# Breaking of the gauge symmetry in lattice gauge theories

Claudio Bonati  Dipartimento di Fisica dell’Università di Pisa and INFN Largo
Pontecorvo 3, I-56127 Pisa, Italy  Andrea Pelissetto  Dipartimento di Fisica
dell’Università di Roma Sapienza and INFN Sezione di Roma I, I-00185 Roma,
Italy  Ettore Vicari  Dipartimento di Fisica dell’Università di Pisa and INFN
Largo Pontecorvo 3, I-56127 Pisa, Italy

###### Abstract

We study perturbations that break gauge symmetries in lattice gauge theories.
As a paradigmatic model, we consider the three-dimensional Abelian-Higgs (AH)
model with an $N$-component scalar field and a noncompact gauge field, which
is invariant under U(1) gauge and SU($N$) transformations. We consider gauge-
symmetry breaking perturbations that are quadratic in the gauge field, such as
a photon mass term, and determine their effect on the critical behavior of the
gauge-invariant model, focusing mainly on the continuous transitions
associated with the charged fixed point of the AH field theory. We discuss
their relevance and compute the (gauge-dependent) exponents that parametrize
the departure from the critical behavior (continuum limit) of the gauge-
invariant model. We also address the critical behavior of lattice AH models
with broken gauge symmetry, showing an effective enlargement of the global
symmetry, from U($N$) to O($2N$), which reflects a peculiar cyclic
renormalization-group flow in the space of the lattice AH parameters and of
the photon mass.

Introduction. Gauge symmetries play a key role in the construction of the
theoretical models of fundamental interactions Weinberg-book ; ZJ-book  and in
the description of emergent phenomena in condensed-matter and statistical
physics Wen-book ; Anderson-15 ; GASVW-18 ; Sachdev-19 ; SSST-19 ; GSF-19 .
They may be exact, as in the Standard Model of fundamental interactions, or
effectively emerge at low energies, as in some many-body systems. Effectively
emergent gauge symmetries have also been discussed in the context of
fundamental interactions, see, e.g., Refs. Wen-book ; Wetterich-17 ; FNN-80 ;
INT-80 . In this case, they may arise from microscopic interactions of
different nature, such as string models Polchinski-book .

To correctly interpret experimental results in terms of models with an
emergent gauge symmetry, a solid understanding of the effects of gauge-
symmetry violations is essential. This issue is crucial in the context of
analog quantum simulations, for example, when controllable atomic systems are
engineered to effectively reproduce the dynamics of gauge-symmetric
theoretical models, with the purpose of obtaining physical information from
the experimental study of their quantum dynamics in laboratory. Several
proposals of artificial gauge-symmetry realizations have been reported, see,
e.g., Refs. ZCR-15 ; Banuls-etal-20  and references therein (see also Refs.
Martinez-etal-16 ; Bernien-etal-17 ; Klco-etal-18 ; Schweizer-etal-19 ; Gorg-
etal-19 ; Mil-etal-20  for some experimental realizations), in which the gauge
symmetry is expected to effectively emerge in the low-energy dynamics. A
possible strategy is that of adding a penalty term to the Hamiltonian, that
suppresses the interactions violating the gauge symmetry. This strategy
assumes that gauge-symmetry breaking (GSB) terms become negligible at low
energies, thereby making the dynamics effectively gauge invariant in this
limit ZR-11 ; ZCR-15 ; BC-20 . In spite of the relevance of these issues,
there is at present little understanding of the effects of GSB perturbations
on the continuum limit of quantum or statistical systems with gauge
symmetries, or equivalently on the critical behavior close to continuous
transitions, where long-range correlations develop, realizing the
corresponding quantum field theory.

In this paper we address this problem by considering three-dimensional (3D)
lattice gauge theories, obtained by discretizing the action of corresponding
quantum field theories. We study the role of GSB perturbations at the critical
transitions of gauge-invariant models, to understand whether and when they are
relevant, i.e. they break gauge invariance in the low-energy or large-distance
behavior (continuum limit). If this is the case, GSB terms may lead to
different continuum limits, as we shall see.

The model. As a paradigmatic model, we consider the 3D scalar electrodynamics
or Abelian-Higgs (AH) field theory, with an $N$-component complex scalar field
$\Phi({\bm{x}})$ coupled to the electromagnetic field $A_{\mu}({\bm{x}})$. Its
Lagrangian density reads ZJ-book

${\cal
L}=|D_{\mu}{\Phi}|^{2}+w\,{\Phi}^{*}{\Phi}+\frac{u}{4}\,({\Phi}^{*}{\Phi})^{2}+\frac{1}{4g^{2}}\,(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})^{2}\,,$
(1)

where $D_{\mu}\equiv\partial_{\mu}+iA_{\mu}$. The AH theory is invariant under
U(1) gauge and SU($N$) global transformations. Its 3D renormalization-group
(RG) flow has a stable charged (with nonzero gauge coupling) fixed point (FP)
for $N\geq N_{c}$ HLM-74 ; MZ-03 , with $N_{c}=7(2)$ BPV-21 ; IZMHS-19 .
According to the RG theory WK-74 ; Fisher-75 ; Wilson-83 ; PV-02 , the charged
FP is expected to describe the critical behavior, and therefore the continuum
limit, of U(1) gauge models with SU($N$) global symmetry.

Lattice representations of the continuum theory (1) differ for the topological
nature of the lattice gauge field. One can either use the real field
$A_{{\bm{x}},\mu}$ as in the continuum theory (noncompact model) or the link
variables $\lambda_{{\bm{x}},\mu}\in{\rm U}(1)$ (compact model, corresponding
to $e^{iA_{{\bm{x}},\mu}}$). In this work we mostly consider the 3D noncompact
AH (ncAH) model defined on cubic lattices of size $L^{3}$, which has a
continuous transition line for $N>N_{c}$, along which the continuum limit is
described by the 3D AH field theory (1) BPV-21 ; PV-19-AH3d ; footnote-
highercha . The fundamental fields are unit-length $N$-component complex
vectors ${\bm{z}}_{\bm{x}}$ ($\bar{\bm{z}}_{\bm{x}}\cdot{\bm{z}}_{\bm{x}}=1$)
defined on the lattice sites ${\bm{x}}$ and real fields $A_{{\bm{x}},\mu}$
defined on the lattice links. The lattice action is

$\displaystyle S_{\rm AH}({\bm{z}},{\bm{A}})$ $\displaystyle=$
$\displaystyle-J\,N\sum_{{\bm{x}},\mu}2\,{\rm
Re}\,(\bar{\bm{z}}_{\bm{x}}\cdot\lambda_{{\bm{x}},\mu}\,{\bm{z}}_{{\bm{x}}+\hat{\mu}})$
$\displaystyle\;+{1\over
4g_{0}^{2}}\sum_{{\bm{x}},\mu\nu}(\Delta_{\mu}A_{{\bm{x}},\nu}-\Delta_{\nu}A_{{\bm{x}},\mu})^{2}\,,$

where $\lambda_{{\bm{x}},\mu}\equiv e^{iA_{{\bm{x}},\mu}}$, $g_{0}$ is the
lattice gauge coupling, $\hat{\mu}$ are unit vectors along the lattice
directions, and
$\Delta_{\mu}A_{{\bm{x}},\nu}=A_{{\bm{x}}+\hat{\mu},\nu}-A_{{\bm{x}},\nu}$.
The action $S_{\rm AH}$ has a global SU($N$) symmetry, ${\bm{z}}_{\bm{x}}\to
V{\bm{z}}_{\bm{x}}$ with $V\in\mathrm{SU}(N)$, and a local U(1) gauge
symmetry, ${\bm{z}}_{\bm{x}}\to e^{i\theta_{\bm{x}}}{\bm{z}}_{\bm{x}}$ and
$A_{{\bm{x}},\mu}\to
A_{{\bm{x}},\mu}+\theta_{\bm{x}}-\theta_{{\bm{x}}+\hat{\mu}}$. We consider
$C^{*}$ boundary conditions BPV-21 ; KW-91 ; LPRT-16  (see also App. A), to
remove the degeneracy under $A_{{\bm{x}},\mu}\to A_{{\bm{x}},\mu}+2\pi
n_{\mu}$ with $n_{\mu}\in\mathbb{Z}$, obtaining well defined expectation
values for gauge-invariant operators $O({\bm{z}},{\bm{A}})$,

$\langle
O({\bm{z}},{\bm{A}})\rangle={\sum_{\\{{\bm{z}},{\bm{A}}\\}}O({\bm{z}},{\bm{A}})\;e^{-S_{\rm
AH}({\bm{z}},{\bm{A}})}\over\sum_{\\{{\bm{z}},{\bm{A}}\\}}e^{-S_{\rm
AH}({\bm{z}},{\bm{A}})}}\,.$ (3)

The phase diagram of the ncAH model (Breaking of the gauge symmetry in lattice
gauge theories) with $N\geq 2$ is characterized by a Coulomb phase for small
$J$ (short-ranged scalar and long-ranged gauge correlations), a Higgs phase
for large $J$ and small $g_{0}$ (condensed scalar-field and gapped gauge
correlations), and a molecular phase for large $J$ and $g_{0}$ (condensed
scalar-field and long-ranged gauge correlations) BPV-21 . They are separated
by three transition lines, which are continuous or of first order depending on
$N$. In particular, for $N>N_{c}=7(2)$, the ncAH model undergoes continuous
transitions between the Coulomb and Higgs (CH) phases, for
$0<g_{0}^{2}\lesssim 4$. The corresponding critical behavior is described by
the charged FP of the 3D AH field theory BPV-21 . For $g_{0}\to 0$, one has
$A_{{\bm{x}},\mu}\to 1$ modulo gauge transformations, so that one recovers the
O($2N$) vector model. We consider the gauge-invariant bilinear operator

$Q_{{\bm{x}}}^{ab}=\bar{z}_{\bm{x}}^{a}z_{\bm{x}}^{b}-{1\over
N}\delta^{ab}\,,$ (4)

which transforms as $Q_{{\bm{x}}}\to{V}^{\dagger}Q_{{\bm{x}}}\,{V}$ under
global SU($N$) transformations. It provides an effective order parameter for
the spontaneous breaking of the global SU($N$) symmetry.

GSB perturbations. We study how perturbations breaking the U(1) gauge symmetry
affect the CH critical behavior. In this exploratory study we consider the
quadratic perturbation

$\displaystyle P_{M}={r\over 2}\sum_{{\bm{x}},\mu}A_{{\bm{x}},\mu}^{2}\,,$ (5)

which can be interpreted as a photon mass term. Such a mass term is generally
introduced as an infrared regulator in perturbative computations in quantum
electrodynamics ZJ-book . We also consider the local quadratic operators

$\displaystyle P_{L}={a\over
2}\sum_{{\bm{x}}}\,(\sum_{\mu}\Delta_{\mu}A_{{\bm{x}},\mu})^{2},\;\;P_{A}={b\over
2}\sum_{{\bm{x}}}\,(\sum_{\mu}n_{\mu}\,A_{{\bm{x}},\mu})^{2},\;\;$ (6)

where $n_{\mu}$ is an arbitrary unit vector. When added to the ncAH action,
i.e., if one considers $S=S_{\rm AH}+P_{\\#}$, all quadratic terms defined in
Eqs. (5) and (6) break gauge invariance, leaving a global U($N$) symmetry
${\bm{z}}_{\bm{x}}\to U{\bm{z}}_{\bm{x}}$, $U\in\mathrm{U}(N)$. However, they
affect the critical behavior quite differently. The mass term (5) is expected
to be relevant at the CH transitions, since it drastically changes the long-
distance properties of the gauge-field correlations. In particular, the
Coulomb phase disappears in the presence of a photon mass. Therefore, as soon
as the perturbation is turned on ($r>0$), the system is expected to flow out
of the charged AH FP. On the other hand, the quadratic terms $P_{L}$ and
$P_{A}$, cf. Eq. (6), may be interpreted as the result of the Fadeev-Popov
procedure for a gauge fixing ZJ-book , being related to the Lorentz
($\partial_{\mu}A_{\mu}=0$) and axial (${\bm{n}}\cdot{\bm{A}}=0$) gauge fixing
footnote-gf , respectively. If they are the only GSB perturbations present in
the model, they are expected to be irrelevant for gauge-invariant correlations
(more precisely, their presence does not change gauge-invariant expectation
values). However, as we shall see below, they play a role, when they are added
to the action together with the mass term (5), as they make the limit $r\to 0$
well defined.

Relevance of the GSB perturbations. To characterize the strength of the
perturbation $P_{M}$, we compute the corresponding RG dimension $y_{r}>0$.
This exponent provides information on how to scale $r$ to keep GSB effects
small. Indeed, when the correlation length $\xi$ increases, approaching the
continuum limit, one should decrease $r$ faster than $\xi^{-y_{r}}$ to ensure
that GSB effects are negligible. We estimate $y_{r}$ by finite-size scaling
(FSS) analyses of Monte Carlo (MC) data. We consider the correlation function
$\langle{\rm Tr}\,Q_{\bm{x}}Q_{\bm{y}}\rangle$ of the operator $Q_{\bm{x}}$
defined in Eq. (4), and the corresponding second-moment correlation length
$\xi$. We consider RG-invariant quantities $R$, such as $R_{\xi}=\xi/L$ and
the Binder parameter
$U={\langle\mu_{2}^{2}\rangle}/{\langle\mu_{2}\rangle^{2}}$, where
$\mu_{2}=\sum_{{\bm{x}},{\bm{y}}}{\rm Tr}\,Q_{{\bm{x}}}Q_{\bm{y}}$. At
continuous transitions driven by the parameter $J$, they are expected to
behave as PV-02

$R(L,J,g_{0})\approx f_{R}(X)+O(L^{-\omega})\,,\;\;X=(J-J_{c})\,L^{1/\nu}\,,$
(7)

where $\nu$ is the length-scale critical exponent, and $\omega>0$ is the
exponent controlling the leading scaling corrections. It is also useful to
consider the FSS relation BPV-19-sqcd

$U=F_{U}(R_{\xi})+O(L^{-\omega})\,,$ (8)

where $F_{U}$ is a universal function independent of any normalization. To
estimate $y_{r}$, we consider the behavior of the RG invariant quantities $R$
in the presence of the GSB term (5). In the large-$L$ limit, we expect Fisher-
MCP

$R(L,J,g_{0},r)\approx{\cal F}_{R}(X,Y)\,,\qquad Y=rL^{y_{r}}\,,$ (9)

which holds provided MCP  that $y_{r}>1/\nu$, where $\nu$ is the thermal
exponent of the gauge model (along the CH transition line we have
$1/\nu=1.387(6)$, 1.247(12) for $N=15$, 25, respectively). Eq. (9) is the
usual FSS relation for a multicritical point in systems with a global
symmetry. However, in the present case its validity is not obvious, given that
the mass term $P_{M}$ is not well defined in the ($r=0$) gauge-invariant
noncompact theory: averages of the mass term can only be computed in the
presence of a maximal gauge fixing Creutz-book ; footnote-gf , such as the
axial (using $C^{*}$ conditions) or Lorentz ones. For these reasons, we
consider three different actions with GSB terms:

$\displaystyle{\rm M}1:\quad$ $\displaystyle S_{1}=S_{\rm AH}+P_{M}\,,$ (10)
$\displaystyle{\rm M}2:\quad$ $\displaystyle S_{2}=S_{\rm AH}+P_{L}+P_{M}\,,$
(11) $\displaystyle{\rm M}3:\quad$ $\displaystyle S_{3}=S_{\rm
AH}+P_{M}\;\;\;{\rm with}\;\;A_{{\bm{x}},3}=0\,,\quad$ (12)

where M2 can be associated with the Lorentz gauge, and M3 is defined imposing
the axial gauge. We expect Eq. (9) to be well defined in models M2 and M3,
while its validity in model M1 is instead not clear.

Numerical estimates of the RG dimensions. We performed MC simulations for
$N=15$ and $N=25$ along the CH transition line (estimates of the critical
points and exponents can be found in Ref. BPV-21 ), for the three models M#,
see App. A for details. The results confirm that $P_{M}$ is relevant. Indeed,
for fixed $r$, there is a clear departure from the gauge-invariant ($r=0$)
critical behavior. In Fig. 1 we show results for $N=25$ at the critical point.
The exponent $y_{r}$ is estimated by fitting the data at $J_{c}$ to Eq. (9),
setting $X=0$. We obtain $y_{r}=1.4(1)$ for M2 [for both $a=1$ and $a=10$, cf.
Eq. (6)] and $y_{r}=2.55(5)$ for M3. We also mention that if we apply Eq. (9)
to $U$ computed in M1 without gauge fixing, we obtain the effective estimate
$y_{r}\approx 1.4$, see top Fig. 1, confirming the relevance of $P_{M}$ along
the CH transition line. Analogous results are obtained for $N=15$, in
particular $y_{r}=2.55(10)$ for M3. The exponent $y_{r}$ turns out to depend
on the gauge fixing, indicating that the gauge fixing influences the RG
properties of the mass perturbation. Apparently, gauge-dependent modes, that
are controlled by the gauge fixing term, are crucial in determining the
effects of the photon mass term. Note that $y_{r}$ is quite large, therefore
the corresponding GSB perturbation must decrease rapidly with $L$ —faster than
$L^{-y_{r}}$—to keep GSB effects under control.

Figure 1: Data of $U$ at the critical point $J_{c}\approx 0.295515$ of the
ncAH model for $N=25$ and $g_{0}^{2}=2.5$, as a function of $Y=rL^{y_{r}}$.
Results for models M1 (without gauge fixing, top) and M3 (axial gauge,
bottom).

Critical behaviors in the presence of finite GSB terms. We now address the
behavior of the ncAH model in the presence of a finite GSB term such as the
photon mass. Also for finite $r$ we expect a transition at a finite value
$J_{c}(r)$, with $J_{c}(r=0)=J_{c}$, where $J_{c}$ is the CH transition point
in the gauge-invariant model. Since the charged fixed point is unstable with
respect to $P_{M}$, we expect the transition to belong to a different
universality class, which should only depend on the global symmetry of the
model. Although the global symmetry group for $r>0$ is U($N$), we will now
argue that continuous transitions at $J_{c}(r)$ are characterized by a larger
O($2N$) invariance group. We note that, since gauge fields are not expected to
be relevant for $r\not=0$, one can use the standard Landau-Ginzburg-Wilson
(LGW) approach WK-74 ; Fisher-75 ; Wilson-83 ; PV-02  to predict the critical
behavior. Since the gauge symmetry is broken, ${\bm{z}}_{\bm{x}}$ represents
the microscopic order-parameter field. Therefore, the LGW basic field is an
$N$-component complex vector $\Psi({\bm{x}})$. The Lagrangian is the sum of
the kinetic term $|\partial_{\mu}\Psi|^{2}$ and of the most general
U($N$)-invariant quartic potential:

$\displaystyle{\cal L}_{\rm
LGW}=\partial_{\mu}\Psi^{*}\cdot\partial_{\mu}\Psi+w\;\Psi^{*}\cdot\Psi+{u\over
4}\,(\Psi^{*}\cdot\Psi)^{2}\,.$ (13)

It is easy to check that ${\cal L}_{\rm LGW}$ is actually O($2N$) invariant.
Indeed, there are no dimension-2 and 4 U($N$) invariant operators that break
the O($2N$) symmetry. The lowest-dimension operators that are not O($2N$)
symmetric have dimension six close to four dimensions—for instance,
$(\mathrm{Im}\,\Phi^{*}\cdot\partial_{\mu}\Phi)^{2}$—and thus they are
expected to be irrelevant at the 3D O($2N$) FP. Therefore, the critical
behavior of generic vector systems with global U($N$) invariance (without
gauge symmetries) is expected to belong to the O($2N$) universality class,
implying an effective enlargement of the global symmetry of the critical modes
(restricted only to the critical region).

The above analysis can be extended to lattice AH models with compact gauge
fields (cAH), using the link variables $\lambda_{{\bm{x}},\mu}\in{\rm U}(1)$
and the pure gauge action
$S_{\lambda}=-g_{0}^{-2}\sum_{{\bm{x}},\mu\neq\nu}{\rm
Re}\,\lambda_{{\bm{x}},{\mu}}\,\lambda_{{\bm{x}}+\hat{\mu},{\nu}}\,\bar{\lambda}_{{\bm{x}}+\hat{\nu},{\mu}}\,\bar{\lambda}_{{\bm{x}},{\nu}}$
in Eq. (Breaking of the gauge symmetry in lattice gauge theories). Unlike ncAH
models, cAH models with $N\geq 2$ present only two phases, separated by a
disorder-order transition line where gauge correlations are not critical
PV-19-AH3d . Since the scalar fields turns out to be the only critical degrees
of freedom, the effective description of the transitions is provided by the
SU($N$)-invariant LGW $\Phi^{4}$ theory with a matrix gauge-invariant order
parameter, corresponding to $Q_{\bm{x}}^{ab}$ in Eq. (4) PV-19-CP ; PV-19-AH3d
. For $N=2$ this LGW theory has a stable O(3) vector FP, thus predicting O(3)
continuous transitions PV-02  for any gauge coupling $g_{0}>0$, including
$g_{0}\to\infty$ [for $g_{0}\to 0$, instead, the model becomes equivalent to
the O(4) vector model]. This has been also confirmed numerically PV-19-AH3d .
Gauge invariance can be broken by adding $P_{M}=-r\sum_{{\bm{x}},\mu}{\rm
Re}\,\lambda_{{\bm{x}},\mu}$, which plays the role of a photon mass for
$\lambda_{{\bm{x}},\mu}$ close to 1. When the gauge symmetry is effectively
broken (as discussed in Ref. BPV-inprep , this requires $r$ to be sufficiently
large), the critical behavior should be described by the LGW theory (13),
which predicts that continuous transitions belong to the O(4) vector
universality class.

Figure 2: Estimates of $U$ versus $R_{\xi}$ for the model M1 with $N=25$,
$g_{0}^{2}=2.5$, $r=1$. We also report results for the gauge-invariant model
($r=0$) and the O(50) vector model (full line, obtained by large-$L$
extrapolations of MC data for the appropriate spin-2 correlations, see Ref.
PV-19-AH3d  and App. A). The results for $r=1$ appear to converge toward the
O(50) universal curve, consistently with $O(L^{-\omega})$ corrections with
$\omega\approx 1$, supporting the RG prediction reported in the text.

The RG predictions at fixed $r$ are confirmed by numerical results for both
ncAH and cAH models. In Fig. 2 we plot $U$ versus $R_{\xi}$ for the ncAH model
with $N=25$ and $r=1$. The data around the critical point $J_{c}(r)$ are
expected to converge to a universal curve, cf. Eq. (8), which can be compared
with the analogous curves of models that belong to known universality classes.
The data approach the O($2N$) vector universal curve (obtained using an
appropriate operator that corresponds to $Q_{\bm{x}}^{ab}$ in the O($2N$)
model PV-19-AH3d ), confirming the LGW RG argument. For the cAH model with
$N=2$, we observe an asymptotic O(4) vector behavior for $r=1$ and $r=2.25$,
in agreement with the general arguments in App. A.

Various classes of GSB perturbations. On the basis of the results presented in
this paper, we may distinguish three classes of GSB perturbations. (i) First,
there are GSB perturbations that are relevant at the stable FP of the lattice
gauge-invariant theory. They drive the system out of criticality and may give
rise to a different critical behavior. The photon mass term (5) plays this
role along the CH line in the ncAH model. (ii) A second class corresponds to
gauge fixings and GSB perturbations like those appearing in Eq. (6). If they
are the only GSB terms present in the model, they are irrelevant: gauge-
invariant observables are unchanged. However, if they are present together
with some relevant GSB perturbation, they play a role: the RG flow close to
the charged FP depends both on the gauge-fixing and on the relevant
perturbation. This may be due to the fact that a gauge fixing is needed to
make non-gauge-invariant correlations well defined in the gauge-invariant
theory or to the role of gauge-dependent modes that are sensitive to gauge
fixings. (iii) GSB perturbations associated with RG operators with negative RG
dimensions, whose effects are suppressed in the critical (continuum) limit.

When the added GSB perturbations are relevant, the lattice system may develop
a different critical behavior or continuum limit. This is the case of the ncAH
model with a photon mass term, which has a global U($N$) invariance. Quite
interestingly, the transitions in this model belong to the O($2N$) vector
universality class, with an effective enlargement of the global symmetry at
the transition. This symmetry enlargement is expected in any model in which
the GSB perturbation is relevant and it preserves the global U($N$) symmetry.

Cyclic RG flow. It is worth noting that the above results lead to a peculiar
RG flow, see Fig. 3 for a sketch in the coupling space $(J,g_{0},r)$. For
$g_{0}^{2}\to 0$, the gauge fields are frozen, and the model is equivalent to
the O($2N$) vector model, whose critical behavior is controlled by the
corresponding O($2N$) FP. If the gauge interactions are turned on, i.e., one
sets $g_{0}>0$ keeping $r=0$, the systems flows towards the charged FP of the
AH field theory, which is stable for any $0<g_{0}^{2}\lesssim 4$. Finally, if
a photon mass is added, i.e., one sets $r>0$, since the charged FP is unstable
under this perturbation, the RG flow goes back to the O($2N$) FP, which is now
stable, independently of $g_{0}$ and $r$. This RG behavior can be hardly
reconciled with an irreversibility of the RG flow, analogous to that generally
associated with the monotonic properties implied by the $c$-theorem of 2D
critical systems Zamolodchikov-1986 ; Cardy-book , see also Refs. Pufu-17 ;
Grover-14 ; CH-12 ; KPSS-12 ; KPS-11 ; MS-11  for similar proposals in 3D
systems footnoteirrflow .

Figure 3: Sketch of the cyclic RG flow of the ncAH model in the space of the
parameters $J$, $g_{0}$ and $r$, showing an unusual loop between the O($2N$)
and the charged FP C($N$).

Conclusions. In conclusion, we have studied the effect of GSB perturbations on
the critical behavior—or, equivalently, the continuum limit—of gauge-invariant
theories. The behavior at charged FP turns out to be more complicated than
that observed when global symmetries are broken. In particular, we observe
apparent violations of universality. For instance, the RG dimension of the
same GSB perturbation appears to depend on local gauge-fixing conditions, a
result that, we believe, should be further investigated. Moreover, GSB
perturbations give rise to unexpected phenomena, like the cyclic RG flow
sketched in Fig. 3.

Several extensions are called for, to achieve a satisfactory understanding of
the problem and to identify its universal features, such as the study of other
lattice gauge theories—in particular, it would be interesting to extend the
analysis to the nonabelian gauge groups— and of other classes of GSB
perturbations, for example preserving residual discrete gauge subgroups (such
approximations may be useful for analog simulations). It would also be
important to rephrase and extend the present results to quantum Hamiltonian
systems Kogut-79  (see Refs. VHH-20 ; ZVHHB-21  for recent works addressing
issues related to GSB effects and the approach to the continuum limit).

###### Acknowledgements.

Numerical simulations have been performed on the CSN4 cluster of the
Scientific Computing Center at INFN-PISA.

##  Appendix A The numerical analyses

In this appendix we provide some details on the numerical computations
reported in the paper.

###  A.1 The models

In most of the simulations we have considered the noncompact Abelian-Higgs
(ncAH) model. The fundamental fields are unit-length $N$-component complex
vectors ${\bm{z}}_{\bm{x}}$ ($\bar{\bm{z}}_{\bm{x}}\cdot{\bm{z}}_{\bm{x}}=1$)
defined on the lattice sites and real fields $A_{{\bm{x}},\mu}$ defined on the
lattice links. The lattice action is

$S_{\rm ncAH}({\bm{z}},{\bm{A}})=S_{z}({\bm{z}},{\bm{A}})+S_{nc}({\bm{A}})\,,$
(14)

where

$\displaystyle S_{z}({\bm{z}},{\bm{A}})$ $\displaystyle=$
$\displaystyle-J\,N\sum_{{\bm{x}},\mu}2\,{\rm
Re}\,(\bar{\bm{z}}_{\bm{x}}\cdot\lambda_{{\bm{x}},\mu}\,{\bm{z}}_{{\bm{x}}+\hat{\mu}})\,,$
(15) $\displaystyle S_{nc}({\bm{A}})$ $\displaystyle=$ $\displaystyle{1\over
4g_{0}^{2}}\sum_{{\bm{x}},\mu\nu}(\Delta_{\mu}A_{{\bm{x}},\nu}-\Delta_{\nu}A_{{\bm{x}},\mu})^{2}\,,$

$\lambda_{{\bm{x}},\mu}\equiv e^{iA_{{\bm{x}},\mu}}$, $g_{0}$ is the lattice
gauge coupling, $\hat{\mu}$ are unit vectors along the lattice directions, and
$\Delta_{\mu}A_{{\bm{x}},\nu}=A_{{\bm{x}}+\hat{\mu},\nu}-A_{{\bm{x}},\nu}$.

We have also considered the lattice compact AH model (cAH), with action

$S_{\rm
cAH}({\bm{z}},{\bm{\lambda}})=S_{z}({\bm{z}},{\bm{\lambda}})+S_{c}({\bm{\lambda}})\,,$
(16)

where $S_{z}({\bm{z}},{\bm{\lambda}})$ is given by Eq. (15), while

$S_{c}({\bm{\lambda}})=-g_{0}^{-2}\sum_{{\bm{x}},\mu\neq\nu}{\rm
Re}\,\lambda_{{\bm{x}},{\mu}}\,\lambda_{{\bm{x}}+\hat{\mu},{\nu}}\,\bar{\lambda}_{{\bm{x}}+\hat{\nu},{\mu}}\,\bar{\lambda}_{{\bm{x}},{\nu}}\,.$
(17)

To simulate the ncAH model, it is not possible to use periodic boundary
conditions, since all gauge-invariant observables associated to loops that
wrap around the lattice are not bounded and their average values are ill-
defined. As in our previous work BPV-21 , we consider $C^{*}$ boundary
conditions. They are used here for the cAH model too, although periodic
boundary conditions would be appropriate, as well. We consider cubic lattices
of size $L$, so that $C^{*}$ boundary conditions amount to the identifications
(see Ref. BPV-21  for a thorough discussion)

$A_{{\bm{x}}+L\hat{\nu},\mu}=-A_{{\bm{x}},\mu}\
,\qquad{\bm{z}}_{{\bm{x}}+L\hat{\nu}}=\bar{\bm{z}}_{\bm{x}}\,.$ (18)

To be consistent with Eq. (18), local gauge transformations are defined by
$A_{{\bm{x}},\mu}\to
A_{{\bm{x}},\mu}+\alpha({\bm{x}}+\hat{\mu})-\alpha({\bm{x}})$, with an
antiperiodic function $\alpha({\bm{x}})$:
$\alpha({\bm{x}}+L\hat{\nu})=-\alpha({\bm{x}})$. As a consequence, observables
that involve a nontrivial wrapping around the lattice are not gauge invariant.

$C^{*}$ boundary conditions are very convenient when implementing axial
gauges. Indeed, it is possible to fix $A_{{\bm{x}},3}=0$—or, more generally,
$\sum_{\mu}n_{\mu}A_{{\bm{x}},\mu}=0$— on all lattice sites, at variance with
the case of periodic boundary conditions. From the explicit construction
discussed in Ref. BPV-21 , it follows that no residual gauge freedom is left
once $A_{{\bm{x}},3}=0$ is enforced on all lattice sites. It is important to
note that also the Lorentz gauge is a maximal gauge, with no residual gauge
freedom. Indeed, suppose the opposite, i.e., that there are two different
gauge configurations $A_{{\bm{x}},\mu}^{(1)}$ and $A_{{\bm{x}},\mu}^{(2)}$
that are related by a gauge transformation,
$A_{{\bm{x}},\mu}^{(2)}=A_{{\bm{x}},\mu}^{(1)}+\Delta_{\mu}\alpha({\bm{x}})$,
and that both satisfy the condition
$\sum_{\mu}\Delta_{\mu}A_{{\bm{x}},\mu}^{(i)}=0$. The function
$\alpha({\bm{x}})$ must satisfy

$\sum_{\mu}\Delta_{\mu}[\Delta_{\mu}\alpha({\bm{x}})]=0\,,$ (19)

which implies that $\alpha$ is a zero eigenmode of the lattice Laplacian. For
$C^{*}$ boundary conditions there is no zero mode, as $\alpha$ is
antiperiodic, proving that $A_{{\bm{x}},\mu}^{(1)}=A_{{\bm{x}},\mu}^{(2)}$.
For periodic boundary conditions, there is one zero mode,
$\alpha({\bm{x}})=C$, where $C$ is space independent, so that also in this
case $A_{{\bm{x}},\mu}^{(1)}=A_{{\bm{x}},\mu}^{(2)}$.

We have considered the ncAH model for $N=15$ and $25$, fixing in both cases
$g_{0}^{2}=2.5$. For this value of $g_{0}$ the ncAH model undergoes a
continuous transition for $J=J_{c}$. As discussed in Ref. BPV-21 , such
transition is controlled by the charged fixed point (FP) of the Abelian-Higgs
field theory. We have performed simulations for $J=J_{c}$ on lattices of size
$L\leq 96$ ($N=25$) and $L\leq 64$ ($N=15$). For $N=15$ we used the estimate
$J_{c}$ reported in Ref. BPV-21 , see Table 1. For $N=25$ we used an improved
estimate. We performed additional simulations for $J\approx J_{c}$ on larger
lattices (while in Ref. BPV-21  we limited ourselves to lattices with $L\leq
64$, here we consider values of $L$ up to $L=96$) and reanalyzed the data. The
result is $J_{c}=0.295515(4)$, which is consistent with the estimate
$J_{c}=0.295511(4)$ reported in Ref. BPV-21 . We have also considered the cAH
model for $N=2$, the only case where a continuous transition is present. The
parameter $g_{0}^{2}$ does not play any role PV-19-AH3d  and we have therefore
set $1/g_{0}^{2}=0$ (no gauge action); the estimate of the corresponding
critical value $J_{c}$ is also reported in Table 1.

Beside simulations of the gauge model, we have also performed simulations of
the O($2N$) spin model with action $S_{z}$ and $\lambda_{{\bm{x}},\mu}=1$,
measuring the same quantities we compute in the gauge model (see Appendix B of
Ref. PV-19-AH3d  for a discussion of the relation between correlation
functions of the CPN-1 order parameter $Q^{ab}$ and spin-two correlations in
the vector O($2N$) spin model). We considered $N=2$ and $N=25$, determining
$U$ and $R_{\xi}$, and in particular the universal curve $U=F(R_{\xi})$.

$N$ | $g_{0}^{2}$ | $J_{c}$
---|---|---
25 | 2.5 | 0.295515(4)
15 | 2.5 | 0.309798(6)
2 | $\infty$ | 0.7102(1)
Table 1: Critical values $J_{c}$ of the coupling $J$ for the values of
$g_{0}^{2}$ used in the present simulations. The value of $J_{c}$ for $N=25$
is an improvement of the estimate of Ref. BPV-21 . Results for $N=15$ and
$N=2$ are taken from Ref. BPV-21  (note that $1/g_{0}^{2}$ was named
$\kappa$), and Ref. PV-19-CP , respectively.

Figure 4: Estimates of $R_{\xi}$ (top) and $U$ (bottom) at the critical point
$J_{c}\approx 0.309798$ of the ncAH model for $N=15$ and $g_{0}^{2}=2.5$, as a
function of $rL^{y_{r}}$, for model M3 (axial gauge).

###  A.2 Technical details: simulations and data analysis

In the Monte Carlo simulations we use an overrelaxation algorithm, obtained by
combining Metropolis updates of the scalar and of the gauge fields and
microcanonical updates of the scalar field. The latter moves are obtained by
generalizing the usual reflection moves used in O($N$) models. We perform a
Metropolis update of the ${\bm{z}}$ and gauge fields every 5 (on the largest
lattices, every 10) microcanonical updates of the scalar field. Trial states
for the Metropolis updates are generated by adding a random number to
$A_{{\bm{x}},\mu}$ and by multiplying ${\bm{z}}_{\bm{x}}$ by a random $2\times
2$ unitary matrix close to the identity; in both cases, we tune the update to
have an acceptance rate of approximately $30\%$.

The typical statistics (discarding thermalization) of each data point varies
in the range $4\times 10^{5}$-$2\times 10^{6}$. Errors are estimated by using
a combination of jackknife and blocking procedures. In all cases, the
autocorrelation times were at most of the order of $10^{2}$ updates.

To compute the estimates of $y_{r}$, we have assumed that $U$ and $R_{\xi}$
satisfy the scaling relation

$R(r,L)=f(rL^{y_{r}})+L^{-\omega}g(rL^{y_{r}})$ (20)

at the critical point $J=J_{c}$. In the fits we have approximated the scaling
functions with polynomials and we have analyzed $U$ and $R_{\xi}$ together,
looking for a value of $y_{r}$ the minimizes the sum of the residuals for the
two observables. The value of $\omega$ is unknown. In the absence of GSB
terms, we expect $\omega$ to be close to 1 (for $N=\infty$ we have
$\omega=1$), but, once GSB terms are added, new irrelevant operators may
appear and $\omega$ may be smaller. For this reason we have determined $y_{r}$
for values of $\omega$ in the interval 0.2-1. Except for model M1, the
$\omega$ dependence is at most of the same order of the statistical errors.

###  A.3 Results for $N=15$

To investigate the $N$ dependence of the exponent $y_{r}$, beside considering
the ncAH model with $N=25$ we have also considered the same model for $N=15$.
We have only studied model M3 (axial gauge), obtaining

$y_{r}=2.55(10)\,,$ (21)

which should be compared with the result for $N=25$, $y_{r}=2.55(5)$. The $N$
dependence is apparently small, much smaller than the statistical errors. In
Fig. 4 we show $R_{\xi}$ and $U$ against $rL^{y_{r}}$ (the analogous plot of
$U$ for $N=25$ is shown in the main text). The ratio $R_{\xi}$ shows a very
nice scaling behavior with small scaling corrections, which increase as
$rL^{y_{r}}$ increases. The Binder parameter shows larger corrections, that
have the opposite behavior: they decrease as $rL^{y_{r}}$ increases.

###  A.4 Results for the compact model with $N=2$

As discussed in the main text, we also performed an exploratory investigation
of the effect of explicit gauge breaking terms using the compact
discretization. We studied the $N=2$ cAH model, which is the only one in which
a continuous transition is present PV-19-AH3d . In this case we defined the
photon mass operator as

$P_{M}=-r\sum_{{\bm{x}},\mu}{\rm Re}\,\lambda_{{\bm{x}},\mu}\,,$ (22)

and performed simulations using the axial gauge. We set
$\lambda_{{\bm{x}},3}=1$ (this the analogue of the condition
$A_{{\bm{x}},3}=0$ used in the noncompact case) on all sites. Note that this
is possible as we use $C^{*}$ boundary conditions also for the compact model.

Figure 5: Estimates of $U$ versus $R_{\xi}$ for the cAH model M1 with $N=2$,
$g_{0}^{2}=\infty$: results for $r=1$ (top panel), and $r=2.25$ (bottom
panel). The critical couplings are $J_{c}\approx 0.552$ and $J_{c}\approx
0.328$ for $r=1$ and $r=2.25$ respectively. We also report results for the
gauge-invariant model ($r=0$) and the O(4) vector model (full line, obtained
by large-$L$ extrapolations of MC data for the appropriate spin-2
correlations, see Ref. PV-19-AH3d ).

As for $N=25$ we studied the behavior of the model with action $S_{cAH}+P_{M}$
for two finite values of $r$, $r=1$ and $r=2.25$. In Fig. 5 we report the
Binder parameter versus $R_{\xi}$. Data for $r=2.25$ are perfectly consistent
with an O(4) behavior, confirming the expected symmetry enlargement. The
results for $r=1$ instead are still quite far from the O(4) curve, although
they show the correct trend. Except for values of $R_{\xi}$ where $U$ has a
maximum, as $L$ increases from 16 to 64, the data move towards the O(4)
universal curve. Close to the maximum, the behavior is nonmonotonic, but the
data start moving towards the O(4) curve for $L\geq 48$. A thorough
investigation of the compact model will be reported in a forthcoming
publication BPV-inprep .

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