Line operators, vortex statistics, and Higgs versus confinement dynamics

We study a $2{+}1$D lattice gauge theory with fundamental representation scalar fields which has both Higgs and confining regimes with a spontaneously-broken $U(1)$ $0$-form symmetry. We show that the Higgs and confining regimes may be distinguished by a natural gauge invariant observable: the phase $\Omega$ of a correlation function of a vortex line operator linking with an electric Wilson line. We employ dualities and strong coupling expansions to analytically explore parameter regimes which were inaccessible in previous continuum calculations, and discuss possible implications for the phase diagram.


Introduction and summary
Gauge theories with fundamental representation matter fields can exhibit both confining and Higgs regimes, but it can be very difficult to construct useful gauge invariant observables that sharply distinguish these two regimes. 1 In these theories it is generally expected that if all global symmetries have identical realizations in both Higgs and confining regimes then the two regimes need not be separated by a phase boundary and that there is no sharp physical distinction between them. 2 In this paper, which revisits and clarifies arguments originally developed in Refs.[15, 16], we explore the distinction between Higgs and confining regimes in gauge theories with global symmetries whose realizations remain unaltered between the two regimes.However, the presence of these global symmetries opens up the possibility to evade the continuity scenario of Fradkin and Shenker [8]. 3 While the realization of the global symmetry is identical in the Higgs and confining regimes in the system we study, the symmetry structure will nevertheless allow us to define a natural, but non-local, order parameter that distinguishes the two regimes.However, truly demonstrating the existence of a Higgs-confinement phase boundary in the present setting requires either finding new theoretical insights and/or performing careful numerical lattice simulations.The tools and analytic calculations presented in the current paper set the groundwork for future numerical investigations of the full phase diagram guided by the sharp change in behavior of our proposed order parameter.
To review the basic questions motivating this paper in a simple setting, consider first a U (1) gauge theory in 2+1 spacetime dimensions coupled to a unit-charge Higgs field ϕ.Suppose that the UV completion of this Abelian-Higgs model admits finite-action monopoleinstanton field configurations, so that there is no nontrivial magnetic symmetry which would act on monopole operators (which are local in 2+1 dimensions).When the Higgs field has a large positive mass-squared, one can integrate it out and the long distance dynamics of local operators is well-described by the Polyakov model, which is famously confining and features a mass gap for the photon [17].It is natural to describe the regime where ϕ is heavy as a confining regime in the usual heuristic sense, despite the fact that sufficiently long confining flux tubes will eventually break due to pair-production of dynamical ϕ particles.In contrast, if ϕ condenses then one enters a Higgs regime where, at least deep in the semiclassical domain, it is clear that the gauge boson gets a mass related to the "expectation value" of ϕ (using perturbative gauge-dependent language).These Higgs and confining regimes are not distinguished by the expectation value of any local gauge invariant operator.Both regimes are gapped, and in neither regime is any global symmetry spontaneously broken.Consequently, one may expect that they are part of the same phase.This is corroborated by the classic result of Fradkin and Shenker exhibiting a smooth path between the Higgs and confining regimes of certain lattice gauge theories with fundamental representation matter and no global symmetries [8].
The situation can change dramatically in systems with global symmetries.One example is furnished by a variant of the 2+1D Abelian-Higgs model discussed above but without finiteaction monopole-instantons, so that there is an exact U (1) m magnetic global symmetry.The 'condensing' unit-charge Higgs field ϕ is neutral under the magnetic symmetry.Nonetheless, the realization of U (1) m changes as one dials the mass-squared parameter of ϕ.In the Higgs phase the U (1) m symmetry is unbroken, while in the phase where ϕ is heavy the U (1) m symmetry is spontaneously broken.The phase with spontaneously broken U (1) m can be called confining, as it features a logarithmic attractive potential between probe electric charges (with opposite charges) for distances up to the 'string-breaking' length scale. 4Therefore, global symmetry considerations imply that the Higgs and confining regimes of this model are distinct phases.
In this paper our interest lies in gauge theories where the Higgs fields are charged under some global symmetry G.In such theories, there can be an obvious distinction between Higgs and confining regimes if the global symmetry G is spontaneously broken in the Higgs regime, but unbroken in the confining regime.If so, then the Higgs and confining regimes must be distinct phases of matter, separated by a phase boundary.We focus on a subtler question: can one distinguish Higgs and confining regimes when the realization of G is the same in both regimes?We will be particularly interested in distinguishing the Higgs and confining regimes when the global symmetry G is spontaneously broken.
Apart from its intrinsic theoretical interest, this question is also motivated by longstanding puzzles concerning the behavior of QCD as a function of density.Low temperature finite-density QCD has a spontaneously broken U (1) B = U (1)/Z Nc baryon-number global symmetry (with N c = 3), indicating that bulk nuclear matter is a superfluid.As the baryon chemical potential is increased, it is natural to regard dense QCD as evolving from a "confining" regime in which bound nucleons provide the natural description, to a "Higgs" regime in which, using gauge-variant language, composite scalar di-quark operators condense. 5It is a long-standing open problem to determine whether these regimes are sharply distinguishable from each other and separated by a thermodynamic phase transition.(See, e.g., Ref. [20] for a review and references on the high-density phase of QCD.)In particular, Schäfer and Wilczek observed that the Higgs and confining regimes of cold dense QCD, near the SU (3) flavor symmetric point, have the same realization of all conventional global symmetries.This motivated their well-known conjecture that the two regimes are smoothly connected [21].Their arguments relied on the traditional Landau paradigm for distinguishing phases of matter based on realizations of (0-form) symmetries, as well as the compatible nature of the spectrum of low-lying excitations, and follow the general expectation of continuity based on the classic work of Fradkin and Shenker [8].
Over the last several decades it has become clear that there can be genuinely distinct phases of matter that are not visible within the Landau paradigm, see e.g., Refs.[22-26].This can, for example, occur when differing phases of matter are distinguished by intrinsically nonlocal observables, rather than the local order parameters which play the starring role in the Landau-Ginzburg approach.With this more general perspective in mind, it is thus natural to ask whether the Higgs and confining regimes of dense QCD can be sharply distinguished.But given the complicated nature of QCD, it is helpful to first examine, carefully, analogs of this question in simpler settings. 6n this paper, following our earlier work [15], we will focus on a U (1) gauge theory in 2+1 dimensions.The gauge theory we study has U (1) F 0-form symmetries, as well as a U (1) (1) 1-form symmetry which results from eliminating all dynamical vortices. 7As will be discussed in Sec.3.1, the U (1) B and U (1) (1) symmetries have a mixed 't Hooft anomaly causing the U (1) B symmetry to always be spontaneously broken, while the U (1) (1) symmetry remains unbroken.As in dense QCD, this field theory features Higgs and confining regimes, all within a gapless U (1) (0) B -broken phase.Though all conventional and generalized symmetries have identical realizations in these two regimes, we argue that they are differentiated by the values of the Aharonov-Bohm phase Ω of electric probe particles encircling global vortices.Physically, this phase appears if the vortices carry magnetic flux in their cores.With insights from a lattice regularization, we write Ω in terms of a correlation function of a Wilson loop W (C) linked with a loop operator V ( Γ) which inserts a vortex, 8 F symmetry implies that the phase Ω must be real.We will analytically compute the Ω correlation function (1.1) in tractable portions of the parameter space of our theory and show that Ω takes distinct values, namely Ω = +1 and Ω = −1 in the Higgs and confining regimes, respectively.The value of Ω thus serves to distinguish the Higgs and confining regimes of the QFTs we study in this paper.The fact that the Higgs and confining regimes can be distinguished in this manner does not necessarily mean that they are separated by a bulk phase boundary.While we think the presence of such a phase boundary is plausible, we have not been able to prove that it must be present.This will be discussed further in Section 5. Fig. 1 gives a schematic picture of the resulting phase diagram of our theory, where the red line on which Ω changes sign may or may not signal a thermodynamic phase transition.
The fact that vortices may carry a quantized magnetic flux in the theory we study is similar to the behavior of vortices in charge N > 1 superconductors, which are gapped phases.There the parallel result can be understood in terms of topological order, characterized by a topological Z N gauge theory at long distances.Here our focus is on a gapless system which does not satisfy any conventional definition of topological order.Nevertheless, we explain 6 For recent progress on distinguishing the Higgs and confining regimes of dense QCD, see Ref. [27] which argued that the two regimes are separated by a non-Landau phase transition associated with a jump in the quantized gravitational theta term.This jump is associated with a phase boundary in the time-reversal symmetric locus of parameter space, and is directly related to the pattern of quark pairing in the high density phase while being insensitive to the explicit breaking of baryon number symmetry.In contrast, the type of transitions we contemplate in this paper would also occur in bosonic theories and, given the role played by superfluid vortices, rely on a spontaneously broken U (1) symmetry. 7Here and henceforth we use the notation G (p) to denote a p-form symmetry based on a group G. 8 The minimal separation between the two loops must be large compared to the vortex core size.See also  B and (Z 2 ) F 0-form symmetries, and a U (1) (1) 1-form symmetry.Due to a mixed 't Hooft anomaly involving U (1) B and U (1) (1) , the U (1) B symmetry is always spontaneously broken.The (Z 2 ) F symmetry is unbroken in regimes connected to controllable limits, and leads to a quantization of the Aharonov-Bohm phase Ω = ±1 acquired by electrically-charged probe particles when they encircle vortices.The value of this phase thus distinguishes the Higgs and confining regimes, as illustrated in the sketch.The axes show parameters of our lattice discretization of the theory, defined in Sec. 3, with β controlling a lattice gauge coupling and κ c a scalar field mass-squared.Indicated in the corners of the sketch are the sections in this paper where we evaluate the phase Ω.As shown, the strongly-coupled region of parameter space may contain a region in which the (Z 2 ) F symmetry is spontaneously broken.
why it is natural to view Ω as a well-defined label for the Higgs and confining regimes for the class of QFTs we study.However, as we will discuss, we are currently unable to rule out a scenario where Ω jumps due to a level-crossing phenomena among multiple types of vortices in a manner which would not involve a genuine phase transition.Future numerical lattice simulations are needed to convincingly demonstrate the presence (or otherwise) of a phase boundary associated with changes in Ω.
The organization of this paper is as follows.In Section 2 we review the continuum model studied in Ref. [15], and highlight some limitations and unanswered questions from our previous work which we now address in Sections 3 and 4. Readers who are happy to accept that Ω jumps from −1 to +1 as one goes from the Higgs to the confining regime can skip ahead to Section 5 for a discussion of the implications of this behavior of Ω for the phase diagram.Readers who are interested in detailed evaluations of Ω can read Section 3, which introduces a lattice discretization of the model we study and defines Ω in terms of line operator correlation functions, and Section 4 which computes Ω in various limiting regimes.Section 6 contains conclusions and comments on future research directions.Some ancillary materials are collected in the Appendices.

Higgs versus confinement in a continuum gauge theory
We consider a 2+1D U (1) gauge theory with monopoles, coupled to two Higgs scalars ϕ + , ϕ − with electric charges ±1, plus a neutral complex scalar ϕ 0 .The action is [15] We assume that the UV completion allows finite-action magnetic monopoles so that there is no magnetic U (1) m symmetry.We also assume that the potential V int is invariant under the U (1) transformation ϕ ± → e iα ϕ ± , ϕ 0 → e −2iα ϕ 0 .The minimal gauge invariant operators such as ϕ + ϕ − and ϕ † 0 have charge 2 under this naive U (1) transformation.The faithfully-acting continuous global symmetry is therefore U (1)/Z 2 , under which ϕ + ϕ − and ϕ † 0 have charge 1.Finally, we assume that the model enjoys a flavor-flip symmetry so that the internal global symmetry is where (Z 2 ) C is the charge conjugation symmetry and acts by a µ → −a µ , ϕ ± → ϕ † ± .In the action (2.1), we write explicitly the symmetry-allowed cubic term ϕ 0 ϕ + ϕ − with coefficient ϵ.If ϵ is set to zero (and V int contains no other terms coupling ϕ + ϕ − to ϕ 0 ), then the internal global symmetry is enlarged to become where the first factor comes from gauge-inequivalent rotations acting on the gauge-charged fields, while the second factor comes from phase rotations of the neutral field.When ϵ ̸ = 0, as we will assume everywhere below, SO(3) × U (1) is broken to (Z 2 ) F × (U (1)/Z 2 ).We will refer to the U (1)/Z 2 symmetry as the 'baryon number' symmetry due to its analogy to the baryon number symmetry in QCD.
One may interpret the gauge-neutral scalar ϕ 0 as an interpolating field for chargedparticle bound states like ϕ † + ϕ † − .The value of including ϕ 0 as an explicit degree of freedom in Eq. (2.1) is that it allows us to access a wider range of regimes of the theory using semiclassical techniques.If not for the inclusion of ϕ 0 , the model would have only two semiclassically-tractable corners: one where m 2 /g 4 is large and negative and the U (1) B symmetry is spontaneously broken, and another where m 2 /g 4 is large and positive and the U (1) (0) B symmetry is not spontaneously broken.These are obviously distinct thermodynamic phases.But can there be a distinction between Higgs and confining regimes where U (1) B is spontaneously broken in both regimes?Including ϕ 0 in the story allows us to explore this question by dialing the mass squared parameter m 2 0 , or rather the dimensionless combination m 2 0 /g 4 .In particular, we can consider parameter regimes where m 2 0 /g 4 is held sufficiently negative while m 2 /g 4 is varied, taking the theory between confining and Higgs regimes while U (1) (0) B remains spontaneously broken.In Ref. [15] we argued that the physics of global vortices sharply distinguish these two U (1) B -broken regimes.We showed that in the Higgs regime unit-winding vortices carry a non-trivial Aharonov-Bohm phase of π, and argued that in the confining regime the Aharonov-Bohm phase becomes trivial.
The argument that the Aharonov-Bohm phase is ±π deep in the Higgs phase, so that Ω = −1, is very simple.The action per unit length of a global vortex running along some large curve Γ, assumed circular for simplicity, grows logarithmically with the size of the curve and is, for sufficiently large curves, dominated by the contributions of scalar kinetic terms.Provided the magnitude of ϕ ± approaches some constant v > 0 far from the vortex, as is the case when −m 2 /g 4 ≫ 1, and denoting the phases of ϕ ± by φ ± , the Aharonov-Bohm phases of vortices are determined by the charged-scalar kinetic terms Thinking of Γ as an infinite straight line, and viewing a vortex V ( Γ) as a field configuration where a unit-charge order parameter winds by 2π far from Γ, then ϕ 0 (which carries charge B ) must go like e −iθ far from the vortex core.Thanks to the cubic ϵ term in the action (2.1), static solutions of the equations of motion must have one of the phases φ ± approach θ far from the vortex, while the other phase approaches 0. And far from Γ the gauge field must approach a pure-gauge form, a θ → Φ 2πr , for some value of the enclosed flux Φ. Minimizing the long-distance energy density coming from the charged scalar kinetic terms immediately implies that the magnetic flux carried by the vortex is Φ = ±π.9Therefore unit-winding vortices carry an Aharonov-Bohm phase factor of e iΦ = Ω = −1 deep in the Higgs regime.
In Ref. [15] we also argued that the vortex magnetic flux vanishes in the confining U (1) Bbroken regime using a combination of effective field theory ideas along with physical arguments involving string-breaking effects.While we believe that these arguments are solid they are not quantitative, unlike the calculations we were able to present deep in the Higgs regime, due to the challenges of discussing confinement and string-breaking effects using continuum effective field theory-based techniques.But taking the result about the triviality of the Aharonov-Bohm phase in the confining regime at face value, we obtained a sharp distinction between the confining and Higgs regimes: in the U (1) (0) B -broken confining regime vortices carry trivial Aharonov-Bohm phases, while in the Higgs regime they carry non-trivial Aharonov-Bohm phases.
In Ref. [15] we argued that the ability to distinguish these two regimes via a sharp distinction in the physics of vortices provides an indication of the presence of a genuine phase boundary separating the Higgs and confining regimes.However, in that prior work we did not address a number of related questions.These include: 1. Can this putative order parameter be expressed as a standard vacuum correlation function?The quantity studied in Ref. [15] involved a Wilson loop expectation value defined by a path integral constrained to contain a prescribed vortex loop.Can one extract the physical observable of interest, namely the mutual statistics between electric probes and global vortices, from the correlation function of a Wilson loop W (C) and some well-defined vortex creation operator V ( Γ) localized on Γ?
2. Under what conditions will any long-distance correlation functions involving the putative vortex-insertion operator V ( Γ) be non-trivial?The issue is that global vortices on large contours Γ of characteristic size L have an action which grows as µ L log L, with µ a renormalization point, or faster than linearly with the loop size.One should worry that explicit insertions of V ( Γ) may get screened by dynamical vortices.If so, then at long distances V ( Γ) would flow to the unit operator, having trivial correlation functions with Wilson loops. 10 Assuming that the issues above are suitably clarified, can one present a fully explicit and controlled computation demonstrating the triviality of the Aharonov-Bohm phases of vortices in a confining U (1) B -broken regime? 11dressing these questions will make a much more convincing case for the existence of a sharp distinction between the Higgs and confining U (1) B -broken regimes based on the physics of vortices.Resolving the issues above is also key for enabling non-perturbative explorations of Higgs-confinement continuity in U (1) (0) B -broken phases using, e.g., numerical lattice simulations.In this paper we address all of the questions above in the context of a Euclidean lattice formulation of the continuum model (2.1), using a version of the Villain formulation for gauge and compact scalar fields.We will see that working on the lattice helps to make the questions above sufficiently concrete that the answers become very clear, and provides a natural starting point for future numerical explorations of the phase diagram using Monte Carlo techniques.As a start, our analysis leads to the reformulation (1.1) of the vortex order parameter discussed in Ref. [15].In Sections 3 and 4 we will discuss a lattice construction that allows us to explore Ω in various regimes of interest.Then in Section 5 we explain why our analysis implies that Ω must change non-analytically as one varies microscopic parameters to get from Higgs to confining regimes within the U (1) (0) B -broken phase.We also discuss whether such a change in Ω should be associated with a bulk phase boundary.

Vortex operators on the lattice
Since vortex operators play a starring role in our analysis, and the familiar classification of vortices involves the notion of winding number, we begin this section by discussing scalar field winding numbers on the lattice.Consider a continuum field theory of a real compact scalar φ ≡ φ + 2π.The winding number ν on a closed curve C of a field configuration of φ can be written as To obtain configurations of φ for which ν(C) ̸ = 0 it is crucial to use the fact that φ is compact, and thus needs to be single-valued along C only up to an integer multiple of 2π.
A common lattice UV completion of a compact scalar involves introducing a real-valued scalar field on sites s, φ s ∈ R, and replacing the continuum kinetic term ∼ d 3 x (∂ µ φ) 2 by the 'XY model' action where κ is the hopping parameter, s labels lattice sites, µ labels the three orthogonal directions on a cubic lattice with unit lattice spacing, µ is a unit vector, and (dφ) s,µ ≡ φ s+ µ −φ s denotes a lattice finite difference.Below we will generally suppresses the individual (s, µ) labels for links and refer to them by ℓ.To reduce clutter we sometimes suppress the lattice labels of fields where it is unlikely to cause confusion.Our lattice conventions are summarized in Appendix A. The naive lattice parallel of Eq. (3.1) is But this expression is identically zero because φ s ∈ R and hence is single-valued, implying that (dφ) ℓ sums to zero around any closed loop.
If one were to project, by hand, dφ onto the range (−π, π] in Eq. (3.3) one could obtain a non-vanishing result for ν(C) [29].But this is somewhat arbitrary and unnatural.For our purposes, it is much nicer to work with a different lattice discretization in which vortex configurations have a particularly clean definition.Specifically, we will use a modified form of the Villain lattice action for scalar fields [30-33].In the Villain formalism, the basic idea is to work with a real-valued field φ s ∈ R at each site, subject to a 2πZ gauge redundancy with an associated discrete gauge field n ℓ ∈ Z.The discrete gauge transformations take the form where k s ∈ Z is an arbitrary site-dependent gauge parameter.As a result, the pair (φ s , n ℓ ) describes a compact scalar field with periodicity 2π, and one can write a lattice scalar kinetic term as with the understanding that φ s and n ℓ are independent fields to be integrated or summed over, respectively.A path integral based on Eq. (3.5) can have a U (1) (0) global symmetry which acts as an overall phase rotation, e iφs → e iφs e iα , with α ∈ [0, 2π).It will be interesting to extend this scalar theory and consider the closely related model: where λ ∈ R + and N ∈ N are new parameters and v l ∈ Z is an auxiliary Lagrange multiplier field living on links l of the dual lattice.As discussed further below, the parameter λ controls the size of fluctuations in the discrete gauge field n, while summing over the Lagrange multiplier v implements a constraint forcing the discrete gauge flux to vanish modulo N on every plaquette, (dn) p = 0 mod N .With this constraint, the model has a Z N 1-form symmetry acting on vortex loop operators in addition to a U (1) (0) 0-form symmetry.
In the Villain formulation, the winding number ν(C) ∈ Z of the compact scalar about some contour C is defined as where the middle equality follows because φ s is single-valued, while the right-most equality follows from Stokes' theorem assuming that the contour C is the boundary of some surface D. Unlike the naive definition (3.3), the winding number (3.7) may take on any integer value and will typically be non-zero on generic field configurations {φ s , n ℓ }.Field configurations for which ν(C) is non-zero can be interpreted as containing ν(C) unit-winding vortices within the contour C.
From the above right-most expression for ν(C) one sees that the λ term in the action (3.6), involving the square of the field strength of the discrete gauge field n ℓ , controls the per-unit-length action cost of vortex excitations.The last term in the action (3.6), involving the Lagrange multiplier v ⋆p , serves to enforce a constraint eliminating from the path integral all configurations with winding ν(C) ̸ = 0 mod N about any (topologically trivial) contour.As a consequence of this constraint, the theory defined by Eq. (3.6) has a Z (1) N symmetry generated by the topological line operators where k ∈ Z and C is a closed curve on the lattice.The 2π/N factor in the exponent makes U k (C) topological because (dn) p = 0 mod N .One can immediately verify that U 1 (C) N is equal to the identity operator, and that these operators obey a Replacing 2πk/N by a generic real number in Eq. (3.8) would produce a non-topological line operator.We now define vortex insertion operators.The simplest vortex operator is with w ∈ Z and Γ a closed curve on the dual lattice (for our purposes Γ must be contractible, i.e., the boundary of a 2-surface in the dual lattice).Inserting V w ( Γ) into the path integral shifts the value of the constraint enforced by the Lagrange multiplier field v; a short computation shows that This justifies the interpretation of Eq. (3.9) as a vortex insertion operator, and implies that V w ( Γ) carries charge w mod N under the N symmetry, so that As reviewed in Appendix B, particle-vortex duality holds exactly in this Villain formulation lattice theory.This duality maps the above theory to a charge-N Abelian-Higgs model without monopoles, with scalar hopping parameter λ/N 2 and the squared gauge coupling (2π) 2 κ.The global vortex operators (3.9) (with slight modifications when λ is finite) map to Wilson lines.The absence of configurations with ν(C) ̸ = 0 mod N in the scalar field theory (3.6) corresponds to the absence of excitations with electric charge q ̸ = 0 mod N in the dual charge-N Abelian-Higgs model.
We now consider promoting the integer-valued Lagrange multiplier v l to a real-valued This action can be thought of as describing the N → ∞ limit of S N (3.6).This model now has a continuous U (1) (1) symmetry generated by topological line operators with α ∈ R.This symmetry acts on vortex operators with winding number w ∈ Z,, in the same manner as (3.11), namely The N → ∞ theory (3.12) is dual to a pure U (1) gauge theory without monopoles, and has a U (1) (1) 1-form symmetry associated with conservation of magnetic flux in addition to a U (1) (0) symmetry.These two symmetries have a mixed 't Hooft anomaly [19, 26], as discussed in Appendix C. The generalized Coleman-Mermin-Wagner theorem for 1-form symmetries [26, 34] says that U (1) (1) symmetries cannot be spontaneously broken in 2+1 dimensions, and thus the 't Hooft anomaly is matched by the spontaneous breaking of the U (1) (0) shift symmetry of (3.12) for all κ [26].
As an aside, it is instructive to consider the special case of of the theory (3.6) when N = 1.Then there is no imposed constraint on (dn) p and field configurations with all winding numbers contribute to the path integral.Our putative vortex insertion operator V w ( Γ) defined in (3.9) becomes the trivial identity operator.Nevertheless, one may attempt to define a non-trivial vortex operator in a different manner by modifying the (dn) 2 p term in the action, so as to preferentially bias the action toward configurations having non-vanishing discrete gauge flux w on plaquettes dual to links in Γ. 12 In fact, this is precisely the particle-vortex dual of a charge-w electric Wilson loop in the charge-1 Abelian-Higgs model, see Appendix B. The above operator creates a per-unit-length incentive for the system to create vorticity around the curve Γ.Although this change (3.16) is non-trivial, viewed as an operator insertion it flows to the identity operator at long distances.To see how this works, note that the leading cost in action of a dynamical vortex of size L grows like L log L. When N = 1, for large enough L (with fixed λ and κ) the L log L cost of a vortex outweighs the perimeter-law incentive for vortex creation produced by the change (3.16).Consequently, for sufficiently large L, the bias in the action produced by the vortex creation operator (3.16) will be insufficient to actually create a vortex.Equivalently but more descriptively, the system will produce a dynamical anti-vortex that screens the vortex we inserted by hand.This results in a vanishing winding number as measured on large contours C which link with Γ.And hence, when N = 1 and there is no 1-form symmetry, all vortex operators have trivial correlation functions in the long-distance limit.
The lesson we take from this discussion is that if we wish to formulate an order parameter which involves long-distance correlation functions of vortex line operators, we should focus on doing so in theories formulated to have a 1-form symmetry which protect vortices from being completely screened, as is the case in our theory (3.6) with N > 1.And the simplest choice is to send N → ∞, producing the theory (3.12) in which no vortex screening whatsoever can take place.
Therefore, in the following sections we will ensure that the more complicated theories we study have a U (1) (1) 1-form symmetry, with one sector looking just like (3.12).As we already mentioned, continuous U (1) (1) symmetries cannot be spontaneously broken in 2+1D, so this U (1) (1) will be a spectator in the dynamics we discuss below, but its presence will allow us to have the cleanest possible definition of the observables we are interested in.This addresses point 2 raised in Sec. 2.

Coupled scalar-Higgs theory
Keeping in mind the preceding discussion, we now construct a lattice field theory analog of the continuum theory (2.1).We write our lattice action as where S QED is a Villain lattice action for two-flavor scalar QED, S neutral is a Villain lattice action for an electrically-neutral scalar field, and S mixing couples the neutral scalar and the charged scalars in such a way that there is a single continuous global symmetry U (1) B .For an appropriate choice of microscopic parameters below there will also be also (Z 2 ) F chargedscalar-exchange symmetry.We assume that the full action S also has a standard coordinate reflection symmetry, which forbids a Chern-Simons term for the gauge field a, as well as a charge conjugation symmetry which will act by flipping the sign of all of the dynamical fields.Finally, there will be a U (1) (1) symmetry that acts on vortex line operators, so that vortex correlation functions with Wilson loops can be non-trivial in the long-distance limit.
The QED action S QED takes the form The pair of fields a ℓ ∈ R and m p ∈ Z describe a compact U (1) gauge field with gauge coupling e 2 ≡ 1/β.The magnetic flux through any closed 2-surface M 2 is quantized, Similarly, the fields (φ ± ) s ∈ R and (n ± ) ℓ ∈ Z describe a pair of charge ±1 compact scalar fields having hopping parameters κ ± ∈ R + .Setting κ + = κ − endows the theory with an additional (Z 2 ) F symmetry (analogous to the flavor flip symmetry discussed in section 2) that interchanges the charged scalars and negates the gauge field, The gauge redundancies associated with S QED are where α s ∈ R, r ℓ ∈ Z and (k ± ) s ∈ Z are arbitrary gauge parameters.
There is no magnetic 0-form symmetry in S QED .Magnetic monopole-instanton configurations are associated with configurations with non-vanishing discrete flux through cubes, (dm) c ̸ = 0.As we sum over all values of m p , monopole-instanton configurations have finite action.(One could control the action of monopole configurations by dialing the coefficient of an additional interaction term in the action of the form c (dm) 2 c .To keep things simple, we do not add such a term.) Our action for the neutral scalar field is where the lattice variables (φ 0 ) s ∈ R, (n 0 ) ℓ ∈ Z, and θ l ∈ R have the gauge redundancies: where (k 0 ) s , h l ∈ Z and σ s ∈ R.This model has a U (1) (1) symmetry generated by topological line operators of the form (3.13) (with n → n 0 ).Once again, this symmetry acts on winding w vortex operators, in the manner shown in Eq. (3.15).To simplify notation, subsequently we will often write just V ( Γ) when discussing minimal winding vortices with w = 1.In the rest of the paper we will analyze the correlation functions of these vortex operators with unit-charge Wilson loops given by Finally, the S mixing term in the action couples the neutral scalar φ 0 to the charged scalars φ + and φ − , via where ϵ > 0 is a real parameter. 13The newly introduced lattice variable t s ∈ Z is a discrete dynamical field whose role is to maintain the 2π periodicity of the compact scalar fields.Accordingly, under the gauge transformations (3.4) and (3.21).This form of S mixing is chosen with two goals in mind.First, just like the ϵ term in the continuum model (2.1), S mixing ensures that there is only one continuous global symmetry, which acts as where the gauge-inequivalent choices of α ∈ [0, π) parameterize U (1) . As a result, excitations created by e iφ 0 will mix with bound states of e iφ + and e iφ − excitations.The second goal ensured by this particular form of S mixing is that this mixing takes place through a quadratic term, which simplifies our later analysis.As discussed above in Sec.3.1, the mixed 't Hooft anomaly between the U (1) (1) and U (1) B symmetry is necessarily broken throughout the parameter domain where κ 0 , κ ± , β, and ϵ are all non-zero and positive.

Line operator correlation functions
In this section we compute the correlation functions of the line operators we use to examine the phase structure of our model.We first discuss symmetry constraints on relevant correlators in Sec. 13 Setting ϵ to zero would cause the theory to acquire an unwanted additional continuous 0-form global symmetry associated with independent shifts of φ0 and φ+ + φ−.Our interest lies in the theory with ϵ > 0.

Symmetry implications
To begin, it will be helpful to consider the constraints on correlators arising from Hermiticity as well as the charge conjugation and (Z 2 ) F symmetries.Because of the presence of the imaginary i θ ⋆p (dn 0 ) p term in the action (3.29), the usual Hermiticity statement that ⟨O⟩ * = ⟨O * ⟩ is only valid for observables O which are independent of the Lagrange multiplier θ l.More generally, ⟨O(θ)⟩ * = ⟨O(−θ) * ⟩.This implies that vortex loop expectations are real, and that Note that ⟨V w ( Γ)⟩ is also necessarily positive since, after integrating over θ ⋆p , the resulting measure is real and positive.Unbroken charge conjugation symmetry 14 implies that vortex expectations are independent of the direction of winding and that Wilson loop expectations are real, The (Z 2 ) F flavor-flip symmetry also converts W (C) to W (C) * while leaving V w ( Γ) unchanged.Hence this symmetry, if not spontaneously broken, also shows that ⟨W (C)⟩ is real and, when combined with (4.2), implies that Consequently, the phase of the w = 1 correlator, which defines the Aharonov-Bohm flux of minimal winding vortices, can only be ±1, showing that the magnetic flux carried by vortices must be either π mod 2π, or 0 mod 2π (provided the (Z 2 ) F symmetry is unbroken).
It will be easy to see that the (Z 2 ) F symmetry of our theory is not spontaneously broken in any of the semi-classically tractable regions of parameter space, so we will find that Ω = ±1 in our analysis below of controllable parameter regions.If (Z 2 ) F were to break spontaneously in some strongly-coupled interior region, as sketched in Fig. 1, this region would necessarily be accompanied by a thermodynamic phase transition, and so would not cause any issue with our goals. 14Because the U (1) B symmetry is spontaneously broken, the charge conjugation transformation which leaves a particular choice of spontaneously broken vacuum invariant is the naive charge conjugation transformation (which flips the sign of all our lattice fields) conjugated by a U (1) phase rotation dependent on the particular choice of vacuum.This extra conjugation is not relevant for vortex or Wilson loop operators.
In the remainder of this section we will focus our attention on the physics of our theory in the limit of κ 0 ≫ 1 while considering various scalings for κ ± and β.Holding κ 0 ≫ 1 simplifies our calculations (and also allows us to take a continuum limit if we scale κ ± , β appropriately), but due to the fact that U (1) B is spontaneously broken for all choices of parameters in our model, decreasing κ 0 to O(1) values is not expected to lead to any significant changes.

Worldvolume representation
It will be useful to introduce dual representations of our model, which we will refer to as "worldvolume representations," in which the gauge field is "integrated out" and replaced by a sum over appropriate worldsheets and (some of) the matter fields are similarly replaced by sums over worldlines.Such representations in Abelian lattice gauge theories have a long history; see e.g., Ref. [35] for a review.Dualizing the gauge field results in a representation where S neutral and S mixing have the same form as in the preceding section, while the gauge theory action S QED is replaced by We relegate the explicit derivation of Eq.
The theory defined by the action (4.7) is exactly dual to our prior theory (3.29) for any choice of parameters, but it is especially useful in the lattice strong-coupling limit, β ≪ 1, where large worldsheets are highly suppressed.So long as β ≪ 1, this representation provides a useful starting point for an analysis of the physics in both regimes κ ± ≪ 1 and κ ± ≫ 1.

U (1)
(0) B -breaking confining regime Consider the parameter regime κ ± ≪ 1 and β ≪ 1. Taking κ ± ≪ 1 means that the φ ± particles are heavy in lattice units, while β ≪ 1 means the lattice gauge dynamics is strongly coupled (i.e., far from continuum perturbative behavior).In this regime, charged test particles are confined in the same heuristic sense as probe quarks are confined in QCD.We emphasize that taking κ ± , β ≪ 1 means that in this section we are examining physics far from the continuum limit.
Given that κ ± ≪ 1 implies that κ c is also tiny, it will be useful to work with a further dual representation of the model in which the composite fields φ c , n c are also dualized in terms of neutral worldlines.This will produce a representation in which, in addition to the electric flux worldsheet Σ one also sums over a collection Ξ of worldlines of the neutral composite scalar φ c .For details, see Eq. (D.11).
It will also be helpful for our following discussion to incorporate in the effective action the effect of inserting a Wilson loop W (C) and a vortex loop V ( Γ) into the original path integral.The discussion in Appendix D shows that inserting a contractible charge-q Wilson loop W q (C) into the path integral corresponds to a change of the dual action where D is some surface which spans the loop C, so ∂D = C.The resulting worldvolume action is where we have also included a minimal winding vortex loop insertion V ( Γ).In this representation, one sums over surfaces Σ and curves Ξ, where Σ is a collection of worldsheets of electric flux whose boundaries ∂Σ represent charged particle worldlines, and Ξ is a collection of worldlines of the electrically-neutral composite particles built out of φ + +φ − .The last term in the effective action indicates that the endpoints of composite worldlines ∂Ξ are charged under U (1) B and must be dressed by insertions of e iφ 0 . 15Given the representation (4.10), it is straightforward to deduce the behavior of the expectations ⟨W (C)⟩, ⟨V ( Γ)⟩ and ⟨W (C)V ( Γ)⟩.Let us start with ⟨W (C)⟩ and, for simplicity, consider the (Z 2 ) F symmetric case κ ± = 2κ c .Since worldsheet boundaries ∂Σ are highly suppressed, one contribution to ⟨W (C)⟩ will arise from the term where Σ is closed and Σ − D is a minimal spanning surface of the contour C, and Ξ is empty.This contribution gives 15 Provided ϵ ≪ |s − s ′ |/(2κc), the two-point function of the composite field is given by ⟨e i(φc)s e −i(φc) s ′ ⟩ = e − 1 ϵ ⟨e i(φ 0 )s e −i(φ 0 ) s ′ ⟩ to leading order in ϵ and κc.This indicates that even in the confining regime the heavy composite field 'tracks' the neutral field due to the ϵ term.This does not, however, imply that the integervalued Villain fields nc and n0 are tightly correlated.
where A[C] is the minimal spanning area of C.This confining area-law behavior will be the dominant contribution for sufficiently small loops C (or arbitrarily large but fixed loops in the limit κ ± → 0), where string-breaking contributions from the matter fields are negligible due to the small values of κ ± ≪ 1, reflecting the fact the electrically-charged fields are very heavy.
One may instead consider a very large contour C when κ ± is small but fixed.For sufficiently large Wilson loops it becomes advantageous to allow Σ to have boundaries.Indeed for asymptotically large contours it is advantageous to minimize the β term in the action by setting Σ = D, so that ∂Σ = C. Therefore in this limit the charged matter worldlines screen the Wilson loop.The effective action evaluated on this leading large-loop screening configuration takes the form Taking Σ to coincide with D means that the would-be confining string is completely broken, and one expects perimeter-law behavior for ⟨W (C)⟩.To see this, and determine the value of the screening mass m which will multiply the perimeter L(C) of the Wilson loop contour C, requires understanding which composite worldlines Ξ will be dominant.To do so, we first consider contributions to the path integral where the composite worldlines Ξ are closed.Then there are two potential dominant contributions: one where Ξ = 0 and another where Ξ = C.For both configurations ∂Ξ = 0 so the last terms in (4.12) do not contribute.These two configurations give identical perimeter-law contributions, so that with an leading order screening mass m = (4κ c ) −1 and the ellipsis (• • • ) denoting further contributions from other worldlines Ξ and worldsheets Σ.These further contributions fall into two basic classes: 1.There are subleading contributions involving fluctuations of Σ surfaces and closed Ξ worldlines, which lead to corrections to the screening mass m and the pre-exponential factor of (4.13) suppressed by factors of e −1/2β and e −1/2κc , respectively.
2. There are corrections from open Ξ worldlines, whose endpoints are sources for the neutral scalar φ 0 .The leading contributions of this type are single line segments stretched along C.These contributions carry no additional κ c -suppression, but each such line segment comes with a suppression factor of e −1/ϵ .
The sum of such corrections from a single open worldline segment connecting sites s, s ′ ∈ C gives The double sum in (4.14) has short-distance contributions, when s and s ′ are close together relative to the loop perimeter L[C], and long-distance contributions when the separation of s and s ′ are of order of the loop size.The short distance contributions will be proportional to the perimeter L[C] and will exponentiate (when worldlines Ξ comprising multiple line segments along C are included), leading to small O(e −1/ϵ ) corrections to the screening mass, For large loops, the long-distance contribution of the above double sum will generate corrections proportional to L[C] 2 , since the correlator ⟨e i(φ 0 )s e −i(φ 0 ) s ′ ⟩ approaches a constant, equal to the absolute square of the (spontaneously broken) vacuum expectation value v ≡ ⟨e iφ 0 ⟩.Consequently, This correction cannot naively exponentiate (as doing so would change perimeter law behavior into unphysical behavior growing exponentially with L[C] 2 ).Rather, it reflects the presence of two differing states with slightly different values of the screening mass, with This shows that the long-distance corrections also simply shift the screening mass by a small O(e −1/2ϵ ) amount, with one of the two terms dominating as L[C] → ∞.
We now consider the vortex correlator ⟨W (C)V ( Γ)⟩.The evaluation of this correlator differs from the above treatment of ⟨W (C)⟩ based on (4.14)only through the change that the expectation value of e iφ 0 will now have a spatially varying phase.This will merely decrease the size of the (already small) long-distance contribution to the double sum in (4.14), leading to a completely analogous result for the vortex-Wilson loop correlator, ). Therefore in the confining regime under consideration, showing that the magnetic flux carried by vortices is 0 mod 2π.

U (1)
(0) B -breaking Higgs regime Now let us suppose that κ ± ≫ 1 as well as κ 0 ≫ 1, putting the theory into a regime where both φ 0 and φ ± are all condensed.But since φ ± carries gauge charge, this more properly means a Higgs regime. 16We hasten to add that at this stage this is only a heuristic statement as, a-priori, it is not clear how to sharply define a Higgs phase.

Strong gauge coupling: β ≪ 1
We will analyze the corners β ≪ 1 and β ≫ 1 separately, and first consider the strongcoupling regime β ≪ 1.We will see that this domain is smoothly connected to the continuum "deep Higgs regime" where κ 0 , κ ± , and β are all large and, e.g., the gauge field develops a parametrically large mass.
In the strong-coupling regime, β ≪ 1, working with the worldvolume action (4.7) for the gauge field but leaving the matter fields in their original form is most convenient.The complete effective lattice action including insertions of W (C) and V ( Γ) takes the form The first two terms in this action are minimized when Σ = D, so that ∂Σ = C, with resulting perimeter-law behavior for the Wilson loop, ⟨W (C)⟩ ∼ e −(κc/2κ + κ − )L(C) .Large κ ± combined with strong coupling, β ≪ 1, implies that screening by the charged fields is highly efficient.The φ 0 , n 0 terms on the second line of (4.21), at large κ 0 , ensure (after integration over θ ⋆p ) that n 0 is non-zero only on a dual sheet ending on the vortex worldline Γ, and that φ 0 jumps by 2π upon crossing this sheet.Minimizing these terms leads to φ 0 having the expected vortex profile with |dφ 0 − 2πn 0 | falling with inverse distance away from the vortex worldline.For a large but finite contractible vortex loop with perimeter L( Γ), the resulting cost in action scales as L( Γ) ln L( Γ), so that ⟨V ( Γ)⟩ ∼ e −κ 0 C L( Γ) ln L( Γ) , with C an O(1) constant dependent on the shape of the vortex loop Γ. 17 Next, the φ c -dependent terms in the third line of above action will be minimized when dφ c ≈ 2πn c and simultaneously φ c ≈ φ 0 mod 2π.For a large vortex loop Γ, any configuration in which the winding of φ 0 on paths linking Γ is not matched by corresponding winding in φ c (mod 2π) will be highly suppressed, with a cost in action growing at least as fast as the area of a surface spanning Γ.Consequently, the action is minimized when n c = n 0 and φ c and φ 0 have identical profiles around the vortex, so that these terms in the action are also of order L( Γ) ln L( Γ).
The last term of the action (4.21) is imaginary and determines the phase of the vortex-Wilson loop correlator.The sum in this term, for the minimal action configuration, is just the linking number of the vortex and Wilson loops, Therefore, in this strong coupling Higgs regime we find that showing that vortices and Wilson loops have non-trivial braiding phases with each other.
In this gapless Higgs regime, in the absence of (Z 2 ) F symmetry (i.e., when κ + ̸ = κ − ) the Aharonov-Bohm phase of a vortex is not quantized and depends on coupling constants of the theory, in contrast to the more familiar examples of analogous braiding phases in conventional topologically-ordered systems with a mass gap.But in the (Z 2 ) F symmetric case where κ + = κ − and κ c /κ − = 1/2, the phase of ⟨W (C)V ( Γ)⟩ does become quantized and equal to ±1, as required by the earlier symmetry analysis, so that Weak gauge coupling: β ≫ 1 We now consider the regime where κ 0 , κ ± , and β are all large compared to 1.This is a semiclassical regime and is connected to the continuum limit of our lattice theory.One could analyze the physics in this regime by directly minimizing the lattice action (3.29), with or without a vortex operator insertion.But since we are only interested in long-distance properties and not details of, e.g., the vortex core structure, it is far more convenient to use a continuum description.To this end, let us introduce "physical" parameters by defining κ 0 = v 2 0 a lat , κ ± = v 2 ± a lat , β = 1/(e 2 a lat ), and ϵ = ϵ a 3 lat .The continuum description involves a U(1) gauge field a, charge ±1 2π-periodic Stueckelberg scalars φ ± , and a 2π-periodic Nambu-Goldstone boson field φ 0 with the action where • • • denotes higher harmonics in the Fourier expansion of the original ϵ-interaction.
In this description, a unit-winding vortex operator V ( Γ) can be defined as a defect living on a closed curve Γ in spacetime around which φ has monodromy 2π.Such a defect has a divergent (i.e., UV-cutoff sensitive) core action, so the vortex operator may be regarded as a non-dynamical probe.Consider a vortex loop for a curve Γ whose radius of curvature is larger than any microscopic scale of the theory, so that in a large neighborhood around some point on Γ (large compared to microscopic scales but small compared to the overall size of Γ) the segment of Γ passing through this neighborhood is essentially straight.Analyzing the behavior of the fields in (4.25) in this neighborhood is then a simple exercise, identical to the treatment in Ref. [15].Taking cylindrical coordinates (r, θ, z) centered on Γ, we can set φ 0 = θ to focus on the case of a minimal vortex of winding number 1. Minimizing the ϵ term contribution to the action then requires that φ + have identical angular dependence while φ − is constant, or vice-versa.Outside the vortex core, the gauge field a must rapidly approach a flat connection, so by cylindrical symmetry a θ ∼ Φ/(2πr) for some total magnetic flux Φ. Minimizing the charged scalar kinetic terms then determines the value of this flux.One finds encircling the vortex acquires an Aharonov-Bohm phase which simply measures this magnetic flux, so to leading order in this weak-coupling regime, with A Wilson loop which does not link the vortex, and is always well outside the vortex core, acquires no Aharonov-Bohm phase.This demonstrates that deep in the κ 0 , κ ± ≫ 1 Higgs regime, Wilson loops and vortex loops have the same linking-dependent braiding phases for weak gauge coupling, β ≫ 1, as they do at strong coupling, β ≪ 1.In fact, there is a simple argument that implies that these two regimes should be smoothly connected.Following Ref. [11], consider taking κ ± → ∞.Then the charged scalar kinetic terms in the lattice action (3.29) become constraints requiring that This implies that (da) p = −2π(dn + ) p = 2π(dn − ) p so that the part of the lattice action involving the gauge field reduces to Now one can perform the field redefinition m p → m p + (dn − ) p , integrate over φ ± (which contributes a β-independent overall factor in the path integral) and arrive at The trivial sum over m p configurations can then be performed, yielding an overall factor in the path integral which is a smooth analytic function of β.Therefore, there is no phase transition as a function of β in the strict κ ± → ∞ limit.And consequently, any putative non-analyticity must lie in the interior of the phase diagram at some finite values of κ ± , with smooth continuity between the weak and strong coupling regimes assured above these finite values.It thus makes sense to regard the entire κ 0 , κ ± ≫ 1 regime, for any value of β, as a single U (1) B -broken Higgs phase.

Phase diagram
We now discuss the implications of our results for the phase diagram of our model.As noted earlier, this theory has a spontaneously broken U (1) B symmetry throughout our parameter region of interest where κ 0 , κ ± , β, and ϵ are all positive.To simplify the discussion we will focus on the parameter slice κ + = κ − , where the theory also enjoys a (Z 2 ) F symmetry.Then, as long as the (Z 2 ) F symmetry is not spontaneously broken, our Aharonov-Bohm phase Ω defined via a correlation function of line operators is quantized and must equal either +1 or −1.This was verified explicitly in the tractable periphery of the phase diagram, including both confining and Higgs regimes.We found that Ω = +1 deep in a calculable corner of the confining regime, but Ω = −1 deep in the calculable corners of the Higgs regime.Since the phase of Ω is quantized, this means that it must change abruptly (and non-analytically) along some curve in the β − κ c plane.This, of course, suggests that Ω is a natural gauge invariant observable that can provide a sharp distinction between Higgs and confining regimes, as illustrated in Fig. 1.
The fundamental question is whether the red "transition" line in Fig. 1 necessarily denotes a genuine bulk phase transition.If it does not, then the question is why not -how might non-analyticity in Ω be understood if this behavior is not associated with non-analyticity in ground state properties, i.e., a thermodynamic phase transition?
One might wonder if Ω could cease to be well-defined everywhere along the red curve in Fig. 1 due to a divergence in the vortex core size. 18However, a divergent vortex core size (or vanishing Meissner mass) would itself signal a bulk phase transition due to the appearance of an additional gapless mode in the bulk.It is important to distinguish this from the distinct phenomenon known to occur on defects with d > 1 worldvolumes where there can be a second-order phase transition on the defect, so that new gapless excitations appear on the defect worldvolume while the bulk remains gapped, see e.g., Refs.[36-38].A diverging vortex core size would reflect a divergence in the correlation length of some mode in a spatial volume whose transverse size from the defect is itself diverging.That is fundamentally different and signals the appearance of a new bulk gapless mode, which would in turn imply a bulk phase transition due to the usual non-analyticities associated with new modes going gapless on subdomains of the phase diagram.So putting aside this hypothetical possibility of a diverging vortex core size, another possibility is that the vortex phase might be non-analytic, without a bulk phase transition, if at some point it ceases to be well-defined due to line operator correlator ⟨W (C)V ( Γ)⟩ passing through zero.This could happen if there are contributions to the correlator from both vortices with flux π (mod 2π) and vortices with flux zero (mod 2π), with the magnitudes of these two contributions crossing on some curve in parameter space, thus producing an exact cancellation in the ⟨W (C)V ( Γ)⟩ correlator. 19In other words, a level-crossing phenomena between flux-0 and flux-π vortices could cause non-analyticity in Ω.Is this plausible?We cannot rule this scenario out using the tools at our disposal.Our calculations for the vortex states are reliable deep in the confining regime and the Higgs regime.Within these calculable regimes, a single type of vortex accounts for the behavior of the vortex expectation value ⟨V ( Γ)⟩ and associated ⟨W (C)V ( Γ)⟩ correlator.We see no evidence of any locally metastable vortex states.However, one can imagine that such metastable states appear only in intermediate regions of parameter space where none of our semiclassical calculations are reliable.Then these putative coexisting vortex states may level-cross, causing the ⟨V ( Γ)W (C)⟩ correlator to pass through zero.If this is the case, then in the parameter space around the level-crossing line the higher energy vortex state must be unstable to decay to the minimum energy vortex.In our previous work [15], we noted that if the higher-energy "wrong flux" vortex were to decay to the lower-energy vortex via some sort of instanton process on the vortex world sheet -i.e., via a process which is well localized on the vortex worldsheet -this would imply the existence of a vortex junction carrying fractional magnetic flux in violation of Dirac flux quantization.In other words, such a world-sheet localized decay process is impossible.However, just as a metal transitioning into a superconducting state can simply expels a portion of an applied magnetic field whose flux is a non-integer multiple of the magnetic flux quantum, the putative higher-energy vortex in the above levelcrossing scenario can shed (or absorb) flux and decay into the lower energy vortex state by producing an outward moving shell of radiation which carries the "unwanted" flux.Within the semiclassically tractable Higgs regime, one may explicitly verify that this decay process of is energetically feasible.
Where does this leave us?On the upper edge of the schematic phase diagram of Fig. 1, the limit β → ∞ effectively turns off the gauge field and theory acquires an extra global U (1) 0-form symmetry and should lie in the same universality class as the 3D XY-model, with a continuous phase as a function of κ ± as in the classic analysis of Ref. [8].Moving inward from this XY transition should be a line of first order transitions, as may be seen in a reliable weak-coupling perturbative analysis of 3D gauge-Higgs systems, see e.g.Ref. [15].In the asymptotically large-β, large-κ ± regime, it is clear that this bulk transition line coincides with the line at which Ω changes sign.
One viable possibility is that this transition line extends all the way to β = 0 (perhaps encountering some intermediate spontaneously broken (Z 2 ) F symmetry zone along the way, as indicated in Fig. 1).Alternatively, this transition line might terminate in a critical point beyond which a level-crossing line emerges across which Ω changes sign.Should this scenario take place, it would be quite interesting in its own right since it would involve the simultaneous existence of two types of vortices but only in a deeply quantum regime.Numerical simulations should be able to reveal whether or not vortices carrying different fluxes can meaningfully coexist in our model, and should be able to map out the phase diagram and the extent to which thermodynamic phase transitions coincide with changes in Ω.
To prove the existence of a phase boundary, and rule out the level-crossing scenario, perhaps one could show that the Higgs regime is some sort of gapless symmetry protected topological (SPT) or symmetry enriched topological (SET) phase, characterized by some non-trivial quantized topological term (involving either background or dynamical fields) in the long-distance effective action. 20Unfortunately gapless topological phases are far less well understood than gapped topological phases of matter, and we have not succeeded in finding such an explanation. 21inally, as noted earlier, one could simply study our lattice model (3.29) numerically.This model is well-suited for exploration using Monte Carlo simulations.It should be quite feasible to perform simulations scanning the physics in the β-κ c plane and testing whether changes in Ω are associated with bulk phase transitions.If a phase boundary exists, it would persist all the way down to β = 0, which is a natural starting point for numerical studies.In this infinite gauge coupling limit, integrating out the gauge field binds the charged fields φ ± 20 For instance, to look for possible SPT phases one could imagine turning on background fields for the (Z2)F symmetry, which is not involved in any 't Hooft anomalies (thereby ensuring that possible SPT terms could not be absorbed by field redefinitions, see e.g.Sec.2.5 in Ref. [27] for a summary of these issues).However, the partition function in the presence of (Z2)F background fields is positive for all values of the couplings, ruling out a non-trivial SPT response.One could also turn on backgrounds for the unbroken U (1) (1) and charge conjugation symmetries, but anomaly considerations make this analysis more subtle. 21For some related attempts in this direction, see Refs.[39-41], which discuss whether topological order can appear in Higgs phases with fundamental representation matter fields.Conventional topological order can be interpreted in terms of spontaneous breaking of a higher-form symmetry.Hirono and Tanizaki identified a putative emergent higher-form symmetry which is not spontaneously broken, and interpreted this result as an argument in favor of quark-hadron continuity [39].However, as discussed in Ref. [18] there are reasons to be skeptical that the "emergent symmetry" contemplated in Ref. [39] is well-defined.Moreover, we do not agree with the starting premise of Ref. [39] that the only way to argue for a phase transition in the context of our results here and in Refs.[15, 16] is to relate the change in vortex phase to the spontaneous breaking of a higher form symmetry, which would gives rise to an essentially standard sort of topological order.
into a single composite φ c = φ + + φ − described by a simple effective action, (5.2) In this limit, the Wilson loop reduces to (5.3b) and the order parameter Ω probes whether inserting a vortex for φ 0 induces a vortex for φ c .We hope to perform such numerical studies in the future.

Conclusions
In this paper we have sharpened the proposal in our earlier papers Refs.[15, 16] that, within "superfluid" (or spontaneously broken U (1)) phases), the Aharonov-Bohm phase Ω of charged particles encircling a minimal vortex can be used to distinguish the Higgs and confinement regimes of gauge theories, despite the absence of any distinguishing local order parameters.We expressed Ω in terms of a standard correlation function of two line operators, where the minimal distance between contours C and Γ is large compared to the vortex core size.A key virtue of our lattice construction is that it allowed us to compute Ω in the confining regime with string-breaking effects fully taken into account.This is something which could not be done quantitatively in previous continuum approaches [15, 16].We took advantage of lattice dualities and strong coupling expansions to compute Ω explicitly in Higgs and confining regimes and found that Ω = +1 , confining regime; −1 , Higgs regime .
We discussed whether this non-analytic change in Ω might result from a level-crossing phenomena without signaling a genuine thermodynamic phase transition separating Higgs and confining regimes.However, at present we can neither demonstrate nor exclude such a levelcrossing scenario as the origin of non-analyticity in Ω. 22 A numerical or analytic demonstration of the coexistence of vortices with distinct fluxes would thus be of independent interest.
In any case, the natural next step from the analysis in this paper is perform numerical simulations of the model we have defined, studying its phase diagram and vortex correlation functions.
A major caveat for phenomenological applications is that our definition of Ω (6.1) is only well-defined in theories, such as the model examined in this paper, possessing a 1-form symmetry that acts on vortex operators V ( Γ).The benefit of using expression (6.1) to define Ω is that it is completely concrete, is not tied to any semiclassical approximations, and should be straightforward to measure in lattice simulations.We believe that doing so would be very worthwhile.It would, of course, be desirable to understand to what extent our conclusions can also apply to superfluid phases in d-dimensional theories which do not have exact (d − 2)form symmetries acting on vortices.In particular, high density QCD lacks such a symmetry.There is a sense in which a spontaneously-broken U (1) (0) symmetry leads to an emergent U (1) (d−2) symmetry in the long distance limit [18, 19], but the precise details are subtle.In such theories, the presence of dynamical vortices can make it difficult to isolate properties of a single vortex, and one cannot define an order parameter -a vacuum correlation functionin the same manner as done in this paper.One way to avoid this issue would be to study finite rotating "superfluid" systems which will have stable unscreened vortices.This, of course, is of direct relevance to neutron star physics.More generally, it seems likely that sufficiently dilute dynamical vortex excitations will be unable to turn a bulk phase transition into a smooth crossover.In particular, in three spacetime dimensions adding dynamical vortices is equivalent to adding massive particles with unit charge under the emergent gauge field on the gauge theory side of particle-vortex duality (as discussed in Appendix B).If there is a phase boundary when these particles are infinitely massive, then it should persist when they are very heavy compared to the energy scales associated with the phase transition, which should be the case in many applications.

B Review of particle-vortex duality on the lattice
To understand how one inserts a global vortex in, for example, the XY model, it is helpful to consider particle vortex duality.The XY model is dual to the Abelian-Higgs theory of charge 1.The duality maps charge-1 Wilson lines in the Abelian-Higgs model to a global vortex on the XY side.Similarly, a particle excitation world-line in the XY model maps to a gauged vortex excitation on the Abelian-Higgs side.Since, the insertion of a Wilson line on the Abelian-Higgs side is standard, this will be our starting point.The application of duality will then reveal the appropriate definition of a global vortex insertion on the XY side.We will also implement the reverse procedure for the gauged vortex on the Abelian-Higgs side, i.e. beginning with a particle world line on the XY side and then follow the duality to reveal the insertion of a gauged Abelian-Higgs vortex.
To keep our discussion somewhat general we begin with a U (1) gauge theory coupled to a charge-N scalar field.We will use a Villain formulation for the lattice action.The action of this theory, formulated on the dual lattice, is given by Here κ and λ are positive parameters, and s, v ∈ Z, φ, b, χ ∈ R are fields to be summed or integrated.Integration of the Lagrange multiplier field φ ensures that there are no dynamical magnetic monopole-instantons.The above action has the gauge redundancy where ω ∈ R, m, h, k ∈ Z.
Performing a Poisson resummation on s, followed by the Gaussian integral over b, and dropping total derivatives, yields another description of the same model involving the scalar field φ x : where n ℓ ∈ Z is an "emergent" integer field Poisson dual to s l.This field transforms as n ℓ → n ℓ + (dk) ℓ under the gauge symmetry (B.2).In addition to the gauge redundancy making the scalar field φ compact, the exponential of the action above is also invariant under the transformation thanks to the summation-by-parts identity (A.6a) together with the fact that d 2 = 0. Finally, S φ enjoys the global symmetry and e iφ 0 is a well-defined local operator that carries unit U (1) charge.The transformation from S AH to S φ is an exact lattice realization of particle-vortex duality.
We can now introduce Wilson loops in the Abelian-Higgs side W w ( Γ) in order to understand how they map to vortex insertions on the dual theory.Tracking through the duality with an insertion of a Wilson loop, we find that inserting W w ( Γ) into the path integral is equivalent to replacing the right-most two terms in (B.3) by The shift of the second term is equivalent to inserting the vortex operator (3.9) while the first term is an additional modification analogous to (3.16) which disappears in the limit λ → ∞.
C The U (1) (0) × U (1) (1) anomaly on the lattice Consider the modified Villain action (3.12) for a compact scalar without dynamical vortices, We use the Einstein summation convention for lattice indices to reduce clutter.This model has two interesting global symmetries, the U (1) (0) shift symmetry and the U (1) (1) vortex symmetry.They can be associated to the currents The U (1) (0) symmetry of the model defined via (3.12) is generated by 23 U α ( Σ) ∼ exp iα p∈ Σ(⋆J (1) ) p , (C.3) while the U (1) (1) symmetry is generated by where Σ is a closed surface on the dual lattice while C is a closed curve on the original lattice.One may couple external gauge fields to these currents and look for anomalies.The background gauge field coupling to the U (1) (0) current consists of a pair (A ℓ , X p ) where A ℓ is R-valued and X p is Z-valued.The background 1-form gauge redundancy acts as A ℓ → A ℓ +2πK ℓ , X p → X p +(dK) p and ensures that this pair describes a compact U (1) gauge field.Similarly, the 2-form background gauge field which couples to the U (1) (1)  The full gauge invariance we would like to impose is The fact that we cannot completely remove the gauge variation using background counterterms signals a genuine 't Hooft anomaly.Finally, we note that the anomaly persists even if we restrict either U (1) (0) or U (1) (1) to discrete subgroups.For instance, we can restrict the background gauge fields for U (1) (1) to discrete background gauge fields for the Z Here B p can be regarded as an integer lift of the Z N gauge field, and should be restricted to be flat modulo N , (d B) c ∈ N Z. 24 The coupling to the Z

D Worldvolume representations
In this Appendix we derive various dual forms of the partition function used in the main text.Our starting point is Eq.(3.18), reproduced here for convenience: We begin by dualizing the gauge field in terms of a sum over worldsheets, and then treat the matter fields.

D.1 Dualizing the gauge field
Dual forms of the action can be obtained directly by applying Poisson summation to the integer-valued fields appearing above.Equivalently, we can linearize each of the terms above by introducing real-valued auxiliary variables σ p and (ρ ± ) ℓ .Up to an overall constant in the partition function, the above action is dual to We can solve this for ρ − to obtain where we have defined the 'composite' gauge invariant field φ c ≡ φ + + φ − , and n c ≡ n + + n − .We also dropped iδσ(dφ − − 2πn − ), since the first term is a total derivative and the other is a multiple of 2πi.Note that the linear combination φ + − φ − is absent from the action, and integrating over it contributes an overall infinite constant to the partition function which we ignore.Gaussian integration over ρ + gives where we dropped a total derivative and defined Finally, the path integral over σ p ∈ Z can be recast as a sum over collections of surfaces Σ by writing σ p = [Σ] p .These surfaces can have boundaries, which should be interpreted as the worldlines of charged particles.This becomes more clear when computing a Wilson loop expectation value using these dual variables.Tracking through the derivation above, a charge-q Wilson loop insertion on the contour C modifies the dual form of the action to Furthermore, when κ + = κ − , the action (with q = 0 above) is invariant mod 2πi under Σ → −Σ.

4 . 1
and then derive a useful dual representation of our model in Sec.4.2.Then we compute the line correlation operator correlation functions, first in the U (1) confining regime in Sec.4.3, and then in the U (1) (0) B -breaking Higgs regime in Sec.4.4.

( 4 . 7 )
to Appendix D.Here Σ represents a worldsheet, that is an arbitrary set of plaquettes which may be viewed as forming one or more twodimensional electric-flux surfaces (open or closed).The path integral now involves a sum over all possible collections of open and closed surfaces Σ.The worldsheet boundaries ∂Σ represent worldlines of the underlying charged particles (generated by a hopping parameter expansion in φ + − φ − ).The fields φ c ∈ R and n c ∈ Z appearing above represent the remaining electricallyneutral composite scalar field, with φ c ≡ φ + + φ − and n c ≡ n + + n − .The composite hopping parameter κ c is given by

( 1 )
N subgroup by setting B p = 2π N B p, where B p ∈ Z, and a discrete gauge redundancy B p → B p + (d β) p + N L p with β, L ∈ Z.