Modular Invariance in the Gapped XYZ Spin-1/2 Chain

Modular invariance in the gapped XYZ spin-1/2 chain

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.

Citation Ercolessi, Elisa, Stefano Evangelisti, Fabio Franchini, and Francesco Ravanini. "Modular invariance in the gapped XYZ spin-1/2 chain." Phys. Rev. B 88, 104418 (September 2013). © 2013 American Physical Society

As Published http://dx.doi.org/10.1103/PhysRevB.88.104418

Publisher American Physical Society

Version Final published version

Citable link http://hdl.handle.net/1721.1/88718

Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW B 88, 104418 (2013)

1 Modular invariance in the gapped XYZ spin- 2 chain

Elisa Ercolessi,1 Stefano Evangelisti,1 Fabio Franchini,2,3,* and Francesco Ravanini1 1Department of Physics, University of Bologna and INFN, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy 2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3SISSA and INFN, Via Bonomea 265, 34136 Trieste, Italy (Received 9 June 2013; revised manuscript received 20 August 2013; published 20 September 2013) We show that the elliptic parametrization of the coupling constants of the quantum XYZ spin chain can be analytically extended outside of their natural domain, to cover the whole phase diagram of the model, which is composed of 12 adjacent regions, related to one another by a spin rotation. This extension is based on the modular properties of the elliptic functions and we show how rotations in parameter space correspond to the double covering PGL(2,Z) of the modular group, implying that the partition function of the XYZ chain is invariant under this group in parameter space, in the same way as a conformal ﬁeld theory partition function is invariant under the modular group acting in real space. The encoding of the symmetries of the model into the modular properties of the partition function could shed light on the general structure of integrable models.

DOI: 10.1103/PhysRevB.88.104418 PACS number(s): 75.10.Pq, 02.30.Ik, 11.10.−z, 03.67.Mn

I. INTRODUCTION the exploiting of modular properties hidden behind the Baxter reparametrization of the XYZ chain can lead, similarly to the The identification of modular properties in integrable aforementioned examples of conformal and gauge theories, to theories has always represented a progress in our understand- new tools in the investigation of crucial properties of the XYZ ing of the underlying physical and mathematical structures chain and its family of related models. of the model. Modular transformations can appear in a In this paper we show that the XYZ partition function physical model in many different ways. For example, in is invariant under a set of transformations that implement a two-dimensional conformal field theory (CFT) the modular in- modular group covering the full plane of parameters of the variance of partition functions on a toroidal geometry1 has led model in a very specific way. This, in our opinion, prepares to the classification of entire classes of theories2 and, in many the way to the application of the modular group to the study cases, has opened the way to their higher genus investigation, of more complicated quantities, that we plan to investigate which is a very important tool in applications ranging from in the near future. For instance, this modular invariance may strings to condensed-matter physics. In a different context, play an important role in a deeper understanding of the Renyi modular transformations are at the root of electromagnetic and von Neumann bipartite entanglement entropies, which duality in (supersymmetric) gauge theories.3 In this case the are well known to be excellent indicators to detect quantum modular transformations act on the space of parameters of the phase transitions. The paper is organized as follows. In Sec. II theory, rather than as space-time symmetries. we recall basic facts about the Hamiltonian formulation of the In integrable two-dimensional lattice models modular prop- XYZ spin chain, the properties of its phase-space diagram erties were crucial in a corner transfer matrix (CTM) approach4 and the symmetry transformations leading one region of the to get the so-called highest weight probabilities.5,6 More diagram into another. In Sec. III we recall Baxter’s mapping of recently, the problem of computing bipartite entanglement the eight-vertex model’s parameters through elliptic functions Renyi entropies (or their von Neumann limit) in integrable and the relation between them and those of the XYZ chain in spin chains has been related to the diagonalization of the CTM the different parts of its phase diagram. Section IV is devoted of the corresponding two-dimensional (2D) lattice integrable to the analytical extension of Baxter solution by switching model.7 The expressions for these quantities often present a to new parameters that make the modular properties more form that can be written in terms of modular functions, like the evident. This shows how one of the phase-space regions can be Jacobi θ functions, suggesting links with a modular theory in mapped into the others by applying a modular transformation. the space of parameters. In the XY chain, for example, modular The generators of these modular transformations are identified expressions have been found for the entanglement entropy in in the next section, Sec. V, where the characterization of the Ref. 8. Similar modular expressions have been obtained in actions of the full modular group is described. Our conclusions the calculation9,10 of entanglement entropies in the XYZ spin and outlooks are drawn in Sec. VI, while the Appendix collects chain, notoriously linked in its integrability to the eight-vertex some definitions and useful identities on elliptic functions. lattice model.4 The XYZ model is one of the most widely studied examples of a fully interacting but still integrable theory, playing the role of a sort of “father” of many other very II. XYZ CHAIN AND ITS SYMMETRIES interesting and useful spin chain models, like the Heisenberg 1 model and its XXZ generalization. Here we would like to The quantum spin- 2 ferromagnetic XYZ chain can be start the investigation of its modular symmetries, which—let described by the following Hamiltonian: us stress it again—are not space-time symmetries, like in CFT, ˆ =− x x + y y + z z but symmetries in the space of parameters. We believe that HXYZ Jx σn σn+1 Jy σn σn+1 Jzσn σn+1 , (1) n

α = = where the σn (α x,y,z) are the Pauli matrices acting on us first look at the Jy Jx line. Here we have the familiar the site n, the sum ranges over all sites n of the chain, and XXZ model and we recognize the critical (paramagnetic) the constants Jx , Jy , and Jz take into account the degree phase for |J˜z| < 1 (bold, red online, continuous line), the of anisotropy of the model. We assume periodic boundary antiferromagnetic Ising phases for J˜z < −1 (dotted line) and conditions. the Ising ferromagnetic phase for J˜z > 1 (dashed line). The The space of parameters (Jx ,Jy ,Jz) forms a projective same physics can be observed for Jy =−Jx . There, one can space, since an overall rescaling only changes the energy units. rotate every other spin by 180◦ around the x axis to recover To visualize the phase diagram of the model we can single out a traditional XXZ model. Note, however, that Jz changes one of them, say Jx , which, without loss of generality we take sign under this transformation and therefore the ferromagnetic as positive, and switch to homogeneous coordinates: and antiferromagnetic Ising phases are reversed. The lines Jz =±Jx correspond to a XYX model, i.e., a rotated XXZ Jy Jz J˜y ≡ and J˜z ≡ . (2) model. Thus the phases are the same as before. Finally, along Jx Jx the diagonals Jy =±Jz we have a XYY model of the form However, as Jx varies, this mapping introduces artificial y z discontinuities when Jx vanishes. The appearance of these sin- ˆ =− x x + y ± z HXYY Jx σn σn+1 Jy σn σn+1 σn σn+1 . (3) gularities is unavoidable: it is just due to the choice of using a n single two-dimensional map to describe the three-dimensional space of parameters and it is a known phenomenon in the Thus the paramagnetic phase is for |J˜y | > 1 and the Ising study of projective spaces. We will soon deal with this branch phases for |J˜y | < 1, with the plus sign (Jy = Jz) for the Ising discontinuities in discussing the symmetries of the model. ferromagnet and the minus (Jy =−Jz) for the antiferromag- Later on in Sec. V, however, we will revert to the full net. three-dimensional parameter space in order to avoid these In this phase diagram we find four tricritical points at singularities and to show more clearly the action of the modular (J˜z,J˜y ) = (±1, ±1). In the vicinity of each paramagnetic group. phase, in a suitable scaling limit,8,11,12 the model renormalizes In Fig. 1 we draw a cartoon of the phase diagram of the to a sine-Gordon theory, with β2 = 8π at the antiferro- Jz Jy 2 → XYZ model in the ( , ) = (J˜z,J˜y ) plane. We divide it into magnetic isotropic point and to a β 0 theory toward Jx Jx 12 regions, whose role will become clear as we proceed. Let the ferromagnetic Heisenberg point. This is the reason why the critical nature of these tricritical points is quite differ- J > C = J˜ ,J˜ = − , C = Jy ent. Assuming x 0, at 1 ( z y ) ( 1 1) and 2 (J˜ ,J˜ ) = (1, −1) we have two conformal points dividing three AFM Jx FM z y equivalent (rotated) sine-Gordon theories with β2 = 8π.At = ˜ ˜ = = ˜ ˜ = − − Principal XY Y E1 (Jz,Jy ) (1,1) and E2 (Jz,Jy ) ( 1, 1), we have 2 = Region two β 0 points which are no longer conformal, since the low-energy excitations have a quadratic spectrum there. The XXZ former points correspond to an antiferromagnetic Heisenberg C1 E1 chain at the Berezinkii-Kosterlitz-Thouless (BKT) transition, while the latter correspond to a Heisenberg ferromagnet at its first-order phase transition. The different nature of the XY X tricritical points has been highlighted from an entanglement Jz entropy point of view in Ref. 10. = =± Jx For Jx Jy Jz, we deal with an (anti)ferromagnetic Heisenberg model. As soon as some anisotropy is introduced, E C its original full SU(2) symmetry is broken: along any of the 2 2 XXZ lines a U(1) symmetry is left, whereas in the most general situation the residual Z2 symmetry group is generated by π α i(π/2)σn = rotations, n e , α x,y,z, of every spin around one of the axes. In fact, as already observed, the partition function generated by Hamiltonian (1) is invariant under π/2 rotations α = i(π/4)σn Rα n e , since the effect of these transformations can be compensated by a simultaneous exchange of two FIG. 1. (Color online) Phase diagram of the XYZ model in ↔ the ( Jz , Jy ) plane. The critical XXZ, XYX,andXYY chains couplings: Eα : Jβ Jγ being (α,β,γ ) a cyclic permutation Jx Jx are represented by bold (red) lines. Dashed (blue) lines are the of (x,y,z). Thus, this Z2 symmetry of the Hamiltonian is continuations of the critical phases into the ferro-/antiferromagnetic enhanced to become the symmetric group (of permutations S (FM/AFM) gapped chain (dashed/dotted), ending at infinity on of three elements) 3 at the level of the partition function. a FM/AFM Ising model. There are four tricritical points where The partition function shows an additional Z2 symmetry, the three gapless phases meet: two of them, C1,2, correspond to generated by a π rotation of every other spin around one of the = α the isotropic AFM Heisenberg chain and relativistic low-energy axes α n σ2n, which can be compensated by the reversal excitations (spinons), while the other two, E1,2, give the isotropic Pα of the signs of the couplings in the corresponding plane. FM chain, whose low-energy excitations have a quadratic dispersion The total symmetry group is generated by these six elements, relation (magnons). which are however not all independent. There are only three

104418-2 1 MODULAR INVARIANCE IN THE GAPPED XYZ SPIN- 2 CHAIN PHYSICAL REVIEW B 88, 104418 (2013)

+ IIIc , but this would interchange the antiferromagnetic nature of the region into a ferromagnetic one, since this transformation would not preserve the parity of signs in the couplings. Thus, in order to project the three-dimensional space of parameter on the two-dimensional plane, we deﬁne Ey Ey ◦ Py when − − acting on Jz < 0, so that it maps Ia correctly into IIIa . Similarly, one can see that the area IIa is reached starting from Ia via the Ez exchange, which in turn can be written as Ey ◦ Ex ◦ Ey . In this way, we covered the whole area above the two diagonals. As said before, we can reﬂect about the diagonal acting with Ex , thus covering the whole region to the left of the antidiagonal quadrant. Finally, the remaining half plane can be reached via a π rotation Px , as explained above. In the following sections we will show that the transformations explained above are equivalent to the action of an extension of the modular group. By taking the principal region Ia as a reference starting point, in Sec. V we collect the action of the different operators in Table I.

III. BAXTER’S SOLUTION The Hamiltonian (1) commutes with the transfer matrices of FIG. 2. (Color online) The phase diagram of the XYZ chain in the zero-field eight vertex model4 and this means that they can Fig. 1 is divided into 12 regions, named Ia,b,c,d,IIa,b,c,d, IIIa,b,c,d. Each region is further separated into the + half lying around one of the FM be diagonalized together and they share the same eigenvectors. − Indeed, as shown by Sutherland,13 when the coupling constants isotropic points E1,2 and the half closer to the AFM conformal of the XYZ model are related to the parameters and of points C1,2. the eight vertex model14 at zero external field by the relations independent generators that we choose: Px ,Ex ,Ey . In fact, in Jx : Jy : Jz = 1: : , (5) choosing homogeneous coordinates (2), we cannot implement P or P , since they would invert the sign of one between J˜ and the row-to-row transfer matrix T of the latter model commutes y z y with the Hamiltonian Hˆ of the former. In the principal regime J˜ , thus changing the parity of the signs of the couplings and z of the eight vertex model it is customary4 to parametrize the hence inverting ferromagnetic and antiferromagnetic phases. constants and in terms of elliptic functions, We may look at the effect of these symmetries in the space of parameters {J ,J ,J }. Explicitly, the generators act as 1 + k sn2(iλ; k) cn(iλ; k)dn(iλ; k) x y z = , =− , (6) 1 − k sn2(iλ; k) 1 − k sn2(iλ; k) Px : {Jx ,Jy ,Jz} →{Jx , −Jy , −Jz}, where sn(z; k), cn(z; k), and dn(z; k) are Jacoby elliptic E : {J ,J ,J } →{J ,J ,J }, (4) x x y z x z y functions of parameter k, while λ and k are the argument { } →{ } Ey : Jx ,Jy ,Jz Jz,Jy ,Jx , and the parameter (respectively) whose natural regimes are and relate the different regions of the phase diagram shown 0 k 1, 0 λ K(k ), (7) in Fig. 2, named Ia,b,c,d,IIa,b,c,d,IIIa,b,c,d. In particular, K(k ) being the complete elliptic integral of the first kind (A7) P corresponds to a π rotation in the plane mapping, for √ x of argument k = 1 − k2. example, region I onto I . Under this transformation, for a c In light of (5), Baxter’s parametrization encourages us instance, the half plane to the right of the antidiagonal to use the homogeneous coordinates (2), although this will J˜ =−J˜ is mapped into the half plane to the left of it. z y introduce artificial discontinuities when we will extend it to The transformation E instead corresponds to a reflection x the whole phase diagram. This is the aforementioned issue in about the main diagonal, mapping, e.g., region I onto I . a b using a two-dimensional map to describe a three-dimensional Combined, these two transformations transform any of the four projective space. In Sec. V, we will give an alternative sectors of the plane lying in between the diagonal and the prescription, which highlights the action of the modular group antidiagonal (that we distinguish with the letters a,b,c,d) one and acts directly on the three-dimensional parameter space, into the others. Inside each of these sectors, we will denote + − thus avoiding any discontinuity. with superscripts and the subregions of I, II, III touching Equations (5) and (6) have the great limitation that, in the the critical points of type E and C respectively, again as − | | + ={ ˜ ˜ } natural regime of the parameters (λ,k), 1, 1, indicated in Fig. 1. For example Ia 0 < Jz < 1,Jy > 1 − ={ ˜ ˜ − } which correspond to the antiferroelectric phase of the eight and Ic 0 < Jz < 1,Jy < 1 . To see how Ey works, let + vertex model. Thus, Eq. (5) only covers region Id of Fig. 2. us start from Ia .NowEy exchanges the role of x and z, ˜ ˜ ∈ + However, using the symmetries of the model and the freedom mapping a point (Jz,Jy ) Ia to the point with coordinates J J˜ under the rearrangement of parameters, one can take the ( Jx , y ) = ( 1 , y ), which belongs to the zone in the first Jz Jz J˜z J˜z antiferroelectric (, )in(6) and use them in the whole phase + − quadrant which we denote with IIIa .Naively,Ey maps Ia into diagram of the XYZ chain, generalizing (5). The result of this

104418-3 ERCOLESSI, EVANGELISTI, FRANCHINI, AND RAVANINI PHYSICAL REVIEW B 88, 104418 (2013)

procedure gives, when |J˜y | < 1 and |J˜z| < 1,

|J˜z − J˜y |−|J˜z + J˜y | 2 = , =− , |J˜y | < 1 and |J˜z| < 1, (8) |J˜z − J˜y |+|J˜z + J˜y | |J˜z − J˜y |+|J˜z + J˜y | while for |J˜y | > 1or|J˜z| > 1, we get |J˜z−J˜y |−|J˜z+J˜y | min 1, 1 |J˜ − J˜ |+|J˜ + J˜ | = 2 | ˜ − ˜ |−|˜ + ˜ | =− z y z y | ˜ | | ˜ | | ˜ − ˜ |−| ˜ + ˜ | sgn( Jz Jy Jz Jy ) , | ˜ − ˜ |−| ˜ + ˜ | , Jy > 1orJz > 1, Jz Jy Jz Jy 2 Jz Jy Jz Jy max 1, 2 max 1, 2 (9)

where, we remind, and are defined by (6). For instance in Using (A14) we have region Id ,for|J˜y | 1 and J˜z −1, we recover J˜y = and ˜ = ˜ | ˜ | 1 − 2 Jz , while in region Ia,forJy 1 and Jz 1, we have l = . (15) 2 − 2 J˜y =− and J˜z =−.InFig.1 we show the phase diagram divided in the different regions where a given parametrization Furthermore, elliptic functions can be inverted in elliptic applies. integrals and from (12) we have (see the Appendix for the While this procedure allows us to connect the solution of definition of the incomplete elliptical integral F ) the eight-vertex model to each point of the full phase-diagram − dt of the XYZ chain, it treats each region separately and iu = introduces artificial discontinuities at the boundaries between −1 (1 − t2)(1 − l 2t2) every two regions, as indicated by the max and min functions = i[K(l ) − F (arcsin ; l )]. (16) and the associated absolute values. In the next section we will show that, with a suitable analytical continuation of Together (15) and (16) invert (11) and (12) and, together with the elliptic parameters, we can extend the parametrization (8) and (9), give the value of Baxter’s parameters for equivalent of a given region to the whole phase diagram. As we points of the phase diagram. It is important to notice that these discussed, some discontinuities will remain unavoidable, but identities ensure that the domain (13) maps into || 1 and their locations will become a branch choice. Moreover, we will −1 and vice versa. show that this extension numerically coincides with Baxter’s The idea of the analytical continuation is to use (15) to prescription. extend the domain of l beyond its natural regime. To this end, let us start from a given region, let say Ia in Fig. 1, with |J˜z| 1 ˜ IV. ANALYTICAL EXTENSION OF BAXTER’S SOLUTION and Jy 1. By (9) we have To streamline the computation, it is convenient to perform a cn(iu; l) J˜y (u,l) =− = , Landen transformation on (6) and to switch to new parameters dn(iu; l) (17) (u,l) instead of (k,λ): 1 √ J˜z(u,l) =− =− , 2 k dn(iu; l) l ≡ ,u≡ (1 + k)λ, (10) + and thus 1 k ˜ so that15 1 − J˜2 Jz dt l = z ,iu= . (18) 2 2 2 1 + k sn (iλ; k) 1 J˜ − J˜ −1 (1 − t2)(1 − l 2t2) = = , (11) y z 1 − k sn2(iλ; k) dn(iu; l) Notice that within region Ia we can recast the second identity cn(iλ; k)dn(iλ; k) cn(iu; l) as =− =− . (12) 1 − k sn2(iλ; k) dn(iu; l) u = K(l ) + F (arcsin J˜z; l ), (19) The natural domain of (λ,k) corresponds to which makes u explicitly real and 0 u 2K(l ). 0 u 2K(l ), 0 l 1. (13) We now take (18) as the definition of l and use it to extend it over the whole phase diagram. For instance, in region IIa We also have the following identities: ˜ ˜2 ˜2 (0 Jy 1, Jz Jy ), using (18) we find l>1. To show 1 − k l = 1 − l2 = ,K(l) = (1 + k)K(k), that this analytical continuation gives the correct results, we 1 + k write l = 1/l˜, so that 0 l˜ 1 and use (A32) in (17), which (14) 1 + k gives K(l ) = K(k ), 2 ˜ ˜ = dn(iu˜; l) ˜ =− 1 from which it follows that the Landen transformation doubles Jy (u,l) , Jz(u,l) , (20) ˜ ˜ = K(k ) = K(l ) = cn(iu˜; l) cn(iu˜; l) the elliptic parameter τ(k) i K(k) 2i K(l) 2τ(l). 104418-4 1 MODULAR INVARIANCE IN THE GAPPED XYZ SPIN- 2 CHAIN PHYSICAL REVIEW B 88, 104418 (2013)

FIG. 3. (Color online) A schematic representation of the analytical extension of Baxter’s parametrization (22) on the full phase diagram of Fig. 1. Left: the directions of the quarter periods K(l)andiK(l ) in I, II, and III regions. Right: the paths a, b, c,andd of the argument of the elliptic functions. The argument always runs along the sides of the same rectangle, but, due to the branch discontinuities introduced in representing a three-dimensional space with two parameters, the path looks different in the three cases. The corners of the rectangle toward which the arrows point represent the tricritical ferromagnetic points, while the corners with the circles from which arrows start are the tricritical conformal points.

˜ =−Jz = where all the elliptic integrals here are taken at their principal which reproduces Baxter’s deﬁnitions in (8): ˜ , Jy value. The choice of either branch discontinuity corresponds − 1 . Note that we also rescale the argument u˜ = lu, so that J˜y to different ways of projecting the 3D projective space of 0 u˜ 2K(l˜ ). This shows that we can use (17) to cover parameters into the 2D phase space. We introduced this both regions Ia and IIa, by letting 0 l<∞ (note that l = 1 phenomenon in the previous section and we will come back corresponds to the boundary between the two regions). to these jumps in a few paragraphs. Close to J˜z 0, the We can proceed similarly for the rest of the phase diagram. ﬁrst line of (21) is continuous, while the second has a jump. However, the second part of (18) needs some adjustment. Since The opposite happens for J˜y 0, but, due to the periodicity elliptic integrals are multivalued functions of their parameters, properties of (11), (12), both expressions are proper inversions to determine the proper branch of the integrand one starts in of (17) valid everywhere. the principal region and follows the analytical continuation. Thus, we accomplished to invert (17) and to assign a pair The result of this procedure can be summarized as of (u,l) to each point of the phase diagram (J˜y ,J˜z), modulo the periodicity in u space. The analysis of these mappings = + ˜ + − ˜ iu i[K(l ) F (arcsin Jz; l )] [1 sgn(Jy )]K(l) shows an interesting structure. From (18) it follows that that

= F (arcsin J˜y ; l) − K(l) + i[1 + sgn(J˜z)]K(l ), (21) regions I’s have 0 l 1, while II’s have l 1, which can

104418-5 ERCOLESSI, EVANGELISTI, FRANCHINI, AND RAVANINI PHYSICAL REVIEW B 88, 104418 (2013)

FIG. 4. (Color online) Contour plots of the (u,l) lines in the different regions of the phase diagram, as given by (22). be written as l = 1/l˜, with 0 l˜ 1. Finally, the regions domain has been observed before by other authors’ analysis, III’s have purely imaginary l = i l/˜ l˜ , with 0 l˜ 1. In each we did not find any reference to it in the literature. We show in region, the argument runs along one of the sides of a rectangle detail how to construct the complete extension to the whole in the complex plane of sides 2K(l˜) and i2K(l˜ ). Thus, we can phase diagram, which is condensed in Table I in the next write section. cn(ζu(z); l) 1 J˜ (u,l) = , J˜ (u,l) = , y dn(ζu(z); l) z dn(ζu(z); l) V. ACTION OF THE MODULAR GROUP (22) 0 z π, In the previous section, we showed that Baxter’s parametrization of the parameters of the XYZ chain can be where ζ = 1,l,˜ l˜ in regions I’s, II’s, and III’s respectively, analytically extended outside of the principal regime to cover and with z defined implicitly as follows. The path of u(z) the whole phase diagram and thus that any path in the real runs along the sides of a rectangle drawn on the half periods two-dimensional (J˜ ,J˜ ) space corresponds to a path of (u,l) 2K(l˜) and 2iK(l˜ ), as shown in Fig. 3. For instance, for type I y z in C2. This extension is due to the fact that the analytical we have u(z) = i(π − z) 2 K(l˜ ) for regions of type a; u(z) = π continuation of elliptic functions in the complex plane can − 2 ˜ = ˜ + 2 ˜ (π z) π K(l) for type b; u(z) 2K(l) izπ K(l ) for regions be related to the action of the modular group. We refer to = ˜ + 2 ˜ of type c; and u(z) i2K(l ) z π K(l) for regions of type d. Refs. 17–19 for a detailed explanation of the relation between Using the same paths for II’s and III’s would reverse the role of elliptic functions and the modular group and to the Appendix the ferro-/antiferromagnetic nature of the phases and thus we for a collection of useful identities. are forced to introduce jump discontinuities in the paths; see Points related by a modular transformation correspond to Fig. 3: these are the aforementioned singularities introduced the same (, ) point in the mapping to the eight-vertex model by the 2D representation of the three-dimensional projective (8) and (9). We now want to show that the modular group can space of parameters. In the next section, we will work directly be connected to the symmetries of the model that we discussed on the latter and we will show that the modular group naturally in the Introduction. Let us start again in region Ia,buttomake generates the regular paths of regions I for all the cases. In the modular structure more apparent, this time let us write the Fig. 4 we draw the contour lines of (u,l)inthe(J˜y ,J˜z) plane, parametrization in terms of θ functions using expression (A6): as given by (22). | − | The rectangular shape for the path of the argument is not Jy θ3(0 τ) θ2[(z π)τ τ] J˜y = = , (23a) surprising, since it is known that only purely real or purely Jx θ2(0|τ) θ3[(z − π)τ|τ] imaginary arguments (modulo a half periodicity) ensure the | − | ˜ = Jz = θ3(0 τ) θ4[(z π)τ τ] reality of an elliptic function (and thus that the coupling Jz , (23b) Jx θ4(0|τ) θ3[(z − π)τ|τ] constants are real and the Hamiltonian is Hermitian). The validity of these analytical continuations can be checked, like where 0 z π, and we used (A4). we just did for region IIa, using the relations connecting elliptic In the previous section, we noticed that using the homoge- functions of different elliptic parameters (see the Appendix neous coordinates (J˜z,J˜y ) introduces artificial discontinuities or Ref. 17). All these relations are rooted in the modular in the phase diagram. Thus, we now revert back to the original invariance of the torus on which elliptic functions are defined. three-dimensional projective space of parameters and write In the next section, we will show that the action of the modular θ [(z − π)τ|τ] group in covering the phase diagram of the XYZ model mirrors J = J 3 , x | (24a) the symmetry of the model. θ3(0 τ) Before we proceed, we remark that, while it is possible that θ2[(z − π)τ|τ] Jy = J , (24b) the validity of Baxter parametrization outside of its natural θ2(0|τ)

104418-6 1 MODULAR INVARIANCE IN THE GAPPED XYZ SPIN- 2 CHAIN PHYSICAL REVIEW B 88, 104418 (2013)

TABLE I. List of the action of modular transformations, according to (26) and (33), in mapping the phase diagram, starting from Ia with the parametrization given by (25).

Region Modular Generator φ Couplings σR σI τlφModular Generator Region

Ia I −πτ {Jx ,Jy ,Jz} 1 ττlπ˜ P Ic − { } 1 − 1 ˜ π Ib S π Jx ,Jz,Jy τ 1 τ l τ SP Id − { } + l˜ IIIa T πτ Jz,Jy ,Jx 1 ττ1 i l˜ π TP IIIc − { } 1 τ−1 1 π IIb TS π Jz,Jx ,Jy τ 1 τ l˜ τ TSP IId −πτ { } 1 τ −1 l˜ π IIIb ST τ+1 Jy ,Jz,Jx τ+1 τ+1 τ+1 i l˜ τ+1 STP IIId −πτ { } 1 τ τ 1 π IIa TST τ+1 Jy ,Jx ,Jz τ+1 τ+1 τ+1 l˜ τ+1 TSTP IIc −πτ 1 τ τ 1 π STS τ−1 τ−1 τ−1 τ−1 l˜ τ−1 STSP

θ4[(z − π)τ|τ] i.e., it expresses the original real and imaginary periods in Jz = J , (24c) θ4(0|τ) terms of the two new periods. To see the action of these transformations in the space of where J is an additional normalization which we will need. parameters, let us start again from the subregion Ia. Applying As the Jacobi elliptic functions are doubly periodic, the the S transformation Sτ = τ =−1 ,wehave torus on which they are defined can be viewed as the quotient S τ C/Z2.InregionI,for0 l 1, the fundamental domain − | a = θ3[z π τS ] is the rectangle with sides on the real and imaginary axis, SJx (SJ ) θ3(0|τS ) of length respectively ω1 = 4K(l) and ω3 = i4K(l ), and the 2 θ [(z − π)τ|τ] path followed by z is a half period along the imaginary = SJ e(iτ/π)(z−π) 3 = J , ( ) | x (28a) axis. A modular transformation turns the rectangle into θ3(0 τ) a parallelogram constructed on the same two-dimensional θ2[z − π|τS ] SJy = (SJ ) lattice, but leaves the path of the argument untouched. To θ2(0|τS ) this end, we extend the representation of the modular group − | (iτ/π)(z−π)2 θ4[(z π)τ τ] to keep track of the original periods and introduce the two = (SJ )e = Jz, (28b) θ (0|τ) vectors σ and σ , which points along the original real and 4 R I − | ω1 θ4[z π τS ] imaginary axis. In Ia, these two vectors are σR = = 1 and = ω1 SJz (SJ ) ω3 θ (0|τ ) σI = = τ (we remind that, when using θ functions, every 4 S ω1 − | length is normalized by the periodicity on the real axis, i.e., (iτ/π)(z−π)2 θ2[(z π)τ τ] = (SJ )e = Jy , (28c) ω1). We will also need to extend the modular group and thus we θ2(0|τ) introduce the additional complex number φ =−πσI , which is a sort of an initial phase in the path of the argument. where we defined With these definitions, we write (24) as − − 2 SJ ≡ e (iτπ)(z π) J. (29) θ [zσ + φ|τ] θ [zσ + φ|τ] J = J 3 I ,J= J 2 I , x θ (0|τ) y θ (0|τ) Equations (28) show that S corresponds to the exchange Ex . 3 2 The other generator of the modular group Tτ = τ = τ + 1 + | (25) T θ4[zσI φ τ] gives Jz = J . θ4(0|τ) θ [(z − π)τ|τ ] TJ = TJ 3 T The two generators of the modular group act on this extended x ( ) | representation as θ3(0τ τT ) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ θ4[(z − π)τ|τ] 1 = TJ = J , τ τ + 1 τ − ( ) | z (30a) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ τ ⎟ θ4(0τ τ) − σR ⎜ σR ⎟ ⎜ σR ⎟ ⎜ σR ⎟ ⎜ ⎟ θ [(z − π)τ|τ ] T ⎜ ⎟ = ⎜ ⎟ , S ⎜ ⎟ = ⎜ τ ⎟ . (26) = 2 T ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ − σI ⎠ TJy (TJ ) σI σI σI τ θ2(0|τT ) φ φ φ − φ − | τ θ2[(z π)τ τ] = (TJ ) = Jy , (30b) + θ (0|τ) A general modular transformation which gives τ = c dτ also 2 a+bτ θ [(z − π)τ|τ ] does TJ = TJ 4 T z ( ) | 1 θ4(0 τT ) σ → = d − bτ , − | R + θ3[(z π)τ τ] a bτ = (TJ ) = Jx , (30c) τ θ3(0|τ) σ → =−c + aτ , (27) I a + bτ which shows that T corresponds to Ey and sets φ φ → , a + bτ TJ ≡ J. (31)

104418-7 ERCOLESSI, EVANGELISTI, FRANCHINI, AND RAVANINI PHYSICAL REVIEW B 88, 104418 (2013)

The normalization constant J is inconsequential, since generic complex argument u would yield a non-Hermitian it just sets the energy scale. However, to make the link Hamiltonian, but would preserve the structure that makes it between the generator of the modular group and the symmetry integrable. We do not have a physical interpretation for such a of the model more apparent and to maintain reality of the nonunitary extension, but its mathematical properties indicates couplings, we choose it such that it cancels the phases of the S that it might be worth exploring this possibility. transformation. Such choice can be accommodated by setting, for instance, VI. CONCLUSIONS AND OUTLOOK J = J (z,τ) The possibility to write an exact solution for the XYZ spin −1/3 ∝ (θ2[(z − π)τ|τ]θ3[(z − π)τ|τ]θ4[(z − π)τ|τ]) . chain is based on its relation with the eight-vertex model. (32) However, the relation between the parameters (, )ofthe latter is not one to one with the couplings (Jx ,Jy ,Jz)ofthe Finally, we introduce an additional operator P: ⎛ ⎞ ⎛ ⎞ former (and with the Boltzmann weights of the classical model τ τ as well): indeed such mapping changes in the different regions ⎜ ⎟ ⎜ ⎟ of the phase diagram; see Eqs. (8) and (9). In this paper, ⎜ σ ⎟ ⎜ σ ⎟ ⎜ R ⎟ = ⎜ R ⎟ we have shown that this re-arrangement procedure can be P ⎝ ⎠ ⎝ ⎠ , (33) σI σI reproduced by means of the natural analytical extension of φ φ + π [σR + σI ] the solution valid in a given initial region. We have also given a prescription that relates the parameter (u,l) of the elliptic which shifts by half a period in both directions the origin of functions and the physical parameter of the XYZ model in the the path of the argument. Its action is given by [see (A4)] whole of the phase diagram and provided an inversion formula θ3[zτ + π|τ] valid everywhere. PJx = (PJ ) = Jx , (34a) θ3(0|τ) Clearly symmetries in the space of parameters become manifest as invariance properties of observables and corre- θ2[zτ + π|τ] PJ = (PJ ) =−J , (34b) lations. For example, in Ref. 9 we have analytically calculated y θ (0|τ) y 2 entanglement entropies for the XYZ model and in particular + | θ4[zτ π τ] obtained an expression for the corrections to the leading terms PJz = (PJ ) =−Jz, (34c) θ4(0|τ) of Renyi´ entropies that emerge as soon as one moves into the massive region, starting from the critical (conformal) line. so that P corresponds to the π-rotation generator Px . The three operation T, S, and P, with Also, by making explicit use of such a kind of parametrization, we have shown that finite-size and finite-mass effects give (S · T)3 = I, S2 = I, P2 = I, rise to different contributions (with different exponents), thus (35) violating simple scaling arguments. This topic deserves further P · S = S · P, P · T = T · P , analytical and numerical studies that we plan to present in a generate a Z2 extension of the modular group that can be future paper. identified20 with PGL(2,Z). We collect in Table I the action An interesting point of the approach presented in this paper of the different transformations generated by T, S, and P for lies in the connection between analytic continuation in the the regions in the positive half plane, the other being given just parameter space and the action of the modular group. This by acting with the π rotation. connection allowed us to show that a certain extension of A few additional comments are in order. In the previous the modular group realizes in parameter space the physical section we gave a prescription for the path of the argument symmetries of the model. This symmetries are in the form of around a rectangle; see (22) and Fig. 4. This structure is dualities, since they connect points with different couplings, explained by Table I:theS generator interchanges the role and thus could be exploited to compute different correlation of the axes and thus the path that runs along the imaginary functions, whose operator content is related by these dualities, axis in the new basis runs along the real one, consistently with potentially simplifying the operation. the prescription given for regions of type b and d. The action Modular invariance plays a central role in the structure and of P is to shift the path and thus interchanges regions of type integrability of conformal field theories in 1 + 1 dimension. a with c and b with d (and vice versa). The role of P can On its critical lines, the XYZ model is described by a c = 1 be thought of as that of promoting the modular group to its CFT and thus, in the scaling limit, its partition function is discrete affine. modular invariant in real space. Moreover, it is known21 that In conclusion, we see that the whole line l2 ∈ (−∞,∞) can modular invariance plays a central role in organizing the be used in (22), which divides in three parts, one for each of the operator content of a CFT. Our results show that, even in the regions of type I, II, or III. It is known, that, for each point on gapped phase, the XYZ chain is also a modular invariant in this line, only if the argument of the elliptic functions is purely parameter space, due to the symmetries of the model. Elliptic imaginary or purely real (modulo a half periodicity) are we structures are common in the solution of integrable models guaranteed that the parameters of the model are real. Hence the and it is tempting to speculate that their modular properties two (four) paths, related by S (and P) duality. The paths of the do encode in general their symmetries and thus the class of argument are compact, due to periodicity. Thus, the mapping integrability-preserving relevant perturbations that drive the 2 between (u,l) and (Jy ,Jz) is topologically equivalent to R → system away from criticality. In fact, this observation might R ⊗ C ⊗ Z2 ⊗ Z2. Extending the mapping by considering a help in interpreting the long-standing mystery for which the

104418-8 1 MODULAR INVARIANCE IN THE GAPPED XYZ SPIN- 2 CHAIN PHYSICAL REVIEW B 88, 104418 (2013) partition function of integrable models in their gapped phases The periodicity properties of the θ functions are can be organized in characters of Virasoro representations, with the finite-size/-temperature scale set, instead, by the θ1(z|τ) =−θ1(z + π|τ) =−λθ1(z + πτ|τ) mass parameter.5,6,22 This is a peculiar observation that has = λθ1(z + π + πτ|τ), (A4a) so far eluded fundamental explanation: our work hints that the | =− + | = + | symmetries of an integrable system can manifest themselves θ2(z τ) θ2(z π τ) λθ2(z πτ τ) with a modular structure in parameter space and this would =−λθ2(z + π + πτ|τ), (A4b) constrain the form of the partition function in the same way as θ3(z|τ) = θ3(z + π|τ) = λθ3(z + πτ|τ) modular invariance in coordinate space operates to organize = + + | the spectrum of critical theories. While some recent results λθ3(z π πτ τ), (A4c) from the study of entanglement entropies already support this θ4(z|τ) = θ4(z + π|τ) =−λθ4(z + πτ|τ) 23,24 line of thinking, we are currently exploring to which extent =−λθ4(z + π + πτ|τ), (A4d) this speculation can hold true in general. where λ ≡ qe2iz. 1 1 Moreover, incrementation of z by the half periods 2 π, 2 πτ, 1 + ACKNOWLEDGMENTS and 2 π(1 τ) leads to F.F. thanks P. Glorioso for interesting and fruitful discus- θ (z|τ) =−θ z + 1 π|τ =−iμθ z + 1 πτ|τ sions. This work was supported in part by two INFN COM4 1 2 2 4 2 =− + 1 + 1 | grants (FI11 and NA41). F.F. was supported by a Marie Curie iμθ3 z 2 π 2 πτ τ , (A5a) International Outgoing Fellowship within the 7th European θ (z|τ) = θ z + 1 π|τ = μθ z + 1 πτ|τ Community Framework Programme (FP7/2007-2013) under 2 1 2 3 2 Grant No. PIOF-PHY-276093. = μθ z + 1 π + 1 πτ|τ , (A5b) 4 2 2 θ (z|τ) = θ z + 1 π|τ = μθ z + 1 πτ|τ 3 4 2 2 2 1 1 = μθ1 z + π + πτ|τ , (A5c) APPENDIX: ELLIPTIC FUNCTIONS 2 2 1 1 θ4(z|τ) = θ3 z + π|τ =−iμθ1 z + πτ|τ Elliptic functions are the extensions of the trigonometric 2 2 = + 1 + 1 | functions to work with doubly periodic expressions, i.e., such iμθ2 z 2 π 2 πτ τ , (A5d) that with μ ≡ q1/4eiz. The Jacobi elliptic functions are then defined as f (z + 2nω1 + 2mω3) = f (z + 2nω1) = f (z), (A1) θ (0|τ) θ (z|τ) sn(u; k) = 3 1 , (A6a) θ (0|τ) θ (z|τ) where ω ,ω are two different complex numbers and n,m are 2 4 1 3 | | + + = θ4(0 τ) θ2(z τ) integers (following Ref. 17,wehaveω1 ω2 ω3 0). cn(u; k) = , (A6b) The Jacobi elliptic functions are usually defined through θ2(0|τ) θ4(z|τ) the pseudoperiodic θ functions: θ (0|τ) θ (z|τ) dn(u; k) = 4 3 , (A6c) θ3(0|τ) θ4(z|τ) ∞ √ n (n+1/2)2 = 2 | = 2 ≡ K(k ) ≡ − 2 θ1(z; q) = 2 (−1) q sin[(2n + 1)z], (A2a) where u θ3 (0 τ) z π K(k) z, τ i K(k) , k 1 k , n=0 and ∞ 2 1 θ z q = q(n+1/2) n + z , dt 2( ; ) 2 cos[(2 1) ] (A2b) K(k) ≡ n=0 0 (1 − t2)(1 − k2t2) ∞ π/2 n2 dθ θ3(z; q) = 1 + 2 q cos(2nz), (A2c) = √ (A7) 2 2 n=1 0 1 − k sin θ ∞ n n2 is the elliptic integral of the first type. Additional elliptic θ4(z; q) = 1 + 2 (−1) q cos(2nz), (A2d) n=1 functions can be generated with the following two rules: denote with p,q the letters s,n,c,d, then where the elliptic nome q is usually written as q ≡ eiπτ in 1 pn(u; k) pq(u; k) ≡ , pq(u; k) ≡ . (A8) terms of the elliptic parameter τ and accordingly, qp(u; k) qn(u; k)

For 0 k 1, K(k)isreal,iK (k) ≡ iK(k ) is purely | ≡ θj (z τ) θj (z; q). (A3) imaginary and together they are called quarter periods.The Jacobi elliptic functions inherit their periodic properties from

104418-9 ERCOLESSI, EVANGELISTI, FRANCHINI, AND RAVANINI PHYSICAL REVIEW B 88, 104418 (2013) the θ functions, thus combination of the two transformations, 01 1 sn(u; k) =−sn(u + 2K; k) = sn(u + 2iK ; k) S = −→ τ =− , (A18) −10 τ =−sn(u + 2K + 2iK ; k), (A9a) 10 =− + =− + T = −→ τ = τ + 1, (A19) cn(u; k) cn(u 2K; k) cn(u 2iK ; k) 11

= cn(u + 2K + 2iK ; k), (A9b) which satisfy the defining relations = + =− + dn(u; k) dn(u 2K; k) dn(u 2iK ; k) S2 = 1, (ST)3 = 1. (A20) =−dn(u + 2K + 2iK ; k). (A9c) The transformation properties for the S transformation are The inverse of an elliptic function is an elliptic integral. The 1 2 θ z|− =−i(iτ)1/2eiτz /π θ (τz|τ), (A21a) incomplete elliptic integral of the first type is usually written 1 τ 1 as 1 1/2 iτz2/π θ2 z|− = (−iτ) e θ4(τz|τ), (A21b) φ dθ τ F (φ; k) = √ = sn−1(sin φ; k), (A10) − 2 2 1 2 0 1 k sin θ θ z|− = (−iτ)1/2eiτz /π θ (τz|τ), (A21c) 3 τ 3 π = and it is the inverse of the elliptic sn. Clearly, F ( 2 ; k) K(k). 1 2 θ z|− = (−iτ)1/2eiτz /π θ (τz|τ), (A21d) Further inversion formulas for the elliptic functions can be 4 τ 2 found in Ref. 17. Additional important identities include from which follows − 1 = − 1 = θ 2(0|τ) θ 2(0|τ) k k (τ),k k(τ), (A22) k = k(τ) = 2 ,k = k (τ) = 4 , (A11) τ τ θ 2(0|τ) θ 2(0|τ) 3 3 and and 1 K − = K (τ) = K(k ), τ 2 + 2 = (A23) sn (u; k) cn (u; k) 1, (A12) 1 iK − = iK(τ) = iK(k). dn2(u; k) + k2sn2(u; k) = 1, (A13) τ Moreover, dn2(u; k) − k2cn2(u; k) = k 2. (A14) sn(iu; k) 1 sn(u; k ) =−i , cn(u; k ) = , Taken together, the elliptic functions have common periods cn(iu; k) cn(iu; k) ω = K k ω = iK k (A24) 2 1 4 ( ) and 2 3 4 ( ). These periods draw a dn(iu; k) lattice in C, which has the topology of a torus. While dn(u; k ) = , .... cn(iu; k) the natural domain is given by the rectangle with corners (0,0),(4K,0),(4K,4iK ),(0,4iK ), any other choice would be For the T transformation we have exactly equivalent. Thus, the parallelogram defined by the half θ (z|τ + 1) = eiπ/4θ (z|τ), (A25a) periods 1 1 iπ/4 θ2(z|τ + 1) = e θ2(z|τ), (A25b) = + = + | + = | ω1 aω1 bω3,ω3 cω1 dω3, (A15) θ3(z τ 1) θ4(z τ), (A25c) θ4(z|τ + 1) = θ3(z|τ), (A25d) with a,b,c,d integers such that ad − bc = 1, can suit as a fundamental domain. This transformation, which also changes and thus the elliptic parameter as k(τ) 1 k(τ + 1) = i ,k (τ + 1) = , (A26) k (τ) k (τ) ω c + dτ τ = 3 = , + (A16) and ω1 a bτ K(τ + 1) = k (τ)K(τ), can be cast in a matrix form as (A27) iK (τ + 1) = k [K(τ) + iK (τ)]. ω ab ω 1 = 1 Finally (A17) ω cd ω 3 3 k sn(u/k ; k) k cn(u/k ; k) sn u; i = k − , cn u; i = , k dn(u/k ; k) k dn(u/k ; k) and defines a modular transformation. The modular transformations generate the modular group PSL(2,Z). Each element k 1 dn u; i = , ... (A28) of this group can be represented (in a not unique way) by a k dn(u/k ; k)

104418-10 1 MODULAR INVARIANCE IN THE GAPPED XYZ SPIN- 2 CHAIN PHYSICAL REVIEW B 88, 104418 (2013)

While it is true that by composing these two transfor- = − 1/2 i[(τ−1)z2/π] where F (1 τ) e .Wehave mations we can generate the whole group, for convenience τ 1 1 we collect here the formulas for another transformation k = = , 1 − τ k(τ) k we use in the body of the paper: τ → τ , corresponding 1−τ (A30) to STS, τ k (τ) k k = i = i , τ 1 − τ k(τ) k θ z| = i1/2Fθ((1 − τ)z|τ), (A29a) 1 1 − τ 1 and τ K = k[K(τ) − iK (τ)], 1 − τ τ θ2 z| = Fθ3((1 − τ)z|τ), (A29b) (A31) 1 − τ τ iK = ikK (τ). 1 − τ τ Moreover, θ z| = Fθ((1 − τ)z|τ), (A29c) 3 1 − τ 2 1 u 1 u sn u; = k sn ; k , cn u; = dn ; k , k k k k τ (A32) θ z| = i1/2Fθ((1 − τ)z|τ), (A29d) 1 u 4 1 − τ 4 dn u; = cn ; k ,.... k k

*[email protected] 13B. Sutherland, J. Math. Phys. 11, 3183 (1970). 1J. L. Cardy, Nucl. Phys. B 270, 186 (1986). 14We adopt the conventions of Ref. 4 on the relation among , and 2A. Cappelli, C. Itzykson, and J.-B. Zuber, Nucl. Phys. B 280, 445 the Boltzmann weights of the eight-vertex model. (1987). 15See, for instance, Table 8.152 of Ref. 16. 3N. Seiberg and E. Witten, Nucl. Phys. B 426, 19 (1994); 430, 16I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and 485(E) (1994). Products (Academic, New York, 2007). 4R. J. Baxter, Exactly Solved Models in Statistical Mechanics 17D. F. Lawden, Elliptic Functions and Applications (Springer- (Academic, London, 1982). Verlag, New York, 1989). 5E. Date, M. Jimbo, T. Miwa, and M. Okado, Phys.Rev.B35, 2105 18N. I. Akhiezer, Elements of the Theory of Elliptic Functions, (1987); M. Jimbo, T. Miwa, and M. Okado, Nucl. Phys. B 275, 517 Translations of Mathematical Monographs Vol. 79 (American (1986); E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, Mathematical Society, Providence, 1990). ibid. 290, 231 (1987). 19H. Bateman and A. Erdelyi, Higher Trascendental Functions 6M. Bauer and H. Saleur, Nucl. Phys. B 320, 591 (1989). (McGraw-Hill, New York, 1953). 7P. Calabrese and F. H. L. Essler, J. Stat. Mech. (2010) P08029; 20G. A. Jones and D. Singerman, Complex Functions (Cambridge P. Calabrese, J. Cardy, and I. Peschel, ibid. (2010) P09003. University Press, Cambridge, England, 1987). 8F. Franchini, A. R. Its, and V. E. Korepin, J. Phys. A: Math. Theor. 21P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field 41, 025302 (2008). Theory (Springer, New York, 1999). 9E. Ercolessi, S. Evangelisti, and F. Ravanini, Phys. Lett. A 374, 22J. L. Cardy, in Conformal Invariance and Statistical Mechanics, 2101 (2010). edited by E. Brezin and J. Zinn-Justin (North-Holland, Amsterdam, 10E. Ercolessi, S. Evangelisti, F. Franchini, and F. Ravanini, Phys. 1990). Rev. B 83, 012402 (2011). 23E. Ercolessi, S. Evangelisti, F. Franchini, and F. Ravanini, Phys. 11A. Luther, Phys. Rev. B 14, 2153 (1976). Rev. B 85, 115428 (2012). 12J. D. Johnson, S. Krinsky, and B. M. McCoy, Phys. Rev. A 8, 2526 24A. De Luca and F. Franchini, Phys.Rev.B87, 045118 (1973). (2013).

104418-11