}(\infty)\propto N_{s}^{-1/\nu}$ to extract
$\beta_{c}=\beta_{c}(\infty)$. The resulting fit is shown in Figure 4(a). The
intercept yields $\beta_{c}=2.2989\pm 0.0009$, supporting our reference
estimate of $\beta_{c}=2.2986(6)$ from [12].

(a)

(b)

Figure 4: (a) Estimating $\beta_{c}$ using a linear fit, assuming known $\nu$.
The pseudo-critical values of $\beta$, obtained as the points where the curves
in Figure 3 cross $0.5$, are fitted to a straight line against
$N_{s}^{-1/\nu}$ with $\nu=0.629971$. Error bars are estimated by
bootstrapping. (b) The curve collapse of $\chi_{k\mathrm{NN}}$ using
$\beta_{c}=2.2988$ and $\nu=0.634$.

Alternatively, we can try to estimate both $\beta_{c}$ and $\nu$
simultaneously via a curve collapse of the variance curves
$\chi_{k\mathrm{NN}}=\langle O_{k\mathrm{NN}}^{2}\rangle-\langle
O_{k\mathrm{NN}}\rangle^{2}$ using a numerical procedure like that in [1]. The
result of using the Nelder-Mead method is shown in Figure 4(b). The obtained
estimates of $\beta_{c}$ and $\nu$

$\displaystyle\beta_{c}$ $\displaystyle=2.2988\pm 0.0007$ $\displaystyle\nu$
$\displaystyle=0.634\pm 0.014$

are consistent with previous estimates.

##  6 Conclusion

We designed a methodology to use persistent homology to detect center vortices
and tested the efficacy of this by using it to distinguish configurations
generated using a twisted action from configurations generated using the usual
Wilson action. We also performed a quantitative analysis of the deconfinement
transition using two different phase indicators derived from the persistent
homology. We argue that, since the methodology summarises center vortices but
is also sensitive to the phase transition, then center vortices must play some
role in the phase transition. For a stronger argument we would need to
consider the sensitivity of our methodology to the objects involved in other
pictures of confinement, e.g., monopoles.

## Acknowledgments

Numerical simulations have been performed on the Swansea SUNBIRD system. This
system is part of the Supercomputing Wales project, which is part-funded by
the European Regional Development Fund (ERDF) via Welsh Government.
Configurations of the $\mathrm{SU}(2)$ lattice gauge theory were sampled using
the HiRep software [5]. Persistent homology calculations were performed using
giotto-tda [20]. Histogram reweighting calculations were performed using
pymbar [15]. NS has been supported by a Swansea University Research Excellence
Scholarship (SURES). JG was supported by EPSRC grant EP/R018472/1. BL received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme under grant agreement No
813942. The work of BL was further supported in part by the UKRI Science and
Technology Facilities Council (STFC) Consolidated Grant ST/T000813/1, by the
Royal Society Wolfson Research Merit Award WM170010 and by the Leverhulme
Foundation Research Fellowship RF-2020-461\9.

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