't Hooft surface in lattice gauge theory

We discuss the lattice formulation of the 't Hooft surface, that is, the two-dimensional surface operator of a dual variable. The 't Hooft surface describes the world sheets of topological vortices. We derive the formulas to calculate the expectation value of the 't Hooft surface in the multiple-charge lattice Abelian Higgs model and in the lattice non-Abelian Higgs model. As the first demonstration of the formula, we compute the intervortex potential in the charge-2 lattice Abelian Higgs model.

One of the well-established approaches is a higher-form symmetry [9][10][11][12]. Order parameters for the generalized global symmetries are extended objects, e.g., loop, surface, and so on. Their topological natures make it possible to classify the topological orders.
The most famous example of the non-local order parameters is a loop operator. For instance, in SU(N ) gauge theory, the Wilson loop [13] is defined by the path-ordered product of the loop integral Since the gauge field A µ couples to electrically charged particles, the Wilson loop describes the world lines of charged particles. The SU(N ) gauge theory also has magnetically charged particles, namely, magnetic monopoles. The world lines of magnetic monopoles define the 't Hooft loop [14]. Using the dual fieldÃ µ , the 't Hooft loop is given bỹ The Wilson and 't Hooft loops are the order parameters for the confinement of electric and magnetic charges, respectively. A more nontrivial example is a surface operator. In the gauge theory coupled to Higgs fields, topological vortices appear. The vortices are one-dimensional defects, so their trajectories sweep two-dimensional planes. The world sheets of the vortices define a dual surface operator, which is called the 't Hooft surface. For instance, in the BF theory, it is given by the closed-surface integral over the dual 2-form field B µν , When the theory has Z N topological order, the vortices have fractional magnetic charge q/N e (q ∈ Z). The 't Hooft surface gives the criterion for the confinement of the fractionally charged vortices. It plays an essential role in the topological order of Cooper pairs in superconductors [15][16][17][18][19] and that of diquarks in color superconductors [20][21][22][23][24][25][26]. These 't Hooft operators have crucial difficulty. The 't Hooft operators can be elegantly formulated in topological quantum field theory, that is, effective theory to reproduce topological properties of the original quantum field theory. The calculation beyond the effective theory is, however, not easy. The dual gauge fields are not fundamental fields but defects or singularities in the original theory. It might seem impossible to calculate the expectation values of the 't Hooft operators in a quantitative manner. Surprisingly, this is possible in lattice gauge theory. The formulation has been known for the 't Hooft loop in the Yang-Mills theory [27][28][29]. It was applied to several lattice simulations [30][31][32][33][34][35][36][37][38][39][40].
In this paper, we study the 't Hooft surface in lattice gauge theory. After introducing the basics of dual variables in Sec. II, we discuss how to compute the 't Hooft surface in lattice simulation. We focus on the lattice gauge Higgs models with Z N topological order: the Abelian Higgs model in Sec. III and the non-Abelian Higgs model in Sec. IV. The simulation results are shown in Sec. V. Finally, Sec. VI is devoted to summary. The Euclidean four-dimensional lattice is considered and the lattice unit is used throughout the paper.

II. DUAL LATTICE AND DUAL VARIABLE
Dual variables live on the dual lattice. The dual lattice is defined by translating the original lattice by a half of lattice spacing in all directions. The schematic figure is shown in Fig. 1. The sites of the dual lattice are in positions of the hypercube centers of the original lattice. In four dimensions, there are the one-to-one correspondences between d-dimensional objects on the original lattice and (4 − d)-dimensional objects on the dual lattice: a bond b is dual to a cube c * , a plaquette p is dual to a plaquette p * , etc. The asterisks denote the objects on the dual lattice.
In lattice gauge theory, there are two ways to introduce dual variables. One is to replace all the integral variables in the path integral by their dual variables. The path integral is completely reformed. For example, the gauge Higgs theory is described by dual plaquette variables and dual cube variables [41] (see also Refs. [42][43][44]). The other is to insert the dual variables without reforming the path integral. For example, when a π-flux exists on p * , the plaquette variable on p changes as U µν → e iπ U µν but the integral variables do not change. This is exactly what is done in the lattice formulation of the 't Hooft loop [27][28][29]. We consider such insertion of the 't Hooft surface in the following sections.

III. ABELIAN HIGGS MODEL
We first consider Abelian gauge theory. When the Higgs field has the multiple electric charge N e (N ∈ Z), the theory has Z N topological order. The magnetic charge of a vortex is fractionally quantized to be 1/N e. The theory is often referred as the charge-N lattice Abelian Higgs model [45]. The path integral is given by with the Abelian gauge field A µ and the complex scalar field φ. Let us define the link variable and the plaquette variable whereμ stands for the unit lattice vector in the µ direction. The Euclidean action is given by with the gauge part the local part of the scalar field and the hopping part The theory is invariant under the local U(1) gauge transformation and the global Z N transformation Because of the Z N symmetry, the vacuum is N -fold degenerate.
We put the 't Hooft surface of magnetic charge q/N e on a two-dimensional closed surface S * . The path integral changes as The hopping part is modified as where B(V * ) is defined by the hopping terms penetrating V * , s.t. S * = ∂V * (see Fig. 2). Therefore, the expectation value of the 't Hooft surface is given by the formula with This formula is the main result of this paper. The explicit derivation is given in Appendix A.
The above formula can be intuitively understood in Fig. 2. The red line is the three-dimensional volume V * on the dual lattice. The hopping terms penetrating V * are multiplied by the Z N element e i2πq/N . The winding number of each plaquette is given by the sum of the angles of the four hopping terms. As shown by the circle arrows in the figure, the Z N element changes the winding number by +q/N at A, and by There are two remarks on the above formula. The first one is that the inserted vortices have fractional winding numbers. They are different from the standard vortices defined by integer winding numbers. The above formula cannot realize the integer winding numbers because ∆S = 0 for q/N ∈ Z. Another formulation is necessary to insert the vortices with integer winding numbers [46]. The second one is that V * is non-unique. In four dimensions, V * can be deformed in the perpendicular direction by integral variable transformation, as long as S * is fixed. The same ambiguity exists in the 't Hooft loop [32]. On the other hand, S * is invariant under the transformation. This means that the positions of the vortices are physical.

IV. NON-ABELIAN HIGGS MODEL
Next let us consider non-Abelian gauge theory. Among the non-Abelian gauge Higgs models, the N -color and N -flavor case is of special importance. When the numbers of color and flavor are equal, the non-Abelian vortex, as well as the Abelian vortex, can exist. The minimal winding number of the non-Abelian vortex is 1/N , while the winding number of the Abelian vortex is integer.
In the N -color and N -flavor non-Abelian Higgs model, the hopping part of the lattice action is This is is almost the same as Eq.
The inserted 't Hooft surface satisfies the same properties as in the Abelian case.

V. SIMULATION
To demonstrate the above formalism, we perform the numerical simulation in the simplest case, the charge-2 lattice Abelian Higgs model. The simulation details are summarized in Appendix B. The Z 2 group has two elements: one trivial state +1 (zero magnetic charge) and one nontrivial state −1 (fractional magnetic charge 1/2e). This means that a vortex and an antivortex are equivalent. This specialty is due to the compactness of the link variable. In general, the lattice Abelian Higgs model sometimes shows different behaviors from the continuous Abelian Higgs model. The results should be interpreted as the properties of lattice superconductors, not of realistic superconductors in continuum space.
We consider the four-dimensional hypercuboid N x × N y × N z × N τ with periodic boundary conditions. As shown in Fig. 3, the 't Hooft surface is inserted on the dual cuboid X × Y × T inside the hypercuboid. We take Y = N y to simplify the analysis. The nontrivial Z 2 element −1 is multiplied to the hopping terms in the z direction. When the time extent T is large enough, the 't Hooft surface asymptotically behaves as where E is the energy of a static and straight vortex-antivortex pair. Since E is proportional to Y because of translational invariance, E/Y is a function of X. Therefore, E/Y can be interpreted the intervortex potential per length. The simulation results are shown in Fig. 4. Changing the scalar self-coupling constant, we calculated the potential in the Higgs phase and the Coulomb phase. In the Higgs phase, the potential is linear. The vortices with fractional magnetic charge are confined. Thus, only the states with zero magnetic charge will appear at low energy. This will be the common property both for lattice and continuous superconductors, apart from the functional form. (The confinement potential will be logarithmic in the continuum.) In the Coulomb phase, the potential is almost flat. As the U(1) symmetry is unbroken, the dual variables are massive, so they cannot propagate to long range. The interaction between the vortices is diminished. The linear potential is equivalent to the volume-law scaling of the 't Hooft surface, which is the criterion for the confinement of magnetic vortices. This is dual to the volume-law scaling of the Wilson surface, which is the criterion for the confinement of electric strings. In this model, however, the volume law is not exactly satisfied. If the linear potential were exactly correct, the potential energy would be very large at long distance. A dynamical vortex-antivortex pair will be created in-between the original vortex and antivortex to lower the total energy. This is called the surface breaking as the analog of the string breaking of the Wilson loop [41]. The surface breaking will happen when the potential energy reaches the mass of the dynamical vortex-antivortex pair. The potential will be flat above a critical distance. This is not seen in Fig. 4. We need the long-distance analysis in larger lattice volume.

VI. SUMMARY
In this paper, we obtained the formula for the t' Hooft surface in the lattice gauge Higgs models with the Z N topological order. The formula is simple; the Z N element is multiplied to the hopping terms inside the 't Hooft surface. We performed the lattice simulation in the charge-2 Abelian Higgs model. We found that the the vortices with fractional magnetic charge are confined in the Higgs phase, and thus they do not emerge in the physical spectrum. This is consistent with our understanding of superconducting vortices.
The advantage of our formulation is that the 't Hooft surface is calculable in the original path integral. It is also attainable in the dual path integral, in which all the variables are dualized. However, the dual approach works only when the duality between the two theories is ensured, e.g., in the London limit of type-II superconductors. Our formulation is, therefore, more useful for the quantitative analysis in the original theory. Let us derive the formula (17). The derivation is parallel to Ref. [28]. We introduce a special notation in this appendix. When a bond b connects x and x +μ, we write as φ ∂b (1) = φ(x), φ ∂b (2) = φ(x +μ), and U b = U µ (x). The hopping part (10) is rewritten as (1) .
(A1) The summation is taken over all bonds b.
The scalar field φ is uniquely decomposed into a Z N part and a U(1) / Z N part. When arg φ(x) ∈ (−π, π], the decomposition is given by where α(x) ∈ {0, 1, · · · , N − 1} and arg ϕ(x) ∈ (−π/N, π/N ]. For a bond b between x and x +μ, we define The hopping part (A1) is rewritten by is the remaining part dependent on b and "c.c." represents the complex conjugation. The path integral becomes With the identity relation for Z N elements we can expand as with the expansion coefficients for all x. In terms of the Z N -valued link variable Eq. (A10) is rewritten as Since bonds are dual to cubes, it is convenient to use the dual cube variables ξ c * ≡ ξ b . The constraint on the original lattice, Eq. (A12), is identical with where h * (x) is the four-dimensional hypercube dual to the site x. In order to satisfy Eq. (A13) we define the dual plaquette variable ζ p * by The summation of l b with the constraint (A10) is replaced by the summation of the unconstrained variables ζ p * . The path integral in the dual representation is given by We define the 't Hooft surface on the dual lattice bỹ where q ∈ {0, 1, . . . , N − 1}. S * represents a twodimensional closed surface on the dual lattice. With the threedimensional volume V * s.t. ∂V * = S * , it also reads The expectation value ofṼ [S * ] is given by We go backward to arrive at the formula (17). Introducing B(V * ), which is the subset of the bonds penetrating V * , and we can rewrite as It follows from Eq. (A14) that