# 't Hooft surface in lattice gauge theory

’’t Hooft surface in lattice gauge theory

Takuya Shimazaki and Arata Yamamoto

Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan

We discuss the lattice formulation of the ’t Hooft surface, that is, the two-dimensional surface operator of adual variable. The ’t Hooft surface describes the world sheets of topological vortices. We derive the formulasto calculate the expectation value of the ’t Hooft surface in the multiple-charge lattice Abelian Higgs model andin the lattice non-Abelian Higgs model. As the ﬁrst demonstration of the formula, we compute the intervortexpotential in the charge-2 lattice Abelian Higgs model.

I. INTRODUCTION

Topological orders are quantum phases beyond the Landautheory of symmetry breaking [1]. Local order parameters failto capture their manifestation. Some topological orders arecharacterized by long-range quantum entanglements exhibit-ing fractional excitations and topological degeneracy [2–8].One of the well-established approaches is a higher-form sym-metry [9–12]. Order parameters for the generalized globalsymmetries are extended objects, e.g., loop, surface, and soon. Their topological natures make it possible to classify thetopological orders.The most famous example of the non-local order parame-ters is a loop operator. For instance, in SU( N ) gauge theory,the Wilson loop [13] is deﬁned by the path-ordered product ofthe loop integral W [ C ] = P e i (cid:72) C A µ ( x ) dx µ . (1)Since the gauge ﬁeld A µ couples to electrically charged parti-cles, the Wilson loop describes the world lines of charged par-ticles. The SU( N ) gauge theory also has magnetically chargedparticles, namely, magnetic monopoles. The world lines ofmagnetic monopoles deﬁne the ’t Hooft loop [14]. Using thedual ﬁeld ˜ A µ , the ’t Hooft loop is given by ˜ W [ C ∗ ] = e i (cid:72) C∗ ˜ A µ ( x ) dx µ . (2)The Wilson and ’t Hooft loops are the order parameters for theconﬁnement of electric and magnetic charges, respectively.A more nontrivial example is a surface operator. In thegauge theory coupled to Higgs ﬁelds, topological vortices ap-pear. The vortices are one-dimensional defects, so their tra-jectories form world sheets. The vortex world sheets are two-dimensional defects with delta function support on the sur-faces. The surface operator to create the vortex world sheetis referred to as the ’t Hooft surface [15]. For instance, in the BF theory, it is given by the closed-surface integral over thedual 2-form ﬁeld B µν , ˜ V [ S ∗ ] = e i (cid:72) S∗ B µν dS µν . (3)When the theory has Z N topological order, the vortices havefractional magnetic charge q/N e ( q ∈ Z ). The ’t Hooft sur-face gives the criterion for the conﬁnement of the fractionallycharged vortices. It plays an essential role in the topologicalorder of Cooper pairs in superconductors [16–18] and that of diquarks in color superconductors [19–23].These ’t Hooft operators have crucial difﬁculty. The ’tHooft operators can be elegantly formulated in topologicalquantum ﬁeld theory, that is, effective theory to reproducetopological properties of the original quantum ﬁeld theory.The calculation beyond the effective theory is, however, noteasy. The dual gauge ﬁelds are not fundamental ﬁelds butdefects or singularities in the original theory. It might seemimpossible to calculate the expectation values of the ’t Hooftoperators in a quantitative manner. Surprisingly, this is possi-ble in lattice gauge theory. The formulation has been knownfor the ’t Hooft loop in the Yang-Mills theory [24–26]. It wasapplied to several lattice simulations [27–37]. This can begeneralized to higher-dimensional defects, say, the ’t Hooftsurface.In this paper, we study the ’t Hooft surface in lattice gaugetheory. After introducing the basics of dual variables inSec. II, we discuss how to compute the ’t Hooft surface inlattice simulation. We focus on the lattice gauge Higgs mod-els with Z N topological order: the Abelian Higgs model inSec. III and the non-Abelian Higgs model in Sec. IV. Thesimulation results are shown in Sec. V. Finally, Sec. VI is de-voted to summary. The Euclidean four-dimensional lattice isconsidered 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 de-ﬁned by translating the original lattice by a half of lattice spac-ing in all directions. The schematic ﬁgure is shown in Fig. 1.The sites of the dual lattice are in positions of the hypercubecenters of the original lattice. In four dimensions, there arethe one-to-one correspondences between d -dimensional ob-jects on the original lattice and (4 − d ) -dimensional objectson 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 objectson the dual lattice.In lattice gauge theory, there are two ways to introducedual variables. One is to replace all the integral variables inthe path integral by their dual variables. The path integral iscompletely reformed. For example, the gauge Higgs theoryis described by dual plaquette variables and dual cube vari-ables [38] (see also Refs. [39–43]). The other is to insert thedual variables without reforming the path integral. For exam-ple, 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 a r X i v : . [ h e p - l a t ] S e p b c* FIG. 1. Original lattice (black solid lines) and dual lattice (redbroken lines). In four dimensions, the bond b (blue thick line) is dualto the dual cube c ∗ (red thick line). change. This is exactly what is done in the lattice formulationof the ’t Hooft loop [24–26]. We consider such insertion ofthe ’t Hooft surface in the following sections. III. ABELIAN HIGGS MODEL

We ﬁrst consider Abelian gauge theory. When the Higgsﬁeld has the multiple electric charge

N e ( N ∈ Z ) , the theoryhas Z N topological order. The magnetic charge of a vortex isfractionally quantized to be /N e . The theory is often referredas the charge- N lattice Abelian Higgs model [44]. The pathintegral is given by Z = (cid:90) DADφ e − S (4)with the Abelian gauge ﬁeld A µ and the complex scalar ﬁeld φ . Let us deﬁne the link variable U µ ( x ) = e iA µ ( x ) (5)and the plaquette variable U µν ( x ) = U µ ( x ) U ν ( x + ˆ µ ) U − µ ( x + ˆ ν ) U − ν ( x ) , (6)where ˆ µ stands for the unit lattice vector in the µ direction.The Euclidean action is given by S = S gauge + S local + S hop (7)with the gauge part S gauge = − e (cid:88) x (cid:88) µ,ν Re U µν ( x ) , (8)the local part of the scalar ﬁeld S local = (cid:88) x (cid:104) φ ∗ ( x ) φ ( x ) + λ (cid:8) φ ∗ ( x ) φ ( x ) − v (cid:9) (cid:105) , (9) and the hopping part S hop = − (cid:88) x (cid:88) µ (cid:26) φ ∗ ( x ) U Nµ ( x ) φ ( x + ˆ µ )+ φ ∗ ( x + ˆ µ ) U − Nµ ( x ) φ ( x ) (cid:27) . (10)The theory is invariant under the local U(1) gauge transforma-tion U µ ( x ) → Λ( x ) U µ ( x )Λ ∗ ( x + ˆ µ ) (11) φ ( x ) → Λ N ( x ) φ ( x ) (12)and the global Z N transformation U µ ( x ) → U µ ( x ) e i π/N . (13)Because of the Z N symmetry, the vacuum is N -fold degener-ate.We put the ’t Hooft surface of magnetic charge q/N e on atwo-dimensional closed surface S ∗ . The path integral changesas Z S ∗ = (cid:90) DADφ e − S (cid:48) (14) S (cid:48) = S gauge + S local + S (cid:48) hop . (15)The hopping part is modiﬁed as S (cid:48) hop = − (cid:88) x,µ ∈ B ( V ∗ ) (cid:26) e i πq/N φ ∗ ( x ) U Nµ ( x ) φ ( x + ˆ µ )+ e − i πq/N φ ∗ ( x + ˆ µ ) U − Nµ ( x ) φ ( x ) (cid:27) − (cid:88) x,µ/ ∈ B ( V ∗ ) (cid:26) φ ∗ ( x ) U Nµ ( x ) φ ( x + ˆ µ )+ φ ∗ ( x + ˆ µ ) U − Nµ ( x ) φ ( x ) (cid:27) , (16)where B ( V ∗ ) is deﬁned by the hopping terms penetrating V ∗ ,s.t. S ∗ = ∂ V ∗ (see Fig. 2). Therefore, the expectation valueof the ’t Hooft surface is given by the formula (cid:104) ˜ V [ S ∗ ] (cid:105) = Z S ∗ Z = (cid:82) DADφ e − ∆ S e − S (cid:82) DADφ e − S = (cid:104) e − ∆ S (cid:105) (17)with ∆ S = S (cid:48) − S = S (cid:48) hop − S hop = − (cid:88) x,µ ∈ B ( V ∗ ) (cid:26) (cid:16) e i πq/N − (cid:17) φ ∗ ( x ) U Nµ ( x ) φ ( x + ˆ µ )+ (cid:16) e − i πq/N − (cid:17) φ ∗ ( x + ˆ µ ) U − Nµ ( x ) φ ( x ) (cid:27) . (18)This formula is the main result of this paper. The explicitderivation 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 duallattice. The hopping terms penetrating V ∗ are multiplied bythe Z N element e i πq/N . The winding number of each pla-quette is given by the sum of the angles of the four hoppingterms. As shown by the circle arrows in the ﬁgure, the Z N element changes the winding number by + q/N at A, and by − q/N at B, and by 0 elsewhere. This means that a vortex andan antivortex are inserted at S ∗ = ∂ V ∗ . BA FIG. 2. Schematic ﬁgure for Eq. (16). The ’t Hooft surface isinserted on the three-dimensional volume V ∗ (red thick line). TheZ N element e i πq/N is multiplied to the hopping terms penetrating V ∗ (blue arrows). There are two remarks on the above formula. The ﬁrst oneis that the inserted vortices have fractional winding numbers.They are different from the standard vortices deﬁned by inte-ger winding numbers. The above formula cannot realize theinteger winding numbers because ∆ S = 0 for q/N ∈ Z .Another formulation is necessary to insert the vortices withinteger winding numbers [45]. The second one is that V ∗ isnon-unique. In four dimensions, V ∗ can be deformed in theperpendicular direction by integral variable transformation, aslong as S ∗ is ﬁxed. The same ambiguity exists in the ’t Hooftloop [29]. On the other hand, S ∗ is invariant under the trans-formation. This means that the positions of the vortices arephysical. IV. NON-ABELIAN HIGGS MODEL

Next let us consider non-Abelian gauge theory. Among thenon-Abelian gauge Higgs models, the N -color and N -ﬂavorcase is of special importance. When the numbers of colorand ﬂavor are equal, the non-Abelian vortex, as well as theAbelian vortex, can exist. The minimal winding number ofthe non-Abelian vortex is /N , while the winding number ofthe Abelian vortex is integer.In the N -color and N -ﬂavor non-Abelian Higgs model, thehopping part of the lattice action is S hop = − (cid:88) x (cid:88) µ (cid:88) i (cid:26) φ † i ( x ) U µ ( x ) φ i ( x + ˆ µ )+ φ † i ( x + ˆ µ ) U − µ ( x ) φ i ( x ) (cid:27) . (19)This is is almost the same as Eq. (10), except that the linkvariable U µ is a U( N ) element and the N -ﬂavor scalar ﬁelds φ i ( i = 1 , · · · , N ) are N -component vectors. The other partsare quite different, but they are irrelevant for the present ar-gument. (See Ref. [46] for the complete form of the latticeaction.) We can easily derive the formula; Eq. (18) is replacedby ∆ S = − (cid:88) x,µ ∈ B ( V ∗ ) (cid:26) (cid:16) e i πq/N − (cid:17) φ † ( x ) U µ ( x ) φ ( x + ˆ µ )+ (cid:16) e − i πq/N − (cid:17) φ † ( x + ˆ µ ) U − µ ( x ) φ ( x ) (cid:27) . (20)The inserted ’t Hooft surface satisﬁes the same properties asin the Abelian case. V. SIMULATION

To demonstrate the above formalism, we perform the nu-merical simulation in the simplest case, the charge- latticeAbelian Higgs model. The simulation details are summarizedin Appendix B. The Z group has two elements: one triv-ial state +1 (zero magnetic charge) and one nontrivial state − (fractional magnetic charge / e ). This means that a vor-tex and an antivortex are equivalent. This specialty is due tothe compactness of the link variable. In general, the latticeAbelian Higgs model sometimes shows different behaviorsfrom the continuous Abelian Higgs model. The results shouldbe interpreted as the properties of lattice superconductors, notof 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 inFig. 3, the ’t Hooft surface is inserted on the dual cuboid X × Y × T inside the hypercuboid. We take Y = N y to simplifythe analysis. The nontrivial Z element − is multiplied to thehopping terms in the z direction. When the time extent T islarge enough, the ’t Hooft surface asymptotically behaves as (cid:104) ˜ V [ S ∗ ] (cid:105) ∝ e − ET , (21)where E is the energy of a static and straight vortex-antivortexpair. Since E is proportional to Y because of translationalinvariance, E/Y is a function of X . Therefore, E/Y can beinterpreted the intervortex potential per length.The simulation results are shown in Fig. 4. Changing thescalar self-coupling constant, we calculated the potential inthe Higgs phase and the Coulomb phase. In the Higgs phase,the potential is linear. The vortices with fractional magneticcharge are conﬁned. Thus, only the states with zero magneticcharge will appear at low energy. This will be the commonproperty both for lattice and continuous superconductors. Inthe Coulomb phase, the potential is almost ﬂat. As the U(1)symmetry is unbroken, the dual variables are massive, so theycannot propagate to long range. The interaction between thevortices is diminished.The linear potential is equivalent to the volume-law scalingof the ’t Hooft surface, which is the criterion for the conﬁne- xy zxyX TY XY FIG. 3. Geometry projected on the xyτ hyperplane (left) and on the xyz hyperplane (right). The two-dimensional ’t Hooft surface (darkred) surrounds the three-dimensional volume V ∗ (light red). E / Y X HiggsCoulomb

FIG. 4. Intervortex potential per length in the charge-2 latticeAbelian Higgs model. The potentials in the Higgs phase ( λ = 4 )and in the Coulomb phase ( λ = 1 ) are shown. ment of magnetic vortices. This is dual to the volume-lawscaling of the Wilson surface, which is the criterion for theconﬁnement of electric strings. In this model, however, thevolume law is not exactly satisﬁed. If the linear potential wereexactly correct, the potential energy would be very large atlong distance. A dynamical vortex-antivortex pair will be cre-ated in-between the original vortex and antivortex to lower thetotal energy. This is called the surface breaking as the analogof the string breaking of the Wilson loop [38]. The surfacebreaking will happen when the potential energy reaches themass of the dynamical vortex-antivortex pair. The potentialwill 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 sur-face in the lattice gauge Higgs models with the Z N topologicalorder. The formula is simple; the Z N element is multiplied tothe 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 chargeare conﬁned in the Higgs phase, and thus they do not emergein the physical spectrum. This is consistent with our under-standing of superconducting vortices.The advantage of our formulation is that the ’t Hooft surfaceis calculable in the original path integral. It is also attainable inthe dual path integral, in which all the variables are dualized.However, the dual approach works only when the duality be-tween the two theories is ensured, e.g., in the London limit oftype-II superconductors. Our formulation is, therefore, moreuseful for the quantitative analysis in the original theory. ACKNOWLEDGMENTS

A. Y. was supported by JSPS KAKENHI Grant Number19K03841. The numerical calculations were carried out onSX-ACE in Osaka University.

Appendix A: DERIVATION OF THE FORMULA

Let us derive the formula (17). The derivation is parallel toRef. [25]. We introduce a special notation in this appendix.When a bond b connects x and x + ˆ µ , we write as φ ∂b (1) = φ ( x ) , φ ∂b (2) = φ ( x + ˆ µ ) , and U b = U µ ( x ) . The hopping part(10) is rewritten as S hop = − (cid:88) b (cid:26) φ ∗ ∂b (1) U Nb φ ∂b (2) + φ ∗ ∂b (2) U − Nb φ ∂b (1) (cid:27) . (A1)The summation is taken over all bonds b .The scalar ﬁeld φ is uniquely decomposed into a Z N partand a U(1) / Z N part. When arg φ ( x ) ∈ ( − π, π ] , the decom-position is given by φ ( x ) = e i πα ( x ) /N ϕ ( x ) (A2)where α ( x ) ∈ { , , · · · , N − } and arg ϕ ( x ) ∈ ( − π/N, π/N ] . For a bond b between x and x + ˆ µ , we de-ﬁne ∆ b α = α ( x + ˆ µ ) − α ( x ) . (A3)The hopping part (A1) is rewritten by S hop [ α, A, ϕ ] = (cid:88) b (cid:16) e i π ∆ b α/N R b [ A, ϕ ] + c . c . (cid:17) (A4)where R b [ A, ϕ ] is the remaining part dependent on b and“c.c.” represents the complex conjugation. The path integralbecomes Z = (cid:90) DADφ e − S = (cid:90) DADϕ e − S gauge [ A ] − S local [ ϕ ] (cid:89) x N − (cid:88) α ( x )=0 e − S hop [ α,A,ϕ ] . (A5)With the identity relation for Z N elements N N − (cid:88) l =0 e − i πnl/N = δ n (mod N ) , (A6)we can expand as e − S hop = (cid:89) b N − (cid:88) l b =0 I l b e i πl b ∆ b α/N (A7)with the expansion coefﬁcients I l b = 1 N N − (cid:88) n b =0 e − i πl b n b /N exp (cid:16) − e i πn b /N R b − c . c . (cid:17) . (A8)It follows that (cid:89) x N − (cid:88) α ( x )=0 e − S hop = (cid:89) x N − (cid:88) α ( x )=0 (cid:89) b N − (cid:88) l b =0 I l b e i πl b ∆ b α/N = N (cid:88) { l b } (cid:48) (cid:89) b I l b . (A9) (cid:80) { l b } (cid:48) denotes the summation over the conﬁgurations whichsatisfy (cid:88) b (cid:51) x l b = 0 (mod N ) (A10)for all x . In terms of the Z N -valued link variable ξ b ≡ e i πl b /N , (A11)Eq. (A10) is rewritten as (cid:89) b (cid:51) x ξ b = 1 . (A12)Since bonds are dual to cubes, it is convenient to use the dualcube variables ξ c ∗ ≡ ξ b . The constraint on the original lattice,Eq. (A12), is identical with (cid:89) c ∗ ∈ h ∗ ( x ) ξ c ∗ = 1 (A13)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 ξ c ∗ ≡ (cid:89) p ∗ ∈ c ∗ ζ p ∗ . (A14)The summation of l b with the constraint (A10) is replaced bythe summation of the unconstrained variables ζ p ∗ . The pathintegral in the dual representation is given by Z = N (cid:90) DADϕ e − S gauge − S local (cid:88) { ζ p ∗ } (cid:89) b I l b . (A15)We deﬁne the ’t Hooft surface on the dual lattice by ˜ V [ S ∗ ] ≡ (cid:89) p ∗ ∈S ∗ ζ qp ∗ (A16)where q ∈ { , , . . . , N − } . S ∗ represents a two-dimensional closed surface on the dual lattice. With the three-dimensional volume V ∗ s.t. ∂ V ∗ = S ∗ , it also reads ˜ V [ S ∗ ] = (cid:89) c ∗ ∈V ∗ ξ qc ∗ . (A17)The expectation value of ˜ V [ S ∗ ] is given by (cid:104) ˜ V [ S ∗ ] (cid:105) ≡ NZ (cid:90) DADϕ e − S gauge − S local (cid:88) { ζ p ∗ } ˜ V [ S ∗ ] (cid:89) b I l b . (A18)We go backward to arrive at the formula (17). Introducing B ( V ∗ ) , which is the subset of the bonds penetrating V ∗ , and k b = b ∈ B ( V ∗ ))0 ( b / ∈ B ( V ∗ )) , (A19)we can rewrite as ˜ V [ S ∗ ] = (cid:89) b ∈ B ( V ∗ ) ξ qb = (cid:89) b e i πl b k b q/N (A20)It follows from Eq. (A14) that (cid:104) ˜ V [ S ∗ ] (cid:105) ≡ NZ (cid:90) DADϕ e − S gauge − S local × (cid:88) { l b } (cid:48) (cid:89) b I l b e i πl b k b q/N = 1 Z (cid:90) DADϕ e − S gauge − S local × (cid:89) b N − (cid:88) l b =0 (cid:89) x N − (cid:88) α ( x )=0 I l b e i πl b ∆ b α/N e i πl b k b q/N ≡ Z (cid:90) DADϕ e − S gauge − S local (cid:89) x N − (cid:88) α ( x )=0 e − S (cid:48) hop (A21)where S (cid:48) hop = (cid:88) b (cid:16) e i π ∆ b α/N e i πk b q/N R b [ A, ϕ ] + c . c . (cid:17) . (A22)Equation (A22) is nothing but Eq. (16). Therefore, (cid:104) ˜ V [ S ∗ ] (cid:105) = Z S ∗ Z . (A23)This is the end of the proof of Eq. (17). Appendix B: SIMULATION DETAILS

We performed the lattice simulation with the hybrid MonteCarlo method. We analyzed two cases: the Higgs phase andthe Coulomb phase. The scalar self-coupling constant was setat λ = 4 for the Higgs phase and λ = 1 for the Coulombphase. The other parameters were ﬁxed at /e = 2 and v =0 . . The lattice volume is N x N y N z × N τ = 10 × and all boundary conditions are periodic. The size of the ’t Hooft sur-face is X = { , , · · · , } , Y = 10 , and T = { , , · · · , } .We obtained the energy E by ﬁtting the data with Eq. (21) ina ﬁnite range of T . We checked that the results are insensitiveto the ﬁtting range of T .From Eq. (17), we see that the overlap between Z S ∗ and Z is exp( − ∆ S ) . The overlap exponentially decreases as thesurface size increases. When the surface size is large, the rel-evant conﬁgurations hardly appear in the Monte Carlo sam-pling. This is called the overlap problem. 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