Discrete aspects of continuous symmetries in the tensorial formulation of Abelian gauge theories
DDiscrete aspects of continuous symmetries in the tensorial formulation of Abeliangauge theories
Yannick Meurice ∗ Department of Physics and Astronomy, 514 Van Allen Hall, The University of Iowa, Iowa City IA 52242 (Dated: July 6, 2020)We show that standard identities and theorems for lattice models with U (1) symmetry get re-expressed discretely in the tensorial formulation of these models. We explain the geometrical analogybetween the continuous lattice equations of motion and the discrete selection rules of the tensors.We construct a gauge-invariant transfer matrix in arbitrary dimensions. We show the equivalencewith its gauge-fixed version in a maximal temporal gauge and explain how a discrete Gauss’s law isalways enforced. We propose a noise-robust way to implement Gauss’s law in arbitrary dimensions.We reformulate Noether’s theorem for global, local, continuous or discrete Abelian symmetries:for each given symmetry, there is one corresponding tensor redundancy. We discuss semi-classicalapproximations for classical solutions with periodic boundary conditions in two solvable cases. Weshow the correspondence of their weak coupling limit with the tensor formulation after Poissonsummation. We briefly discuss connections with other approaches and implications for quantumcomputing. I. INTRODUCTION
Tensor Field Theory (TFT) is a recently developed ap-proach of models studied in the context of lattice gaugetheory [1–26]. The basic idea is to rewrite the partitionfunction of lattice models as a product of tensors whereall the indices are contracted. Many lattice models have compact fields. This feature appears naturally when weintegrate over compact unitary groups attached to linksin gauge models or over the Nambu-Goldstone modes of O ( N ) symmetric scalar models in nonlinear sigma mod-els. Functions over compact groups can be expanded interms of discrete sums of characters for Abelian groups[27] or more generally of group representations [28]. Thisproperty was exploited to calculate strong coupling ex-pansions [29] and introduce dual variables [30–32] for thetype of lattice models mentioned above.These group theoretical methods were used in a sys-tematic way to build the tensors [7, 8] of the spin andgauge models reviewed in [31], and to rewrite partitionfunctions and averages of observables in a way that issuitable for exact coarse-graining or sampling of tensorconfigurations similar to the worm algorithm [33–35]. Itis also important to realize that TFT remains useful andaccurate in regimes that are completely beyond the rangeof validity of the strong coupling expansion even whenphase transitions are present [36]. In addition, the dis-creteness of TFT formulations also makes them a nat-ural starting point for building approximate forms ofknown lattice models suitable for quantum computationsor quantum simulation experiments [37, 38].Symmetry considerations have played a crucial role inuncovering the subconstituents of matter and their inter-actions. A key result is Noether’s theorem which asso-ciates a conserved charge to a global continuous symme-try. Is there a way to re-express Noether’s theorem in a ∗ [email protected] completely discrete TFT formulation? In the followingwe will show show that the answer is affirmative in thecase of a continuous and compact U (1) symmetry. Wewill also discuss the effect of approximations and varioustypes of noise, which are unavoidable in practical TFTimplementations, on these symmetry properties.In the conventional formulation of field theory, a global U (1) symmetry results in a conserved Noether current atthe classical level. The use of the continuous symme-try and the classical equations of motion, which resultfrom local continuous variations of the action, are crucialsteps of the derivation. At the quantum level, the invari-ance under a local continuous shift of the field variablesgenerates Schwinger-Dyson equations which are quantumversions of the equations of motion. For local U (1) sym-metries, Ward-Takahashi identities, or more complicatedidentities if gauge fixing is involved, can be found in quan-tum field theory textbooks such as Ref. [39]. These re-markable theorems and identities rely on the fact thatthe field variables are continuous.In this article we show that the basic features of contin-uous Abelian symmetries in the conventional formulationof field theory have discrete counterparts in TFT. Thearticle is organized as follows. In Sec. II, we review thetensorial formulation of models with continuous Abeliansymmetries in arbitrary Euclidean space-time dimension D . We start with the compact Abelian Higgs models(CAHM) and then consider the pure gauge limit and theO(2) spin model limit. In Sec. III, we establish a precisecorrespondence between the classical equations of motionof the lattice action with the selection rules. They haveidentical grading and geometrical interpretation in termsof inside/outside features. In Ref. [40], it was noted thatthese selection rules, a discrete divergenceless condition,could be interpreted as a discrete version of Noether’stheorem and would also extend to discrete symmetries.The selection rules for the CAHM are a discrete versionof Maxwell’s equations with charges and currents. In par-ticular, Gauss’s law which has complicated aspects in the a r X i v : . [ h e p - l a t ] J u l conventional Hamiltonian formulation appears in a trans-parent way in TFT. The questions of gauge-invarianceand gauge-fixing are discussed in Sec. IV. We show thatlocal selection rule redundancies observed in Ref. [40] canbe reinterpreted in terms of a gauge fixing that removesthe integration over the fields leading to the redundantselection rules. In Sec. V, we show that the mechanismcan be extended to global symmetries and discrete sym-metries. Noether’s theorem for Abelian symmetries canbe re-expressed in the tensor reformulation context as:for each symmetry, there is a corresponding tensor re-dundancy. This applies with a remarkable generality tolocal, global, continuous or discrete Abelian symmetries.The transfer matrix of the CAHM is constructed in ar-bitrary dimensions in Sec. VI. It is made out of electricand magnetic “layers” and is automatically gauge invari-ant. It defines a Hilbert space over which Gauss’s lawis implemented when we apply the transfer matrix on anarbitrary state. We use “Gauss’s law” in a context depen-dent manner. In the CAHM context, there are chargesand currents and Gauss’s law means that the quantumnumbers associated to the matter fields are completelyfixed by the quantum numbers associated with the gaugefields. This is because there are infinitely many possibleFourier modes for bosonic fields in contrast to fermionsthat only allow a finite number of possibilities as, forinstance, for the Schwinger model [41–46].However, when we take the pure gauge limit by de-coupling the matter fields, we obtain a restriction on thegauge quantum number which is a discrete version of ∇ · E = 0. In the pure gauge limit, we discuss the equiv-alence with a gauge-fixed version and propose a way toimplement Gauss’s law with unrestricted variables whichcan be used in any dimension. In Sec. VII, we take thetime-continuum limit in the same way as in [31] and geta similar Hamiltonian formulation.The correspondence between the continuous classicalequations of motion and the discrete selection rules al-lows us to connect topological solutions that appear withperiodic boundary conditions, and not with open bound-ary conditions, to tensor assemblies that are allowed orforbidden under the same periodic or open boundary con-ditions. In Sec. VIII, we show that this correspondencecan be made precise using Poisson summation for twomodels that are exactly solvable. The practical conse-quences of the results for coarse graining, the continuumlimit and quantum computations are briefly discussed inthe conclusions. II. ABELIAN LATTICE MODELS
In this section we introduce the compact Abelian Higgsmodel (CAHM) and situations where it can be reducedto the pure gauge U (1) model or the O(2) spin model. Inits original form, the Abelian Higgs model has also a non-compact scalar field which can be decoupled by a strongcoupling limit discussed in [37] and will not be considered in the following. The main purpose of this section is tointroduce models, notations and symmetries.In the following, we use a D -dimensional (hyper) cubicEuclidean space-time lattice. The space-time sites aredenoted x = ( x , x , . . . x D ), with x D = τ , the Euclideantime direction. Lattice units are implicit and the space-time sites are labelled with integers. We use the boldnotation x for the D − x and x + ˆ µ are labelled by ( x, µ ) and the plaquettesdelimited by four sites x , x + ˆ µ , x + ˆ µ + ˆ ν and x + ˆ ν are labelled by ( x, µ, ν ). By convention, we start withthe lowest index when introducing a circulation at theboundary of the plaquette. The total number of sites isdenoted V . Periodic boundary conditions (PBC) or openboundary conditions (OBC) will be considered.Our main object is the CAHM partition function Z CAHM = (cid:89) x (cid:90) π − π dϕ x π (cid:89) x,µ (cid:90) π − π dA x,µ π e − S gauge − S matter , (1)with S gauge = β pl. (cid:88) x,µ<ν (1 − cos( A x,µ + A x +ˆ µ,ν − A x +ˆ ν,µ − A x,ν )) , (2)and S matter = β l. (cid:88) x,µ (1 − cos( ϕ x +ˆ µ − ϕ x + A x,µ )) . (3)The CAHM is a gauged version of the O(2) modelwhere the global symmetry under a ϕ shift becomes local ϕ (cid:48) x = ϕ x + α x (4)and these local changes in S matter are compensated bythe gauge field changes A (cid:48) x,µ = A x,µ − ( α x +ˆ µ − α x ) , (5)which also leave S gauge invariant.The matter fields can be decoupled by simply setting β l. = 0. As they don’t appear in the action, their inte-gration yields a factor 1 and we are left with the puregauge (PG) U (1) lattice model with partition function Z P G = (cid:89) x,µ (cid:90) π − π dA x,µ π e − S gauge . (6)The decoupling of the gauge fields is less straightfor-ward. Strictly speaking, the O(2) spin model is obtainedby removing the gauge fields introduced to make theglobal symmetry a local one and the partition functionof the O(2) model reads Z O (2) = (cid:89) x (cid:90) π − π dϕ x π e − S O (2) , (7)with S O (2) = β l. (cid:88) x,µ (1 − cos( ϕ x +ˆ µ − ϕ x )) . (8)It is tempting to consider this model as the weak gaugecoupling limit ( β pl. → ∞ ) of the CAHM. However forcompact gauge fields, this limit involves subtleties whendefined in the context of the infinite volume and contin-uum limit. In addition, the O(2) model has charge sectorslabelled by integers and it is possible to select a specificcharge sector by tuning the gauge boundary conditions.This is discussed in Ref. [38] in 1+1 dimensions.Most of the results presented in the rest of the paperalso hold for finite subgroups of U (1). If we consider the“clock” restriction to angles ϕ x and A x,µ taking values πq (cid:96) for (cid:96) = 0 , , . . . , q −
1, the values of (cid:96) are addedmodulo q and form the additive group Z q . III. TENSOR SELECTION RULES ANDLATTICE EQUATIONS OF MOTION
In this section, we reformulate the CAHM in arbitrarydimensions using the tensor formalism developed in Refs.[8, 37]. We point out and explain the geometrical analogybetween the tensor selection rules and the lattice equa-tions of motion. We then discuss the pure gauge and spinlimits.
A. General case
The basic ingredients of the tensor reformulation arethe Fourier expansions for the linkse β l. cos( ϕ x +ˆ µ − ϕ x + A x,µ ) = + ∞ (cid:88) n x,µ = −∞ e in x,µ ( ϕ x +ˆ µ − ϕ x + A x,µ ) I n x,µ ( β l. ) , (9)and the plaquettese β pl. cos( A x,µ + A x +ˆ µ,ν − A x +ˆ ν,µ − A x,ν ) = (10) + ∞ (cid:88) m x,µ,ν = −∞ e im x,µ,ν ( A x,µ + A x +ˆ µ,ν − A x +ˆ ν,µ − A x,ν ) I m x,µ,ν ( β pl. ) , where the I n ( β ) are the modified Bessel functions of thefirst kind. Notice that in both cases, the argument ofthe cosine function and the expression multiplying theFourier indices are identical and, in particular, their signsare identical. These signs can be interpreted as forminga binary grading. This grading depends on the link orplaquette to which the fields belong.The classical lattice equations of motions are obtainedby setting the derivatives of the action with respect tothe fields to zero. In general one obtains a sum of sineswith relative signs corresponding to the grading. The tensors to be traced in the reformulation of the parti-tion function, are obtained by integrating over the fields.When the Fourier indices corresponding to a given fieldare collected they appear with relative signs that corre-spond to the same grading and can be interpreted geo-metrically. We now discuss the scalar and gauge deriva-tion/integration separately.For the scalar fields, we first introduce the notation d x,µ ≡ ϕ x +ˆ µ − ϕ x + A x,µ (11)which approximates the covariant derivative of ϕ . Theequation of motion ∂S/∂ϕ x = β l. (cid:88) µ [ − sin( d x,µ ) + sin( d x − ˆ µ,µ )]= 0 . (12)On the other hand the integration with respect to ϕ x implies (cid:88) µ [ − n x,µ + n x − ˆ µ,µ ] = 0 . (13)It is clear that the geometrical structure of the two aboveequations are identical and that the equations can beobtained from each other by the substitution β l. sin( d x,µ ) ↔ n x,µ . (14)The geometrical interpretation is simple: in Eq. (13), the n x,µ come with a minus and correspond to links coming“out ” of x in the positive directions, while the n x − ˆ µ,µ come with a plus and correspond to links coming “in”the site x from the negative direction. Notice that thisfeature is completely dictated by the sign convention ap-pearing in the Fourier expansion Eq. (9). More specifi-cally, fields appearing with a minus (plus) sign inside thecosine belong to an out (in) link, respectively.Notice that Eq. (13) is a discrete version of Noether’stheorem which is a divergenceless condition. This im-plies [40] a discrete version of Gauss’s theorem which ispreserved when truncations are applied. In addition, theequations of motions are satisfied in average, when in-serted in the path integral. This is a consequence of theinvariance under a local shift for each integral which isused in the derivation of Schwinger-Dyson equations. Ifsuch a shift is applied after expansion in Fourier modesin the functional integral, then Eq. (13) follows, makingthe connection between the two sets of equations clear.In a similar manner, we can assign in and out featuresto the plaquettes attached to a link in a way consistentwith the Fourier expansion Eq. (10). For µ < ν , m x,µ,ν are in and m x − ˆ ν,µ,ν out, while for µ > ν , m x,ν,µ are outand m x − ˆ ν,ν,µ in. Using the obvious analogy with thecontinuum, we define the standard lattice field strengthtensor f x,µ,ν ≡ A x,µ + A x +ˆ µ,ν − A x +ˆ ν,µ − A x,ν . (15)As in the continuum they are gauge invariant.With these notations, ∂S/∂A x,µ = β pl. (cid:88) ν>µ [sin( f x,µ,ν ) − sin( f x − ˆ ν,µ,ν )]+ β pl. (cid:88) ν<µ [ − sin( f x,ν,µ ) + sin( f x − ˆ ν,ν,µ )]+ β l. sin( d x,µ )= 0 . (16)On the other hand, the integration over A x,µ yields theselection rule (cid:88) ν>µ [ m x,µ,ν − m x − ˆ ν,µ,ν ]+ (cid:88) ν<µ [ − m x,ν,µ + m x − ˆ ν,ν,µ ]+ n x,µ = 0 . (17)We see again geometric similarities between Eq. (16)with continuous variable and Eq. (17) with integer vari-ables. They both have the tensor structure of ∂ µ F µν = J ν . (18)They can be mapped into each other using Eq. (14) andin addition the substitution β pl. sin( f x,µ,ν ) ↔ m x,µ,ν . (19)Eq. (17) means that the link indices n x,µ can be seen asdetermined by unrestricted plaquette indices m x,µ,ν . Wewrite this dependence as n x,µ ( { m } ) which is shorthandfor Eq. (17). Note that for n x,µ ( { m } ) Eq. (13) holds[40] and as long as the gauge fields are present, there isno need to enforce Eq. (13).Each integration provides a tensor with the selectionrules discussed above. For convenience we factorize allthe I ( β ) factors which dominate the small β regime anddefine the ratios t n ( β ) ≡ I n ( β ) I ( β ) (cid:39) (cid:40) − n β + O (1 /β ) , for β → ∞ β n n n ! + O ( β n +2 ) , for β → . (20)Their limiting behavior at weak and strong coupling willbe used often.The four tensor legs attached to a given plaquette( x, µ, ν ) must carry the same index m . For this purposewe introduce the “ B -tensor” as in [8] B ( x,µ,ν ) m m m m = (cid:40) t m ( β pl. ) , if all m i are the same0 , otherwise . (21)These are assembled (traced) together with “ A -tensors”attached to links with 2( D −
1) legs orthogonal to thelink A ( x,µ ) m ...m D − = t n x,µ ( β l. ) δ n x,µ ,n x,µ ( { m } ) . (22) Notice that in contrast to Ref. [8], the weight of the pla-quettes is carried by the B -tensor. The partition functionwith PBC can now be written as Z = ( e − β pl. I ( β pl. )) V D ( D − / ( e − β l. I ( β l. )) V D × Tr (cid:89) l. A ( l. ) m ,...m D − (cid:89) pl. B ( pl. ) m m m m , (23)where the trace means index contraction following thegeometric procedure described above. The tensor assem-bly is illustrated in Fig. 1 for D = 2. Illustrations for D = 3 will be provided in Sec. VI. FIG. 1. A and B tensors assembled in D = 2. Small circles(blue online) are used for the A - tensors and large circles (redonline) for the B -tensors. It is clear that for PBC, we have a discrete translationinvariance and the tensor assembly is the same every-where. We can introduce OBC by starting with PBCand setting β l. and β pl. to zero on the links and plaque-ttes at the boundary. Since I n (0) = δ n, , (24)this forces the indices at the boundary to be zero withan associated weight 1. B. The O(2) model
Part of the results of subsection III A extend in astraightforward way to the O(2) model. We just needto set A x,µ = 0 in Eqs. (11) to (13). It was pointedout [40], that Eq. (13) is a discrete version of Noether’stheorem associated with the global O(2) symmetry. Adiscrete version of Gauss theorem holds and guaranteesa global neutrality for PBC and OBC.It is also interesting to keep the gauge fields and takethe β pl. → ∞ limit where the weights of the B -tensorsare 1. The relation n x,µ ( { m } ) of Eq. (17) remains validand guarantees that the discrete divergenceless conditionEq. (13) is obeyed for arbitrary plaquette configurations { m } . For D = 2, this corresponds to the dual construc-tion [32], however, the gauge procedure described hereextends to any dimension without requiring the use ofthe dimension-dependent Levy-Civita tensor. C. Pure gauge limit
We now consider the pure gauge limit by setting β l. =0. Eq. (24) imposes the constraint n x,µ = 0 and Eq.(17) reduces to a discrete version of ∂ µ F µν = 0. We canmake this statement more precise by introducing sugges-tive notations. We define the electric integers e x,j ≡ m x,j,D , (25)with j = 1 , . . . , D −
1, the integers associated with timeplaquette and which can be interpreted as electric fields.Eq. (17) for µ = D reads D − (cid:88) j =1 ( e x,j − e x − ˆ j,j ) = 0 . (26)This is a discrete form of Gauss’s law in the pure gaugelimit ∇ · E = 0.For D ≥
3, we can introduce magnetic fields in a di-mension dependent way. For D = 3, we define b x ≡ m x, , . (27)Eq. (17) for µ = 1 and 2 are e x, − e x − ˆ τ, = − ( b x − b x − ˆ2 ) ,e x, − e x − ˆ τ, = ( b x − b x − ˆ1 ) . (28)These are a discrete version of the D = 3 EuclideanMaxwell’s equations ∂ B = ∂ τ E ∂ B = − ∂ τ E , (29)with B = F . However, there is no discrete equa-tion corresponding to Maxwell equation for the dual fieldstrength tensor ∂ µ (cid:15) µνσ F νσ = 0 . (30)Example of legal configurations violating the discrete ver-sion of Eq. (30), also written ˙ B = − ∇ × E , can beconstructed.For D = 4, we can introduce b x,j ≡ (cid:15) jkl m x,k,l , (31)and obtain a discrete version of ∂ τ E = − ∇ × B , (32) with the Euclidean magnetic field F jk = + (cid:15) jkl B l . (33)Note that Eq. (32) implies ∂ τ ( ∇ · E ) = 0 , (34)even if we don’t impose Gauss’s law. Again there is nodiscrete version of the homogeneous equations for thedual field strength ˙ B = − ∇ × E and ∇ · B = 0. Notethat the sign in Eq. (32) is different in Euclidean andMinkowskian spaces. It can be traced to the minus signin the Minkowskian Klein-Gordon equation. D. Restrictions to Z q Some of the results of this section hold in an obviousway for the Z q restrictions. The infinite sums in theFourier expansions are replaced by finite sums with q values. The modified Bessel functions are replaced bytheir discrete counterparts: I n ( β ) → I ( q ) n ( β ) ≡ (1 /q ) q − (cid:88) (cid:96) =0 e β cos( πq (cid:96) ) e in πq (cid:96) , (35)which in the large q limit turns into the usual integralformula. In the Ising case ( q = 2), we have I ( β ) → cosh( β ) , and I ( β ) → sinh( β ) . (36)The selection rules Eqs. (13) and (17) remain valid mod-ulo q . The infinitesimal variations of the action can bereplaced discrete variations by an amount πq , but thesine functions should be replaced by finite differences ofcosine functions. IV. LOCAL GAUGE INVARIANCE,SELECTION RULE REDUNDANCY ANDGAUGE FIXING
Sec. III A makes clear that selection rule Eq. (13) dueto the ϕ integration is redundant and a consequence ofEq. (17). This forces a local neutrality . If we insert e iϕ x in the partition function Eq. (13) is modified and clasheswith the original form of Eq. (13) which follows from Eq.(17), forcing the functional integral to be zero.Similarly, it was shown [40] that in the pure gaugelimit the set of equations (17) with n x,µ = 0 are notindependent. If we pick a site, we can construct a in-out partition for the legs attached to links coming outof this site, the sum of “in” indices is the same as thesum of the “out” indices, and if we assemble them on theboundary of a D -dimensional cube, as illustrated in Fig.2 for D = 3, one of the divergenceless condition followfrom the other 2 D − FIG. 2. Illustration that one divergenceless condition isredundant for D = 3. Imagine the tensor assembled on thesurface of a cube, remove the A -tensor on the top: the sum ofthe in indices equals the sum of the out indices at the missingtensor because it holds at the 17 other vertices. To be completely specific, we review the details of thisin-out partition [40]. For a given pair of directions µ and ν , there are 8 types of legs for the A -tensors on links con-nected the site x that we label [( x, µ ) , ± ˆ ν ], [( x − ˆ µ, µ ) , ± ˆ ν ],[( x, ν ) , ± ˆ µ ], and [( x − ˆ ν, ν ) , ± ˆ µ ]. The pair of indices ap-pearing first refers to the links where the A -tensor is at-tached and the second index to the direction of the legwhich can be positive or negative. The [( x, µ ) , ˆ ν ] with µ < ν are given an out assignment. There are threeoperations that swap in and out: changing ( x, µ ) into( x − ˆ µ, µ ), changing ˆ µ into − ˆ µ and interchanging µ and ν . This redundancy can be rephrased in a more enlighten-ing way in the discrete electric/magnetic language devel-oped in Sec. III C: if Gauss’s law is satisfied for a A -tensorattached to the (( x , τ ) , D ) time link which is assembledwith the divergenceless A -tensors attached to the 2( D − x , τ + 1) , j ) and (( x − ˆ j, τ + 1) , j ) with j = 1 , . . . , D −
1, then the A -tensors attached to the timelink (( x , τ + 1) , D ) is forced to obey Gauss’s law becauseof its connection to the other tensors.We can now see that gauge fixing is equivalent to re-moving these redundant conditions. For the CAHM, wecan go to the unitary gauge where the ϕ can be removedeverywhere and the redundant Eq. (13) obtained fromthe ϕ integration disappears independently of boundaryconditions. For the pure gauge case, we can try to usethe temporal gauge to set A x,D to zero. For OBC, thiscan be accomplished for all time links. From Eq. (24),OBC imply e x,i = 0 on the two time slices at the bound-ary (one below the initial time and one above the finaltime). In other words, it corresponds to a transition fromthe state where there is no electric field into itself. AsGauss’s law is satisfied by the trivial configuration at thetime slices at the boundaries it is also satisfied on every time slice. For PBC, one link remains to be integratedfor each closed time line (Polyakov loop) attached to anygiven spatial site. Putting these unintegrated time linksat the same time, we get a time layer where after inte-grating over the leftover time links, Gauss’s law is satis-fied. Again, Gauss’s law is then propagated to the entirelattice for the reason explained above. V. A REFORMULATION OF NOETHER’STHEOREM
The discussion of Sec. IV clarifies that redundant se-lection rules are in one-to-one correspondence with ir-relevant integrations. In other words, we can skip theintegrations that produce redundant selection rules andreplace these integrated fields by arbitrary values. Thisis exactly what gauge-fixing does. With our normaliza-tion of each integration over the circle to one, this doesnot cost extra factors.The argument can be extended to global symmetries.In the case of the O(2) model, it follows from the discus-sion of Ref. [40] that in-out assignments for the 2 D legsof the divergenceless tensor attached to sites imply that one of the divergenceless conditions is a consequence ofall the other ones. This requires the whole tensor networkto be isolated. For PBC, there are no boundaries. ForOBC, the boundaries carry 0 indices which are neutral(neither in or out). It is clear that the global O(2) sym-metry allows us to fix one of the ϕ fields to an arbitraryvalue.The redundancy argument extends to discrete Z q subgroups of U (1) where the divergenceless conditionis expressed modulo q and the infinite set of Bessel func-tions are replaced by the q discrete ones.In view of this discussion, we suggest that Noether’stheorem can be expressed in the tensor formulation con-text as: for each symmetry, there is a corresponding ten-sor redundancy. This applies to global, local, continuousand discrete Abelian symmetries. VI. TRANSFER MATRIX
In Eq. (23), the partition function is written as thetrace of a product of tensors attached to links and plaque-ttes. We can organize this trace by assembling “time lay-ers” corresponding to “magnetic” time slices and “elec-tric” slices half-way between the magnetic time slices.This construction singles out a time direction as for theHamiltonian treatment. The case D = 2 is discussed inRef. [37] and the pure gauge D = 3 case in Ref. [22]. For D = 3, this construction can be visualized as a “lasagna”.We first discuss the general CAHM case and then the twolimits. A. General case
For the CAHM, the magnetic time slices contain B -tensors on space-space plaquettes and the A -tensors at-tached to their space links. These A -tensors have 2( D − D = 3. Seen “from above”, in other words without thetime legs, this looks like the full D = 2 assembly shownin Fig. 1 FIG. 3. Magnetic layer of the transfer matrix for D = 3 ona time slice. Small circles (blue online) are used for the A -tensors and large circles (red online) for the B -tensors. In between the magnetic time slices we have electriclayers with B -tensors on space-time plaquettes labelledby e ( x ,τ ) ,j with a fixed τ , and the A -tensors attached totheir time links. These A -tensors have 2( D −
1) legs all inspatial directions. This is illustrated in Fig. 4 for D = 3.Seen “from above”, in other words without the time legsof the B -tensors, this looks like the full D = 2 assemblyfor the O(2) model.We want to represent these two types of layers as ma-trices. It is convenient to think of these two types oflayers as matrices connecting electric states |{ e }(cid:105) = ⊗ x ,j | e x ,j (cid:105) . (37)This is a natural choice because the B -tensors on thespace-time plaquettes have two legs in the time direction.In this basis, the electric layer can be expressed as adiagonal matrix T E with matrix elements (cid:104){ e (cid:48) }| T E |{ e }(cid:105) = δ { e } , { e (cid:48) } T E ( { e } ) , (38)where T E ( { e } ) can be written with some implicit nota-tions as a traced product of A tensors on time links with FIG. 4. Electric layer of the transfer matrix for D = 3 betweentwo time slices (top), small circles (blue online) are used forthe A - tensors and large circles (red online) for the B -tensors,and “from above” (bottom). B tensors on space-time plaquettes T E ( { e } ) = Tr (cid:89) time l. A ( l. ) m ,...m D − (cid:89) sp. − time pl. B ( pl. ) ( e ) . (39)Similarly, we can define a magnetic matrix T M with ma-trix elements (cid:104){ e }| T M |{ e (cid:48) }(cid:105) with the indices e and e (cid:48) carried by the time legs of the A -tensors. (cid:104){ e (cid:48) }| T M |{ e }(cid:105) =Tr (cid:89) sp. l. A ( l. ) m ,...m D − ( e , e (cid:48) ) (cid:89) sp. − sp. pl. B ( pl. ) . (40)All the traces are over the spatial legs of the tensors,while the time legs are left open and carry the the in-dices e and e (cid:48) . Figs. 3 and 4 should help visualizingthese matrix elements: the horizontal lines correspondto traced indices while the vertical indices carry the { e } indices.We can now define the transfer matrix T as T ≡ ( e − β pl. I ( β pl. )) ( V/N τ ) D ( D − / ( e − β l. I ( β l. )) ( V/N τ ) D × T / E T M T / E , (41)with N τ the number of sites in the temporal direction.With this definition, we can reexpress the partition func-tion as Z = Tr T N τ , (42) B. O(2) limit
For the O(2) model, the transfer matrix can be con-structed by taking all the O(2) tensors on a time sliceand tracing over the spatial indices. This is discussed indetail in Ref. [35] in the case D = 2 and the extensionto arbitrary dimension is straightforward. In the CAHMreformulation the implicit O(2) tensors attached at eachsite are divergenceless. By implicit, we mean that a linkindex n x,µ carries a weight t n x,µ ( β l. ) as in the O(2) model(see Eq. (22)).From the perspective of quantum simulations, thegauge parametrization insures that the divergencelesscondition is automatically satisfied and insensitive tonoise. However the Hilbert space becomes larger for D ≥ C. Pure gauge limit: a robust way to implementGauss’s law
The restricted electric Hilbert space of the pure gaugecompact U (1) model in D dimensions is equivalent to theset of legal tensor configurations of a D − e x,j are like the link variables n x,j forO(2). Both sets are divergenceless. For the pure gaugemodel, the divergenceless condition is Gauss’s law (with-out charge density).In Sec. VI B we presented the O(2) model as a weakgauge coupling limit of the compact Abelian Higgs model.As seen in Eq. (17), integration over the gauge fieldsprovides an automatic divergenceless condition for thelink quantum numbers. This is a discrete version of ∂ µ ∂ ν F µν = ∂ µ J µ = 0. The argument does not involvethe dimensional-dependent Levy-Civita tensor.We can insure that Gauss’s law is automatically satis-fied by introducing a new set of quantum numbers c x ,j,k ,associated with the plaquettes of a D − { c x ,j,k } , we impose e x,j = (cid:88) k>j [ − c x,j,k + c x − ˆ k,j,k ]+ (cid:88) k 1. It is possible to introduce dimension-dependent “magnetic” notations such as G = (cid:15) kl C kl for D = 3 and G j = (cid:15) jkl C kl for D = 4.For a D = 3 pure gauge theory we can visualize theelectric Hilbert space as a D = 2 O(2) model being on aplane between two time slices, as at the bottom of Fig. 4.We can further imagine the auxilliary variables located inthe middle of the plaquettes of this “horizontal” plane,which means in the center of the D = 3 cubes of theoriginal lattice. This is equivalent to the dual formulationdiscussed in Ref. [22].For D = 4, this reparametrization is a discrete equiv-alent of setting E = ∇ × G . (45)This guarantees Gauss’s law, but ∇ × E is in general non-zero so we don’t use this trick for conventional electro-statics because one of the homogeneous Maxwell’s equa-tion ( ˙ B = − ∇ × E ) implies that the magnetic fieldchanges with time.This method is very efficient for D = 3, because it re-duces the dimensionality of the Hilbert space. There isone index per site ( c x, , ) rather than 2 ( e x , and e x , ).For D = 4, there are 3 indices per sites in both case,because c x,j,k is only defined up to a gradient. However,the robustness against noise is an important advantage.At the end of Sec. VII, we briefly discuss possible opti-mizations. VII. HAMILTONIAN LIMIT For lattice models at Euclidean time, the transitionfrom the Lagrangian to the Hamiltonian formalism is astandard procedure [30, 31]. The central idea is to deformthe isotropic formulation by increasing the β variablesassociated with time directions and to decrease those as-sociated with space directions. In the tensor language,examples involving the transfer matrix in 1+1 dimen-sions [35, 37, 38] and 2+1 dimensions [22, 47] providesteps that will be followed below.The crucial features of T E is that it only involves timelinks and plaquettes having one direction in time. Weintroduce separate β τ couplings for T E and use redefini-tions in terms of the time lattice spacing a τ : β τpl. = 1 a τ g pl. , and β τl. = 1 a τ g l. . (46)Given the weak coupling (large β ) behavior of t n ( β ) givenin Eq. (20), at first order in a τ , we get “rotor” ener-gies (1 / g pl. m for the plaquettes and (1 / g l. n for thelinks.On the other hand, T M only involves space links andspace-space plaquettes and we redefine β s pl. = a τ J pl. , and β s l. = a τ h l. . (47)Given the strong coupling (small β ) behavior of t n ( β )from Eq. (20), at first order in a τ , the contribution to T M involve a single link or plaquette with quantum num-ber ± e x ,j over a link ( x , j ) or raise over two links andlower over the two other links of a plaquette.We define the Hamiltonian H as the order a τ correctionto the identity in the transfer matrix: T = − a τ H + O ( a τ ) (48)After introducing the operators [48] ˆ e x ,j and ˆ U x ,j suchthat ˆ e x ,j | e x ,j (cid:105) = e x ,j | e x ,j (cid:105) ˆ U x ,j | e x ,j (cid:105) = | e x ,j + 1 (cid:105) (49)ˆ U † x ,j | e x ,j (cid:105) = | e x ,j − (cid:105) , the discussion of the first order behavior of T E and T M allows us to write H = 12 g pl. (cid:88) x ,j (ˆ e x ,j ) + 12 g l. ( (cid:88) x ,j (ˆ e x ,j − ˆ e x − ˆ j,j )) − h l. (cid:88) x ,j ( ˆ U x ,j + h.c. ) (50) − J pl. (cid:88) x ,j 1. For in-stance, ∆ c x, , = 1 generates the following changes:∆ e x, = − , ∆ e x +ˆ2 , = 1 , ∆ e x, = 1 , ∆ e x +ˆ1 , = − . (52)This change can be visualized as an electric field circu-lating clockwise on a plaquette in the 1-2 plane and itclearly satisfies Gauss’s law. The changes correspond tothe U † U † U U term in the Hamiltonian (50). For D = 3,this is the end of the story and we can efficiently replacethe term with two raising and two lowering operators by aterm with a single raising or lowering operator [22]. Theconstruction can be repeated for any pair of directionsin higher dimensions, but as discussed in Sec. VI the c x,j,k have some redundancy. For D = 4, the geometricinterpretation is easy with three spatial dimensions: wecan combine 6 plaquettes on a cube in such a way thatall the electric quantum numbers cancel. In other words,the effect of one of the c x,j,k can also be obtained withfive others. For OBC, we could remove this redundancyby eliminating, for instance, all the c x, , except for thoseon a 2-3 plane at the boundary. For PBC, other sectorsshould be added in order to allow electric configurationswrapping around the spatial directions. VIII. TOPOLOGICAL SOLUTIONS ANDSEMI-CLASSICAL APPROXIMATIONS In Sec. III, we found a direct similarity between thecontinuous lattice equations of motion and the discretetensor selection rules. In this section we discuss the ef-fect of periodic boundary conditions on both sets of equa-tions. We will limit ourselves to the solvable cases: the D = 1 O(2) spin model and the D = 2 pure gauge U (1)model.For the D = 1 O(2) spin model with PBC and N τ sites, the equations of motion (12) with A x,µ = 0 areequivalent to the statement that sin( ϕ x +ˆ1 − ϕ x ) takesthe same value on every link. These equations have many0solutions and we will focus our attention on the ones thatcan be interpreted as continuous topological solutions inthe continuum limit for PBC. If we impose that ϕ x +ˆ1 − ϕ x is a small constant, we can obtain a solution that meetsthis requirement. Given any choice for the constant, wecan then “integrate” the equations: starting with some ϕ , we obtain ϕ , and so on until, due to PBC, we getan independent value for ϕ which should be consistentwith the initial value modulo an integer multiple of 2 π .This approximately corresponds to a smooth mapping ofthe circle into itself provided that the successive changescan be made arbitrarily small. This can be accomplishedby requiring that for all links ϕ x +ˆ1 − ϕ x = 2 πN τ (cid:96), (53)for a given integer (cid:96) . By taking, N τ large with fixed (cid:96) weobtain a solution which can be interpreted as a topologi-cal solution with winding number (cid:96) . In the limit (cid:96) (cid:28) N τ ,these solutions have classical action S (cid:96) (cid:39) β πN τ (cid:96) ) N τ . (54)We can calculate the quadratic fluctuations with re-spect to this solution. We can first use the global O(2)symmetry to set ϕ = 0. Other values of ϕ are takeninto account by performing the integration over ϕ whichwith our normalization of the measure yields a factor 1.By construction, the linear fluctuations vanish becausethe first derivatives are zero and all we need to calculateare the quadratic fluctuations∆ = N τ − (cid:89) x =1 (cid:90) π − π dϕ x π e − S quad.(cid:96) , (55)with S quad.(cid:96) = β πN τ (cid:96) )( ϕ +( ϕ − ϕ ) + · · · + ϕ N τ − ) (56)Following the standard quadratic path integral proce-dure, we find∆ = N − / τ (2 πβ cos( 2 πN τ (cid:96) )) − ( N τ − / . (57)We can now attempt to re-sum the topological con-tributions. This is delicate because we have assumed (cid:96) (cid:28) N τ , however if β is large enough, the terms withlarge (cid:96) are exponentially suppressed. In the same spirit,we will ignore the (cid:96) dependence of ∆ and use the Poissonsummation formula ∞ (cid:88) (cid:96) = −∞ e − B (cid:96) = (cid:114) πB ∞ (cid:88) n = −∞ e − (2 π )22 B n , (58)with B = β (2 π ) /N τ . Putting everything together, weget a semi-classical approximation of the partition func-tion in the large β limit Z (cid:39) (2 πβ ) − N τ / ∞ (cid:88) n = −∞ ( e − n β ) N τ . (59) We now consider the solutions of the discrete Eq. (13).The solution is that n x, should be constant. With PBC,this implies the exact expression: Z = ∞ (cid:88) n = −∞ ( e − β I n ( β )) N τ , (60)which can be compared to the semi-classical expressionEq. (59). Using the large β approximations e − β I ( β ) (cid:39) √ πβ (1 + O (1 /β )) , (61)and Eq. (20) in the same limit, we see the approximatecorrespondence between the two expressions.A similar construction can be carried for the D = 2pure gauge U (1) model with PBC. We consider a rectan-gular N s × N τ lattice. The equation of motion requiresthat sin( f x, , ) is constant. Following the analogy withthe O(2) case, we start with f x, , ≡ A x, + A x +ˆ1 , − A x +ˆ2 , − A x, = δ, (62)with δ a constant to be determined with PBC. As seen inSec. IV, we can gauge fix the temporal links with a givenspatial coordinate x to the identity with the exceptionof one time layer. For definiteness, we take this layer ofnontrivial time links to be between τ = N τ − N τ which is identified with 0 due to PBC. The space linkswith a given spatial coordinate, which can be visualizedas a vertical ladder can be treated as the indices of a D = 1 O(2) model changing by − δ at each step untilwe get to the “last” rung and temporal links are present.The constancy of the “last” plaquette requires that A ( x +1 ,N τ − , − A ( x ,N τ − , = N τ δ. (63)Iterating in the spatial direction, we obtain PBC in thespatial direction provided that δ = 2 πN s N τ (cid:96). (64)The action for this topological solution is S U (1) (cid:96) (cid:39) β πN s N τ (cid:96) ) N s N τ . (65)Note that we could have obtained another periodic solu-tion by setting all the time links to 1 and imposing PBCin time for N s independent D = 1 O(2) models, however,the action for these configurations is larger by a factor N s .The quadratic fluctuations can be calculated as in theO(2) case but with extra complications due to the specialtime layer. Keeping track of all the 2 π factors and usingPoisson summation for the winding numbers, we obtainthe semi-classical approximation Z U (1) (cid:39) (2 πβ ) − N s N τ / ∞ (cid:88) n = −∞ ( e − n β ) N s N τ , (66)1which agrees with the exact expression at leading order.As a test of the semi-classical picture we can calculatethe topological susceptibility. For this purpose we firstcalculate Z ( β, θ ) = (cid:89) x,µ (cid:90) π − π dA x,µ π e − S gauge − iθQ , (67)with the topological charge Q defined as Q = 12 π (cid:88) x sin( A x, + A x +ˆ1 , − A x +ˆ2 , − A x, ) . (68)The topological susceptibility is defined as χ = − d dθ ln( Z ) | θ =0 . (69)It can be calculated using the exact resummation [56] Z ( β, θ ) = ∞ (cid:88) n = −∞ [ e − β I n ( (cid:114) β − ( θ π ) ) × ( β − θ π β + θ π ) n/ ] N s N τ . (70)If χ is dominated by configurations corresponding towinding number ± | Q | (cid:39) β estimate χ (cid:39) (0) exp( − β π ) (1) N s N τ )+ ( − exp( − β π ) N s N τ ( − ) . (71)Fig. 5 shows that this estimate is reasonably good when β is large enough. - - - - β Ln ( χ ) FIG. 5. Logarithm of the topological susceptibility using theexact formula for N s = N τ = 8 expanded up to order 5 (dots)and the semi-classical approximation Eq. (71) (continuousline). As a remark, it is a common misconception to identifythe Fourier mode indices n in Eq. (60) as “topological sectors”. They are rather labelling “rotor energy lev-els” n / 2. The fact that Poisson summation interchangesthese energy levels with the correctly identified topolog-ical sectors was observed in Ref. [57] in a version of theO(2) model where the fluctuations are limited. Note alsothat it is possible to construct models where the approx-imations Eqs. (59) and (66) are exact. The questions oftopological configurations and duality are discussed forAbelian gauge models of this type in various dimensionsin Refs. [58, 59]. IX. CONCLUSIONS In summary, we have shown that some standard the-orems and identities associated with the U (1) symmetrythat can be derived in the conventional formulation offield theory have a discrete counterpart in TFT. Thisincludes the equations of motion, Noether’s theorem,Maxwell’s equations with charges and currents, Gauss’law, gauge-fixing and effects of boundary conditions.We have constructed gauge-invariant transfer matricesby reorganizing the partition function obtained integrat-ing over all the fields without gauge fixing. We also ex-plained how an equivalent partition function is obtainedby a gauge-fixing which removes intermediate projectionsinto the sector of the Hilbert space which satisfies Gauss’slaw. These projections are only useful in the case of anoisy evolution. We proposed a reparametrization of thethe sub Hilbert space satisfying Gauss’s law in arbitrarydimension which generalizes dual construction in D = 3[22].Practical implementations of TFT require finite trun-cations. They provide numerically accurate approxima-tions at finite volume [26, 35, 37, 38]. The results derivedhere depend only on the selection rules which completelycapture the symmetry and not on the numerical valuesof the Bessel functions appearing in Fourier expansions.This confirms the observation that truncations preservethe symmetries [40]. The class of universality is encodedin the selection rules of the tensors and it is expectedthat in the continuum limit, results should not dependon microscopic details. Similar expectations are foundin the quantum link approach [49–52]. One advantageof TFT is that it connects smoothly the Lagrangian andHamiltonian approaches in a way that allows testing us-ing standard importance sampling methods. This allowscomparisons with Hamiltonian based quantum simula-tions proposals for Abelian gauge models [60–65].The discrete nature of TFT formulations makes it ageneric tool to setup quantum computing protocols. Itprovides an alternative to field discretization [66–68].Motivations for quantum computing include doing ab-initio real-time calculations relevant to fragmentationprocesses and parton distribution functions [69]. In or-der to work on these ambitious and high-impact projects,we need to move up the steps of a “ladder” of models[31, 70] that has been proven effective to deal with the2static properties of hadrons. The first steps are the spinand gauge Ising models. Practical implementations arediscussed in Refs. [71–73]. The next steps are their coun-terparts with a continuous and compact Abelian U (1)symmetry [22, 52, 74, 75]. In this context, the Euclideantransfer matrix could also be used to prepare initial statesfollowing the suggestion of Ref. [76].Models with Wilson [13–15, 19, 21, 23] and staggered[26] fermions have also been reformulated using TFT.The Schwinger model is of great interest in this context.This model and its Z q approximations have been studieddirectly with the Hamiltonian formalism [41–46], provid-ing useful comparisons for future TFT calculations. ACKNOWLEDGMENTS We thank the late D. Speiser and C. Itzykson fortheir teaching on Pontryagin duality, Peter-Weyl theoremand strong coupling expansions. We thank J. Unmuth-Yockey for many discussions on Abelian gauge theories.We thank R. Edwards and JLab for the invitation togive lectures that helped sharpen some of the argumentsmade in the paper. We thanks M. Vander Linden for helpwith figures. 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