Ising model as a U(1) Lattice Gauge Theory with a θ -term
IIsing model as a U (1) Lattice Gauge Theory with a θ -term Tin Sulejmanpasic ∗ Department of Mathematical Sciences,Stockton Road, Durham University,DH1 3LE Durham, United Kingdom
We discuss a gauged XY model a θ -term on an arbitrary lattice in 1+1 dimensions, and showthat the theory reduces exactly to the 2d Ising model on the dual lattice in the limit of the stronggauge coupling, provided that the topological term is defined via the Villain action. We discuss thephase diagram by comparing the strong and weak gauge coupling limits, and perform Monte Carlosimulations at intermediate couplings. We generalize the duality to higher-dimensional Ising modelsusing higher-form U(1) gauge field analogues. The Ising model is one of the most important statis-tical mechanical models. Its simplicity, universality andexact solvability in 1d and 2d are just some of the rea-sons in its pervasiveness in physics. On the other hand,gauge theory is best known as the theory of light, whichcan be described by a gauge field space-time vector field A µ . The fundamental principle underlying U (1) gaugetheories is gauge invariance, i.e. the statement that allobservables are invariant under the gauge transformation A µ → A µ + ∂ µ ϕ , where ϕ is an arbitrary angle-valuedfunction of space-time coordinates (i.e. the target spaceof ϕ is U (1), hence the name). Crucially the k-chargeWilson loops e ik (cid:72) C dx µ A µ , with a closed spacetime con-tour C are invariant under gauge transformations only ifthe charge k is an integer. The space-time contour C hasan interpretation of a probe particle worldline, carrying k units of U (1) charge.In (1+1)d the connection of the gauge theory with theIsing model can be made evident by considering a sin-gle scalar field coupled to the U (1) gauge field and wasnoted before (see e.g. [1, 2]). The phenomenology of thismodel was discussed by Coleman long ago [3]. Firstlythere exist two, potentially different, regimes : the con-fining regime and the Higgs regime. In the deep confiningregime, the mass-squared M of the scalars is positiveand large, and the theory is very nearly a pure gaugetheory. Such a theory has excitations of φ , which can bevisualized as worldlines, wrapping in the Euclidean com-pact time direction. However the gauge field fluctuationsimpose a confining potential on these excitations, andthe excitations have to pay the energy price of the stringattached to the wordline. In addition the ensemble canbe thought of as consisting of tiny loops of scalar matter,which renormalize the string tension (see Fig. 1).One can then think about placing the system in theexternal electric field, which is equivalent to inserting anon-zero θ -term [3]. This setup corresponds to the Eu- The two regimes are in fact not separated by a phase transition,unless the θ -angle is set to π . FIG. 1. A cartoon of the 1+1 gauge theory ensemble. Thecircle is the compact Euclidean time, and spatial directionextends from left to right. The blue areas are the “vacuum”,while the red areas are vacuum excitations (i.e. strings) cost-ing finite energy per area due to the electric field between thesources. If the external electric field is introduced (i.e. the θ -term, the vacuum energy goes up, while the string tension, forappropriately oriented electric dipoles, goes down. At θ = π they become degenerate, and the elementary charges are nolonger confined. clidean action S θ = i θ π (cid:90) d xF , (1)where F = (cid:15) µν F µν = F is the (Euclidean) fieldstrength. As θ moves from zero to 2 π , the vacuum pairof positive and negative particles can move to infinity,exactly cancelling the background electric field. Exactlyat the midpoint, i.e. θ = π , the background electric fieldcorresponds to exactly half of the electric-string-flux. Forlarge and positive scalar M the, vacuum is twice de-generate, corresponding to the positive and negative di-rections of the half-electric flux, and hence breaks thecharge-conjugation symmetry C spontaneously. Chargedparticles can be thought of as changing the electric fluxby one unit, so half of the electric flux directed to theright can be absorbed by a negative charge, leaving theleft-pointing half-electric-flux vacuum on the right. How-ever if the charge was positive, it would produce a 3 / We will be interested only in the Euclidean formulation of Quan-tum Field Theories here. a r X i v : . [ h e p - l a t ] S e p excited vacuum. Their presence is hence energetically pe-nalized by the electric charge e and the size of the string.In the limit of e → ∞ , only domain-wall excitations re-main, whose statistical weight is penalized only by thelength of their wordlines, not the area they enclose. As M is decreased, the domain walls become more commonin the ensemble, mixing the two vacuua and causing C tobe restored. The nature of the transition is the Isingtransition [1, 2, 4].But the ensemble picture, in the limit e → ∞ , lookscompellingly similar to the Ising model itself, which canbe thought of as the ensemble of domain walls connectingthe two Ising vacua. Indeed we will see that there is alattice gauge formulation of the gauged XY model wherethis identification becomes exact. Moreover the formu-lation allows for a higher-dimensional generalization interms of higher-form lattice gauge theories. We focus onthe 2d model first for simplicity, which will render thehigher-dimensional generalizations straightforward. Wediscuss these at the very end. The 2d gauged XY-model an Ising model:
Letus consider a 2d lattice Λ which is made out of sites x ,the bonds or links l and faces or plaquettes p . The XYmodel can be defined by the phases ϕ x ∈ [0 , π ) livingon lattice sites. It is useful to define the derivative livingon the oriented link l ( x, y ) as( dϕ ) l ( x,y ) = ϕ y − ϕ x . (2)We can write the partition function of the XY-model as (cid:89) x (cid:18)(cid:90) π dϕ x (cid:19) e J xy (cid:80) l cos (cid:0) ( dϕ ) l (cid:1) , (3)where the sum in the exponent is over links of fixed ori-entation. To gauge the model we introduce a link gaugefield A l ∈ R . We further define F p = ( dA ) p ≡ A l + A l + · · · + A l i . (4)where the links l , l , . . . l i make the boundary of the pla-quette p .We take the action for the gauge fields to be S gauge = β (cid:88) p ( F p + 2 πn p ) − iθ (cid:88) p n p , (5)where n p are integer variables on plaquettes, and wherethe orientation of the plaquettes is fixed in advance. Thecoupling to the XY-model is made by promoting ( dϕ ) l → ( dϕ ) l + A l in the exponent of (3), so the partition functionis now Z = (cid:32)(cid:89) x (cid:90) dϕ x (cid:33) (cid:32)(cid:89) l (cid:90) dA l e J xy cos (cid:0) ( dϕ ) l + A l (cid:1)(cid:33) × (cid:89) p (cid:88) n p e − β ( F p +2 πn p ) + iθn p (6) The model has a 1-form gauge symmetry which allowsus to set A l ∈ [ − π, π ] [4, 5], so that the kinetic termis just the Villain discretization of the U (1) gauge fields[6]. The the θ -term was introduced in [4, 5]. Noting that (cid:80) p F p = 0, we can perform the Poisson resummation foreach plaquette (cid:88) n p ∈ Z e − β ( F p +2 πn p )+ iθ (cid:16) n p + Fp (2 π ) (cid:17) == 1 √ πβ (cid:88) m p ∈ Z e − ( mp − θ π ) β + iF p m p . (7)The RHS above is nothing but the Fourier expansion ofthe LHS, given that the LHS is periodic in F p → F p +2 π .Upon summing over all plaquettes, it is not difficult toshow that (cid:80) p F p m p = (cid:80) l A l ( δm ) l , where( δm ) l = m p − m p , (8)with p , p being the plaquettes which share the link l (the sign indicates that the plaquettes sharing the samelink have opposite orientations).If we ignore the coupling to the XY model, integratingover A l will impose a constraint that m p = m is constantfor all plaquettes.On the other hand we have that e J xy cos(( dϕ ) l + A l ) = (cid:88) k l ∈ Z I k l ( J xy ) e i ( dϕ l ) k l + iA l k l , (9)which is just a Fourier expansion of the LHS. I k ( J ) isthe modified Bessel function. Upon doing this for ev-ery link, the first term in the exponent can be “partiallyintegrated”, i.e. (cid:88) l ( dϕ ) l k l = − (cid:88) x ϕ x ( δk ) x , (10)where ( δk ) x = k l + k l + k l + · · · + k l i , (11)with l , l , . . . , l i being the links oriented away from thevertex x . Integrating over ϕ x , we have that ( δk ) x =0 , ∀ x ∈ Λ . This is nothing but the current conservationlaw, demanding that the net current k l flowing out/in of x is zero.The partition function is now made out of closed loopsof current k l . By integrating over A l , we further imposethe constraint k l = ( δm ) l . (12)Note that ( δk ) x is automatically satisfied given the aboveconstraint because δ = 0. The partition function is Z = (cid:18) πβ (cid:19) P/ (cid:88) { m } (cid:32)(cid:89) l I ( δm ) l ( J xy ) (cid:33) × (cid:32)(cid:89) p e − e ( m p − θ π ) (cid:33) (13)where we have labeled e = β , (cid:80) { m } indicates the sumover all plaquette variables m p , and P is the total numberof plaquettes on the lattice.Now consider the limit of e → ∞ , and θ = π . Theexponent in the 2nd line above suppresses all configura-tions for which m p is not equal to 0 or 1. Therefore, upto exponentially small corrections in e , the only allowedplaquette variables are m p = 0 ,
1. These will play therole of Ising spins. Let us label σ p = 2 m p −
1. More-over, note that since I n ( x ) = I − n , the dependence on( δm ) l = 2( δσ ) l = 2( σ p − σ p ) , where p and p are pla-quettes which share a common link l . Further, since σ p only take values ±
1, we can write I ( σ p − σ p ) / ( J xy ) = (cid:113) I ( J xy ) I ( J xy ) e − σp σp log (cid:16) I J xy) I J xy) (cid:17) . (14)Since the plaquettes p are dual to the dual lattice sites˜ x , the idenity above reveals that the model in questionis really the Ising model on the dual lattice, with thecoupling J I = − log (cid:16) I ( J xy ) I ( J xy ) (cid:17) h = π − θ πβ plays the role of themagnetic field. To get the Ising model at finite h , onemust take the double scaling limit θ → π, β → h is finite.Several comments are in order • If in (15) we take J xy > J I >
0, so themodel maps the ferromagnetic gauged XY-modelto the ferromagnetic Ising model. If J xy < A l → A l + π to transform J xy → − J xy . Nowif the original lattice Λ consists of only plaquetteswhich have an even number of links in their bound-ary (e.g. a square or a honeycomb lattice), then theshift can be absorbed by the shift of the integers n p in (6). If on the other hand hand there exist pla-quettes which have an odd number of links in theirboundary, it is not difficult to see that the resultingferromagnetic Ising model partition function con-tains a term e i σpπ , which can be interpreted as theimaginary magnetic field h = i π . • What about antiferromagnetic Ising model on afrustrated lattice? Does there exist a U (1) gauge-theory, whose dual lattice is frustrated (e.g. a hon-eycomb lattice), which is dual to an antiferromag-netic Ising model? For real J xy of the XY model,the answer is no. However one can always come upwith a complex value of J xy in (15) which wouldproduce a negative value of J I , so that the analyt-ical continuation of the U (1) gauge theory to com-plex J xy corresponds to an antiferromagnetic Isingmodel. • While we have assumed that the coupling J xy is thesame for all links, we could make them different.The relationship (3) would than be valid link-wise. • If we did not take the limit e → ∞ , the XY modelis still dual to a kind of generalized Ising model,with the spin σ ˜ x = 2 m ˜ x − • There is nothing particularly special about the formfor the XY model (3). Indeed we could have takenthe action to be an arbitrary periodic function of( dϕ ) l , i.e. S = (cid:80) l f (( dφ ) l ), where f ( x + 2 π ) = f ( x ). Then the Bessel functions I k should be re-placed by the Fourier modes of e − f ( x ) . The Isingcoupling would still be given by (15) with this re-placement.Could we imagine a generalized XY model describedabove, with purely real f ( x ) which corresponds to an an-tiferromagnetic Ising model? For that to happen we musthave that the 1st Fourier mode of e − f ( x ) is larger thanthe 0th mode, so that the logarithm in (15) is negativei.e. (cid:90) π − π dx e − f ( x ) cos( x ) > (cid:90) π − π dx e − f ( x ) . (16)However the above can never be satisfied for real f ( x ),and so we conclude that the antiferromagnetic Isingmodel on a frustrated lattice cannot be obtained froma gauged generalized XY model with real couplings. The finite coupling: from Ising to Berezinskii-Kosterlitz-Thouless (BKT) transition
As we saw when the gauge coupling tends to infinity, e → ∞ the ferromagnetic XY model is an Ising model.Let us focus on the square lattice for concreteness, whosedual lattice is also square. We know that the Ising modelon the square lattice has a transition at J c I = log(1 + √ ⇒ J c xy ( e = ∞ ) ≈ . , (17)where the XY coupling at e = ∞ was obtained with theuse of (15). On the other hand we know that if e → A l = 0, reducing the model to an ordinary XYmodel, which has a BKT transition at J c xy ( e = 0) = 1 . . (18) Actually the zero coupling condition forces ( dA ) p = 0, but therecan still be a residual nonzero holonomy in case of space-timewhich has incontractible loops (i.e. a nontrivial 1st cohomologygroup) e.g. a torus. In this case the holonomies label superse-lection sectors of the flat-connection gauged XY model. C-broken (ordered)C-restored (disordered)BKT transition pointIsing transition lineDeconfined crtical phase
FIG. 2. The phase diagram of the gauged XY model. Thediagram shows the transition line in the J xy v.s. e plane.The critical points were determined from the intersectionsof the rescaled susceptibility L νγ χ t ( J xy ) obtained by MonteCarlo simulations of the model (22) on the square latticeswith sizes L = 40 , ,
80. The inlay shows a typical rescaledsusceptibility for e = 6 (including L = 20), which clearlyintersect very close to a single point – the Ising transitionpoint. The two transitions are quite close together, differingby only ∼ e = 0 (i.e. β = ∞ ), we seethat in (22) the proliferation is in terms of interface-linesbetween different values of m p -variables, which are nolonger constrained to be m p = 0 ,
1. Both of these prolif-erations are controlled by the ratio of the Bessel function I ( δm ) l ( J xy ) /I ( J xy ), which tends to suppress the jumpsin m p for smaller values of J xy , and lets them proliferatefor large J xy . For intermediate 0 < e < ∞ , the typicalarea of a loop bounding the region of constant m p (cid:54) = 0and m p (cid:54) = 1 are exponentially suppressed with e , andsuch domains will tend to renormalize the Ising transi-tion, but the effect must be exponential in e − e ( ... ) (seeeq. (22)). On the other hand let us consider the limit e → J xy (cid:46) . e to be nonzero, 1 /e will dictate the typical length-scaleof gauge fluctuations in lattice units, and so it cannotinduce a phase transition until e is of the order of the XY mass-gap, which is exponentially small for the cou-pling close enough to J xy = 1 . J xy v.s. e tobe slowly changing as e is lowered from infinity, keepingclose to the J c xy ( e = ∞ ) = 0 . J c xy ( e = 0) = 1 . e on a square latticeand for the linear system sizes L = 20 , ,
60 and 80. Wedefine the topological susceptibility as χ t = 1 L ∂ log( Z ) ∂θ + e (2 π ) . (19)The shift by the constant above is to match the definitionof the magnetic susceptibility in the Ising limit . At finitevolume we expect χ t = L γν F ( tL ν ) (20)where for 2d Ising ν = 1 and γ = 7 / t is the parameter driving the tran-sition, and F is the universal function. So if we plot L νγ χ t against a parameter driving the transition, we ex-pect that, at the phase transition point t = 0 the curveswill cross. Indeed, plotting L νγ χ t against J xy shows thatall the curves intersect pretty closely at a single point,as can be seen in the inlay of Fig. 2 where simulationsfor e = 6, are shown for the four volumes. We repeatedthe simulations for values of e ranging from e = 0 . L = 20 to minimize power corrections to the scaling.In addition the Ising scaling, discussed above does notset in at L = 20 for the smallest values of e . This isexpected as the dominant fixed point for small enoughvolumes should be of the BKT nature. Generalizations to higher dimensions:
Generaliza-tion to higher dimensional cases is now straightforward.First we define the lattice Λ in terms of p -cells c p . A0-cell is a vertex. We then connect vertices with 1-cells(links), and 1-cells with 2-cells (plaquettes), etc. In D -dimensions we define a ( D − U (1),which will naturally live on ( D − B c D − . This is the generalization of A l for thespacetime dimension D = 2. In addition we introduce( D − A c D − , living on c D − . Similarto before we define the derivatives d and δ which map a p -form field to a p +1 and p − (cid:88) c D β dB ) c D + 2 πn c D ] + iθn c D − J (cid:88) c D − cos[( dA ) c D − + B c D − ] . (21) Since the susceptibility diverges at the transition point, the con-stant shift affects the finite volume corrections only. This par-ticular shift makes these correction small.
The action is just the generalization of the exponent in(3). Note that the θ -angle has a similar interpretation asbefore: a D − U (1) gauge field B has a naturaltopological charge given by π (cid:82) dB . An example of sucha gauge field is the nonabelian Chern-Simons 3-form in 4spacetime dimensions. Similar reasoning as before leadsto the dual partition function Z = (cid:18) πβ (cid:19) C ( D ) / (cid:88) { m } (cid:89) c D − I ( δm ) cD − ( J ) × (cid:32)(cid:89) c D e − e ( m cD − θ π ) (cid:33) , (22)where again e = β , and C ( D ) is the number of D -cellson the lattice. Now we identify c D with the site of a duallattice ˜ x , and define σ ˜ x = 2 m ˜ x − θ = π , in the limit e → σ = ± D -dimensional Ising modelwith the coupling given by (15), with J xy replaced by J .Let us briefly discuss the phase diagram as a functionof e . At e → ∞ , we have that the model undergoes aphase transition at some value of J c , which correspondsto the Ising transition via the duality relation. Just likebefore, as e is reduced, the phase transition is expectedto raise to slightly larger values of J c , similar to Fig. 2.However in the limit e →
0, the model in question is the( D − D = 3, it is just the usual lattice gaugetheory, which is well known to always be in the gappedphase [7, 8], because the theory always has monopoles.However these are expected to be suppressed exponen-tially with J , and so for very large values of J , the massgap M will be exponentially small. The introduction ofnonzero e , where e again has an interpretation as thelength of the B -field fluctuations, will therefore be ableto induce a transition only when e is of the order of M , which is tiny. So the qualitative picture is very similar toFig. 2, except that the phase-transition boundary shootsup to infinity for e → Acknowledgments:
I would like to thank BernardPiette for helping me understand how to use the Con-dor computer cluster at the Department of Mathemat-ical Sciences, Durham University. I would also like tothank Christof Gattringer for useful comments on themanuscript, and, together with Daniel G¨oschl and NabilIqbal for input with regards to the Monte Carlo erroranalysis. This work is supported by the Royal Society ofLondon University Research Fellowship. ∗ [email protected][1] I. Affleck, “Nonlinear sigma model at Theta = pi:Euclidean lattice formulation and solid-on-solid models,” Phys. Rev. Lett. (1991) 2429–2432.[2] Z. Komargodski, A. Sharon, R. Thorngren, and X. Zhou,“Comments on Abelian Higgs Models and PersistentOrder,” arXiv:1705.04786 [hep-th] .[3] S. R. Coleman, “The Uses of Instantons,” Subnucl. Ser. (1979) 805.[4] D. G¨oschl, C. Gattringer, and T. Sulejmanpasic, “Thecritical endpoint in the 2-d U(1) gauge-Higgs model attopological angle θ = π ,” PoS
LATTICE2018 (2018)226, arXiv:1810.09671 [hep-lat] .[5] T. Sulejmanpasic and C. Gattringer, “Abelian gaugetheories on the lattice: θ -terms and compact gaugetheory with(out) monopoles,” arXiv:1901.02637[hep-lat] .[6] J. Villain, “Theory of one-dimensional andtwo-dimensional magnets with an easy magnetizationplane. 2. The Planar, classical, two-dimensional magnet,” J. Phys.(France) (1975) 581–590.[7] A. M. Polyakov, “Quark Confinement and Topology ofGauge Groups,” Nucl. Phys. B (1977) 429–458.[8] A. M. Polyakov,