# Phase Transitions in Abelian Lattice Gauge Theory:
Production and Dissolution of Monopoles and Monopole–Antimonopole Pairs

Loris Di Cairano<EMAIL_ADDRESS>Matteo Gori<EMAIL_ADDRESS>Matthieu Sarkis<EMAIL_ADDRESS>Alexandre Tkatchenko
<EMAIL_ADDRESS>Department of Physics and Materials Science,
University of Luxembourg, L-1511 Luxembourg City, Luxembourg

###### Abstract

We combine the microcanonical formulation of lattice gauge theories (LGTs)
developed by Callaway and the microcanonical inflection point analysis (MIPA)
proposed by Bachmann et al. to achieve a systematic characterization of phase
transitions (PTs) in $U(1)$ lattice electrodynamics. Besides identifying the
well-known deconfinement PT (DPT) due to the neutral pair dissolution, which
we classify as a first-order PT, we unequivocally detect three higher-order
PTs. According to MIPA, we observe two independent third-order PTs in the
confined phase; instead, in the deconfined (Coulomb) phase, we observe a
dependent third-order PT. For a deeper understanding of the physical meaning
of these PTs, we numerically compute the average number density of monopolar
and pair defects as a function of energy. Our analysis reveals that DPT is
only one of the major mechanisms observable in LGT. The independent third-
order PTs are associated, respectively, to the first occurrence of monopolar
topological defects and to the production of pairs.

Lattice gauge theories, Phase transitions, Microcanonical ensemble, higher-
order phase transitions

††preprint: APS/123-QED

Phase transitions (PTs) are collective phenomena that are omnipresent in
nature, from protein folding in biology to the formation of starts in
astrophysics, or from the superconducting PT in solid state physics to the
generation of masses for the weak gauge bosons and fermions known as Anderson-
Higgs-Kibble (AHK) mechanism [1, 2, 3, 2] in quantum field theory (QFT). In
QFT, a deep understanding of PTs would allow to better comprehend how the
emergence of order at macroscopic scale is generated from the microscopic
scale of elementary quantum components. The origin of this change in scale has
been understood in terms of a condensation mechanism of the Nambu-Goldstone
boson quanta in the ground state of the system [4, 5]. However, such a
mechanism holds only for those theories where PTs are triggered by a
spontaneous symmetry breaking à la Landau [6, 7], that is, when PTs are
signaled by non-vanishing values of an order parameter. Although, Landau’s
mechanism is at the core of many PTs such as, for instance, the AHK mechanism;
it is not all-encompassing. Indeed, PTs observed in lattice gauge theories
(LGTs), for example, the deconfinement PT (DPT), are not entailed by a
symmetry breaking since the gauge symmetry cannot be spontaneously broken due
to Mermin-Wagner theorem [8, 9]. Also in this case, much work has been done to
date in LGTs in order to understand the origin of these PTs and, as shown by
Banks et al. [10], the DPT is guided by the emergence of topological defects
known as magnetic monopoles. First, they interact forming monopole-
antimonopole pairs that, in turn, dissociate giving rise to a phase transition
[11, 11, 12, 13, 14, 15, 16] similar to the vortex-antivortex pairs
dissociation in the Berezinskii-Kosterlitz-Thouless PT [17, 18, 19]. It is
worth noting that the thermodynamic properties of LGTs and consequently the
characterization of PTs have so far been performed within the canonical
ensemble. The choice of adopting such an ensemble is motivated by the fact
that, after Wick rotation [20], the path-integral representation of the
vacuum-to-vacuum transition amplitudes [21, 22] is formally identified with
the canonical partition function [23]. Despite the success of this approach in
providing a physical description of PTs in QFT, an important conceptual
inconsistency is inherent in the canonical ensemble. Due to the Lee-Yang
theorem [24, 25] and Ehrenfest classification [26], PTs in (gran)canonical
ensemble are signaled by nonanalyticities of thermodynamic observables that
can only manifest in the thermodynamic limit. However, the central goal of
LGTs [27] is to provide physically meaningful results for the vacuum-
expectation values of observables on a finite space-time lattice without
employing the continuum limit [28]. Therefore, it would be preferable to
develop a method which enables to systematicaly classify PTs for finite-size
systems.

In this Letter, we address this challenge by combining the microcanonical
formalism developed by Callaway [28, 29, 30, 31, 32, 32, 33, 34] and the
microcanonical inflection point analysis (MIPA) proposed by Bachmann et al.
[35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 38]. We remark that MIPA offers a
complete classification of PTs that can be applied to systematically identify
PTs in the microcanonical ensemble also in finite-size systems, thus
dispensing with the need of reaching the thermodynamic limit. MIPA also
enables a consistent identification of high-order PTs. For example, MIPA was
recently used to discover a new higher-order PT in the 2D Ising model [45].
Morever, it has also proven to be a powerful method in statistical lattice
field theories [46, 47, 48, 47] and for polymers [37, 42, 49]. Therefore, this
fact strongly motivates the application of MIPA in the investigation of the
thermodynamics of QFT systems on a lattice. Specifically, we study one of the
fundamental LGTs given by (compact) electrodynamics $U(1)$ on a 4D lattice.
Our numerical results allow to classify the DPT as a first-order transition,
and two novel (independent) third-order PTs in the confined phase and a
(dependent) third-order PT in the deconfined (Coulomb) phase. We show that the
physical mechanisms which lead to these transitions are strictly related to
peculiar behaviors of topological defects. In order to deeper understand the
physical nature of these transitions, a study of the density of isolated
monopoles and monopole-antimonopole pairs as functions of energy is provided.

Figure 1: Microcanonical observables for the 4D $U(1)$-LGT with lattice size
$N=12^{4}$. Left plot: First-order derivative of microcanonical entropy,
$\partial_{E}S(E)\equiv\beta_{\mu c}$. In the inset, the back-bending region
(or S-shape) is shown. The dot-dashed red line passes through the positive-
valued minimum of $\beta_{\mu c}$ allowing the identification of the energy
value where the first-order transition occurs, i.e.,
$\epsilon^{ind}_{1}\approx 1.197$. The horizontal continuous red line
corresponds to $\beta_{1}=\partial_{\epsilon}S(\epsilon^{ind}_{1})$. Right
plot: Average plaquette as a function of inverse temperature,
$\langle\bar{\mathscr{V}}_{\square}\rangle_{\mu c}(\beta)$. The inset shows
the S-shape of the average plaquette that represents the fingerprint of a
first-order PT in the standard description based on the canonical ensemble.
The vertical continuous red line indicates the value of the inverse
temperature, $\beta_{1}=\partial_{\epsilon}S(\epsilon^{ind}_{1})$
corresponding with the deconfinement PT.

Let us consider an Abelian gauge field $A_{\mu}(\bm{x})$ discretized over a
$d=4$ dimensional Euclidean lattice with $N$ sites per side and a lattice
spacing $a$, so that the configuration degrees of freedom are
$\phi_{\bm{n},\mu}:=ag_{0}\,A_{\bm{n},\mu}\in[-\pi/a,\pi/a]$ where $g_{0}$ is
the bare coupling constant. Each field variable $\phi_{\bm{n},\mu}$ is related
to the link connecting the lattice point $\bm{n}\in[1,N]^{4}$ to its nearest
neighbor in the space-time direction $\mu$, i.e. $\bm{n}+\mu$. We employ the
gauge-invariant Euclidean Wilson’s action [27]

$\mathscr{S}_{W}(\bm{\phi},\beta):=\beta\sum\limits_{\square}\left[1-\cos(\Theta_{\mu\nu}(\bm{n}))\right]=\beta\sum\limits_{\square}\mathscr{V}_{\square}(\bm{\phi}),$
(1)

where the sum over the lattice plaquettes, $\square=\\{\bm{n},\mu,\nu\\}$, is
uniquely identified by a site and two independent directions on the lattice.
Then,
$\Theta_{\mu\nu}(\bm{n}):=\phi_{\bm{n},\mu}+\phi_{n+\mu,\nu}-\phi_{n+\nu,\mu}-\phi_{n,\nu}$
is the plaquette angle, $\mathscr{V}_{\square}$ is the so-called plaquette
potential, and $\beta=g_{0}^{-2}$ is a parameter that can be interpreted as an
effective inverse temperature. The microcanonical formulation of an Abelian
LGT proposed by Callaway [28, 29, 30] consists in adding a kinetic term
$K(\pi)=\sum_{n,\mu}\pi_{n,\mu}^{2}/2$ to the Wilson’s action (1) after the
introduction of the fictitious momenta, $\pi_{n,\mu}$, conjugated to
$\phi_{n,\mu}$. This manipulation does not alter the expectation value of
quantum observables, but allows to associate a dynamical system to any LGT
whose Hamiltonian function is given by

$\mathscr{H}(\pi,\phi):=K(\bm{\pi})+\beta^{-1}\mathscr{S}_{W}(\bm{\phi}),$ (2)

and we naturally identify a microcanonical ensemble. In order to study the
thermodynamics, we introduce the entropy function which is the fundamental
microcanonical thermodynamic potential that generates all the observables.
Hence, the specific entropy function is defined by

$\begin{split}S_{N_{ind}}(E)&:=\dfrac{1}{N_{ind}}\log\Omega_{N{{}_{ind}}}(E)\\\
&=\dfrac{1}{N_{ind}}\log\int\delta(\mathscr{H}[\pi,\phi]-E)\delta(\mathscr{C}(\pi))~{}D\pi
D\phi\,\,,\end{split}$ (3)

where $\Omega_{N{{}_{ind}}}(E)$ is the microcanonical partition function and
$N_{ind}$ is the number of independent degrees of freedom [28, 29] due to the
gauge-fixing constraint $\mathscr{C}(\pi)=0$ (See Supplemental Material (SM)
for further details). Thus, the expectation value of an LGT-observable,
$\mathscr{O}[\pi,\phi]$, can be computed by averaging such a function with
respect to the microcanonical measure, i.e.:

$\langle\mathscr{O}\rangle_{\mu
c}=\Omega_{N_{ind}}^{-1}\int\mathscr{O}[\pi,\phi]\delta(\mathscr{H}[\pi,\phi]-E)\delta(\mathscr{C}(\pi))~{}D\pi\,D\phi\,.$
(4)

where $E$ is the energy of the system which is conserved by the Hamiltonian
flow generated by (S.1) and defined by Hamilton’s equations of motion
$d\phi_{n,\mu}/d\tau=\pi_{n,\mu}$
$d\pi_{n,\mu}/d\tau=-\beta^{-1}\partial\mathscr{S}_{\rm
W}[\phi]/\partial\phi_{\bm{n},\mu}$. In applications, one numerically solves
the Hamilton equations of motion getting a solution
$(\pi^{S}_{n,\mu},\phi^{S}_{n,\mu})$; then, assuming in bona fide the
ergodicity of the trajectory, one recasts Eq. (4) into a time average:

$\langle\mathscr{O}\rangle_{\mu
c}\equiv\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\mathscr{O}[\pi^{S}_{n,\mu}(\tau),\phi^{S}_{n,\mu}(\tau)]~{}d\tau\,.$
(5)

It should be noticed that the introduction of fictitious momenta leads
inevitably to introduce a fictitious time, $\tau$, along which the system
evolves. This construction makes the (4+1)D microcanonical formulation
equivalent to the stochastic quantization [50].

Having defined a microcanonical description, we can investigate the
thermodynamics properties of the $U(1)$-LGT adopting the microcanonical
inflection point analysis (MIPA) [37, 35, 38]. According to MIPA, PTs are
split into two classes: independent and dependent. Independent PTs represent
the major change in the system’s phase, whereas a dependent PT can only occur
if an independent one already manifested; thereby, they are considered the
precursors of a major transition. An independent PT of even (odd) order $2n$
($2n-1$) occurs at $E_{c}$ if $\partial^{2n-1}_{E}S_{N}$
($\partial^{2n-2}_{E}S_{N}$) admits an inflection point at $E_{c}$ and
$\partial^{2n}_{E}S_{N}$ ($\partial^{2n-1}_{E}S_{N}$) combined with a
negative-valued maximum (positive-valued minimum) at $E_{c}$. A dependent PT
of even (odd) order $2n$ $(2n+1)$ is detected at $E_{c}$, if
$\partial_{E}^{2n-1}S$ ($\partial_{E}^{2n}S$) admits a least-sensitive
inflection point at $E_{c}$ combined with positive-valued minimum (negative-
valued maximum) at $\partial_{E}^{2n}S$ ($\partial_{E}^{2n+1}S$) [38]. We
present here the application of the MIPA to the previously mentioned
microcanonical formulation of the $U(1)$-LGT. In the left panel of Fig. 1, the
first-order derivative of entropy $\partial_{\epsilon}S$ (coinciding with the
inverse of microcanonical temperature $\beta_{\mu c}$ ) as a function of the
energy density, $\epsilon=E/N_{ind}$ is presented(see SM for details about
such a computation). In order to calculate the derivative of microcanonical
entropy, we adopted a method introduced by Pearson et al. [51] allowing to
estimate the $n$-order derivative of microcanonical entropy at given energy as
a function of the time averages (S.4) of the first $n$ positive integer powers
of the inverse of the kinetic energy, for instance,
$\partial_{E}S(E)=(N_{ind}/2-1)\langle K^{-1}\rangle_{\mu c}$ (see SM for
higher order derivatives). We can easily identify a back-bending region
(S-shape) in the inverse of microcanonical temperature $\partial_{\epsilon}S$
entailing a positive-valued minimum of at $\epsilon^{ind}_{1}\approx 1.197$.
Such a minimum corresponds to a positive-valued maximum in
$\partial^{2}_{\epsilon}S$ (see Fig. 2), thus to an independent first-order
PT, according to the MIPA. Such a transition corresponds to the deconfinement
phase transition (DPT). Indeed, the value of $\beta_{\mu c}$ at the transition
point $\epsilon^{ind}_{1}$ corresponds to
$\partial_{\epsilon}S(\epsilon^{ind}_{1})\equiv\beta(\epsilon^{ind}_{1})=1/T_{1}\approx
1.00872$ in unit of $\hbar=k_{B}=1$ which coincides with the transition value
of $\beta$ predicted by Monte Carlo simulations [52, 53, 54, 55, 28, 29]. It
should be stressed that the DPT has sometimes been classified as a second-
order PT [56, 52, 30, 57, 53, 58, 59], but many evidences, such as the
emergence of metastable states, led to conclude that the DPT is of first-order
[55, 60, 61, 54, 14, 15, 62]. In order to confirm the identification of such a
first-order PT with the DPT, we provide the microcanonical computation of the
average plaquette, $\langle\bar{\mathscr{V}}_{\square}\rangle_{\mu
c}=\langle\mathscr{V}\rangle_{\mu c}/N_{ind}$ as a function of the
microcanonical temperature, as suggested in Ref. [28]. As shown in the plot on
the right in Fig. 1, the average plaquette becomes a multi-valued function
that admits an S-shape around $\beta_{1}\approx 1.00872$ (see the inset); this
is a typical signature of a first-order PT. It has to be noticed that such an
S-shape is a direct consequence of the inflection point in
$\partial_{\epsilon}S$. In particular, the local minimum associated with the
DPT ($\partial_{\varepsilon}S|_{\varepsilon_{1}}=0$) is the deep origin of the
vertical tangent point indicated by the vertical red line on the right panel
of Fig. 1, i.e.

$\lim_{\beta_{\mu
c}\rightarrow\beta_{1}^{+}}\dfrac{\partial\langle\mathscr{\bar{V}}_{\square}\rangle_{\mu
c}}{\partial\beta_{\mu c}}=\lim_{\beta_{\mu
c}\rightarrow\beta_{1}^{+}}\dfrac{\partial\langle\mathscr{\bar{V}}_{\square}\rangle_{\mu
c}}{\partial\varepsilon}\dfrac{\partial\varepsilon}{\partial\beta_{\mu
c}}=-\infty,$ (6)

as $\partial_{\varepsilon}\langle\bar{\mathscr{V}}_{\square}\rangle$ is
positive and bounded on the considered range of energies (see Fig. 2 of SI)
and $\lim_{\beta_{\mu c}\rightarrow\beta_{1}^{+}}\partial_{\beta_{\mu
c}}\varepsilon=-\infty$. Following the classification scheme associated with
MIPA, we can identify a further third-order independent PT, associated with an
inflection point in $\partial^{2}_{\epsilon}S$ (see the vertical blue line in
Fig. 2) together with a positive-valued minimum in $\partial^{3}_{\epsilon}S$
around $\epsilon^{ind}_{3\,(b)}\approx 0.93$ (see Fig. 3). In addition, we
have good evidence that encourages the identification of two other third-order
PTs; the first is independent, whereas the second is dependent.

Figure 2: Second-order derivative of the microcanonical entropy for the 4D
$U(1)$-LGT with lattice size $N=12^{4}$. All of the PTs discussed in the main
text are represented in the plot by vertical lines. Independent third-order
PT, $\epsilon^{ind}_{3\,(a)}$: black. Independent third-order PT,
$\epsilon^{ind}_{3\,(b)}$: blue. Independent first-order PT,
$\epsilon^{ind}_{1}$: red. Dependent third-order PT, $\epsilon^{dep}_{3}$:
green. The panel shows the behavior of the second-order derivative around the
deconfinement PT. $\partial_{\epsilon}^{2}S$ vanishes in correspondence of
$\epsilon^{ind}_{1}$ with a subsequent maximum as expected in a first-order
PT.

The former (see the vertical black line in Fig. 3) occurs around
$\epsilon^{ind}_{3\,(a)}\approx 0.57$ and the latter (see the vertical green
line in Fig. 3) around $\epsilon^{dep}_{3}\approx 1.375$. To understand the
physical meaning of the _large_ number of PTs identified by MIPA and never
observed so far, we analyze and compare the third-order derivative of the
microcanonical entropy with the average number density of topological defects.
The current knowledge of the role of topological defects during the DPT is
summarized in Ref. [10]: at $T<T_{1}$, there are few monopole loops, and they
will be small in spatial extent. Increasing $T$ increases the density and size
of loops. At $T_{1}$, the monopole-antimonopole pairs dissolve, and the
theory’s vacuum becomes a gas of monopoles and antimonopoles without strong
correlations. In light of our results, we numerically demonstrate that the
spectrum of transitional/collective phenomena occurring in $U(1)$-LGT is
actually richer than the expected one. Indeed, the pairs dissolution is only
one of the mechanisms occurring in an Abelian LGT. For this aim, we consider
the 3D monopoles (topological defects) of the gauge field defined according to
[12, 63] (see SM for more details). In our simulations, we compute the average
densities of monopole-antimonopole pairs at the nearest-neighbors distance,
$\langle\rho_{pair}\rangle$, and of isolated monopoles,
$\langle\rho_{mon}\rangle$. The yellow and violet curves in Fig. 3 represent
$\langle\rho_{pair}\rangle$ and $\langle\rho_{mon}\rangle$, respectively, as
functions of the energy. At $\epsilon\leq\epsilon^{ind}_{3\,(a)}$, monopoles
are not detected, and $\langle\rho_{mon}\rangle=\langle\rho_{pairs}\rangle=0$.
The first appearance of monopoles, i.e., non-vanishing values for
$\langle\rho_{mon}\rangle$ and $\langle\rho_{pairs}\rangle$ occurs around
$\epsilon_{first}\approx 0.62$ (see the inset in Fig. 3). This effect is
reflected in $\partial^{3}_{\epsilon}S$ with the presence of a positive-valued
minimum of at $\epsilon^{ind}_{3\,(a)}$; this corresponds to an independent
third-order PT, according with the more recent version of MIPA [38]. It should
be mentioned that a non-negligible mismatch is found between
$\epsilon_{first}$ and $\epsilon^{ind}_{3\,(a)}$. A discussion about the
reasons at the ground of this discrepancy is provided in the SI. For energies
slightly larger than $\epsilon^{ind}_{3\,(a)}$, only a small number of
monopoles is found in bonded pairs; the largest amount of monopoles is in the
isolated configuration, i.e., the free states. Increasing the energy, the
density of isolated monopoles $\langle\rho_{mon}\rangle$ reaches a local
maximum around $\epsilon_{peak}\approx 0.78$. Thus, the production of isolated
monopoles is favored over that of pairs up to $\epsilon_{peak}$ where such a
production ends. For $\epsilon_{peak}\leq\epsilon\leq\epsilon^{ind}_{3\,(b)}$,
the process reverses: the isolated monopoles begin to interact _leaving_ the
isolated configuration to form pairs as one deduces from the subsequent drop
of $\langle\rho_{mon}\rangle$. At this stage, the production of pairs
increases, reaching a peak about $\epsilon\simeq\epsilon^{ind}_{3\,(b)}$
corresponding to the independent third-order PT. Therefore, such a transition
corresponds to the saturation of the pairs production process; the system
reaches a new phase, that is, a new vacuum state of the theory mostly
populated by pairs. Increasing the energy, the density of pairs
$\langle\rho_{pairs}\rangle$ decreases crossing over $\epsilon^{ind}_{1}$ and
reaching a plateau in correspondence of $\epsilon^{dep}_{3}$. The drop in the
density of pairs is the physical process originating from the pair-dissolution
mechanism and leading to the DPT. It is worth noting that, after the pair
dissolution (see the vertical red line in Fig. 3), the slope of
$\langle\rho_{pairs}\rangle$ ($\langle\rho_{mon}\rangle$) passes from being
negative (positive) to being ideally constant for any
$\epsilon>\epsilon^{dep}_{3}$. Such a change corresponds to the dependent PT
of third-order associated with a negative-valued maximum in
$\partial^{3}_{\epsilon}S$ around $\epsilon^{dep}_{3}$ (see the vertical green
line in Fig. 3 and the inset within) signaling the transition of the system to
the last vacuum state represented by a gas of monopoles and pairs.

Figure 3: Comparison between third-order derivative and density of monopoles
for the 4D $U(1)$-LGT with lattice size $N=12^{4}$. The black curve represents
$\partial_{\epsilon}^{3}S$ whereas the yellow and violet curves are,
respectively, the number density of monopole-antimonopole pairs and isolated
monopoles. The vertical colored lines indicate the energy value of each PT as
indicated in Fig. 2. Inset on the top: zoom on the energy range where the
first appearance of monopoles is observed. Inset on the bottom: zoom on the
negative-valued maximum associated to the dependent PT.

In summary, we have shown that the application of MIPA to a microcanonical
formulation of the $U(1)$-LGT allowed us to identify two independent (major)
third-order PTs and a dependent (minor) third-order PT, never observed before
through canonical ensemble-based approaches. In practice, microcanonical PTs,
corresponding to inflection points of entropy or its derivatives, have been
interpreted as qualitative changes in the behavior of topological defects
(monopoles) when the energy is varied. We found that all the transitional
phenomena are driven by the occurrence of topological defects or by their
interactions. In particular, we have observed an independent third-order PT at
$\epsilon^{ind}_{3(a)}$ at very low energy associated with the first
appearance of monopoles. Increasing the energy, the monopoles interact among
them, forming pairs interpreted as bound states of the theory and reaching a
saturation point corresponding to a further independent third-order PT around
$\epsilon_{3(b)}^{ind}$. At higher energies, a dissociation-pairs process has
been observed in correspondence to the well-known deconfinement transition
that is an independent first-order PT according to the MIPA scheme. During
this transition, we are in the presence of a coexistence of two phases of the
theory: pairs (bounded states) and isolated monopoles (free states). At large
energy, the ground state of the theory corresponds to a gas of monopoles, and
it is signaled by a dependent third-order PT. It should be stressed that
adopting Kogut’s continuum-time formulation of QFT [11, 34, 64], MIPA can be
easily extended to systems of fermions interacting with a gauge field as well
as to the study of PTs for the Higgs field.

We conclude by discussing an intriguing similarity between the $U(1)$-LGT, in
Villain approximation given by Banks et al. [10], and many-body systems in
condensed matter. Indeed, the Wilson action appearing in the partition
function can be recast into an equivalent action where the new degrees of
freedom are represented by a monopole density at each lattice site
$\\{m(\bm{n})\\}_{\bm{n}}$. Thus, the action describes a collection of
monopoles (many-body system) interacting with each other through the lattice
Coulomb potential, i.e., $m(\bm{n})v(\bm{n}-\bm{u})m(\bm{u})$. Since the PTs
observed in LGT occur due to the emergence of monopoles and their
interactions, by analogy, we could expect new transition phenomena also in
condensed-matter systems where the charge density plays a similar role to the
monopole density.

L.D.C is grateful with Dr. Dario Consoli and Dr. Alfredo Grillo for their
helpful suggestions and discussions.

##

## References

  * P.W. Higgs [1966] P.W. Higgs, Phys. Rev. 145, 1156 (1966).
  * P.W. Higgs [1964] P.W. Higgs, Phys. Rev. Lett. 13, 508 (1964).
  * F. Englert and R. Brout [1964] F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964).
  * M. Blasone, P. Jizba, and G. Vitiello [2011] M. Blasone, P. Jizba, and G. Vitiello, (World Scientific, 2011).
  * H. Umezawa [1995] H. Umezawa, (Springer, 1995).
  * L.D. Landau [1937a] L.D. Landau, Zh. Eksp. Teor. Fiz. 11, 19 (1937a).
  * L.D. Landau [1937b] L.D. Landau, Zh. Eksp. Teor. Fiz. 11, 627 (1937b).
  * N.D. Mermin and H. Wagner [1966] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
  * N.D. Mermin [1967] N.D. Mermin, J. Math. Phys. 8, 1061 (1967).
  * T. Banks, R. Myerson, and J.B. Kogut [1977] T. Banks, R. Myerson, and J.B. Kogut, Nucl. Phys. B 129, 493 (1977).
  * J.B. Kogut [1983] J.B. Kogut, Rev. Mod. Phys. 55, 775 (1983).
  * T.A. DeGrand and D. Toussaint [1980] T.A. DeGrand and D. Toussaint, Phys. Rev. D 22, 2478 (1980).
  * J.S. Barber and R.E. Shrock [1985] J.S. Barber and R.E. Shrock, Nucl. Phys. B 257, 515 (1985).
  * J.S. Barber [1984] J.S. Barber, Phys. Lett. B 147, 330 (1984).
  * V. Grösch, K. Jansen, J. Jersak, C.B. Lang, T. Neuhaus and C. Rebbi [1985] V. Grösch, K. Jansen, J. Jersak, C.B. Lang, T. Neuhaus and C. Rebbi, Phys. Rev. B 162, 171 (1985).
  * W. Kerler, C. Rebbi and A. Weber [1994] W. Kerler, C. Rebbi and A. Weber, Phys. Rev. D 50, 6984 (1994).
  * J.M. Kosterlitz [1974] J.M. Kosterlitz, J. Solid State Phys. 7, 1046 (1974).
  * J.M. Kosterlitz and D.J. Thouless [1973] J.M. Kosterlitz and D.J. Thouless, J. Phys. C: Solid State Phys. 6, 1181 (1973).
  * V.L. Berezinskii [1971] V.L. Berezinskii, Sov. Phys. JETP 32, 493 (1971).
  * G.C. Wick [1954] G.C. Wick, Phys. Rev. 96, 1124 (1954).
  * R.P. Feynman [2005] R.P. Feynman (World Scientific, 2005) pp. 71–109.
  * R.P. Feynman and A.R. Hibbs [1965] R.P. Feynman and A.R. Hibbs, (1965).
  * M.E. Peskin and D.V. Schroeder [1995] M.E. Peskin and D.V. Schroeder, (1995).
  * C.N. Yang and T.D. Lee [1952] C.N. Yang and T.D. Lee, Phys. Rev. 87, 404 (1952).
  * T.D. Lee, Tsung-Dao and C.N. Yang [1952] T.D. Lee, Tsung-Dao and C.N. Yang, Phys. Rev. 87, 410 (1952).
  * P. Ehrenfest [1933] P. Ehrenfest, (NV Noord-Hollandsche Uitgevers Maatschappij, 1933).
  * K.G. Wilson [1974] K.G. Wilson, Phys. Rev. D 10, 2445 (1974).
  * D.J.E Callaway and A. Rahman [1982] D.J.E Callaway and A. Rahman, Phys. Rev. Lett. 49, 613 (1982).
  * D.J.E Callaway and A. Rahman [1983] D.J.E Callaway and A. Rahman, Phys. Rev. D 28, 1506 (1983).
  * D.J.E Callaway [1985a] D.J.E Callaway, Contemp. Phys. 26, 23 (1985a).
  * D.J.E Callaway [1985b] D.J.E Callaway, Contemp. Phys. 26, 95 (1985b).
  * T.D Andersen [2019] T.D Andersen, Phys. Rev. D 99, 016012 (2019).
  * J.B. Kogut, J. Polonyi, H.W. Wyld, J. Shigemitsu, and D.K. Sinclair [1985] J.B. Kogut, J. Polonyi, H.W. Wyld, J. Shigemitsu, and D.K. Sinclair, Nucl. Phys. B 251, 311 (1985).
  * J. Polonyi and H.W. Wyld [1983] J. Polonyi and H.W. Wyld, Phys. Rev. Lett. 51, 2257 (1983).
  * M. Bachmann [2014a] M. Bachmann, (Cambridge University Press, 2014).
  * M. Bachmann [2014b] M. Bachmann, J. Phys. Conf. Ser. (IOP Publishing, 2014) p. 012013.
  * S. Schnabel, D.T Seaton, D.P. Landau, and M. Bachmann [2011] S. Schnabel, D.T Seaton, D.P. Landau, and M. Bachmann, Phys. Rev. E 84, 011127 (2011).
  * K. Qi and M. Bachmann [2018] K. Qi and M. Bachmann, Phys. Rev. Lett. 120, 180601 (2018).
  * T. Koci and M. Bachmann [2017] T. Koci and M. Bachmann, Phys. Rev. E 95, 032502 (2017).
  * K. Sitarachu, R.K.P. Zia and M. Bachmann [2020] K. Sitarachu, R.K.P. Zia and M. Bachmann, J. Stat. Mech. Theory Exp. 2020, 073204 (2020).
  * K. Sitarachu and M. Bachmann [2020] K. Sitarachu and M. Bachmann, J. Phys. Conf. Ser. (IOP Publishing, 2020) p. 012009.
  * D. Aierken and M. Bachmann [2020] D. Aierken and M. Bachmann, Polymers 12, 3013 (2020).
  * K. Qi, B. Liewehr, T. Koci, B. Pattanasiri, M.J. Williams and M. Bachmann [2019] K. Qi, B. Liewehr, T. Koci, B. Pattanasiri, M.J. Williams and M. Bachmann, J. Chem. Phys. 150, 054904 (2019).
  * T. Koci and M. Bachmann [2015] T. Koci and M. Bachmann, Phys. Rev. E 92, 042142 (2015).
  * K. Sitarachu and M. Bachmann [2022] K. Sitarachu and M. Bachmann, Phys. Rev. E 106, 014134 (2022).
  * B. Ghofrane, M. Gori, V. Penna, G. Pettini, and R. Franzosi [2020] B. Ghofrane, M. Gori, V. Penna, G. Pettini, and R. Franzosi, Entropy 22, 380 (2020).
  * G. Pettini, M. Gori, R. Franzosi, C. Clementi and M. Pettini [2019] G. Pettini, M. Gori, R. Franzosi, C. Clementi and M. Pettini, Phys. A: Stat. Mech. Appl. 516, 376 (2019).
  * L. Di Cairano [2022] L. Di Cairano, J. Phys. A Math. Theor. (2022).
  * L. Di Cairano, R. Capelli, B. Ghofrane and M. Pettini [2022] L. Di Cairano, R. Capelli, B. Ghofrane and M. Pettini, Phys. Rev. E 106, 054134 (2022).
  * D.J.E. Callaway [1984] D.J.E. Callaway, Phys. Lett. B 145, 363 (1984).
  * E.M. Pearson, T. Halicioglu, and W. A. Tiller [1985] E.M. Pearson, T. Halicioglu, and W. A. Tiller, Phys. Rev. A 32, 3030 (1985).
  * M. Creutz, L. Jacobs, and C. Rebbi [1979] M. Creutz, L. Jacobs, and C. Rebbi, Phys. Rev. D 20, 1915 (1979).
  * G. Bhanot [1981] G. Bhanot, Phys. Rev. D 24, 461 (1981).
  * D.R. Stump and J.H. Hetherington [1987] D.R. Stump and J.H. Hetherington, Phys. Lett. B 188, 359 (1987).
  * J. Jersák, and T. Neuhaus and P.M. Zerwas [1983] J. Jersák, and T. Neuhaus and P.M. Zerwas, Phys. Lett. B 133, 103 (1983).
  * K.J.M Moriarty [1982] K.J.M Moriarty, Phys. Rev. D 25, 2185 (1982).
  * M. Creutz, L. Jacobs, and C. Rebbi [1983] M. Creutz, L. Jacobs, and C. Rebbi, Phys. Rep. 95, 201 (1983).
  * T.A. DeGrand and D. Toussaint [1981] T.A. DeGrand and D. Toussaint, Phys. Rev. D 24, 466 (1981).
  * D.G. Caldi [1983] D.G. Caldi, Nucl. Phys. B 220, 48 (1983).
  * S. Sarkar [2021] S. Sarkar, Sci. Rep. 11, 1 (2021).
  * B. Lautrup and M. Nauenberg [1980] B. Lautrup and M. Nauenberg, Phys. Lett. B 95, 63 (1980).
  * H.G. Evertz, T. Jersak, T. Neuhaus and P.M. Zerwas [1985] H.G. Evertz, T. Jersak, T. Neuhaus and P.M. Zerwas, Nucl. Phys. B 251, 279 (1985).
  * J. Tobochnik and G.V. Chester [1979] J. Tobochnik and G.V. Chester, Phys. Rev. B 20, 3761 (1979).
  * J.B. Kogut [1987] J.B. Kogut, Nucl. Phys. B 290, 1 (1987).
  * L. Casetti [1995] L. Casetti, Phys. Scr. 51, 29 (1995).

SUPPLEMENTAL MATERIAL

##  I Hamiltonian model for the U(1) compact model on 4D Euclidean lattice:
numerical algorithm

The investigation of the thermodynamic properties of the $U(1)$-LGT is based
on the molecular dynamics simulation of the Hamiltonian system defined by

$\mathscr{H}(\pi,\phi):=\sum_{\mu,n}\frac{\pi_{n,\mu}^{2}}{2}+\sum\limits_{\square_{\mu,n}}\left[1-\cos(\phi_{\bm{n},\mu}+\phi_{n+\mu,\nu}-\phi_{n+\nu,\mu}-\phi_{n,\nu})\right].$
(S.1)

where $\square_{\mu,\nu,n}$ is a representative plaquette. We then numerically
solved the Hamilton equations of motion

$\frac{d\phi_{\mu,n}}{d\tau}=\pi_{\mu,n},\qquad\frac{d\pi_{\mu,n}}{d\tau}(\tau)=-\frac{\partial\mathscr{H}}{\partial\phi_{\mu,n}},$
(S.2)

adopting a second order bilateral symplectic algorithm [65] since it conserves
the symplectic structure of the Hamiltonian flow. Such an algorithm reads
($\tau_{n/2}:=\tau_{0}+n\Delta\tau/2$):

$\begin{split}\phi_{\mu,n}(\tau_{1/2})&=\phi_{\mu,n}(\tau_{0}),\\\
\pi_{\mu,n}(\tau_{1/2})&=\pi_{\mu,n}(\tau_{0})-F_{\mu,n}[\phi(\tau_{1/2})]\Delta\tau/2,\\\
\phi_{\mu,n}(\tau_{1})&=\phi_{\mu,n}(\tau_{1/2})+\pi_{\mu,n}(\tau_{1/2})\Delta\tau,\\\
\pi_{\mu,n}(\tau_{1})&=\pi_{\mu,n}(\tau_{1/2})-F_{\mu,n}[\phi_{1}]\Delta\tau/2,\\\
\pi_{\mu,n}(\tau_{3/2})&=\pi_{\mu,n}(\tau_{1}),\\\
\phi_{\mu,n}(\tau_{3/2})&=\phi_{\mu,n}(\tau_{1})+\pi_{\mu,n}(\tau_{3/2})\Delta\tau/2,\\\
\pi_{\mu,n}(\tau_{2})&=\pi_{\mu,n}(\tau_{3}/2)-F_{\mu,n}[\phi(\tau_{3/2})]\Delta\tau,\\\
\phi_{\mu,n}(\tau_{2})&=\phi_{\mu,n}(\tau_{3/2})+\pi_{\mu,n}(\tau_{3/2})\Delta\tau,\end{split}$
(S.3)

where
$F_{\mu,n}(\tau_{n/2}):=\partial_{\phi_{\mu,n}}\mathscr{V}[\phi(\tau_{n/2})]$.
The time step has been chosen to be $\Delta\tau=0.01$ and it leads to a
conservation of energy around $10^{-6}$-$10^{-7}$. The initial conditions have
been chosen randomly for $\phi_{\mu,n}$ and zero for $\pi_{\mu,n}$ (as
explained later). In this way, we obtained a certain energy value, $E_{in}$,
and we allowed the system to evolve for $10^{6}$ steps so as to equilibrate
the trajectory. After this procedure, if $E_{in}$ was not the desired energy,
say $E_{des}$, we reached the energy value $E_{des}$ by choosing a suitable
positive (negative) constant $\alpha$ such that, multiplying each momentum
$\pi_{\mu,n}$ by $\alpha$, the system was heated (cooled). In practice,
starting from $E_{in}$, one guesses a $\alpha_{guess}$ and measures
$E_{guess}=\mathscr{H}[\alpha\pi_{\mu,n},\phi_{\mu,n}]$. This procedure is
repeated as long as $|E_{guess}-E_{des}|$ is smaller than the chosen
precision. Once the desired energy has been reached, the system is
equilibrated again for $10^{6}$ iterations before saving the trajectory used
for computing averages. Each thermodynamic observable has been evaluated on
$N_{step}=131071$ measurements, and the definition of the microcanonical
average is based on the ergodic assumption that, adapted to simulations, reads

$\langle
f\rangle_{\varepsilon}=\frac{1}{N_{step}}\sum_{n=1}^{N_{step}}f(\tau_{n}).$
(S.4)

The constraint $\pi_{\mu,n}=0$ for the initial values has been proposed by
Callaway in Refs. [28, 29] to eliminate the extra degrees of freedom coming
from the gauge invariance. This procedure is equivalent to imposing a gauge
such as, for instance, the Lorentz gauge or the axial one.

##  II Calculation of Microcanonical Entropy Derivatives

The Microcanonical Inflection Point Analysis (MIPA) requires to calculate the
derivatives of the specific microcanonical entropy defined as

$S_{N_{\rm ind}}(E)=\log\Omega_{\rm
ind}(E)=\log\int\delta(H(\mathbf{\pi},\mathbf{\phi})-E)\delta(\mathscr{C}(\mathbf{\pi}))\,\,\,\mathrm{D}\mathbf{\pi}\mathrm{D}\mathbf{\phi}$
(S.5)

where the condition $\mathscr{C}(\mathbf{\pi})=0$ is equivalent to a gauge-
fixing condition (see the previous section) and $N_{\rm ind}=3N_{\rm lat}^{4}$
is the number of independent momenta [28, 29]. The specific microcanonical
entropy has been introduced so as to extract extensive quantities and compare
systems of different sizes: its form is $s_{N_{\rm ind}}(\varepsilon)=N_{\rm
ind}^{-1}S_{N_{\rm ind}}(N_{\rm ind}\varepsilon)$.

###  II.1 First- and second-order derivative of microcanonical entropy

As shown in Ref. [51], in the microcanonical ensemble, given any fixed value
of the specific energy, the derivatives of specific entropy can be expressed
in terms of the moments of the distribution of the inverse specific kinetic
energy $k=K(\bm{\pi})/N_{\rm ind}$. In this framework, the first- and second-
order derivative of specific microcanonical entropy can be written as

$\begin{split}&\partial_{\varepsilon}s_{N_{\rm
ind}}(\varepsilon)=\left(\frac{1}{2}-\frac{1}{N_{ind}}\right)\langle
k^{-1}\rangle_{\varepsilon}\\\ &\partial^{2}_{\varepsilon}s_{N_{\rm
ind}}(\varepsilon)=N_{ind}\bigg{[}\bigg{(}\dfrac{1}{2}-\dfrac{1}{N_{ind}}\bigg{)}\bigg{(}\dfrac{1}{2}-\dfrac{2}{N_{ind}}\bigg{)}\langle
k^{-2}\rangle_{\varepsilon}-\bigg{(}\dfrac{1}{2}-\dfrac{1}{N_{ind}}\bigg{)}^{2}\langle
k^{-1}\rangle^{2}_{\varepsilon}\bigg{]}\\\ \end{split}$ (S.6)

where the microcanonical averages have been estimated by Eq.(S.4). It is worth
noting that the standard thermodynamic observables as the microcanonical
temperature, $T(\varepsilon)$, and the microcanonical specific heat,
$C_{v}(\varepsilon)$, given by

$\begin{split}&T_{\mu c}(\varepsilon)=\beta_{\mu c}=(\partial
s/\partial\varepsilon)^{-1}\\\ &C_{v}(\varepsilon)=(\partial
T(\varepsilon)/\partial\varepsilon)^{-1}\equiv-(\partial
s/\partial\varepsilon)^{2}(\partial^{2}s/\partial\varepsilon^{2})^{-1}\,\,.\end{split}$
(S.7)

are connected to the derivatives of microcanonical entropy and, thereby, they
can be computed by means of Eqs. (S.6).

Figure 4: Microcanonical thermodynamic observables in U(1) LGT. The
microcanonical temperature $T_{\mu c}$ (black circles) and the microcanonical
specific heat $C_{v}$ (purple circles) vs. the specific energy $\varepsilon$
have been reported for the $U(1)$-LGT with a cubic lattice with $N=12$ sites
for side.

###  II.2 Third-order derivative of microcanonical entropy

In principle, the third-order derivative of microcanonical entropy can be
estimated using the same formalism presented in the previous subsection.
However, the poor convergence of the third-order moment of kinetic energy
distribution does not allow calculating the third-order derivative of
microcanonical entropy with such a method. Thus numerical derivative of the
second-order derivative $\partial_{\varepsilon}^{2}s$ of specific
microcanonical entropy has been estimated.

##  III On the origin of the S-shape in the plaquette potential vs
microcanonical entropy

We report in this section the behavior of the plaquette potential
$\langle\mathscr{V}_{\square}\rangle_{\mu c}$ and of its energy derivative
$\partial_{\varepsilon}\langle\mathscr{V}_{\square}\rangle_{\mu c}$ regarded
as functions of specific energy (see Fig. 5 below). Observing such behaviors,
we can conclude that in a neighborhood of the deconfinement phase transition,
the energy derivative of the average plaquette potential is bounded.

Figure 5: Microcanonical plaquette potential vs specific energy. The
microcanonical average of the plaquette potential (black line) and its
derivative (blue line) are plotted as functions of the specific energy. The
dashed-dotted red line indicates the critical energy, $\epsilon^{ind}_{1}$,
for the deconfinement transitions

##  IV Computation of monopoles

The computation of the topological defects used in our work is based on Ref.
[12]. In particular, we studied 3D monopoles redefining the 3D plaquette angle
as $\bar{\Theta}_{ij}=\Theta_{ij}-2\pi K_{ij}$ where
$K_{ij}:=\text{mod}(\Theta_{ij},2\pi)$. Then, using Gauss’ law, we get:

$N(\bm{n})=\sum_{i,j,k=1}^{3}\epsilon_{ijk}(K_{jk}(\bm{n}+\bm{e}_{i})-K_{jk}(\bm{n})).$
(S.8)

Here, $\epsilon_{ijk}$ is the Levi-Civita antisymmetric tensor. By definition,
$\phi_{\bm{n},\mu}\in[-\pi,\pi]$ and $\Theta_{ij}\in[-4\pi,4\pi]$, then,
$N=0,\,\pm 1,\,\pm 2$. If $N=0$ no topological defects are present in the
plaquette defined by the site $\bm{n}$ and the three directions $\bm{e}_{x}$,
$\bm{e}_{y}$ and $\bm{e}_{z}$ whereas if ($N<0$) $N>0$ we are in presence of
(anti)monopoles with charge 1 or 2. We study the average density of isolated
monopoles $\langle\rho_{mon}\rangle$ and the total number of monopole-
antimonopole pairs $\langle\rho_{pairs}\rangle$. The calculation of the former
is summarized in the following procedure. At any lattice site, $\bm{n}$, we
identify $N_{iso}(\bm{n})$ to be the number computed through Eq. (S.8) such
that (i) $N(\bm{n})\neq 0$, (ii)
$N(\bm{n}\pm\bm{e}_{x})=N(\bm{n}\pm\bm{e}_{y})=N(\bm{n}\pm\bm{e}_{y})=0$.
Proceeding in this manner for each lattice site, we define

$\langle\rho_{mon}\rangle=\frac{1}{V}\sum_{\bm{n}}N_{iso}(\bm{n}),$ (S.9)

where $V=N^{4}$ is the volume of the lattice. In our case, $N=12$. For what
concerns the number density of pairs, we proceed similarly as before but we
identify $N_{pairs}(\bm{n})$ with the number computed in Eq. (S.8) if (i)
$N(\bm{n})\neq 0$, (ii) $N(\bm{n}+\bm{e}_{i})=-N(\bm{n})$ only for a single
$i=x,\,y,\,z$. Thus, the associated observable is

$\langle\rho_{pairs}\rangle=\frac{1}{V}\sum_{\bm{n}}N_{pairs}(\bm{n}).$ (S.10)

##  V Apparent discrepancy for the first pairs-appearance phase transition

As mentioned in the main text, we associated the independent third-order PT
occurring at $\epsilon^{ind}_{3(a)}\approx 0.57$ to the first appearance of
pairs and monopoles. This conceptual link is suggested by the presence of a
positive-valued minimum in the third-order derivative of entropy and by non-
vanishing values of the number density of pairs $\langle\rho_{pairs}\rangle$
around $\epsilon_{first}\approx 0.62$. Nevertheless, as one can see in Fig. 3,
these two manifestations do not occur exactly at the same energy and a
mismatch is found $|\epsilon_{first}-\epsilon_{3\,(a)}^{ind}|\approx 0.05$. We
believe that the presence of this mismatch has actually a numerical origin due
to two possible reason: large average waiting time for the occurrence of
monopoles and the definition of monopoles. For what concerns the first, we
notice that the estimation of the energy associated to the first appearance of
topological defects can be refined by means of longer simulations at low
energies. Indeed, we note that at low energies the system is “frozen”; roughly
speaking, the representative point of the trajectory evolves very slowly from
its initial configuration. Thus, we can image that the average waiting time,
$\tau_{wait}(E)$, for the occurrence of a monopole which depends on energy
decreases as the energy increases. In practice, in our simulation we fixed the
time length of the trajectories to be $T_{sim}=N_{step}\cdot\Delta t=1310.71$.
At very low energies, $E\approx 0$, we have that
$T_{sim}/\tau_{wait}(E)\approx 0$. Hence, we should extend the simulations, in
principle, with $T_{sim}\to\infty$. Increasing the energy, we found that
$\tau_{wait}(E)$ becomes smaller then $T_{sim}$ only from energies
$\epsilon\geq\epsilon_{first}\approx 0.62$ where the first appearance of
monopoles has been detected. In conclusion, we expect that for $T_{sim}$
larger than that used now monopoles can be detected earlier. On the other
hand, we also expect that there exists a certain threshold indicated by the
thermodynamic observable below which monopoles are detected only after an
infinite average waiting time. Regarding the second reason that we mentioned
above, we notice that the definition of monopoles is based on the integers
$K_{ij}=\text{mod}(\Theta_{ij},2\pi)$ where $\Theta_{ij}$ is the plaquette
angle that is a real number. Thus, for any value of $\Theta_{ij}$ that is not
greater or exactly equal to $2\pi$, $K_{ij}$ will be always zero. In other
words, Eq. (S.8) does not recognize any monopole arising from those plaquette
angles such that $\Theta_{ij}=2\pi-\delta$ with an arbitrary small $\delta>0$.
In contrast to that, the calculation of kinetic energy used for computing the
thermodynamics observables is not subjected to any constraint expected those
given by the equations of motion. Thus, the signals in the thermodynamic
observable can appear at the correct energy value.