Nonlinear Sigma Models with Compact Hyperbolic Target Spaces
Steven Gubser, Zain H. Saleem, Samuel S. Schoenholz, Bogdan Stoica, James Stokes
PPrepared for submission to JHEP
CALT-TH 2015-019PUPT-2487
Nonlinear Sigma Models with Compact HyperbolicTarget Spaces
Steven Gubser a Zain H. Saleem b,d
Samuel S. Schoenholz b Bogdan Stoica c JamesStokes b a Joseph Henry Laboratories, Princeton University,Princeton, NJ 08544, USA b Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, PA 19104 c Walter Burke Institute for Theoretical Physics,California Institute of Technology, 452-48, Pasadena, CA 91125, USA d National Center for Physics,Quaid-e-Azam University Campus, Islamabad 4400, Pakistan
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We explore the phase structure of nonlinear sigma models with target spacescorresponding to compact quotients of hyperbolic space, focusing on the case of a hyperbolicgenus-2 Riemann surface. The continuum theory of these models can be approximated bya lattice spin system which we simulate using Monte Carlo methods. The target space pos-sesses interesting geometric and topological properties which are reflected in novel featuresof the sigma model. In particular, we observe a topological phase transition at a criti-cal temperature, above which vortices proliferate, reminiscent of the Kosterlitz-Thoulessphase transition in the O (2) model [1, 2]. Unlike in the O (2) case, there are many differenttypes of vortices, suggesting a possible analogy to the Hagedorn treatment of statisticalmechanics of a proliferating number of hadron species. Below the critical temperature thespins cluster around six special points in the target space known as Weierstrass points.The diversity of compact hyperbolic manifolds suggests that our model is only the sim-plest example of a broad class of statistical mechanical models whose main features can beunderstood essentially in geometric terms. a r X i v : . [ h e p - t h ] O c t ontents The breaking of continuous symmetries is accompanied by the appearance of masslessGoldstone modes. Fluctuations of these modes destroy long-range order at any finite tem-perature in dimensions d ≤
2; this is the statement of the Mermin-Wagner theorem. Thisdoes not exclude the possibility of “quasi long-range order”, however, in which correlatorsexhibit power-law, rather exponential decay. This behavior is associated with a continuousphase transition driven by the proliferation of topological defects [3].We can expect topological transitions to occur in a spin system whenever the homotopygroup of the spin space is non-trivial. The XY model is the simplest example in whichthe spin space of S has homotopy group π ( S ) = Z . This is to be compared with the O (3)-invariant Heisenberg model which has π ( S ) = 0 and thus does not exhibit quasi-long-range order in d ≤
2. The first non-trivial example with two-dimensional spin spaceis the torus T = S × S which is essentially two decoupled copies of the XY model. Amore interesting possibility is to consider the two-handled double torus which we study inthis paper.The novelty of the double torus is that, as a consequence of Gauss-Bonnet theorem, itadmits a metric with constant negative curvature, (cid:90) √ gR = 2 − g < , (1.1)since g = 2. The double torus can be obtained as a quotient of the two dimensional hyper-bolic plane H by a discrete subgroup of SO (2 , g = 0 and g = 1 counterparts is the existence of preferred pointsin the manifold, an example of which are the Weierstrass points. To define a Weierstrasspoint, first consider the set of all meromorphic functions which are holomorphic away froma specified point P . P is a Weierstrass point if this set contains more meromorphic func-tions with poles at P of some specific order than are guaranteed by the Riemann-Rochtheorem [4]. We will work with a particularly simple double torus corresponding to a tilingof the hyperbolic plane by regular octagons in which opposite sides are identified. TheWeierstrass points corresponding to this particular identification are known [4] to be thecenter of the octagon together with the midpoints of the sides, as well as the point deter-mined by the eight identified vertices. We will refer to this manifold as the regular doubletorus.In this paper we will perform Monte-Carlo simulations of the double torus model ona two dimensional square lattice with periodic boundary conditions. We will study theimpact of the non-standard topological and geometric properties of this model on thephase transition. Our results can be summarized as follows: • There is a phase transition at a finite temperature T c . Numerical results are consistentwith a second order phase transition, but do not exclude the possibility that the phasetransition is of infinite order as in the XY model. Our numerical results also do notexclude the possibility of additional phase transitions. • For temperatures slightly below T c , the spins cluster around one of six special pointson the regular double torus, which can be defined as fixed points of the discreteautomorphism group. This is quite unlike the XY model, in which there are nopreferred points in the target space. • For temperatures somewhat above T c , vortices of many different topological typesappear, their numbers following thermal distributions as one would predict fromtreating them as free, independent excitations.The organization of the rest of this paper is as follows. In section 2, we explain the generalframework of lattice models with hyperbolic quotients as target spaces and specify theprecise model we are interested in. In section 3 we briefly describe our numerical methodsand then explain our results, focusing on the points just summarized. We end with adiscussion including future directions in section 4. The n -dimensional hyperbolic space H n is a maximally symmetric Euclidean manifoldof constant negative curvature. It can be embedded in R ,n in a manifestly SO (1 , n )symmetric manner, 1 = X − X − · · · − X n . (2.1)– 2 –ore precisely, H n is the upper sheet of the two-sheeted hyperboloid described by Eq.(2.1). We consider non-linear sigma models with target space given by the quotient space H n / Γ where Γ is a discrete subgroup of the orientation-preserving isometries SO + (1 , n ). Inthe ‘upstairs’ picture we can think of the covering space H n as being tiled or tessellated bycells, each of which is related to the fundamental cell by the action of a particular elementof the group Γ. We only consider orientation-preserving isometries so that the resultingtopological space is orientable. Moreover, we assume that Γ is freely acting so that theresulting quotient space does not have fixed points.The Poincar´e ball model maps the hyperboloid H n to the unit ball where the inducedmetric is given by ds = 4(1 − r ) ( dr + r d Ω n − ) . (2.2)We will mostly focus on the n = 2 case where the isometries SO + (1 ,
2) are realized asfractional linear transformations acting on the unit disc in C , z −→ az + ¯ ccz + ¯ a , (2.3)with a, b ∈ C such that | a | − | c | = 1. The geodesics in the Poincar´e coordinates areeither diameters or arcs of circles intersecting orthogonally with the boundary of the disc.These geodesics form the edges of the regular hyperbolic polygons which tile H . A tiling byregular p -gons with q polygons meeting at each vertex exists provided that 1 /p +1 /q < / p even so that the sides of each polygon can be paired. Thesubset γ ⊂ Γ of group elements which pair the sides of the polygon are the generators ofΓ. Note that γ is not a group and the group Γ is obtained by multiplying the elements of γ in all possible ways, Γ = { g · · · g n | g i ∈ γ } . (2.4)In this way we construct a compact orientable surface with constant negative curvature.Note that this procedure will not always lead to a smooth surface. For example in the caseof the { , } tessellation we obtain the genus-2 hyperbolic Riemann surface, i.e. the regulardouble torus, but the surface obtained from the { , } tessellation must necessarily havecusps in order to be consistent with the Gauss-Bonnet theorem. The general structure of the Hamiltonians we consider is H = (cid:88) (cid:104) x,y (cid:105) h ( s x , s y ) . (2.5)Here x and y label sites on the square lattice, and the sum is over nearest neighbors. s x and s y are the ‘spins,’ in other words s x ∈ M for each point x on the lattice. For the XYmodel M = S and the standard approach is to replace it with a real variable θ x , with θ x and θ x + 2 π identified. The function h ( s x , s y ) is a map M × M → R which is bounded– 3 – (cid:45) (cid:45) (cid:45) Figure 1 . The fundamental domain and neighboring cells for the { , } tessellation. below, usually with its lower bound attained precisely when s x = s y . For the XY model,the standard choice is h ( θ x , θ y ) = 1 − cos( θ x − θ y ) . (2.6)A convenient choice for some purposes is the so-called ‘Villain approximation’ [5], h ( θ x , θ y ) = min n ∈ Z ( θ x + 2 πn − θ y ) , (2.7)where the sum over n enforces 2 π periodicity of θ x and θ y . The natural generalization ofEq. (2.7) to the quotient H / Γ is h ( s x , s y ) = min γ ∈ Γ [ γ ( s x ) − s y ] , (2.8)with s x and s y points on H ⊂ R , . The right-hand side is non-negative because any twopoints on the hyperboloid H are spacelike separated.An important feature of the Villain energy function (2.7) is that it is continuous butonly piecewise smooth: there is a discontinuity in its first derivative along the locus where θ x − θ y ≡ π mod 2 π . Likewise, the generalization (2.8) is continuous but only piecewisesmooth: For example, if s y is at the origin of the fundamental octagon, then h ( s x , s y ) hasdiscontinuities in one of its first derivatives at the boundaries of all images of that octagon.An intrinsic coordinate system with periodically defined coordinates is not known, so itis non-trivial to give an explicit, smooth map analogous to (2.6). We will therefore workstrictly with the ‘Villain’ form (2.8), which we may equivalently define as h ( s x , s y ) = min γ ∈ Γ [ − γ ( s x ) · s y ] , (2.9)where the dot product is in the standard mostly plus flat metric on R , .The equivalent forms (2.8) and (2.9) are not suited to computation unless we canefficiently restrict the minimization to a small subset of the elements in Γ. For the XY– 4 –odel, this is easy to do: one requires θ x , θ y ∈ ( − π, π ), and then the only images one needsto consider are θ x and θ x ± π . We must ask: If O ⊂ H is the fundamental octagon, andwe require s x , s y ∈ O , then what is the analogous subset of images γ ( s x ) that we mustminimize over to be sure of finding the minimum value of − γ ( s x ) · s y ?A sufficiently large subset of Γ for the regular double torus is the identity elementtogether with elements γ such that γ ( O ) touches the fundamental octagon either along aside or at a corner. This subset, call it Γ , has 49 such elements, which can be constructedas follows. A standard basis { γ , γ , γ , γ , γ , γ , γ , γ } = { α , β , α , β , α − , β − , α − , β − } (2.10)for Γ satisfies the identity (cid:81) i =0 γ i = 1. The 49 group elements of interest are 1 togetherwith γ jk ≡ k (cid:89) i = j γ i mod 8 , (2.11)where j < k . Counting the distinct γ jk is straightforward if we consider how they move usalong the dual graph to the octagonal tiling of H , a subgraph of which is shown in Fig. 2.A simple topological way to define Γ is that it is the minimal set of generators such thatan open set S ⊂ H can be found satisfying (cid:83) γ ∈ Γ γ ( O ) ⊃ S ⊃ O . Figure 2 . The subgraph of the { , } tiling of H corresponding to the group elements γ jk . Eachnode of the graph corresponds to an octagon in the tiling. Connected nodes correspond to octagonswhich share an edge. The octagons corresponding to group elements γ , γ , γ , γ , and γ allshare a vertex with the 1, γ , and γ octagons. A simulation of the torus model amounts to designing a Markov chain process whichresults in random sampling of configurations from a probability distribution proportionalto e − H/T , where H is given by (2.5) and h is given equivalently by (2.8) or (2.9). Tobuild such a process, one must be able to choose spins s x in the fundamental octagon withuniform probability with respect to the natural measure inherited from H and one must– 5 –e able to evaluate all instances of h ( s x , s y ), which in practice is done by restricting theminimization in (2.8) or (2.9) to γ ∈ Γ . For temperatures T slightly smaller than the critical temperature where the specific heatis maximized (Fig. 3), we notice a surprising clustering of spins around one of the sixWeierstrass points (Fig. 4). Infrared fluctuations must eventually cause the system toexplore all possible regions of phase space, so for sufficiently large lattices we would expectto see a distribution of spins which is democratic among the Weierstrass points at any fixedtemperature. Thus the interesting point is that a spatial correlation length is large enoughnear the critical temperature so that essentially our whole lattice clusters in the vicinity ofone Weierstrass point. E / L T L = 50 L = 100 C V / L T L = 50 L = 100 E / L T L = 50 L = 100 C V / L T L = 50 L = 100 Figure 3 . Average energy (left) and heat capacity (right) for the { , } model in the Villainapproximation. There is clear evidence of a phase transition at T (cid:39) . Above the critical temperature a topological phase transition occurs as a result ofof vortex proliferation. Unlike the XY-model where the winding of a loop of spins ischaracterized by an integer, the winding of spins in the double torus model is characterizedby an operator, which can be determined in the following way: For any consecutive pair ofspins s and s in the loop we find the image of s that is closest to s . This amounts tofinding the operator O s,t ∈ Γ that minimizes the distance s · ( Ot ). If the loop consists ofspins s , . . . , s n , the operator corresponding to the loop is O loop = O s ,s · · · O s n − ,s n O s n ,s . (3.1)We map this operator to a number by taking the matrix trace. We observe only one pronounced local maximum for the specific heat. Thus we only have evidencefor one critical temperature where a phase transition may occur. It is interesting, however, that there aremany types of vortices, and it is not impossible that there could be many phase transitions as a result. – 6 – igure 4 . Preferred spin configurations below critical temperature.
The fact that the winding is characterized by a non-abelian matrix rather than anumber suggests that we should consider loops which are larger than the elementary 1 × O loop but we may see a bigger set of group elementsif we consider larger loops. Indeed, this intuition is corroborated by simulations (see Fig. 5)in which we see that the number of vortices of fixed trace increases as we increase the loopsize from 1 × ×
3. The logarithmic scale on these plots indicates that vortices areproliferating approximately according to Boltzmann statistics; that is, N V = N exp( − E/T ) (3.2)for some N and E . Moreover, the vortices appear for all temperatures, not just for T > T c .A closer examination of the occurrence of vortices is worthwhile. We refer to Fig. 5and Table 1. The main features to note are: • The exponential law (3.2) persists up to temperatures comparable to T c . At largertemperatures, we see some evidence of saturation, where the number of vorticesper site becomes greater than 1 /
10, and continuing to follow (3.2) would eventuallyconflict with the limit N V /L <
1. Provisionally then, we think of (3.2) as a dilutegas approximation. – 7 – N V / L /T N V / L /T N V / L /T Figure 5 . Number vortices of fixed trace using plaquettes of size 1 × × × • We found it useful to distinguish vortices based on the trace of their monodromymatrix O loop . Bigger tr O loop presumably means a larger vortex with bigger energy.But we cannot conclude that all vortices with the same tr O loop have the same energy.Most likely, each trace class includes vortices of different energies, and the vorticeswith the lowest energies dominate N V in that particular trace class. To see thisexplicitly we plot in Fig. 6 the number of different O loop observed with constanttr O loop as a function of temperature for 1 × O loop =22 .
37 begins to proliferate the number of observed O loop increases in turn as expected.However, we also see that as tr O loop = 134 . O loop = 22 .
37 vortices also occurs. We interpretthis increase to hint that a population of tr O loop = 22 .
37 have higher energy. Thustr O loop is correlated with energy but does not uniquely determine it. • The energies E determined by fitting N V in a given trace class to the Boltzmann– 8 – × N E Trace25.0 3.09 22.37.51 6.05 135 . . · − . × N E Trace58.5 3.11 22.3558. 6.62 135.249. 6.74 341.13.0 5.93 453.100. 7.36 791.12.4 7.41 1 . · . · . · . · . · × N E Trace78.9 3.06 22.33 . · . · . · . · . · . · . · . · . · . · . · . · . · Table 1 . Fit parameters for plaquettes of size 1 ×
1, 2 × ×
3. The emphasized valuescorrespond to the data in Fig. 5. form (3.2) do not change much when we go from 1 × × × O loop = 340 . × ×
2, but in this case the sample size for1 × E encourages the view that we cantreat vortices in a dilute gas approximation. But we should be a little cautious aboutvalues of E since the fits are sometimes shaky due to noisy data and a limited rangeof 1 /T . • The prefactor N , which we think of as related to the exponential of the fugacity, isconspicuously different for some trace classes from 1 × × × N rises drastically with plaquette size, as it does for trace 340 . × × N is to beexpected as plaquette size increases, because if the vortex is small, a large plaquettewill enclose it completely at several different positions, all of which contribute to N .We suspect that a low-temperature expansion in terms of a dilute gas of vortices– 9 – V o r t e x T y p e s T Figure 6 . The number of different O loop elements for fixed trace observed in the system as afunction of temperature from trace 22.37 (blue) to trace 340 (red). of many types can be used to account for much of the dynamics up to T c . It would beinteresting to pursue this further because it could be a low-dimensional analog to the hadrongas treatment of the low-temperature phase of QCD, inspired originally by Hagedorn’sstatistical bootstrap approach [6]. Optimistically, one might try to estimate T c in termsof the growth in the number of different vortices with energy. The number of differentvortex types increases exponentially with the length on H between an initial point s andits image O loop s . (This is essentially the statement that the area enclosed by a circle in H increases exponentially with its radius.) So it is not implausible that the number ofdifferent vortex types also increases exponentially with energy, facilitating a Hagedorn-styleargument where a “vortex gas” eventually reaches a maximum possible temperature, whichis T c . However, with our present numerical results, we cannot go very far in developing sucha vortex gas model, for two main reasons. First, we don’t have a clear notion of the energyof a vortex; certainly the trace tr O loop is only very loosely correlated with the energy E as obtained from a fit of N V to the Boltzmann form (3.2) for vortices in a given traceclass. Moreover, the standard Kosterlitz-Thouless treatment of the XY model discouragesus from thinking that the energy of a single vortex is well-defined in isolation. Second, thecoefficients N are not really accessible from numerics. Ideally they should correlate withthe number of vortices of a given energy.A simple alternative point of view is that T c occurs naturally when the dilute gasapproximation for the smallest vortices breaks down, and the larger vortices have at mosta modest effect on the thermodynamics. Referring to the 1 × T only modestlybelow T c . – 10 – Discussion and future directions
We have considered one of an infinite number of tilings of the hyperbolic plane H . The { , } is simplest tiling which yields a smooth target manifold but it is straightforward togeneralize to hyperbolic Riemann surfaces of higher genera. It will also be interesting toconsider quotients of hyperbolic space H n with n >
2. In addition, it would be interestingto consider deformations in the moduli space of the double torus which correspond tochanging the geodesic lengths of the various cycles. Under such conditions one can expecta splitting of the phase transition into different critical temperatures corresponding to eachcycle.A natural question to ask is whether the sigma models studied in this paper have well-defined UV completions. A general nonlinear sigma model is a theory of maps R −→ T with Euclidean action functional given by S = 12 α (cid:90) d x g µν ∂ α X µ ∂ α X ν . (4.1)The beta function for this theory can be computed perturbatively in the coupling parameter α . Assuming that T is an n -dimensional maximally symmetric space for simplicity, β µν = α (cid:48) R µν + α (cid:48) R µλσρ R λσρν + O ( α (cid:48) ) , α (cid:48) ≡ α π , (4.2) β ( α (cid:48) ) = ∂α (cid:48) ∂ log µ = − α (cid:48) n g µν β µν . (4.3)Here µ and ν are curved indices for the target manifold T and α is a flat index for theworldsheet R . Taking T = H n , the one-loop beta function is given by β ( α (cid:48) ) = ( n − α (cid:48) ,which naively suggests a Landau pole in the UV. It is conceivable, however, that thehigher derivative terms can balance the one-loop contribution and drive the theory to aconformal fixed point [7, 8]. There also exist arguments [9] that models defined on compacttargets with negative curvature H n / Γ should possess conformal fixed points which arisefrom the competition between infrared freedom at weak coupling and the discrete spectrumof the Laplacian at strong coupling. It is therefore an important open problem to verifyif the transition seen in simulations is of the second-order type, and to ascertain if criticalfluctuations are described by a conformal field theory. Answering this delicate questionmay require moving beyond the Villain approximation, by finding an exact embedding ofhyperbolic double torus.In this paper we have focused on the models with constant negative curvature andcompact target H n / Γ but it is also interesting to consider models on the non-compact space H n . The non-compactness of the target makes it difficult to define correlation functionsand a non-standard basis of observables may be required [10].Finally, the modern conformal bootstrap techniques have so far only been applied totheories with compact symmetry groups. It would be very interesting to extend them tothe non-compact groups. – 11 – Acknowledgements
J.S., Z.H.S. and S.S.S. would like thank Randall Kamien, Hernan Piragua and AlexanderPolyakov for discussions. B.S. would like to thank Hirosi Ooguri for useful discussions,and the Institute for Advanced Study, Princeton University, and the Simons Center forGeometry and Physics for hospitality. B.S. also gratefully acknowledges support from theSimons Summer Workshop 2015, at which part of the research for this paper was performed.J.S. is supported in part by NASA ATP grant NNX14AH53G. Z.H.S. is supported in partby DOE Grant DOE-EY-76-02-3071. S.S.S. is supported by DOE DE-FG02-05ER46199.B.S. is supported in part by the Walter Burke Institute for Theoretical Physics at Caltechand by U.S. DOE grant DE-SC0011632.
A Coordinate systems
The following coordinate system covers the upper sheet of H in R , , X = cosh ρ , (A.1) X = sinh ρ cos φ , (A.2) X = sinh ρ sin φ . (A.3)The induced metric and area element are given by ds = dρ + sinh ρdφ , dA = sinh ρdρdφ . (A.4)The form of the area element implies that we should choose φ uniformly in [0 , π ] andchoose the cumulative distribution function v to be uniformly distributed in [0 , v = (cid:82) ρ sinh ρ (cid:48) dρ (cid:48) (cid:82) ρ c sinh ρ (cid:48) dρ (cid:48) = cosh ρ − ρ c − . (A.5)Therefore ρ = cosh − (1 − v + v cosh ρ c ) . (A.6)Defining x = sinh ρ cos φ and y = sinh ρ sin φ we obtain the alternative parametrization X = (cid:112) x + y , (A.7) X = x , (A.8) X = y . (A.9)The Poincar´e disc model is obtained by defining r = tanh( ρ/ ds = 4(1 − r ) ( dr + r dθ ) . (A.10)and the relationship with the rectangular coordinates is r = tanh (cid:20)
12 cosh − ( (cid:112) x + y ) (cid:21) , (A.11) θ = tan − (cid:16) yx (cid:17) . (A.12)– 12 – Parametrization of hyperbolic polygons
The solutions to the geodesic equation on the Poincar´e disc are given by [11] x ( s ) = cos φ cosh s − R sin φ sinh s √ R cosh s + R ,y ( s ) = R cos φ sinh s + sin φ cosh s √ R cosh s + R , (B.1)where s ∈ ( −∞ , + ∞ ). These functions define arcs on the Poincar´e disc with the radius R and the centers at the point x = √ R cos φ , y = √ R sin φ , lying beyond the unitdisc. In order to draw a the fundamental polygon corresponding to the { p, q } tessellation,we need to fix the angle α between any two neighboring lines from the center of the disk tothe vertices to be α = 2 π/p and the distance a of all the vertices from the center of the diskto be such that the interior angles are 2 π/q . We will assume the corners of the polygon areat the points a exp ( ikπ/ a exp i ( α + kπ/ p geodesics correspondingto the p exterior circles are completely specified by radii R ± and angles φ ± + kπ/ R ± = 12 a (cid:113) T ± + (1 − a ) , φ ± = arctan (cid:34)(cid:18) T ± a (cid:19) ± (cid:35) , T ± = a ± tan( α − π/ . (B.2)For the { , } tessellation we require a = 2 − / and φ ± with k = 0 , , References [1] V. L. Berezinskii, Sov. Phys. JETP , 610 (1972).[2] J. M. Kosterlitz and D. J. Thouless, J. Phys. C: Solid State Phys. , 1181 (1973).[3] D. R. Nelson, B. I. Halperin, Phys. Rev. B. , 2457 (1979)[4] T. Kuusalo and M. N¨a¨at¨anen, Quart. J. Math. , 355 (2003).[5] W. Janke and H. Kleinert, Nucl. Phys. B , 135 (1986).[6] R. Hagedorn, Nuovo Cim. Suppl. , 147 (1965).[7] J. J. Friess and S. S. Gubser, Nucl. Phys. B , 111 (2006), hep-th/0512355.[8] G. Michalogiorgakis and S. S. Gubser, Nucl. Phys. B , 146 (2006), hep-th/0605102.[9] A. M. Polyakov, Nucl. Phys. B , 199 (2008), arXiv:0709.2899.[10] A. Polyakov, Z. H. Saleem and J. Stokes, Nucl. Phys. B , 54 (2015), arXiv:1412.1437.[11] A. V. Nazarenko, Int. J. Mod. Phys. B, , 3415 (2011)., 3415 (2011).