# Multicomponent gauge-Higgs models with discrete Abelian gauge groups

Giacomo Bracci Testasecca1 and Andrea Pelissetto1,2 1Dipartimento di Fisica,
Università di Roma “La Sapienza", P.le Aldo Moro 2, I-00185 Roma, Italy  2
INFN, Sezione di Roma, P.le Aldo Moro 2, I-00185 Roma, Italy

(August 27, 2024)

###### Abstract

We consider a variant of the charge-$Q$ compact Abelian-Higgs model, in which
an $N_{f}$-dimensional complex vector is coupled with an Abelian
${\mathbb{Z}}_{q}$ gauge field. For $N_{f}=2$ and $Q=1$ we observe several
transition lines that belong to the O(4), O(3), and O(2) vector universality
classes, depending on the symmetry breaking pattern at the transition. The
universality class is independent of $q$ as long as $q\geq 3$. The
universality class of the transition is uniquely determined by the behavior of
the scalar fields; gauge fields do not play any role. We also investigate the
system for $N_{f}=15$ and $Q=2$. In the presence of U(1) gauge fields, the
system undergoes transitions associated with charged fixed points of the
Abelian-Higgs field theory. These continuous transitions turn into first-order
ones when the U(1) gauge fields are replaced by the discrete
${\mathbb{Z}}_{q}$ fields: in the present compact model charged transitions
appear to be very sensitive to the nature of the gauge fields.

##  1 Introduction

Symmetries have always played a central role in the development of physical
models. Global symmetries are commonly used to characterize the different
phases of matter [1] and to understand the nature and properties of the
transition lines between them. In the absence of local symmetries, the
universal features of critical transitions can be understood using the
traditional Landau-Ginzburg-Wilson (LGW) approach to critical phenomena [2, 3,
4, 6, 5]. It assumes that universal properties can be explained by a
statistical field theory, in which a local order parameter plays the role of
fundamental field. The universality classes are uniquely determined by the
breaking pattern of the global symmetry, i.e., by the symmetry of the two
phases coexisting at the transition, and by the transformation properties of
the order parameter under the symmetry group.

Local gauge symmetries also play a fundamental role, both in high-energy
physics [7], and in condensed-matter physics, where their applications span
from superconductivity [8] to topological order and quantum phase transitions,
see, e.g., Refs. [9, 10, 11]. The critical behavior of three-dimensional
systems with global and local symmetries is, at present, not fully understood.
First of all, in the presence of gauge degrees of freedom, one may have
topologically ordered phases, that cannot be characterized in terms of local
order parameters, whose existence is the basic tenet of the LGW approach [14,
15, 12, 13]. Second, it is unclear under which conditions the gauge degrees of
freedom play a role at transitions where global symmetries are broken and
matter fields show long-range correlations.

Critical transitions in three-dimensional lattice gauge theories can be
classified in three different groups. First, there are topological
transitions, where only the gauge degrees of freedom play a role and the
global symmetry of the coexisting phases is the same. Matter fields are
insensitive to the behavior of the gauge degrees of freedom and the critical
behavior of the gauge modes is the same as in pure gauge models. Transitions
of this type have been observed, e.g., in the three-dimensional Abelian Higgs
(AH) model, both with noncompact fields [16] and with charge-$Q$ scalar fields
[17, 18].

Second, one should consider transitions in which only matter correlations are
critical. At the transition gauge modes do not display long-range
correlations, but are still crucial in determining the universality class of
the critical behavior. Indeed, the gauge symmetry prevents non-gauge-invariant
matter observables from developing long-range order: the gauge symmetry
hinders some matter degrees of freedom—those that are not gauge invariant—from
becoming critical. In this case, the gauge symmetry plays a crucial role in
determining the order parameter of the transition and its transformation
properties under the global symmetry group. However, in the absence of long-
range gauge modes, one can still use the LGW approach, without including the
gauge fields, to predict the critical behavior. The lattice CPN-1 model is an
example of a U(1) gauge model that shows this type of behavior [19, 20].
Transitions of this type also occur in several non-Abelian gauge models [21,
22, 23, 24]. The finite-temperature transition of massless quantum
chromodynamics has also been assumed to have these features [25].

Finally, there are transitions at which both matter and gauge correlations are
critical. In this case an appropriate effective field-theory description of
the critical behavior requires the inclusion of explicit gauge fields.
Transitions of this type have been observed in very few cases; for instance,
in the lattice AH model with noncompact gauge fields [16] (along the
transition line that separates the Coulomb and the Higgs phase), and in the
lattice AH model with compact gauge fields and scalar fields with charge
$Q\geq 2$ [18] (along the transition line between the confined and the
deconfined phase), when the number of components is sufficiently large
(numerical simulations find the charged transition for $N_{f}\geq 10$). In the
non-Abelian case, at present, there is some evidence of this type of behavior
only in SU(2) gauge theories coupled with scalar matter in the fundamental
representation [22].

In this work we consider charge-$Q$ $N_{f}$-component scalar fields
interacting by means of a U($N_{f}$)-invariant Hamiltonian and couple them
with ${\mathbb{Z}}_{q}$ compact gauge fields. The model represents a
generalization of the compact charge-$Q$ AH model [26, 18], in which the gauge
group U(1) is replaced by its subgroup ${\mathbb{Z}}_{q}$. We determine the
phase diagram of the model and the nature of the critical lines, with the
purpose of verifying whether the previous considerations also apply to systems
with discrete gauge groups.

We will study the model for two different values of the charge and number of
components, which have been chosen on the basis of the results for U(1) gauge
models. First, we perform a numerical study of the model for $N_{f}=2$ and
$Q=1$. In the presence of compact U(1) gauge fields (it represents the
standard compact lattice AH model), it shows a single critical line associated
with the breaking of the global SU(2) symmetry [27]. The transitions have an
effective LGW description that predicts O(3) vector behavior. In the LGW
approach to gauge systems, the nature of the gauge group is not relevant, and
thus we expect the same critical behavior in ${\mathbb{Z}}_{q}$ gauge models,
provided the breaking pattern of the global symmetry is the same. Our
numerical results for $q=3$ and $q=4$ are in full agreement with the LGW
predictions: also for these values of $q$, transitions associated with the
breaking of the SU(2) global symmetry belong to the O(3) vector universality
class. In the discrete gauge model, the global symmetry group is larger. The
model is also invariant under U(1)$/{\mathbb{Z}}_{q}$ global transformations.
For $q=3$ and 4, we find a second transition line where this symmetry breaks.
Also along this transition line the gauge group does not play any role: the
transitions belong to the O(2)/XY vector universality class, as expected on
the basis of the LGW analysis.

The second case of interest corresponds to systems with $N_{f}=15$ and $Q=2$.
The corresponding U(1) gauge-invariant model was studied in Ref. [18], finding
a charged transition line, where both scalar and gauge degrees of freedom play
a role. Gauge fields must be included in the effective description of the
critical behavior and indeed the observed behavior is consisted with that
predicted by the AH field theory [28, 29, 30, 31, 32]. A natural question is
whether the charged transition survives if one replaces the continuous gauge
group U(1) with the discrete ${\mathbb{Z}}_{q}$ group. In the case of a global
U(1) symmetry, it is well known that systems with microscopic
${\mathbb{Z}}_{q}$ symmetry may have transitions in the U(1)/O(2) universality
class if $q\geq 4$ (see Refs. [33, 34] and references therein). At the
transition one observes an enlargement of the global symmetry: The large-
distance behavior is invariant under transformations that are not symmetries
of the microscopic theory. Here, we investigate whether a similar phenomenon
occurs for gauge symmetries, i.e., whether it is possible to have a gauge
symmetry enlargement. Simulations with $q=6$ and 10 indicate that no such
symmetry enlargement occurs. We identify a transition line that separates two
phases that have the same features as those coexisting along the charged
transition line in the U(1) gauge model, but in ${\mathbb{Z}}_{q}$ gauge
compact models these transitions turn out to be of first order for both values
of $q$. Apparently, the microscopic model should be exactly U(1) gauge
invariant to allow the system to develop critical charged transitions.

The paper is organized as follows. In Sec. 2 we define the model, while in
Sec. 3 we specify the quantities that are determined in the Monte Carlo
simulations. Some limiting cases, useful to understand the general featurs of
the phase diagram, are discussed in Sec. 4, while the numerical results are
presented in Sec. 5 (results for $N_{f}=2$) and in Sec. 6 (results for
$N_{f}=15$). Finally, in Sec. 7 we summarize the results and draw our
conclusions.

##  2 The model

We consider a ${\mathbb{Z}}_{q}$ gauge model coupled with an $N_{f}$-component
charge-$Q$ complex scalar field defined on a cubic lattice. The fundamental
fields are complex $N_{f}$-dimensional vectors ${\bm{w}}_{\bm{x}}$, satisfying
$|w_{\bm{x}}|=1$, associated with the sites of the lattice, and phases
$\sigma_{{\bm{x}},\mu}$, $|\sigma_{{\bm{x}},\mu}|=1$, associated with the
lattice links. The gauge variables can only take $q$ values. More precisely,
we set

$\sigma=\exp(2\pi in/q),$ (1)

where $n=0,\ldots,q-1$.

The Hamiltonian is

$H=H_{\rm kin}+H_{\rm g}.$ (2)

The first term is

$H_{\rm kin}=-J\
\hbox{Re}\sum_{{\bm{x}},\mu}\left(\overline{\bm{w}}_{\bm{x}}\cdot{\bm{w}}_{{\bm{x}}+\hat{\mu}}\sigma_{{\bm{x}},\mu}^{Q}\right),$
(3)

where the sum is over all lattice sites ${\bm{x}}$ and directions $\mu$
($\hat{\mu}$ are the corresponding unit vectors), and $Q$ is the integer
charge of the scalar field. Since $\sigma_{{\bm{x}},\mu}^{q}=1$, we can limit
ourselves to charges $Q$ satisfying $1\leq Q\leq q/2$.

The second term is

$H_{\rm g}=-g\sum_{{\bm{x}},\mu>\nu}\hbox{Re }\Pi_{{\bm{x}},\mu\nu},$ (4)

where the sum is over all lattice plaquettes, and the plaquette contribution
is given by

$\Pi_{{\bm{x}},\mu\nu}=\sigma_{{\bm{x}},\mu}\sigma_{{\bm{x}}+\hat{\mu},\nu}\overline{\sigma}_{{\bm{x}}+\hat{\nu},\mu}\overline{\sigma}_{{\bm{x}},\nu},$
(5)

The partition function is

$Z=\sum_{\\{w,\sigma\\}}e^{-H/T}.$ (6)

In the following we will use $\beta=J/T$ and $\kappa=g/T$ as independent
variables.

The model is invariant under the local ${\mathbb{Z}}_{q}$ transformations

${\bm{w}}_{\bm{x}}\to\lambda_{\bm{x}}^{Q}{\bm{w}}_{\bm{x}}\qquad\sigma_{{\bm{x}},\mu}\to\lambda_{\bm{x}}\sigma_{{\bm{x}},\mu}\overline{\lambda}_{{\bm{x}}+\hat{\mu}},$
(7)

where $\lambda_{\bm{x}}$ are phases satisfying $\lambda_{\bm{x}}^{q}=1$. It is
also invariant under the global U($N_{f}$) transformations

${\bm{w}}_{\bm{x}}\to V{\bm{w}}_{\bm{x}},$ (8)

where $V\in U(N_{f})$.

For $q/Q=2$, the global symmetry is larger. Indeed, in this case
$\sigma_{{\bm{x}},\mu}^{Q}$ is real (it takes the values $\pm 1$) and this
allows us to rewrite the Hamiltonian as a vector Hamiltonian. We define a real
field $\Phi^{A}$ with $2N_{f}$ components

$\Phi^{a}_{\bm{x}}=\hbox{Re
}w^{a}_{\bm{x}}\qquad\Phi^{a+N_{f}}_{\bm{x}}=\hbox{Im }w^{a}_{\bm{x}},$ (9)

where $1\leq a\leq N_{f}$. Since

$\hbox{Re
}\overline{\bm{w}}_{\bm{x}}\cdot{\bm{w}}_{\bm{y}}=\Phi_{\bm{x}}\cdot\Phi_{\bm{y}},$
(10)

the scalar Hamiltonian can be rewritten as

$H_{\rm
kin}=-J\sum_{x\mu}\sigma_{x,\mu}^{Q}\Phi_{\bm{x}}\cdot\Phi_{{\bm{x}}+\hat{\mu}}.$
(11)

This Hamiltonian is invariant under the transformations $\Phi_{\bm{x}}\to
V\Phi_{\bm{x}}$, where $V$ is an $O(2N_{f})$ matrix. For $Q=1$ and $q=2$, the
scalar Hamiltonian corresponds to a particular RPN-1 vector model
($N=2N_{f}$), in which the vector fields take values in the RPN-1 manifold.
For $N=3$ this model is relevant for liquid crystals [35, 36] and it has been
studied, also in the presence of gauge interactions, in Refs. [37, 38].

##  3 The observables

We simulate the system using a combination of standard Metropolis updates of
the scalar and gauge fields and of microcanonical updates of the scalar field.
We compute the energy densities and the specific heats

$\displaystyle E_{k}=-{1\over VJ}\langle H_{\rm kin}\rangle,\qquad
C_{k}={1\over VJ^{2}}\left(\langle H^{2}_{\rm kin}\rangle-\langle H_{\rm
kin}\rangle^{2}\right),$ $\displaystyle E_{g}=-{1\over Vg}\langle H_{\rm
g}\rangle,\qquad C_{g}={1\over Vg^{2}}\left(\langle H^{2}_{\rm
g}\rangle-\langle H_{\rm g}\rangle^{2}\right),$ (12)

where $V=L^{3}$.

We consider the two-point correlation function of the gauge-invariant tensor
combination

$T^{ab}=\overline{w}^{a}w^{b}-{1\over N_{f}}\delta^{ab},$ (13)

defined as

$G_{T}({\bm{x}},{\bm{y}})=\sum_{ab}\langle
T^{ab}_{\bm{x}}T^{ba}_{\bm{y}}\rangle.$ (14)

The correlation function is invariant under local ${\mathbb{Z}}_{q}$
transformations and global U($N_{f}$) transformations. We also consider the
scalar-field combination

$\Sigma_{k}^{a_{1},\ldots,a_{k}}=w^{a_{1}}\ldots w^{a_{k}}.$ (15)

If $kQ$ is a multiple of $q$, this quantity is gauge invariant.
Correspondingly, we consider the charge-$k$ correlation function

$\displaystyle G_{k}({\bm{x}},{\bm{y}})$ $\displaystyle=$
$\displaystyle\hbox{Re}\,\langle\overline{\Sigma}_{k,\bm{x}}\cdot{\Sigma}_{k,\bm{y}}\rangle=\hbox{Re}\,\langle(\overline{\bm{w}}_{\bm{x}}\cdot{\bm{w}}_{\bm{y}})^{k}\rangle.$

The two local order parameters $T^{ab}_{\bm{x}}$ and
$\Sigma_{{\bm{x}},k}^{a_{1},\ldots,a_{k}}$ belong to two different
representations of the global symmetry group U($N_{f}$). In particular,
$T^{ab}_{\bm{x}}$ is invariant under transformations belonging to the Abelian
subgroup U(1), while $\Sigma_{{\bm{x}}k}^{a_{1},\ldots,a_{k}}$ transforms
nontrivially. For $q=2Q$, the symmetry group is O($2N_{f}$). In this case, the
gauge-invariant order parameters $T^{ab}$ and $\Sigma_{2}^{ab}$ are
equivalent, being both related to

$Q^{AB}=\Phi^{A}\Phi^{B}-{1\over 2N_{f}}\delta^{AB},$ (16)

where $\Phi$ is defined in Eq. (9). If

$G_{\Phi}({\bm{x},\bm{y}})=\sum_{AB}\langle
Q_{\bm{x}}^{AB}Q_{\bm{y}}^{BA}\rangle,$ (17)

one easily derives

$\displaystyle G_{T}({\bm{x},\bm{y}})$ $\displaystyle=$
$\displaystyle{2(N_{f}-1)\over 2N_{f}-1}G_{\Phi}({\bm{x},\bm{y}}),$
$\displaystyle G_{2}({\bm{x},\bm{y}})$ $\displaystyle=$
$\displaystyle{2N_{f}\over 2N_{f}-1}G_{\Phi}({\bm{x},\bm{y}}).$ (18)

The calculation of $G_{k}({\bm{x},\bm{y}})$ is particularly time-consuming
when $k$ and $N_{f}$ are large. In some cases, we have used the following
identity that relies on the $U(N_{f})$ invariance of the theory:

$G_{k}({\bm{x},\bm{y}})={(k+N_{f}-1)!\over
k!(N_{f}-1)!}{\langle({\bm{a}}\cdot\overline{\bm{w}})^{k}({\bm{b}}\cdot{\bm{w}})^{k}\rangle\over({\bm{a}}\cdot{\bm{b}})^{k}},$
(19)

where ${\bm{a}}$ and ${\bm{b}}$ are arbitrary $N_{f}$-dimensional vectors.
Choosing vectors ${\bm{a}}={\bm{b}}=(0,\ldots,0,1,0,\ldots)$, we obtain

$G_{k}({\bm{x},\bm{y}})={(k+N_{f}-1)!\over
k!N_{f}!}\sum_{\alpha}\langle\overline{\bm{w}}^{k}_{\alpha}{\bm{w}}^{k}_{\alpha}\rangle.$
(20)

To define the corresponding correlation lengths, we define the Fourier
transform

$\widetilde{G}_{\\#}({\bm{p}})={1\over
V}\sum_{{\bm{x}},{\bm{y}}}e^{i{\bm{p}}\cdot({\bm{x}}-{\bm{y}})}G_{\\#}({\bm{x}},{\bm{y}})$
(21)

($V$ is the volume) of the two correlation functions. The corresponding
susceptibilities and correlation lengths are defined as

$\displaystyle\chi_{\\#}=\widetilde{G}_{\\#}({\bm{0}}),$ (22)
$\displaystyle\xi^{2}_{\\#}\equiv{1\over
4\sin^{2}(\pi/L)}{\widetilde{G}_{\\#}({\bm{0}})-\widetilde{G}_{\\#}({\bm{p}}_{m})\over\widetilde{G}_{\\#}({\bm{p}}_{m})},$
(23)

where ${\bm{p}}_{m}=(2\pi/L,0,0)$. Note that, if one uses open boundary
conditions, the choice of ${\bm{p}}_{m}$ is somewhat arbitrary. Other choices,
as long as they satisfy $|p_{m}|\sim 1/L$, would be equally valid.

In our finite-size scaling (FSS) analysis we use renormalization-group
invariant quantities. We consider

$R_{\xi,\\#}=\xi_{\\#}/L$ (24)

and the Binder parameters

$U_{\\#}={\langle\mu_{2,\\#}^{2}\rangle\over\langle\mu_{2,\\#}\rangle^{2}},\qquad\mu_{2,\\#}=\sum_{{\bm{x}}{\bm{y}}}G_{\\#}({\bm{x}},{\bm{y}}).$
(25)

In the disordered phase, we have

$U_{T}={N^{2}_{f}+1\over N^{2}_{f}-1},\qquad
U_{k}=1+{k!(N_{f}-1)!\over(N_{f}+k-1)!},$ (26)

while in the ordered phase, all these quantities converge to 1.

To determine the nature of the transition, one can consider the $L$ dependence
of the maximum $C_{\rm max}(L)$ of one of the specific heats. At a first-order
transition, $C_{\rm max}(L)$ is proportional to the volume $L^{3}$, while at a
continuous transition it behaves as

$C_{\rm max}(L)=aL^{\alpha/\nu}+C_{\rm reg}.$ (27)

The constant term $C_{\rm reg}$, due to the analytic background, is the
dominant contribution if $\alpha<0$. The analysis of the $L$-dependence of
$C_{\rm max}(L)$ may allow one to distinguish first-order and continuous
transitions. However, experience with models that undergo weak first-order
transitions indicates that in many cases the analysis of the specific heat is
not conclusive [39, 40]. The maximum $C_{\rm max}(L)$ may start scaling as
$L^{3}$ at values of $L$ that are much larger than those at which simulations
can be actually performed. A more useful quantity is a Binder parameter $U$.
At first-order transitions, the maximum $U_{\rm max}(L)$ of $U$ at fixed size
$L$ increases with the volume [39, 40]. On the other hand, $U$ is bounded as
$L\to\infty$ at a continuous phase transition. In this case, in the FSS limit,
the Binder parameter as well as any renormalization-group invariant quantity
$R$ scales as

$R(\beta,L)\approx f_{R}(X)+L^{-\omega}f_{c,R}(X)\quad
X=(\beta-\beta_{c})L^{1/\nu},$ (28)

where $\omega$ is the leading correction-to-scaling exponent. Thus, a first-
order transition can be identified by verifying that $U_{\rm max}(L)$
increases with $L$, without the need of explicitly observing the linear
behavior in the volume.

In the case of weak first-order transitions, the nature of the transition can
also be understood from the combined analysis of $U$ and $R_{\xi}$. At a
continuous transition, in the FSS limit any renormalization-group invariant
quantity $R$ scales as

$R(\beta,L)=F_{R}(R_{\xi})+L^{-\omega}F_{c,R}(R_{\xi})+\ldots$ (29)

where $F_{R}(x)$ is universal and $F_{c,R}(x)$ is universal apart from a
multiplicative constant. The Binder parameter $U$ does not obey this scaling
relation at first-order transitions, because of the divergence of $U$ for
$L\to\infty$. Therefore, the absence of data collapse in plots of $U$ versus
$R_{\xi}$ is an early indication of the first-order nature of the transition.

##  4 Some limiting cases

The phase diagram of the compact model with U(1) gauge fields, i.e., in the
limit $q\to\infty$, has been discussed at length in Refs. [26, 27, 17, 18],
while systems with discrete gauge groups are discussed in Refs. [26, 41, 37,
38, 42]. To derive the phase diagram of the present model, it is useful to
discuss some limiting cases.

In the limit $\kappa\to\infty$, the gauge degrees of freedom freeze and one
can set $\sigma_{{\bm{x}},\mu}=1$ on all links (when open boudary conditions
are used, this is also true in a finite volume). Using the mapping (10), the
scalar Hamiltonian becomes equivalent to that of the O($2N_{f}$) vector model,
which undergoes a standard finite-$\beta$ continuous transition for any
$N_{f}\geq 1$.

For $\beta=0$, there are no scalar fields and one obtains a pure gauge
${\mathbb{Z}}_{q}$ model, that can be related by duality [51] to a
${\mathbb{Z}}_{q}$ spin model, with a global ${\mathbb{Z}}_{q}$ symmetry. The
${\mathbb{Z}}_{q}$ gauge theory undergoes a topological transition at
$\kappa_{c}$, which belongs to the same universality class as the
corresponding transition in the ${\mathbb{Z}}_{q}$ spin clock model. For $q=2$
and 4 it belongs to the Ising universality class, for $q=3$ it is of first
order, while for $q\geq 5$ the critical behavior is the same as in the XY
model, see Refs. [33, 44, 43]. Estimates of $\kappa_{c}$ can be found in Ref.
[45]. For $q=2$, we can use duality and the results Ref. [46] for the standard
Ising model to estimate $\kappa_{c}=0.761413292(12)$. For $q\to\infty$, one
has [45, 18]

$\kappa_{c}\simeq\kappa_{gc}\,q^{2},$ (30)

where $\kappa_{gc}=0.076051(2)$ is the critical coupling of the inverted XY
model [47].

For $\kappa=0$, there is no plaquette contribution. We can, first of all,
simplify the scalar-field interaction term, defining

$\tau_{{\bm{x}},\mu}=\sigma_{{\bm{x}},\mu}^{M},$ (31)

where $M$ is the greatest common divisor of $Q$ and $q$. The new field takes
$p=q/M$ different values, i.e., it satisfies $\tau_{{\bm{x}},\mu}^{p}=1$. If
$r=Q/M$, the Hamiltonian becomes

$H_{\rm kin}=-J\
\hbox{Re}\sum_{{\bm{x}}\mu}\overline{\bm{w}}_{\bm{x}}\cdot{\bm{w}}_{x+\hat{\mu}}\tau_{{\bm{x}},\mu}^{r}.$
(32)

Note that $p$ is always larger than 1, since $M\leq Q<q$. Thus, we obtain an
effective model for charge-$r$ scalars with ${\mathbb{Z}}_{p}$ local gauge
invariance.

For $p/r=q/Q=2$ (in this case we have necessarily $p=2$ and $r=1$), the global
invariance group of the model is the O($2N_{f}$) group and one obtains an
effective RP${}^{2N_{f}-1}$ model. The critical behavior of RPN-1 models is
well known: for any $N\geq 3$ they are expected to undergo a first-order
transition [35, 36, 48], in agreement with standard LGW arguments.

The behavior of the model (32) for $p/r\not=2$ is less clear. Since gauge
fields are not dynamical for $\kappa=0$—they can be integrated out—the
critical behavior is completely determined by an effective model in terms of a
gauge-invariant order parameter. In this effective description only the global
symmetry group is relevant. The gauge invariance group has the only role of
selecting the appropriate gauge-invariant order parameter. In the present case
the global symmetry group of the model is U($N_{f}$)/${\mathbb{Z}}_{p}$, which
is not a simple group. Thus, it is possible to have transitions where only the
SU($N_{f}$) symmetry is broken, transitions where only the Abelian subgroup
U(1)/${\mathbb{Z}}_{p}$ is broken, and transitions where both subgroups
condense simultaneously. Thus, one or two transitions may occur as a function
of $\beta$. In the following we will discuss the behavior of the model for
$Q=1,2$ and two values of $N_{f}$, $N_{f}=2$ and $N_{f}=15$. In all cases
numerical results show that two transitions occur: a first one at $\beta_{c1}$
and a second one at $\beta_{c2}>\beta_{c1}$. At $\beta_{c1}$ the order
parameter $T^{ab}$ condenses, while $\Sigma_{q}$ is still disordered: the
transition is clearly associated with the breaking of the SU($N_{f}$)
symmetry. At $\beta_{c2}$, $\Sigma_{q}$ correlations become critical,
signalling the breaking of the Abelian symmetry. The small-$\beta$ transition
has the same symmetry-breaking pattern of the small-$\kappa$ transitions
observed in the model with U(1) gauge symmetry [27]. Under the assumption that
the nature of the transition only depends on the global symmetry breaking
pattern, we can predict that the critical behavior is the same as that
observed in Ref. [27]. For $N_{f}\geq 3$, only first-order transitions are
possible, under the usual assumption that a cubic term in the Landau-Ginzburg-
Wilson (LGW) Hamiltonian signals discontinuous transitions (for a discussion
of the validity of this assumption see Ref. [27] and references therein). For
$N_{f}=2$, continuous transitions are possible: they belong to the O(3) vector
universality class. The second, large-$\beta$ transition is associated with
the breaking of the Abelian U(1) subgroup. Again, assuming that gauge fields
are not dynamical, we predict that the transition belongs to the O(2)/XY
universality class, if it is continuous.

Let us finally consider the limit $\beta\to\infty$, In this case the scalar
field orders so that

${\bm{w}}_{\bm{x}}=\sigma_{{\bm{x}}+\hat{\mu}}^{Q}{\bm{w}}_{{\bm{x}}+\hat{\mu}},$
(33)

on each link. This relation implies

${\bm{w}}_{\bm{x}}=\Pi_{{\bm{x}},\mu\nu}^{Q}{\bm{w}}_{\bm{x}}\qquad\Rightarrow\qquad\Pi_{{\bm{x}},\mu\nu}^{Q}=1$
(34)

on all plaquettes, which, in turn, implies that $\Pi_{{\bm{x}},\mu\nu}$
belongs to the ${\mathbb{Z}}_{M}$ subgroup of ${\mathbb{Z}}_{q}$ ($M$ is the
greatest common divisor of $Q$ and $q$). Thus, modulo gauge transformations,
for $\beta\to\infty$ we can take $\sigma_{{\bm{x}},\mu}=\exp(2\pi in/M)$, with
$n=0,\ldots M$. If $M=1$, the gauge fields are completely ordered on the line
$\beta=\infty$. If, instead, $M\geq 2$, there is still a nontrivial gauge
dynamics and the system behaves as a ${\mathbb{Z}}_{M}$ gauge model, which
undergoes a finite-$\kappa$ topological phase transition.

##  5 Numerical results: Critical behavior for $N_{f}=2$ and $Q=1$

|  |
---|---|---
Figure 1: Expected phase diagram of the model for $Q=1$. Left: $q\geq 3$;
Right: $q=2$. For $q\geq 3$ (left) there are four different phases. Scalar
fields are disordered in phases 1 and 3, while they are ordered in phases 2
and 4. Transition lines 1-2 and 2-4 are associated with the breaking of the
global symmetry SU($N_{f}$) and of the global symmetry
U(1)/${\mathbb{Z}}_{q}$, respectively; along the line 3-4 the simultaneous
breaking of both global symmetries occurs. A topological ${\mathbb{Z}}_{q}$
gauge transition occurs along the line 1-3. If $q=2$ (right), phase 2 is
missing and there is a single low-$\kappa$ transition line 1-4 along which the
global vector O($2N_{f}$) symmetry is broken.

In this section we discuss the phase diagram of the two-component model
($N_{f}=2$) with charge-one ($Q=1$) scalar fields. We consider three values of
$q$, $q=2$, 3, and 4. The results that we shall discuss below are consistent
with the phase diagrams reported in Fig. 1. As expected—the global symmetry
group is different in the two cases—the phase diagram for $q=2$ is different
from that of systems with $q\geq 3$.

To identify the different transition lines, we have performed runs at fixed
$\kappa$, varying $\beta$. To determine the nature of the transitions that
separate phases 1 and 2, and phases 2 and 4 (see Fig. 1) we have fixed
$\kappa=0.4$ for all $q$ values. This value of $\kappa$ has been chosen on the
basis of the estimates of the critical $\kappa_{c}$ for the pure gauge model
($\beta=0$). Indeed, we have $\kappa_{c}=0.761413292(12)$ (using duality and
the results of Ref. [46]), 1.0844(2), 1.52276(4) [45] for $q=2,$ 3, and 4,
respectively. The value of $\kappa$ we use is significantly smaller than the
values of $\kappa_{c}$ reported above, guaranteeing the we are indeed studying
the small-$\kappa$ transition lines. To investigate the large-$\kappa$
behavior we have instead fixed $\kappa=1.5$ and 2 for $q=2$ and 3,
respectively.

In the runs at fixed $\kappa=0.4$, we consider cubic lattices of linear size
$L$ with periodic boundary conditions. On the other hand, to determine the
large-$\kappa$ critical behavior, we consider open boundary conditions, to
avoid slowly-decaying dynamic modes that are present in systems with periodic
boundary conditions. Indeed, in the latter case, the Polyakov loops (the
product of the gauge compact fields along nontrivial lattice paths that wrap
around the lattice) have a very slow dynamics, if one uses algorithms with
local updates. For open boundary conditions, Polyakov loops are not gauge
invariant and thus their dynamics is not relevant for the estimation of gauge-
invariant observables. A local algorithm is therefore efficient. Of course,
open boundary conditions give rise to additional scaling corrections, due to
the boundary, and thus larger systems are needed to obtain asymptotic results.

###  5.1 Small-$\kappa$ transitions

Figure 2: Histograms of the scalar energy density $E_{k}/V$ for the model with
$q=2$, $N_{f}=2$ along the line $\kappa=0.4$. Left: results for $L=16$; right:
results for $L=20$. The $\beta$ values are reported in the legend.

Figure 3:  Plot of $C_{k}$ (left) and $C_{g}$ (right) versus $\beta$ for the
two-flavor model ($N_{f}=2$) with $q=3$ and $Q=1$, along the line
$\kappa=0.4$.

Figure 4:  Plot of $U_{T}$ versus $R_{\xi,T}$ for the two-flavor model
($N_{f}=2$) along the line $\kappa=0.4$ (small-$\beta$ transition): (left)
results for $q=3$ (we only include data with $\beta<2.3$); (right) results for
$q=4$. The continuous line is the universal curve for vector correlations in
the O(3) vector model (see Appendix of Ref. [49]).  Figure 5:  Plot of
$L^{-2+\eta}\chi_{T}$ versus $R_{\xi,T}$ for the two-flavor model ($N_{f}=2$)
along the line $\kappa=0.4$ (small-$\beta$ transition): results for $q=3$ (we
only include data with $\beta<2.3$) and $q=4$. We use the O(3) value [50]
$\eta=0.03624$.

We have first considered the $q=2$ model, which has an enlarged O(4) global
symmetry. Along the line $\kappa=0.4$, simulations show the presence of a
single first-order transition. The discontinuous nature is apparent from plots
of the data as a function of the Monte Carlo time, which show the typical see-
saw behavior, and from the histograms of the energy, which have a clear
bimodal structure, see Fig. 2. This result is in agreement with the general
theory. Indeed, for $q=2$ the model is equivalent to an RP3 model, which is
expected to undergo a first-order transition [35, 36, 48].

Let us now discuss the behavior for $q\geq 3$. As we have discussed in Sec. 4,
for $\kappa=0$, and therefore also for small values of $\kappa$, it is
possible to have two different transitions as a function of $\beta$,
associated with different breakings of the global symmetry. To verify this
possibility, we have performed a scan in $\beta$. In Fig. 3, we report the
specific heats as a function of $\beta$ for $q=3$. There is a clear evidence
of two transitions, one at $\beta\approx 2.2$, weakly coupled with the gauge
degrees of freedom—the gauge specific heat is almost constant in the
transition region— and a second one at $\beta\approx 2.45$. For $q=4$, we
again observe two transitions, at $\beta\approx 2.3$ and 4.35, respectively.

We first focus on the transition at $\beta\approx 2.2$ for $q=3$. We find
that, while tensor correlations are critical ($\xi_{T}$ increases rapidly at
the transition), the gauge-invariant charge-3 correlations are always short-
ranged. This clearly indicates that the transition is associated with the
breaking of the SU(2) global symmetry. In Fig. 4 we report $U_{T}$ versus
$R_{\xi,T}$ and compare the results with the universal curve appropriate for
the O(3) universality class, reported in the Appendix of Ref. [49]. Data fall
quite precisely onto the O(3) curve, confirming the LGW prediction that the
transition belongs to the O(3) universality class. Also for $q=4$ the data
close to the small-$\beta$ transition fall on top of the O(3) curve,
indicating that the transition belongs to the O(3) universality class for any
$q>3$, a result which is not surprising as the same O(3) transition occurs in
the model with U(1) symmetry, i.e., for $q\to\infty$.

As an additional check we have analyzed $R_{\xi,T}$ and $U_{T}$, fitting the
Monte Carlo data in the small-$\beta$ transition region ($\beta<2.3$ for
$q=3$) to Eq. (28). Scaling corrections have been neglected—as apparent from
Fig. 4, they are small. The scaling function $F_{R}(x)$ has been parametrized
with a polynomial. We obtain $\nu=0.716(13)$ and 0.72(2) for $q=3,4$,
respectively, in good agreement with the O(3) estimate $\nu=0.71164(10)$ of
Ref. [50]. To estimate the critical coupling $\beta_{c}$ we have repeated the
same fits fixing $\nu$ to the O(3) value. We obtain $\beta_{c}=2.2155(3)$ and
2.3175(5) for $q=3,4$. Note that $\beta_{c}$ depends weakly on $q$, indicating
that the nonuniversal features of the transitions are only slightly dependent
on $q$.

Finally, we have checked the behavior of the tensor susceptibility that is
expected to scale as $L^{2-\eta}f(R_{\xi,T})$, where $f(x)$ is universal apart
from a multiplicative rescaling. If the transition belongs to the O(3)
universality class, data should scale provided we set $\eta=\eta_{O(3)}$,
where $\eta_{O(3)}=0.03624(8)$ [50] is the value it takes in the O(3) model.
Results are shown in Fig. 5. We observe an excellent scaling. Moreover, on the
scale of the figure, we do not see any dependence of the scaling curve on $q$.
As observed for $\beta_{c}$, the $q$-dependence of nonuniversal amplitudes is
small.

Figure 6: Scaling plots at the large-$\beta$ critical transition, at fixed
$\kappa=0.4$, for $N_{f}=2$ and $Q=1$. Left: plot of $U_{3}$ versus
$R_{\xi,3}$ for $q=3$; right: plot of $U_{4}$ versus $R_{\xi,4}$ for $q=4$. We
also report the universal scaling curve for vector correlations in the O(2)
vector model (see Appendix of Ref. [49]).

Figure 7: Scaling corrections at the large-$\beta$ critical transition, at
fixed $\kappa=0.4$, for $N_{f}=2$ and $Q=1$. Plot of
$L^{\omega}\Delta(R_{\xi,q})$ versus $R_{\xi,q}$ for $q=3$ (left) and $q=4$
(right), where $\Delta(R_{\xi,q})$ is defined in Eq. (35). We set
$\omega=\omega_{XY}$, where $\omega_{XY}$ is the XY value of the correction-
to-scaling exponent: $\omega_{XY}=0.789$ [44].

Let us now focus on the large-$\beta$ transition. For both $q=3$ and 4,
charge-$q$ correlations diverge, Since the $SU(N_{f})$ symmetry is broken, the
transition is described by an effective scalar model in which the fundamental
field is a phase and the interaction is ${\mathbb{Z}}_{q}$ gauge invariant.
The behavior of this model has been discussed in Ref. [42]. For small values
of $\kappa$, the transition belongs to the XY universality class, if it is
continuous. Moreover, the universal scaling function that expresses $U_{q}$ in
terms of $R_{\xi,q}$ is the same as the scaling fuction $U=f_{U}(R_{\xi})$ for
vector correlations in the XY model. To verify this prediction, we plot
$U_{q}$ versus $R_{\xi,q}$ for $q=3,4$ and compare the data with the universal
XY vector curve reported in Ref. [49], see Fig. 6. Data are consistent with
the XY curve (as $L$ increases, data move closer to the XY curve), although
significant corrections are apparently present, especially for $q=3$, probably
because of the presence of the nearby O(3) transition. In order to have a
stronger check of the correctness of the prediction, we have analyzed the
scaling corrections. We consider the deviations

$\Delta(R_{\xi,q})=U_{q}-f_{XY}(R_{\xi,q}),$ (35)

as a function of $R_{\xi,q}$; here $f_{XY}(R_{\xi})$ is the universal XY curve
for vector correlations. If $f_{XY}(R_{\xi,q})$ is the correct asymptotic
behavior and $\omega$ is the leading correction-to-scaling exponent,
$L^{\omega}\Delta(R_{\xi,q})$ should scale for large $L$. We assume that
$\omega$ is the same as the leading exponent $\omega_{XY}$ in the XY
model.111A priori it is possible that $\omega<\omega_{XY}$, since in our model
there are additional RG subleading operators due, e.g., to the gauge
interactions, the interactions with the frozen SU$(N_{f})$ modes, etc.. They
may be more relevant than the RG operator that controls the scaling
corrections in the XY model . Our numerical results are in agreement with the
assumption $\omega=\omega_{XY}$. In Fig. 7 we report the numerical results.
Data scale reasonably, confirming that the transition belongs to the XY
universality class.

Assuming an XY behavior, we have estimated the location of the critical
transition. We have performed combined fits of $U_{q}$ and $R_{\xi,q}$ to Eq.
(28), including scaling corrections. Fixing $\omega$ and $\nu$ to the XY
values, $\omega=0.789$, $\nu=0.6717$ [44], we obtain $\beta_{c}=2.4769(4)$ and
$\beta_{c}=4.3370(15)$ for $q=3$ and 4, respectively.

###  5.2 Large-$\kappa$ transition line

Figure 8:  Plot of $U_{T}$ versus $R_{\xi,T}$ for the two-flavor model
($N_{f}=2$) along the large-$\kappa$ transition line (open boundary
conditions). Left: results for $q=2$ (at fixed $\kappa=1.5$); right: results
for $q=3$ (at fixed $\kappa=2.0$). The results for the gauge model are
compared with the results for the O(4) vector model.  Figure 9:  Plot of
$U_{3}$ versus $R_{\xi,3}$ for the two-flavor model ($N_{f}=2$) along the
large-$\kappa$ transition line, for $q=3$ (at fixed $\kappa=2$). The results
for the gauge model are compared with the results for the O(4) vector model.

We have studied the large-$\kappa$ behavior of the system for $q=2$ and $q=3$,
performing runs at fixed $\kappa=1.5$ and $\kappa=2$, respectively. As we
discussed above, we expect the same behavior as in the O(4) vector model, for
all gauge-invariant observables. Note that, for $q=3$ this represents an
enlargement of the global symmetry at the transition. In the presence of O(4)
symmetry, tensor and charge-$q$ correlations are related, see Sec. 3, and thus
they should be both critical along the large-$\kappa$ transition line.

The numerical results are in agreement with these predictions. In Fig. 8 we
compare the behavior of the tensor Binder parameter $U_{T}$ as a function of
$R_{\xi,T}$ in the gauge models and in the O(4) model. Data scale onto the
same universal curve, as expected. We have also verified that charge-$q$
correlations behave as in the O(4) model. For $q=2$ this is a consequence of
the exact O(4) symmetry of the model. For $q=3$ this is confirmed by the
results shown in Fig. 9.

###  5.3 Phase diagram

The numerical results presented above confirm that, for $N_{f}=2$, the model
has the phase diagram reported in Fig. 1. We expect the same phase diagram for
any $N_{f}$ as long as $Q=1$. As far as the nature of the transition lines,
for $q\geq 3$ we expect the small-$\kappa$, large-$\beta$ transition, where
the U(1)/$\mathbf{Z}_{q}$ symmetry is broken, to belong to the XY universality
class in all cases. Analogously, the large-$\kappa$ transitions should always
belong to the O($2N_{f}$) universality class. For small $\kappa$ and small
$\beta$, transitions should be analogous to that observed in the
CP${}^{N_{f}-1}$ model, and therefore they should be of first order for any
$N_{f}\geq 3$.

If $Q\not=1$ but $M$ is 1 ($M$ is the greatest common divisor of $Q$ and $q$),
the model should also have the phase diagram reported in Fig. 1. Instead, for
$M\geq 2$ the phase diagram is more complex (it will be discussed in the next
section), because of the presence of a new topological transition line for
large values of $\beta$.

##  6 Numerical results: Critical behavior for $N_{f}=15$ and $Q=2$

|  |
---|---|---
Figure 10: Left: sketch of the phase diagram of the model for $Q=2$ and
$q=10$. We expect the same phase diagram for any even $q\geq 6$. For $q=4$,
the system has a larger O($2N_{f}$) global symmetry and phases $2a$ and $2b$
are missing. In the right panel we report the lines (red thick lines) along
which runs have been performed. We have performed runs at fixed $\kappa$, at
$\kappa_{1}=0.4$, $\kappa_{2}=1.2$, and $\kappa_{3}=10$, and a series of runs
at fixed $\beta$, at $\beta=\beta_{1}=13$. The tensor correlation length
diverges in phases $2a$, $2b$, $4a$, $4b$, signalling the breaking of the
SU($N_{f}$) symmetry; the correlation length $\xi_{5}$ is finite in phases
$2a$, $2b$, and diverges in phases $4a$, $4b$.

In this Section, we consider the model with charge $Q=2$ and $N_{f}=15$ for
different values of $q$ with the purpose of understanding whether it exhibits
a continuous transition in the same universality class as the charged
transition that occurs in U(1) gauge invariant models [16, 17, 18]. We only
consider even values of $q$. If $q$ is odd, gauge fields are completely
ordered for $\beta\to\infty$. In this case, the phase diagram should be
analogous to that obtained for $Q=1$, see Fig. 1, without the large-$\beta$
topological transition line, whose presence is necessary to observe the
charged transition in U(1) gauge invariant models [16, 17, 18]. In most of the
simulations we set $q=10$. The numerical results are consistent with the phase
diagram reported in Fig. 10. Comparing with Fig. 1, one observes the presence
of a new line that starts at $\beta=\infty$ and that is associated with a
${\mathbb{Z}}_{Q}$ (in our case Ising) topological transition. It intersects
the two phases labelled 2 and 4 in Fig. 1. The additional topological
transition is irrelevant for the breaking of the global symmetries: the global
SU($N_{f}$) symmetry is broken in phases $2a$, $2b$, $4a$, and $4b$, while the
U(1)/${\mathbb{Z}}_{q}$ symmetry is broken in both phases $4a$ and $4b$.

For $q=4$ and $Q=2$ the enlarged symmetry forbids the presence of phases $2a$
and $2b$. The only difference with the case $Q=1$ is the presence of the
topological transition line that divides phase 4 in two different phases. The
behavior of the phase diagram as $q$ increases can be easily inferred. Since
phases $4a$, $4b$ and 3 are not present in the U(1) gauge model, the
multicritical point $M_{1}$ should move towards larger $\beta$ values, while
the multicritical point $M_{3}$ should move towards $\kappa=\infty$. For
$q=\infty$ continuous transitions controlled by the charged fixed point are
observed on the line starting at $M_{2}$ and ending at $\kappa=\infty$. Thus,
if this transition survives for finite values of $q$, it should be observed on
the line connecting $M_{2}$ with $M_{3}$, which separates phase 1 from phase
$2b$.

###  6.1 Critical behavior for $q=10$

Figure 11: Plot of $U_{T}$ versus $R_{\xi,5}$ for the model with $N_{f}=15$,
$Q=2$, and $q=10$. Results have been obtained varying $\beta$ along the line
$\kappa=10$. The results are compared with analogous data for the O(30) vector
model.

Figure 12: Plots of $R_{\xi,5}$ versus $X=(\beta-\beta_{c})L^{1/\nu}$ for the
model with $N_{f}=15$, $Q=2$, and $q=10$. Results have been obtained varying
$\beta$ along the line $\kappa=0.4$ (left) and $\kappa=1.2$ (right). We set
$\nu=0.6717$ (the XY-model value [44]), $\beta_{c}=14.57$ ($\kappa=0.4$) and
$\beta_{c}=13.21$ ($\kappa=1.2$).

For $q=10$, we have first performed runs along lines with fixed $\kappa$. We
have chosen three different values $\kappa_{i}$, and we have performed runs
varying $\beta$ at fixed $\kappa=\kappa_{i}$. The values $\kappa_{i}$ have
been chosen such as to probe the transitions $1\to 2a\to 4a$ (runs at fixed
$\kappa_{1}$), the transitions $1\to 2b\to 4b$ (runs at fixed $\kappa_{2}$),
and the transitions $3\to 4b$ (runs at fixed $\kappa_{3}$), see the right
panel of Fig. 10. The appropriate $\kappa_{i}$ values have been determined by
looking at the behavior for $\beta=0$ and $\infty$ (in these two cases the
system corresponds to a pure gauge model, as discussed in Sec. 4). For
$\beta=\infty$, the ${\mathbb{Z}}_{2}$ topological transition line starts at
$\kappa\approx 0.76$, while, for $\beta=0$, the ${\mathbb{Z}}_{10}$
topological transition line starts [45] at $\kappa\approx 7.86$. Therefore, we
have chosen $\kappa_{1}=0.4$, $\kappa_{2}=1.2$, and $\kappa_{3}=10$. For the
runs at $\kappa=\kappa_{1}$ and $\kappa_{2}$ we have used periodic boundary
conditions, while open boundary conditions have been used at $\kappa_{3}$.

For $\kappa=10$, a single transition is observed as $\beta$ is increased, as
expected. To confirm that the transition belongs to the O(30) universality
class we have determined the universal scaling curve of $U_{T}$ versus
$R_{\xi,T}$. The data are presented in Fig. 11, together with the analogous
data for the O(30) vector model. The results for the gauge model and for the
vector model fall on top of each other, confirming that the transition is in
the O(30) universality class.

For both $\kappa=0.4$ and $\kappa=1.2$, we find two different transitions as
$\beta$ is increased. For $\kappa=0.4$ we observe a very strong first-order
transition for $\beta\approx 11$. The energy shows a strong hysteresis for
$10.9\lesssim\beta\lesssim 11$ and $11\lesssim\beta\lesssim 11.3$ for $L=6$
and 8 (for each $\beta$ we perform runs of a few million iterations).
Equilibrated results are only obtained on lattices of size $L=4$. At the
transition the SU($N_{f}$) symmetry is broken— $\xi_{T}$ is large and
increases rapidly with $L$ on the large-$\beta$ side of the transition—while
the U(1)/${\mathbb{Z}}_{q}$ symmetry is preserved—the correlation length
$\xi_{5}$ is small and independent of $L$ at the transition. At the second
transition, observed at $\beta\approx 14.6$, $\xi_{5}$ diverges, signalling
the breaking of the $U(1)/{\mathbb{Z}}_{q}$ global symmetry. We expect the
transition to belong to the XY universality class. In Fig. 12 (left panel) we
plot $R_{\xi,5}$ versus $X=(\beta-\beta_{c})L^{1/\nu}$ using the XY estimate
[44] for the critical exponent $\nu$, $\nu=0.6717$. We observe a reasonable
agreement, indicating that data are indeed consistent the XY behavior.

Figure 13: Plot of the probability distribution of $E_{\rm kin}$ for the model
with $N_{f}=15$, $Q=2$, and $q=10$. Results have been obtained for
$\kappa=1.2$ and a few values of $\beta$ (they are reported in the legend in
the figure).  Figure 14: Scaling plot of $M_{3g}L^{3-3/\nu}$ versus
$X=(\kappa-\kappa_{c})L^{1/\nu}$ for the model with $N_{f}=15$, $Q=2$, and
$q=10$. Results have been obtained varying $\kappa$ at fixed $\beta=13$. We
set $\nu=0.629971$ (the Ising model value, see Ref. [52]) and
$\kappa_{c}=0.817878$.

The same analysis has been repeated for $\kappa=1.2$. As $\beta$ increases,
one first observes a transition at $\beta\approx 10$. The transition is of
first order. The scalar-field energy $E_{\rm kin}$ shows a clear bimodal
structure already for $L=8$, see Fig. 13. The transition has the same features
observed at $\kappa=0.4$, indicating that it is associated with the breaking
of the SU($N_{f}$) global symmetry. As $\beta$ is further increased, a second
transition occurs at $\beta\approx 13.2$, where the correlations $\xi_{5}$
diverges. The numerical data are again consistent with an XY behavior, see the
right panel of Fig. 12.

Finally, to obtain an unambiguous check that the runs at $\kappa=1.2$ are
indeed probing the transitions $1\to 2b\to 4b$, we have performed runs at
fixed $\beta=13$, varying $\kappa$. This value of $\beta$ has been chosen on
the basis of the results for $\kappa=1.2$. Since the XY transition is located
at $\beta\approx 13.2$, if the runs at $\kappa=1.2$ were probing the
transitions $1\to 2a\to 4a$, we would find the topological ${\mathbb{Z}}_{2}$
transition at $\kappa>1.2$ when varying $\beta$ (see Fig. 10). Instead,
starting from $\kappa=1.2$, we find a transition at a smaller value of
$\kappa$, $\kappa\approx 0.81$, confirming the correct identification of the
transition lines for $\kappa=1.2$. The transition at $\kappa\approx 0.81$ has
a clear topological nature: in the two coexisting phases, the SU($N_{f}$)
symmetry is broken ($U_{T}\approx 1$ across the transition), while the U(1)
scalar modes are always disordered, $\xi_{5}\lesssim 1$ in the transition
region. Clearly, we are observing the transition line that separates phases
$2a$ and $2b$. Since the transition is associated with the gauge modes, there
are no local order parameters. Therefore, to characterize the critical
behavior, we consider moments of the gauge energy. In particular, we consider
the third cumulant [51]

$M_{3g}=-{1\over Vg^{3}}\left\langle(H_{\rm g}-\langle H_{\rm
g}\rangle)^{3}\right\rangle.$ (36)

If the transition is continuous, this quantity should scale as

$M_{3g}=L^{3/\nu-3}f(X)\qquad X=(\kappa-\kappa_{c})L^{1/\nu}.$ (37)

If we take $\kappa_{c}=0.8179$ and we use the Ising estimate [52]
$\nu=0.629971$, data scale nicely, see Fig. 14. The transition is therefore
continuous in the Ising universality class.

The results of the runs at $\beta=13$ confirm that the runs at
$\kappa=1.2>\kappa_{c}(\beta=13)=0.8$ are indeed probing the transition lines
$1\to 2b\to 4b$. Since transitions along the 1-$2b$ line are of first order,
there is no evidence of the charged universality class. At variance with what
happens with the global symmetry group—systems with ${\mathbb{Z}}_{q}$
symmetry behave as U(1) systems if $q\geq 4$—in the case of the U(1) gauge
symmetry, no gauge symmetry enlargement occurs: the charged universality class
can apparently be observed only for the U(1) model, i.e., for $q=\infty$.

The runs at $\beta=13$ also show that, along the line running from
$\beta=\infty$ up to the multicritical point $M_{2}$, the transitions are
always topological in the Ising universality class. Scalar fields play no
role, so that the same critical behavior is observed on the whole line
starting at $\beta=\infty$ and ending at the multicritical point $M_{2}$, see
Fig. 10.

###  6.2 Phase behavior for $q=4$ and $q=6$

Figure 15: Plot of the probability distribution of $E_{\rm kin}$ for the model
with $N_{f}=15$ and $Q=2$, for $\kappa=1.2$ and different values of $\beta$
(they are reported in the legend). Results for $q=4$ and $q=6$ are reported in
the left and right panel, respectively. Periodic boundary conditions are used.

Beside the extensive simulations for $q=10$, we also performed some
simulations for $q=4$ and $q=6$. We chose $\kappa=1.2$ in both cases and
varied $\beta$, in order to identify the transitions 1-$2b$, $2b$-$4b$ for
$q=6$ and the single transition 1-$4b$ for $q=4$ (the enlarged O(30) global
symmetry forbids phases $2a$ and $2b$), see Fig. 10. For $q=4$ numerical
simulations confirm the existence of a single transition. Using both open and
periodic boundary conditions, we verify that the transition is of first order:
the energy has a bimodal distribution already on small lattices, see the left
panel of Fig. 15.

For $q=6$ we would expect the same phase diagram as that shown in Fig. 10.
Surprisingly, our data at fixed $\kappa=1.2$ are consistent with a single
first-order transition. The latent heat is large—the distribution of the
energy is bimodal already for $L=4$, see the right panel of Fig. 15—and we are
not able to obtain equilibrated results for $L\geq 6$. On the large-$\beta$
side of the transition, both $\xi_{T}$ and $\xi_{3}$ increase rapidly with $L$
(comparing data for $L=8$ and 16, we estimate an effective behavior $\xi\sim
L^{1.6}$), which would apparently indicate that both the SU($N_{f}$) and the
U(1)/${\mathbb{Z}}_{6}$ symmetry are broken. We do not see any reason why the
model with $q=6$ should have a phase diagram different from that of the model
with $q=10$, presented in Fig. 10. A possible explanation of the observed
behavior that is consistent with the general picture obtained from the
simulations with $q=10$ is the following. It is possible that $\xi_{3}$ on the
large-$\beta$ side of the transition is finite (as expected in phase $2b$),
but large ($\xi_{3}\gg 10$). In this case, our numerical simulations on
lattices of size of order 10 would be unable to distinguish phase $2b$ from
phase $4b$.

##  7 Conclusions

In this paper we consider a variant of the usual three-dimensional compact AH
model in which a complex $N_{f}$ component scalar field of charge $Q$ is
coupled with a U(1) Abelian gauge field. Here, gauge fields are phases that
take only a set of $q$ discrete values so that the model is ${\mathbb{Z}}_{q}$
gauge invariant. Because of the smaller gauge group, the theory is globally
invariant under a larger set of symmetry transformations. Beside SU($N_{f})$
transformations, there is also a nontrivial invariance under global
U(1)/${\mathbb{Z}}_{q}$ phase changes of the scalar fields. The larger
invariance group allows for a richer phase diagram, with additional transition
lines associated with the breaking of the U(1)/${\mathbb{Z}}_{q}$ symmetry.

By means of numerical Monte Carlo simulations, we determine the phase diagram
and investigate the nature of the different transition lines in two different
cases. First, we consider the model with charge-1 two-component scalar fields;
second, we study the model with scalar fields of charge $Q=2$ and $N_{f}=15$
components. These two sets of parameters have been chosen on the basis of the
results for the U(1) gauge AH model [18], which show that in these two cases
gauge fields play a completely different role.

For U(1) gauge fields, the transitions observed for $N_{f}=2$ and $Q=1$ are
uniquely characterized by the critical fluctuations of the scalar fields.
Gauge fields are only relevant in selecting the physical observables (i.e.,
gauge-invariant quantities), but do not determine the nature of the critical
behavior, which can be predicted by using an effective LGW theory for a gauge-
invariant order parameter, with no explicit gauge fields. Therefore, the
universality class of the transition depends only on the transformation
properties of the order parameter under the global SU(2) symmetry and the
symmetry breaking pattern at the transition. In the SO(3) language [note that
SU(2)$/{\mathbb{Z}}_{2}$ = SO(3)], the order parameter is an SO(3) vector and
the symmetry breaking pattern corresponds to $SO(3)\to SO(2)$. Therefore, one
predicts the transition to belong to the SO(3) vector universality class, a
result that is fully supported by simulations (see, e.g., [27]). In the
discrete gauge model, along the line where the SU(2) symmetry is broken, we
observe the same SO(3) vector behavior for any $q\geq 3$. This result further
confirms correctness of the effective LGW effective theory and the irrelevance
of the gauge fields along the O(3) transition line.

As we already mentioned, the discrete model is also invariant under the
U(1)$/{\mathbb{Z}}_{q}$ group. We find transition lines where this Abelian
global symmetry is broken. They occur within the SU(2) ordered phase. Again,
gauge fields are irrelevant for the determination of the critical behavior:
transitions belong to the XY universality class, as in systems without gauge
fields. We also observe the presence of lines where both the SU(2) group and
the U(1)$/{\mathbb{Z}}_{q}$ group break simulteneuosly. On these lines the
critical behavior belongs to the O(4) universality class. We observe here an
enlargement of the global symmetry group: the O(4) group is the smallest
simple group that has U(2) has one of its subgroups. The symmetry enlargement
is strictly related to the irrelevance of the ${\mathbb{Z}}_{q}$ gauge
interactions, as the model we consider is O(4) invariant in the absence of
gauge fields: the addition of ${\mathbb{Z}}_{q}$ gauge fields represents an
irrelevant perturbation (in the renormalization-group sense) of the O(4)
vector fixed point.

We have then considered the model with $Q=2$ and $N_{f}=15$. In the compact
U(1) gauge model, for these values of the parameters, there is a transition
line where the SU($N_{f}$) symmetry is broken and transitions belong to the
charged AH universality class [17, 18]. Gauge fields are crucial at the
transition and the effective field theory model that describes the universal
features of the transition is the continuum AH model with explicit gauge
fields. We have studied whether these charged transitions also occur when the
U(1) group is replaced by one of its ${\mathbb{Z}}_{q}$ subgroups, i.e., if
there is a gauge symmetry enlargement, as it occurs for global U(1)
symmetries. Our numerical Monte Carlo results for $q=6$ and 10 give a negative
answer. In the compact models with discrete gauge groups that we consider, the
transitions along the line where the SU($N_{f}$) symmetry is broken are always
of first order. Apparently, in the compact AH model it is crucial that the
microscopic model is U(1) gauge invariant to observe the charged behavior, at
variance with what happens along the uncharged transition lines, where the
exact gauge symmetry plays little role.

Simulations were performed at the INFN Computation Center of the INFN, Sezione
di Roma.

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