Magnetic operators in 2D compact scalar field theories on the lattice

In lattice compact gauge theories, we must impose the admissibility condition to have well-defined topological sectors. The admissibility condition, however, usually forbids the presence of magnetic operators, and it is not so trivial if one can study the monopole physics depending on the topological term, such as the Witten effect, on the lattice. In this paper, we address this question in the case of 2D compact scalars as it would be one of the simplest examples having analogues of the monopole and the topological term. To define the magnetic operator, we propose the ``excision method,'' which consists of excising lattice links (or bonds) in an appropriate region containing the monopole and defining the dual lattice in a particular way. The size of the excised region is $O(1)$ in lattice units so that the monopole becomes point-like in the continuum limit. We give the lattice derivation of the 't~Hooft anomalies between the electric and magnetic symmetries and also derive the higher-group-like structure related to the Witten effect.


Introduction
When studying quantum field theories (QFTs), we extract their dynamics from the correlation functions of local operators.In the path integral formulation, we can construct various operators from the fundamental variables of path integral, and we refer to them as electric operators.Interestingly, this is not the whole story, and we can introduce other kinds of local operators as the defect of the path integral especially when the target space enjoys nontrivial topology [1].By their nature, such operators are called defect operators or magnetic operators.In the perturbative regime of QFTs, magnetic operators are typically quite heavy and do not affect the dynamics at all, but they can play a significant role in the nonperturbative dynamics.Furthermore, the spectrum of electric and magnetic operators are constrained by the generalization of Dirac quantization condition [2,3], and it is now thought of as a part of the defining data of QFTs [4].
Lattice regularization provides the rigorous foundation for studying nonperturbative aspects of QFTs, and, moreover, lattice discretization of the spacetime gives a natural setup to introduce the magnetic operators.On the other hand, the naive lattice regularization spoils the notion of the continuity of fields, which is essential ingredients to define the magnetic operators in the continuum formulation.To reinstate the topological structure, we can impose the "admissibility condition" on the lattice field configurations [5].In this paper, we shall construct magnetic operators on the lattice field theories with admissibility constraints and discuss their properties.To concretely present our ideas, we mainly focus on the two-dimensional (2D) compact bosons.
The magnetic operator M(x * ) can also be defined by imposing the boundary condition so that φ(x) has the winding number 1 around x * , and this is an analogue of the 't Hooft loop in the 4D gauge theories.In Sect.2, we define the lattice counterpart of M(x * ) when the lattice configurations satisfy the admissibility condition by excising a finite region from the lattice ("excision method").This theory has the U(1) (e) × U(1) (m) symmetry that acts on these electric and magnetic operators, and we derive the mixed 't Hooft anomaly between them in the lattice formalism.
In Sect.3, we consider the 2D theory with two compact bosons φ 1 and φ 2 .This theory has the continuous θ parameter, θ (2π) 2 dφ 1 ∧ dφ 2 , with the periodicity θ ∼ θ + 2π.Like the 4D Maxwell theory, this theory does not have the instanton solution, so one may think that the θ angle does not affect the local dynamics at all.Although this observation is somewhat true for electric operators, the magnetic operators are significantly affected by the θ angle: The analogue of the Witten effect [6] occurs, that is, the magnetic operators acquire the fractional electric charge.We first show that these phenomena can be understood as the higher-group-like structure [7][8][9][10][11][12] in the language of background gauge fields using the continuum description.After that, we discuss its lattice counterpart by extending the discussions in Sect. 2.
Let us comment on other related studies, and this would clarify our motivation more.In this paper, we use the Wilson-type lattice regularization that uses the compact variables as the fundamental variables, and the topology of field space appears by imposing the admissibility constraint.For Abelian theories, there is another approach called the Villain-type formulation, which uses the R-valued n-form field coupled with Z-valued (n + 1)-form gauge field as fundamental variables to represent the U(1)-valued n-form gauge field.A modified version of the Villain formulation is highly developed in Refs.[13][14][15][16][17][18][19] recently, and it can give the lattice derivation of our observations more transparently.Still, we will stick to the Wilson-type formulation in this paper because we would like to uncover the properties of magnetic operators in an extendable way for non-Abelian theories.

Single compact scalar field
In this section, we discuss the lattice formulation of the 2D compact boson.We first give a review on its continuum description emphasizing the role of symmetries and mixed 't Hooft anomaly.After the brief review, we consider the topology of lattice bosons by imposing the admissibility condition, and we reproduce the continuum observations in the lattice description.

Review on 2D compact boson in the continuum description
Let us start from the case of a single compact real scalar field on 2D closed Riemannian manifold M 2 .Its continuum action is given by where R > 0 denotes the compact-boson radius, and φ(x) enjoys the identification This system possesses U(1) zero-form "electric" and "magnetic" global symmetries, and their Noether currents are given by respectively.In the electric picture (2.1), the conservation of the former, dj (e) = 0, is nothing but the equation of motion, and the one for the latter, dj (m) = 0, follows from the Bianchi identity.The charged object for U(1) (e) is the vertex operator e iφ(x) , and the one for is realized as the defect operator; one should remove a small neighborhood of the given point p and impose the boundary condition so that S 1 p dφ 2π = 1 for small S1 surrounding p. Notable feature of the U(1) (e) × U(1) (m) symmetry is the existence of the mixed 't Hooft anomaly.To see this, we introduce background gauge fields A (e) and A (m) that couple to the global symmetries. 1 The gauged continuum action is given by This gauged action is manifestly invariant under the electric gauge transformations, φ → φ − Λ (e) and A (e) → A (e) + dΛ (e) , where the gauge transformation parameter Λ (e) is also a 2π-periodic scalar.Under the magnetic gauge transformation, A (m) → A (m) + dΛ (m) , the gauged action transforms as Defining the partition function, we then find the mixed 't Hooft anomaly m) ]. (2.7) We note that there are no 2D local counter terms that cancel this anomaly, so this is a genuine anomaly.This anomaly can be cancelled by the anomaly inflow from the 3D topological action, with ∂M 3 = M 2 , so that Z[A (e) , A (m) ] exp(−S 3D ) is manifestly gauge invariant.

Lattice formulation with the admissibility condition
Let us take M 2 = T 2 and approximate it as the 2D square lattice Γ = (Z/LZ) 2 of size L with the periodic boundary condition.On the lattice, the basic dynamical variable is defined where n denotes the lattice sites, and one can take the lattice counterpart of the Euclidean action (2.1) as where β = R 2 /(2π) and μ denotes the unit vector in the µth direction.
Let us define the variable φ(n) itself on the lattice by taking the logarithm of e iφ(n) with the principal branch, We also define a directional difference of φ(n) in the same way as ∂φ(n, µ) ≡ 1 i ln e −iφ(n) e iφ(n+μ) , −π < ∂φ(n, µ) ≤ π.
To put it the other way around, the Wilson and Villain formulations share the similar properties up to the above difference.Therefore, most of our discussion, except the one for magnetic defects, can be applied in the same way for both formulations.
is always an integral multiple of 2π; where the sum is taken over links belonging to the loop C. Throughout this paper, we understand that the summand of the directional line sum is ∂φ(n, µ) when the path C goes through the link (n, µ) in the direction from n to n + μ, while −∂φ(n, µ) when C goes though (n, µ) in the opposite direction, i.e., this is a lattice analogue of the line integral.
Equation (2.14) holds because ∆ µ φ(n) does not contribute to the directional line sum along a closed loop; note that the field φ(n) ∈ (−π, π] is single-valued on the lattice. We would like to identify 1 2π ∂φ(n, µ) as the lattice counterpart of the magnetic symmetry generator j (m) = 1 2π dφ.However, Q mag (C) on the lattice suffers from the discontinuous change under the deformation of C in general, and we need a remedy to correctly define the topological sectors.To this end, we impose the "admissibility condition" on allowed or, equivalently, for all the links (n, µ) Let A ǫ denote the set of admissible lattice fields, and the path integral is performed only on A ǫ .Due to this restriction, Q mag (C) turns out to be topological, i.e., it does not change under any continuous deformations of C.
Let us show that Q mag (C) is topological.We first note that the directional line sum of ℓ µ (n) along the boundary of a single plaquette p is bounded as where we note that there are 4 links belonging to a 2D minimal loop.Since the most lefthand-side of this equation is a sum of integers, we obtain3 Since any deformation of the loop C can be realized by repeatedly adding or removing a single plaquette from the loop, we see that Q mag (C) defined by (2.14) is invariant under a change of C, once the admissibility is imposed.We note that Eq. (2.18) can be written as which corresponds to dj (m) = 0 in the continuum theory.
The admissibility condition decomposes the field space A ǫ into distinct topological sectors.
For example, the following configuration ) and it has the winding numbers w 1 , w 2 ∈ Z along the 1, 2 directions, respectively.This corresponds to H 1 (T 2 ; Z) ≃ Z ⊕2 ∋ (w 1 , w 2 ) in the continuum formulation, and the configurations with different (w 1 , w 2 ) cannot be continuously connected without violating the admissibility condition.

Magnetic defect operators on the lattice with admissibility
The admissibility condition allows us to define the conserved current j (m) on the lattice, and the field configurations are also topologically classified as in the case of the continuum theory.To establish the U(1) (m) symmetry, we introduce the charged object for the current j (m) in this subsection.
The admissibility (2.15) tells Q mag (C) = 0 for any contractible loops C on the lattice Γ, and thus we cannot naively introduce the magnetically charged operator.This would remind us of the fact that the magnetic operators are introduced as the defects in the continuum description.Therefore, let us pick a certain 2D region D and remove all lattice points and links contained in D (see Fig. 1).We then define the magnetic operator of charge m ∈ Z by imposing the boundary condition (2.21) We note that the size of the excised region D need to be sufficiently large for the magnetic defect to be well-defined: As the admissibility is applied to links belonging to ∂D, for a given Q mag (∂D) = m, the 1D size of the region must be at least ∼ |m|π/(2ǫ) ( m) in lattice units.As the size of D can be determined independently from the coupling constants including the lattice constant, the magnetically charged object becomes point-like in the continuum limit.This localized magnetic object can be identified with the vertex operator e im φ(x) in the continuum theory, where φ(x) is the dual scalar field identified as ν ε µν ∂ ν φ(x) ∼ D Fig. 1: Excised region D on Γ.As the lattice points and links inside D are completely eliminated, the topological charge Q mag (∂D) around D can take nonzero value in Z.
In the continuum theory, the correlation functions containing both e inφ(x) and e im φ(y) are well-defined as a single-valued function when n, m ∈ Z.One can readily confirm this result in the present lattice formulation as follows: We first note that where C x ′ x is a path connecting sites x ′ and x.Moving e iφ(x) around a magnetically charged object with the magnetic charge m ∈ Z once thus results e iφ(x) → e iφ(x) e 2πim = e iφ(x) .
(2. 23) This shows that the correlation functions of e iφ(x) are single-valued even in the presence of magnetically charged objects defined by our excision method; e iφ(x) and the magnetically charged object with m ∈ Z are mutually local.

Background gauging and 't Hooft anomaly on the lattice 2.4.1 Introduction of background gauge fields
We have shown that the lattice theory has both the electric and magnetic U(1) symmetries, and thus we can now ask if it also has the correct 't Hooft anomaly.To see this, let us consider the coupling to external gauge fields as Eq.(2.4).For this, we introduce two U(1) link variables, U (e) (n, µ), U (m) (ñ, µ). (2.24) Note that we put magnetic variables on the dual lattice whose site is defined from n ∈ Γ by (2.25) The link (ñ, µ) is connecting two sites on the dual lattice, ñ and ñ + μ.The lattice electric gauge transformation is then given by and the magnetic gauge transformation is We define the covariant difference with respect to the electric gauge symmetry by which is invariant under the electric gauge transformation.Instead of Eq. (2.15), we now impose the admissibility of the form Although the above link variables are the fundamental degrees of freedom for lattice gauge fields, we can equivalently describe lattice Abelian gauge theories in terms of gauge potentials, such as the corresponding continuum theory.Let us set (2.30) The corresponding field strengths are defined by and the covariant difference as where µ (n) ∈ Z.Under the electric or magnetic gauge transformation, we find that

.36)
To show the topological nature even on the lattice, we have imposed the admissibility condition for the lattice boson φ(n) in Eq. (2.29).We further need a lattice counterpart of the Bianchi identity dj (m) = 0 in the continuum theory, that is, a constraint corresponding to µ,ν ε µν ∆ µ ℓ ν (n) = 0 in Eq. (2.17) with the background gauge fields.To see this, we assume that the external gauge fields are admissible as 4sup n,µ,ν Then, noticing that µ,ν we have

Computation of the 't Hooft anomaly on the lattice
Now, with the above preparations, we take the following lattice action: The first line of the action corresponds to Eq. (2.4), and is manifestly invariant under the electric gauge transformation (2.26).We note that the second and third terms of the action, the magnetic couplings, have the structure depicted in Fig. 2. The second line, which is a local counter term as we will discuss later, is not invariant under the electric gauge transformation so that we see the 't Hooft anomaly as µν (ñ)Λ (e) (n + μ + ν) . (2.42) Under the magnetic gauge transformation (2.33), the action changes as In the first term in the square brackets on the right-hand side, paying attention to the terms containing Λ (m) (ñ) with a particular ñ, we see the structure µ (n)), disappears in the expression thanks to the third term in the action (2.41), this can be regarded as a lattice counterpart of the mixed anomaly (2.7) in terms of the background fields. 5  It is interesting to ask what happens if we put a magnetic object in the system (2.41) by our excision method; recall Fig. 1.For such a lattice with some region excised, it turns out that the dual lattice, on which the magnetic gauge field A (m) µ (ñ) is residing, should be defined as depicted in Fig. 4, where the dual site ñ * is defined inside the excised region.The rule is that the product of A (m) µ and Dφ is defined in the way depicted in Fig. 2; links on Γ and links on the dual lattice cross in that way.With these understandings, we thus consider where D is the region excised to represent the magnetic object.Also in this setting we assume the condition in Eq. ( 2.39) on Γ − D, and then, Eq. (2.40) follows.As Eq. (2.43), under the 5 Let us discuss some consequences of the 't Hooft anomaly in this lattice formulation.First, the violation of the magnetic gauge invariance suggests that the partition function should vanish if dA (e) = 0.In the lattice realization, there is no configuration (φ, ℓ µ ) that satisfies the admissibility condition when n F (e) 12 (n) = 0, and the path integral vanishes as expected.Inserting a magnetic operator that compensates for the monopole flux of F (e) , the unnormalized path integral can have a nontrivial value.
As the dual of this observation, the path integral should also vanish when dA m = 0.In this case, the admissible lattice configuration exists, but the integration over φ's zero mode gives the complete cancellation as we added the term N (m) 12 φ in the local counter term.As a result, the path integral vanishes, which is consistent with the anomaly requirement.
Even with the presence of the excised region as Fig. 4, we find that the argument is almost the same as above.The first term in the square brackets, however, produces the line sum of ℓ (e) µ along ∂D and, noting the structure of the dual lattice within D, we find where m ≡ (n,µ)∈∂D ℓ The last factor in Eq. (2.48) gives rise to a breaking of the magnetic gauge symmetry owing to the presence of the magnetic object.This breaking however may be cured by connecting an "open 't Hooft line" in the dual lattice to the magnetic object.In fact, by supplementing the phase factor exp where P denotes a path on the dual lattice ending at n * , in the functional integral.With this understanding, under the magnetic gauge transformation, This completes our discussion on the single scalar case.We observed that the excision method to define a magnetic object on the lattice works quite well to reproduce phenomenon expected in the continuum theory.
3 The case of two compact scalar fields In this section, we extend the previous discussion to the case with two compact scalars.As H 2 (S 1 × S 1 ; Z) ≃ Z, one can introduce the continuous θ angle, and the analogue of the Witten effect occurs.This is quite natural since we can obtain this model by putting the Maxwell theory on M 4 = M 2 × T 2 .We first discuss these properties in the continuum description and then give the lattice reformulation.

Continuum description of the θ angle, 't Hooft anomaly, and Witten effect
Having two periodic scalar fields, φ a (x) (a = 1 and 2), we can define the topological charge by which corresponds to H 2 (S 1 × S 1 ; Z).The action is then given by where G ab is a positive symmetric matrix.As a simplest choice, one may take it as 4π δ ab .We note that θ can be regarded as the 2π periodic variable, θ ∼ θ + 2π.This theory has the U(1) (e,a) × U(1) (m,a) symmetry for each φ a , and we write the background gauge fields as A (e,a) and A (m,a) with a = 1, 2. The gauged action is then given by As we have discussed in Eq. (2.4), this is manifestly invariant under the electric gauge transformations φ a → φ a − Λ (e,a) and A (e,a) → A (e,a) + dΛ (e,a) , but the magnetic gauge transformation, A (m,a) → A (m,a) + dΛ (m,a) , has an anomaly.This 't Hooft anomaly can be cancelled by regarding this theory as the boundary theory of the 3D topological action, with ∂M 3 = M 2 .As in the case of Eq. (2.8), the gauged partition function Z θ [A (e,a) , A (m,a) ] cannot respect the background gauge invariance, but Z θ [A (e,a) , A (m,a) ] exp(−S 3D ) does.
The presence of the continuous θ angle provides a richer structure to the global symmetry.To see this, we first note that the 2π periodicity of θ is explicitly broken by the introduction of the background gauge fields in Eq. (3.3): 2) .(3.5) The first term on the right-hand side is quantized as 2πiZ, and thus it does not affect the path-integral weight.The last term does not cause the serious problem as it depends only on the background gauge fields.The serious violation of the θ periodicity comes from the mixed term, A (e,a) ∧ dφ b , and we shall find its remedy by considering the higher-group-type extension of the symmetry [7][8][9][10][11][12].The key observation is that the mixed terms in Eq. (3.5) can be cancelled by the shift of the magnetic gauge fields: 2) , A (m,2) → A (m,2) + A (e,1) .( As a result, we find that ,a) , A (m,1) − A (e,2) , A (m,2) + A (e,1) ] ,a) , A (m,1) , A (m,2) ]. (3.7) The phase factor on the right-hand side is called the global inconsistency or also called the generalized anomaly of higher-group-like structure involving the (−1)-form symmetry [20][21][22][23][24][25][26][27].We note that this anomaly (3.7) is also cancelled by the 3D topological action (3.4): 2) ∧ dA (e,1) + A (e,1) ∧ dA (e,2) 2) . (3.8) In the following subsection, we are particularly interested in how the structure (3.7) is implemented on the lattice with the presence of the magnetically charged object.
Before moving on to the discussion on the lattice regularization, let us discuss the physical meaning of Eq. (3.6):The higher-group structure (3.6) detects the Witten effect [6] for the magnetic defects.As we have discussed, θ has the periodicity 2π as the consequence of the quantization Q ∈ Z.This suggests that the QFTs at θ and θ + 2π are unitary equivalent, but the unitary transformation may have nontrivial actions on the energy levels and/or operator spectrum.For example, when we compute the correlation function that contains M 1 (x) (the magnetic defect for φ 1 ), we have the relation up to the renormalization procedure.Similarly, M 2 (x) should be replaced by M 2 (x)e −iφ 1 (x) when relating θ + 2π and θ.This is nothing but the analogue of the Witten effect in the present system, and the higher group (3.6) captures this phenomenon.
3.2 Lattice formulation of the θ angle and the Witten effect

Definition of the topological charge on the lattice
Now, for the case of two scalar fields, we will find that the topological charge on the lattice possesses a better behavior if the field φ 2 (x) is put on the dual lattice instead of the original lattice Γ.Thus, corresponding to Eq. (2.12), we define As Eq. (2.13), we then have where ℓ 1,µ (n) and ℓ 2,µ (ñ) are integers.The admissibility is set as Then, we have the Bianchi identities, µ,ν Now, as a lattice counterpart of the topological charge (3.1), we adopt We note that this topological charge is given by the sum of terms represented by Fig. 2, where the solid line represents the original link and the broken line the dual link because ∂φ 2 (ñ, µ) and ∂φ 1 (n + μ, ν) are put on those links, respectively.
Let us first confirm that Q (3.14) takes integral values.By substituting Eq. (3.11) into Eq.(3.14), we have For the first and second terms on the right-hand side, we can repeat the argument in Eq. (2.44); recall Fig. 3a.By replacing Λ (m) (ñ) → φ 2 (ñ) and Dφ(n, ν) → ∆ ν φ 1 (n) or ℓ 1,ν (n), we observe that these identically vanish because of the first of the Bianchi identities (3.13).The situation is similar for the third term on the right-hand side of Eq. (3.15) by exchanging the role of the original lattice and the dual lattice; see Fig. 3b.Because of the second of the Bianchi identities (3.13), this also identically vanishes.
In this way, we find Since ℓ 2,µ (ñ) and ℓ 1,µ (n) are integers, the lattice topological charge (3.14) is manifestly an integer.We emphasize that for this, the admissibility condition which ensures the Bianchi identities are crucial.Actually, since all field configurations of φ 1 (n) and φ 2 (ñ) on the lattice are connected, it is impossible to assign an integer topological charge to configurations in a well-defined way without excluding some (non-smooth) field configurations; the admissibility does this.

Witten effect on the lattice
Since we have an integer topological charge on the lattice, we may consider a lattice action containing the θ term corresponding to Eq. (3.2): Moreover, since we can introduce a magnetically charged object by the excising method, we may study an analogue of the Witten effect in the form of Eq. (3.9) in the present lattice formulation.
Let us thus consider a magnetic object which possesses the magnetic charge m 1 with respect to the magnetic symmetry associated with φ 1 (n).This implies that we excise a region, D, in the original lattice Γ as Fig. 1.The magnetic charge is given by m = Q mag (∂D) (2.21) with φ(n) → φ 1 (n).The dual lattice, on which the field φ 2 (ñ) is residing, is defined in the previous section; see Fig. 4. The rule is similar to that in the previous section and the product of φ 1 (n) and φ 2 (ñ) is defined in the way depicted in Fig. 2; links on Γ and links on the dual lattice cross in that way.With these understandings, we consider  As we have analyzed, even with the presence of the excised region D as in Fig. 4, the same argument goes well for most of the parts to identically vanish; compare Figs.3b and 4. where we have used Eq.(2.21) (with m → m 1 and φ → φ 1 ).Regarding ñ * as the position of the magnetic object, this precisely reproduces the Witten effect in Eq. (3.9).We emphasize that our particular discretization made the electric charge of the magnetic object induced by θ → θ + 2π precisely quantized even with finite lattice spacings.

Summary and discussions
In this paper, we have studied the properties of magnetic operators on the lattice field theories with the admissibility condition.Such defect operators often play important roles to characterize the phases of QFTs, and thus it should be useful to understand their properties at the finite lattice spacing.As the simplest model, we focus on the 2D compact bosons and give the lattice derivation of the mixed 't Hooft anomaly between the electric and magnetic U(1) symmetries.When there are several compact bosons, the model admits the continuous θ angle as in the case of 4D Maxwell theory.By introducing the magnetic operators, we can observe the analogue of the Witten effect, and the lattice theory can also derive such a phenomenon.
To observe these phenomena, the most important task is to introduce the magnetic operators.When we impose the admissibility condition to reinstate the topological feature, the lattice configurations no longer accept the magnetic operators in a naive way, and thus we propose the excision method that removes a small region of the lattice and imposes the boundary condition.This method works so nicely for 2D compact bosons that we can reproduce the Witten effect in an ultra-local way at finite lattice spacings.At the formal level, we can apply this excision method for any lattice theories, including higher-dimensional non-Abelian theories, to introduce the magnetic operators.It would be an interesting future study to uncover if this method can derive nontrivial properties of magnetic operators for such theories, such as the Witten effect of 4D SU(N)/Z N gauge theories with the lattice θ angle [28,29].

F
32) with −π < F (e,m) µν ≤ π.Under the electric or magnetic gauge transformation in Eqs.(2.26) and (2.27), we have A the integer fields L (e) µ (n), L (m) µ (ñ) ∈ Z are necessary because of our definition of the gauge potentials; L (e) µ (n) and L (m) µ (ñ) are local functionals of Λ (e) (n) and Λ (m) (ñ), respectively.We rewrite the following gauge-invariant variables in terms of A (e,m) µ ; the field strength as

Fig. 4 :
Fig. 4: Dual lattice in the presence of the excised region.

µ
(n) can be regarded as the magnetic charge in view of Eqs.(2.21) and (2.14).