Quantum transfer-matrices for the sausage model

Journal of High Energy Physics, Jan 2018

In this work we revisit the problem of the quantization of the two-dimensional O(3) non-linear sigma model and its one-parameter integrable deformation — the sausage model. Our consideration is based on the so-called ODE/IQFT correspondence, a variant of the Quantum Inverse Scattering Method. The approach allowed us to explore the integrable structures underlying the quantum O(3)/sausage model. Among the obtained results is a system of non-linear integral equations for the computation of the vacuum eigenvalues of the quantum transfer-matrices.

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Quantum transfer-matrices for the sausage model

JHE Quantum transfer-matrices for the sausage model Vladimir V. Bazhanov 0 1 2 5 Gleb A. Kotousov 0 1 2 3 5 Sergei L. Lukyanov 0 1 2 3 4 Australian National University 0 1 2 0 grable Models , Sigma Models 1 Piscataway , NJ 08855-0849 , U.S.A 2 Canberra , ACT 2601 , Australia 3 NHETC, Department of Physics and Astronomy, Rutgers University 4 L.D. Landau Institute for Theoretical Physics 5 Department of Theoretical Physics, Research School of Physics and Engineering In this work we revisit the problem of the quantization of the two-dimensional O(3) non-linear sigma model and its one-parameter integrable deformation | the sausage model. Our consideration is based on the so-called ODE/IQFT correspondence, a variant of the Quantum Inverse Scattering Method. The approach allowed us to explore the integrable structures underlying the quantum O(3)/sausage model. Among the obtained results is a system of non-linear integral equations for the computation of the vacuum eigenvalues of the quantum transfer-matrices. Field Theories in Lower Dimensions; Integrable Field Theories; Lattice Inte- - HJEP01(28) 1 Introduction 2 Chiral transfer-matrices for the cigar 2.1 Wilson loops for the long sausage 2.2 De nition and basic properties of the chiral transfer-matrices 2.3 Basic facts about the quantum cigar 3 Chiral transfer-matrix for Zn parafermions 3.1 Bosonization of Zn parafermions 3.2 Discretization of the chiral transfer-matrix 4 Spectrum of the chiral transfer-matrix 4.1 4.2 4.3 4.4 5.1 5.2 5.3 Operators ( ) ODE/IQFT correspondence for the vacuum eigenvalues ODE/IQFT correspondence for the full spectrum Operators ( ) and ( ) 4.5 NLIE for the vacuum eigenvalues 5 Integrable structures in the sausage Basic facts about the quantum sausage NLIE for the k-vacuum eigenvalues in the sausage model A, B and T 5.4 ODE/IQFT for the sausage model 6 Discussion A Scaling behaviour of discretized chiral transfer-matrices B Scaling behaviour of Bethe roots for the vacuum state C Derivation of NLIE D Modi ed NLIE for oscillating asymptotics E Two-particle S-matrix for the sausage model { 1 { Introduction Integrability is a traditional area of mathematical physics having a long history. The notion is relatively well understood in nite dimensional systems: it requires the system to have exactly n isolating and commuting integrals of motion where n is the number of degrees of freedom. In the context of 1+1 dimensional eld theory, where the continuous number of degrees of freedom makes the traditional de nition insu cient, a suitable paradigm of integrability was also discovered. The key ingredient in this case is a Lie algebra-valued world sheet connection involving an analytic spectral parameter, whose atness condition is equivalent to the classical equations of motion. Since the Wilson loops T = Tr P exp Z C A remain unchanged under continuous deformations of the integration contour (see gure 1), they generate an in nite family of conserved quantities which can be used to solve the eld theory within the framework of the inverse scattering method. At the end of the seventies the Quantum Inverse Scattering Method (QISM) was proposed [1]. The approach was inspired by the pioneering works of Baxter on lattice statistical systems [2, 3] and based on the study of the common spectrum of the transfer-matrices (T -operators) | the quantum counterpart of the classical Wilson loops. The original formulation of the QISM was restricted to the so-called \ultralocal" models where the elementary transport matrices M n = P exp Rxxnn+1 A commute for di erent segments of the discretized path. The study of the algebra of the ultralocal operators M n led to the discovery of new remarkable mathematical structures collectively known today as the Yang-Baxter algebras. The most studied class of integrable models is the one where the Yang-Baxter algebra of the ultralocal operators M n admits a nite-dimensional representation. In this case the discretized quantum system can be interpreted as an exactly soluble statistical model whose solution can be obtained by means of the Bethe ansatz method. The solution of the continuous QFT is achieved by taking a proper scaling limit. An archetype of this scenario is the sine-Gordon model, while the corresponding statistical system is known as the inhomogeneous 6-vertex model [4]. However the QISM fails when it is applied to many interesting systems including classically integrable Nonlinear Sigma Models (NLSM). The origin of the problem can be illustrated by the O(3) NLSM governed by the Lagrangian relations [5, 6] [ta; tb] = i "abc tc : { 2 { (1.1) where the 3-dimensional vector n is subject to the constraint n n = 1. In this case the light-cone components of the at connection, A 21 (At Ax), are given by the 1 2 A = n = nata 2 su(2) ; @x) while n is built from the components of the unit vector n and ta: x x + R where C(i)(x) are local elds built from na and @tna and take values in the tensor product su(2) su(2). Because of the derivative of the -function, relations of this type are usually referred to as non-ultralocal. The presence of such terms creates serious problems with the rigorous proof of Poisson commutativity of the conserved charges (1.1) for di erent values of the spectral parameter. This, in turn, hampers the rst-principles quantization of the model. The problem with non-ultralocality is a general issue for integrable NLSM and is a major obstacle which prevents the incorporation of these models into the framework of the QISM. This was rst observed in attempts to quantize the principal chiral eld and O(N ) models. In the case of the O(4)-model (i.e., the model (1.2) with n 2 S3), Polyakov and Wiegmann [7] and later Faddeev and Reshetikhin [8] managed to bypass the problem with non-ultralocality. In both works, the NLSM was replaced by a di erent model satisfying the ultralocality condition; Polyakov and Wiegmann considered a model with Nf fermion avors, whereas Faddeev and Reshetikhin focused on a certain spin-S chain. They studied the thermodynamics using the Bethe ansatz technique and gained valuable results for the O(4) sigma model through the large Nf and S ! 1 limit, respectively. Both limiting procedures yielded the same system of Thermodynamic Bethe Ansatz Equations (TBA), which was then justi ed by a comparison with perturbative calculations and the exact results from the S-matrix bootstrap [9]. In the next 30 years an enormous amount of TBA systems were discovered. However, in spite of signi cant achievements in their study, the original problem of the construction of quantum transfer-matrices (including the calculation of their spectrum) has remained unsolved even for the O(3) and O(4) NLSM. { 3 { HJEP01(28) found that such a gauge also exists for the theory described by the Lagrangian This theory also belongs to the class of NLSM, and, assuming 0 < < 1, the corresponding target space is topologically the two-sphere. As ! 0 one can neglect the second term in the denominator which results in the O(3) Lagrangian multiplied by . Hence, the model (1.6) is the one parameter deformation of the O(3) sigma model. It is colloquially known as the \sausage model" since for ! 1 the target manifold can be pictured as a long sausage with length / log( 11+ ) (see gure 2). The sausage model was introduced in the work [14] where strong evidence for its quantum integrability was presented. Its zero curvature representation was found in ref. [15]. Remarkably, with the proper gauge transformation, the at connection can be chosen to satisfy the ultralocal Poisson bracket relations A (xj ) ; A (x0j 0) = A (xj ) A (x0j 0); r( = 0) (x x0) 1 (1.6) (1.7) (1.8) (1.9) Here A (xj ) ; A (x0j 0) = 0 : r( ) = 1 1 1It is worth noting that a transition to an ultralocal gauge has previously been utilised for some other It is straightforward to check now that the formulae2 a( ) = f ( ) t3 + i g( ) t1 ; f ( ) = g( ) = (1 (1 (1 ) ) ) p ) canonically conjugate to (Q; Q ) are given by the relations tanh 12 (Q + Q ) . Let us also introduce the shortcut notations 3 = = 2( 1 1 3 and the eld independent element of sl(2; C) HJEP01(28) de ne the at connection, i.e., satisfying the ultralocality conditions (1.7). Having an explicit formula for the ultralocal at connection we may turn to the construction of the time-independent Wilson loops. In fact their construction requires some assumptions to be made. First of all in writing eq. (1.1) it is assumed that the connection is single valued on the space-time cylinder. In the case under consideration this can be achieved by imposing periodic boundary conditions on the unit vector n. Such boundary conditions look natural for the O(3) NLSM, since they preserve the global invariance of the Lagrangian. In the case of the sausage model, the O(3) symmetry is broken down to U(1) and we will consider more general boundary conditions n3(t; x + R) = n3(t; x) ; n (t; x + R) = e 2 ik n (t; x) (n n1 in2) (1.16) 2Notice that the connection in the ultralocal gauge takes values in the complexi cation of the Lie algebra su(2). This should not worry us because only reality conditions imposed on gauge invariant quantities are a subject of interest and importance. In what follows we'll prefer to use the Cartan-Weyl generators h = 2t3; e = t1 it2: instead of ta (1.4). [h; e ] = 2 e ; [e+; e ] = h { 5 { which depend on the twist parameter k k + 1. In terms of the real and imaginary part of the complex eld Q, Q = + i ; the twisted boundary conditions are equivalently given by (t; x + R) = (t; x) ; (t; x + R) = (t; x) + 2 k : As it follows from eq. (1.14) the connection is not single valued on the space-time cylinder now, charges A (t; x + R) = ei kh A (t; x) e i kh ; (1.19) however with a little modi cation one can still introduce the time independent conserved Tj ( ) = Trhe i kh M j ( ) i ; Z C M j = j P exp dx Ax : (1.20) Notice that the de nition of the Wilson loop requires a choice of the representation of the Lie algebra and the subscript j in (1.20) labels the representations j of sl(2). In our case we will focus on the nite dimensional irreps, so that in the standard convention j = 12 ; 1; : : : . The ultralocal Poisson structure (1.7) implies that the monodromy matrix generates the classical Yang-Baxter Poisson algebra: M j ( ) ; M j0 ( 0) = M j ( ) M j0 ( 0) ; rjj0 ( = 0) ; where rjj0 = j j0 [ r ]: This, supplemented by the easily established property implies the in nite set of relations e i kh e i kh ; r( = 0) = 0 ; Tj ( ); Tj0 ( 0) = 0 : Thus, we have arrived at formulae (1.21) and (1.23) which are key ingredients in the Hamiltonian approach to the inverse scattering method [16, 17]. When faced with the problem of quantizing the sausage NLSM, a simple idea that may come to mind is to discretize the path-ordered integral onto N small segments. Due to ultralocality, the N elementary transport matrices j hP exp Rxxnn+1 dx Axi satisfy the same type of Poisson bracket relation as (1.21) and Poisson commute for di erent segments. These relations can be formally quantized leading to a certain quantum Yang-Baxter algebra. The major problem now is to construct a suitable representation of this abstract algebraic structure. In the case under consideration, the representation is, in all likelihood, in nite dimensional even for nite N . At this moment, it is not clear for us how to construct and handle such representations, let alone take the scaling limit with N ! 1. In this work, we will try to avoid discretization as much as we can and mostly follow the so-called BLZ approach | the variant of the QISM developed in the series of works [11, 18, 19]. For integrable Conformal Field Theories (CFT), it was demonstrated { 6 { (1.17) (1.18) (1.21) (1.22) (1.23) that the T -operators can be constructed without any discretization procedure. Later it was observed that many deep properties of representations of Yang-Baxter algebras in integrable CFT can be encoded in the monodromies of certain linear Ordinary Di erential Equations (ODE) [20{26]. These results were extended to massive Integrable Quantum Field Theories (IQFT) [27] (for recent developments, see also refs. [28{35]). The general relation of this type will be referred to in the paper as the ODE/IQFT correspondence. Broadly speaking, the ODE/IQFT correspondence means that for a given IQFT the eigenvalues of the quantum T -operators are identi ed with certain connection coe cients for the system of equations, D( ) = 0 ; D( ) which is found to be a function of the original spectral parameter from the quantum theory. The system of ODE can be then interpreted as an auxiliary linear problem, whose compatibility condition, [D( ); D( )] = 0, coincides with the zero-curvature representation for some classically integrable eld theory. Thus the ODE/IQFT correspondence reduces the calculation of the spectrum of quantum transfer-matrices to a certain problem in the theory of classical integrable equations. The latter can be e ectively treated by the inverse scattering transform method. This makes the ODE/IQFT correspondence a very powerful tool. In particular, it gives a practical way to make progress in the conceptual long standing problem of the quantization of integrable NLSM. The ultimate goal of this work is to demonstrate this for the case of the quantum sausage model. 2 Chiral transfer-matrices for the cigar The BLZ approach [11, 18, 19] begins with an analysis of the RG xed point which controls the ultraviolet behaviour of the integrable QFT. With this in mind, let's take a quick look at the sausage NLSM (1.6). In the traditional path-integral quantization, the model should be equipped with a UV cuto . A consistent removal of the UV divergences requires that the \bare" coupling in the Lagrangian (1.6) be given a certain dependence on the cuto momentum, i.e., = ( ). To the rst perturbative order the RG ow equation is given by [14] (2.1) (2.2) where ~ stands for the (dimensionless) Planck constant. Integrating this equation leads to where changing = ~ + O(~2). The energy scale E is an RG invariant (i.e., it's kept xed with ), so that ! 1 as ! 1. Having in mind the quantization of the model, this simple analysis shows that the classical eld theory (1.6) deserves special attention when is close to one. 1 + (1 2) + O(~2) ; = (E = ) ; { 7 { t 0 0 R x HJEP01(28) replaced by an integration contour along the characteristics: x arrow) and x+ = t0 + R with t0 < x < t0 R (blue arrow). = t0 with 0 < x+ < t0 + R (red 2.1 Wilson loops for the long sausage Consider the conserved charges Tj ( ) (1.20) for 1 1. If the integration contour appearing in the de nition (1.20) is chosen at the time slice t = t0, Tj ( ) are expressed in terms of the pair of real variables ( ; ) (1.17) and the corresponding canonically conjugate momenta. We can use the magic of the zero-curvature representation to re-express Tj ( ) in terms of the values of the elds (x; t) and (x; t) on the characteristics x+ t + x = t0 + R and x t x = t0, +(x+) = jx =t0 ; (x ) = jx+=t0+R ; +(x+) = (x ) = jx =t0 jx+=t0+R : Indeed, the original integration along the time slice t = t0 in (1.20) can be replaced by the path-ordered integral over the contour glued from two light-cone segments as shown in gure 3. Denoting the light-cone values of the connection as A+(x+) = A+(t; x)jx =t0 ; A (x ) = A (t; x)jx+=t0+R ; and taking into account the boundary conditions (1.18), one can rewrite eq. (1.20) in the form where Tj Tr e i kh M j = Tr Z t0 R t0 j (M) ; This formula is a convenient starting point for the quantization procedure, though one should have in mind that the light-cone elds are complicated functionals of the canonical set at t = t0. In general, they are not independent variables and their Poisson brackets (2.3) (2.4) (2.5) (2.6) { 8 { Here we use the notations Lj( +) = j P exp j( +) 1 i + 2 + where and Notice that the r-matrix in the classical Yang-Baxter Poisson algebra (1.21) depends only on the ratio of the spectral parameters and is independent of . Hence, we can conclude that +li!m1xed Tj = Trh Lj( +) e P1h i : P1 = +(t0 + R) +(t0) : where the variables are de ned by the relations HJEP01(28) + = += and (h; e+; e ) are the Cartan-Weyl generators (see footnote 2 on page 5). Then the connection component A+(x+) only depends on the parameter +, while A (x ) only depends on assume that . According to (2.8) at least one of the ( +; ) vanishes as ! 0 and + is kept xed. Then, it is easy to see that ! 1. Let us are not known explicitly. Fortunately, we can overcome this problem when considering the limit quantities + light-cone elds a( 1) (1.12) in the form and ! 1. In this limit the at connection (1.14) simpli es considerably, since the 12 @xQ are then linearly expressed through the . It is convenient to write the constant Lie algebra elements a Lj( +) ; Lj0( 0+) = Lj( +) Lj0( 0+) ; rjj0( += 0+) ; and also that the j are in involution: f j( +); j0( 0+)g = 0 : For 1 1 the target space of the sausage model consists of two cigars glued together, whose tips are separated by a distance / log( 11+ ) (see gure 2). In the ! 1 limit, the sausage is broken down into two disjoint half-in nite Hamilton's cigars [36]. Let us focus on one of the cigars, say the left one. The Lagrangian of the NLSM which has this target space follows from the sausage Lagrangian (1.6): one should express the sausage Lagrangian in terms of the elds and , perform the constant shift and nally take the limit ! 1. As a result one nds [37, 38] ! L = e 2 2(1 + e2 ) (2.15) This suggests that the j introduced through the limiting procedure (2.9), can be interpreted as conserved quantities in the cigar NLSM (2.15). The elds and which appear in the construction of Lj (see eqs. (2.11), (2.10)), should be now understood as the light-cone values of the elds in the cigar NLSM, in the case when the eld becomes large, i.e., in the asymptotically at domain of the target manifold. The cigar equations of mo0 0, whose general solution can be written as +(x+) + (x ) and +(x+) + (x ). Taking the time slice t = t0 where the eld becomes large, one nds 1 2 1 2 f (x); 0 (y)g = (x y) ; f 0 (x); 0 (y)g = 0(x y) (2.16) f 0 (x); 0 (y)g = f 0 (x); 0 (y)g = f (x); 0 (y)g = f (x); 0 (y)g = 0 ; where the prime stands for the derivative w.r.t. the argument. Recall that the j are time independent charges, and their value does not depend on the choice of t0. Hence, we come to the conclusion that the matrix Lj (2.10), built from the elds +(x), +(x) satisfying the Poisson bracket relations (2.16), will obey the \r-matrix" Poisson bracket algebra (2.13). The following comment is in order here. It is not di cult to see that the matrix Lj does not change under the constant shifts + ! relations (2.16) can be lifted to + + const so that the Poisson bracket f +(x); +(y)g = (x y) ; 1 4 (2.17) where (x) = 2n + 1 for nR < x < (n + 1)R; n 2 Z. The later are consistent with the quasiperiodicity condition (2.12). 2.2 De nition and basic properties of the chiral transfer-matrices We may now turn to the problem of quantization. The quantum counterpart of the conserved charges (2.9) will be referred to as the chiral transfer-matrices and in what follows, will be denoted by the same symbol j. Their construction almost identically follows the steps elaborated in refs. [11, 19] in the context of the quantum KdV theory. Here we present them very brie y, referring the reader to those works for detailed explanations. First of all we should \quantize" the Lie algebra sl(2), so that h, e are understood now as the generators of the quantum universal enveloping algebra Uq sl(2) : [h; e ] = 2 e ; [e+; e ] = (2.18) q h q q h q 1 ; i~ where q = e 2 . Consequently the symbol j will stand for the (2j + 1)-dimensional representation of the quantum algebra. Instead of the Planck constant ~, for convenience we will use the parameter n: matrix ~ 2 n ; q = e n : i The quantum counterpart to Lj is the following (2j + 1) (2j + 1) operator valued Lj ( +) = j P exp i + dx V + q 2h e+ + V q 2h e e P1h : (2.20) where c are some constants and with V (x) = '+(x) = Q1 + #+(x) = Q2 + R R 2 x p n P1 + i X am e 2 Rim x 2 x p n + 2 P2 + i X bm e 2 Rim x ; Fp (\Fock space") be the highest weight module of the Heisenberg algebra (2.23) with the highest weight vector j p i de ned by the equations P1 j p i = pn1 j p i ; P2 j p i = p2 n + 2 j p i : V (x) : Fp1;p2 7! Fp1 i;p2 ; and therefore the matrix elements of Lj ( ) are operators in sion (2.20) contains the ordered exponential which can be formally written in terms of a 1 m= 1Fp1+im;p2 . The expres(2.19) (2.21) (2.22) (2.23) (2.24) (2.25) (2.27) (2.28) It is easy to see that power series in as Lj ( +) = j where However, since 1 X (i +)m Z m=0 dxm dx1 K(xm) K(x1) e P1h ; (2.26) t0+R>xm>:::x1>t0 K(x) = V + q 2h e+ + V q 2h e : V (x2)V (x1) x2!x1+0 a commutation relations 2) c+c , the integrals in (2.26) diverge. As explained in [19], the V 1 (x1)V 2 (x2) = q2 1 2 V 2 (x2)V 1 (x1) ; x2 > x1 ( 1;2 = ) (2.29) allow one to re-express the integrals in (2.26) in terms of two basic contour integrals X0 = q 1 This procedure yields an unambiguous de nition of the ordered exponential in (2.20) for n 6= 2; 4; 6 : : : The case of even n needs some special attention and we will return to it later. Notice that the above analytical regularization of the P-ordered exponential (2.20) is not applicable at n ! 1. This limit can be studied within the renormalized perturbation theory in ~ = 2n . As usual, the perturbative expansion will involve the counterterms depending on the UV regulator. At ~ = 0 the counterterms give rise to anomalous contributions which remain nite in the limit where the UV regulator goes away. This makes the classical limit not entirely straightforward, however if the anomalous terms are properly taken into account, the formula (2.20) precisely reduces to its classical version (2.10) in the limit ~ ! 0. The relevant calculations will be presented elsewhere. The operator valued matrices Lj (2.20) are designed in such a way that, for arbitrary chosen constants c and t0, they obey the quantum Yang-Baxter algebra Rjj0 ( 0+= +) L( +) where the matrix Rjj0 ( ) is the trigonometric solution to the Yang-Baxter equation which acts in the space j j0 . In particular R 1 1 ( ) = BBB 2 2 1 q 1 0 0 The proof of eq. (2.31) follows that from the work [19]. The chiral transfer-matrices, de ned similar to (2.9), satisfy the commutativity condition as a simple consequence of (2.31). Notice that the chiral transfer-matrices act inside a single Fock space, whereas the same is not true for an arbitrary element of Lj ( ). Furthermore, the Fock space Fp naturally splits into the nite dimensional \level subspaces" Fp = L1=0 Fp(L) : L Fp (L) = L Fp(L) ; where the grading operator is given by h j ( +) = Tr Lj ( +) e P1h i ; [ j ( +); j0 ( 0+) ] = 0 ; 1 X m=1 L = 2 a mam + b mbm : (2.32) (2.33) (2.34) (2.35) (2.36) Using the relation, one can show (see appendix C from [19]) that the j ( +) commute with the grading operator, and therefore, act invariantly in each nite-dimensional level subspace: j ( +) : Fp 7! Fp(L) : (L) ( +) y = ( +) : The Fock space Fp can be equipped with an inner product consistent with the Hermiticity conditions aym = a m ; bym = b m imposed on the Heisenberg operators (2.23). It is not di cult to show that for real p21; p22 and 2 , ( +) is a Hermitian operator and + Notice that the commutativity with the grading operators can be interpreted as the independence of the chiral transfer-matrix on the arbitrary chosen constant t0. It turns out that they further do not depend on the constants c appearing in the de nition of the vertex operators V (2.21). Also, a simple dimensional analysis shows that the spectral parameter + and R occur in the chiral transfer-matrix through the combination 2+R n2 only. It is convenient to introduce a dimensionless spectral parameter by means of the relation 2 = converges in the whole complex plane of 2 and de nes an entire function with an essential singularity at 2 = 1. The asymptotic expansion near the essential singularity is of primary interest. It can be written as ( ) = exp 2 sin( 2n ) ( 2) n2 ~ i ( 2 n ) 2(n+2) ; j qj+ 21 1 ( ) = j+ 12 2 q j + j 12 qj+1 ; supplemented by the condition 0 = 1. In what follows, we will mostly focus on the fundamental transfer-matrix and use the notation 2 1 . The integrable structures associated with the commuting family of operators j ( ) were already studied in the context of the so-called paperclip model | an integrable model with boundary interaction [43]. Here for convenience we make a short summary of some basic properties of the operator ( ). For arbitrary complex p = (p1; p2), the operator ( ) 2 End Fp is an entire function of 2 in the sense that all its matrix elements and eigenvalues are entire functions of this variable. Thus the power series and treat the chiral transfer-matrices as functions of this variable rather than the dimensionful +. so-called fusion relation [39{41] The chiral transfer-matrices are not independent operators for di erent values of j = where ~ is a formal power series of the form ~(~) 2 cos 2 p2 1 m=1 + X ~tm ~2m : (2.44) This asymptotic expansion can be applied for arbitrary complex p = (p1; p2) and n 6= 2; 4; 6 : : : . Furthermore, in the case n 1 it holds true for j arg( 2 )j < . The expansion coe cients in (2.42) and (2.44) form two in nite sets of mutually commuting operators. Using the terminology of the work [18], we will refer to tm ~ tm and Remarkably, the formal power series ~(~) can be written in a form similar to (2.33). 1 m=1 as the nonlocal and dual nonlocal Integrals of Motions (IM), respectively. 1 m=1 Namely [43], + 1 2 i 2) are the conventional Pauli matrices and the vertex operators (x) = '0+(x) + is equal to 1 + n+12 and we assume here that they are normalized (2.47) (2.48) (2.49) where 3 ; in such a way that t1(p1; p2) = and t~1(p1; p2) = (2.46) Because of the divergencies, the path ordered exponential in (2.45) should be understood in the same manner as (2.20), i.e., the formal expansion in a power series of ~ + should be rewritten in terms of the basic contour integrals similar to (2.30). With this analytical 2 regularization the r.h.s. of eq. (2.45) becomes a formal power series in ~2+R n+2 with unambiguously de ned expansion coe cients. Up to a factor similar to that in (2.40), this combination can be identi ed with ~2 in eq. (2.44): ~2 = For future reference we present here explicit formulae for the \vacuum" eigenvalues of the operators t1 and ~t1 corresponding to the highest weight vector j p i 2 Fp (2.24): ( 12 + n1 ) (1 + n1 ) n + 2 n 2 + 4p22 1 + 4p21 2 ( 1+2ip1 ) ( n 1 2ip1 ) n n + 2 n+2 2 p ( 1 2 (1 n+12 ) n+12 ) n n + 4 1 4p21 4p22 2 ( 1n+2p22 ) ( 1n++2p22 ) : An e cient integral representation for calculating the vacuum eigenvalue t~2(p1; p2) can be found in appendix A of ref. [43]. For even n, the chiral transfer-matrices require some careful handling. In this case ( ) can be de ned through the limiting procedure ( )jn=2l = lim exp !0 4 2l ( ) n=2l+ so that the asymptotic formula (2.43) should be substituted by (l = 1; 2; 3 : : :) ; (2.50) ( )jn=2l = exp 2 n log( 2 ) ~ i ( 2 ) 2(l+1) l n=2l : (2.51) The formulae (2.43), (2.44) are not valid for positive real 2. In order to describe the asymptotic behaviour for 2 through the relation 1 IM (2.44) by the set g~m m=1 which are algebraically expressed in terms of the former ! +1, it is convenient to substitute the set of dual nonlocal 2 cos(2 P2) + X ~tm zm = 2 cos(2 P2) exp 1 X g~m zm : m=1 1, as 2 ! +1 ; nm z n+2 + O(z 1) (2.52) (2.53) (2.54) Then, for arbitrary complex p = (p1; p2), n 6= 2; 4; 6 : : : , n ( ) = 4 cos eH( 2) cos G( 2) where 1 m=1 2 p2 n + 2 n 2 1 X m=1 H(z) 2 cot G(z) n 2 z 2 + n z 2 + g~m sin 1 X m=1 g~m cos 2 m n + 2 2 m n + 2 nm z n+2 + O(z 1) : For even n, the rst term in the formal power series H(z) should be replaced by 2z n2 log(z). 2.3 Basic facts about the quantum cigar In the previous subsection, we described the formal algebraic construction of the chiral transfer-matrices. Here we brie y discuss how j ( ) can be understood as operators in the quantum cigar NLSM (for more details on the quantum cigar see, e.g., ref. [44]). The cigar NLSM was introduced before at the classical level by means of the Lagrangian (2.15). In the classical eld theory, it is natural to consider the following scattering problem. Suppose that at t ! 1 we are given the eld con guration within the asymptotically at domain of the target manifold, i.e., (t; x)jt! 1 (t; x)jt! 1 0 (in) + 0 (in) + 4 R 2 R 1 P (in) t + (kx + k~t) + X m6=0 i m X m6=0 i m a(min) e 2 Rim (t+x) + a(min) e 2 Rim (t x) (2.55) b(min) e 2 Rim (t+x) + b(min) e 2 Rim (t x) : P (in) 1 the string approaches the tip, scatters and then escapes back to the at region. After the scattering process the zero mode momentum changes sign. In writing this equation, we took into account the boundary conditions (1.18). Also, the constant k~ is the conserved charge for the Noether U(1)-current associated with the Lagrangian (2.15). The set, A (in) = f (0in); P (in); 0(in); k~; a(min); b(min)g, can be interpreted 1 as a classical \in-state" for a string propagating on the target manifold (see gure 4). The nontrivial interaction occurs at some nite time when the elds take values in the vicinity of the tip of the cigar. After scattering at the tip, as t ! +1, the eld con guration returns to the asymptotically at domain and takes the same form as in the r.h.s. of (2.55) with the in-state A (in) replaced by the out-state A (out) = f 0 (out); P1(out); (out); k~; a(mout); b(mout) .3 0 g The classical scattering problem can be formulated as the problem of nding the canonical transformation which maps A (in) to A in nite sets of left- and right-currents [45], i.e., (out). It turns out that the theory possesses two HJEP01(28) (2.56) (2.57) (2.58) k(in) 2 Z. (s = 2; 3; : : :) ; so that the classical dynamics of the elds are strongly constrained. In particular, the magnitude of the zero-mode momentum remains unchanged after the scattering (see gure 4), 1 P (out) = P (in) : 1 Consider now the quantum theory. First of all we note that the value of the U(1) charge is quantized so that (n + 2) k~ = m 2 Z. Thus the space of states of the quantum theory splits into orthogonal subspaces Hk;m labeled by the twist parameter k and the integer m. The quantum theory still possesses the chiral currents satisfying eqs. (2.56). As a result, Hk;m can be decomposed into the highest weight irreps of the W -algebra, W W generated by the elds Ws and Ws [45]. Let Vh (Vh) be the highest weight representation of the chiral W -algebra W (W) labeled by the highest weight h (h). Then, schematically, The highest weight h can be chosen to be a pair of numbers ( ; w), where coincides with the conformal dimensions of the highest weight vector, while w is the eigenvalue of 3Strictly speaking, the winding number k is only conserved modulo an integer, i.e., k(out) Here we ignore this and assume that k(out) = k(in) 2 ( 12 ; + 12 ]. the dimensionless conserved charge R2 R0R dx W3(x), and similar for h. Let us rst focus on the \left" component Vh in the tensor product Vh Vh. It should be clear that the quantum counterpart to the left components of the inasymptotic elds (2.55) can be identi ed with the elds '+ and #+ given by (2.22). Since the quantum elds Ws are chiral currents, i.e. Ws(t; x) = Ws(t + x), they can be expressed in terms of the asymptotic elds '+ and #+. Indeed, for given s, Ws is a certain order-s homogeneous polynomial with constant coe cients w.r.t. the elds '0+, #0+ and their higher derivatives (in other words, any monomials appearing within Ws contains exactly s derivative symbols). This implies that the Fock space Fp, which is the space of representation for the elds '0+, #0+, possesses the structure of the highest weight representation of the chiral W -algebra. It turns out that for real p, the Fock space Fp coincides with irrep Vh as a linear space, provided that h = ( ; w) is related to p = (p1; p2) as follows 4n p21 + 3n + 2 3(n + 2) 2 p 2 2n + 1 12 : Z p1<0 In fact, one can use these formulae to conveniently parameterize the highest weight h by the pair (p1; p2): Vh Vp1; p2 . With this notation, a more accurate version of eq. (2.58) reads as 1 2 1 2 p2 = m + (n + 2) k ; p2 = m (n + 2) k : The direct integral in (2.60) does not include the domain with positive p1, since, as follows from eqs. (2.59), Vp1; p2 V p1; p2 . A basis of in-asymptotic states in Hk;m is formed by a(inm)1 : : : a(in) mN a(inm)1 : : : a(in) mN b(inm)1 : : : b(in) mM b(inm)1 : : : b(in) mM j vac i and can be identi ed with the states from the tensor product of the Fock space Fp1; p2 a m1 : : : a mN a m1 : : : a mN b m1 : : : b mM b m1 : : : b mM j p1; p2 i j p1; p2 i : (2.63) Similarly for the out-states, one has a(omut1) : : : a(omutN) a (omut1) : : : a(omutN) b (omut1) : : : b(omutM) b(omut1) : : : b(omutM) j vac i a m1 : : : a mN a m1 : : : a mN b m1 : : : b mM b m1 : : : b mM j p1; p2 i j p1; p2 i : Usually, the S-matrix is introduced as a unitary operator which relates the in- and outasymptotic bases. In the case under consideration, the S-matrix can be interpreted as the intertwiner acting between the Fock spaces: S^ : Fp1; p2 Fp1; p2 7! F p1; p2 F p1; p2 : (2.59) (2.60) (2.61) (2.62) (2.64) (2.65) It turns out that the operator S^ has the following structure S^ = S0(p) S^L S^R ; (L) (L) where S^L intertwines the level subspaces, S^L : Fp1; p2 7! F p1; p2 , and is normalized by the condition S^L j p1; p2 i = j p1; p2 i, and similarly for S^R. construction of the operators S^L;R is a straightforward algebraic task. The more delicate For a given level `, the problem is nding the overall scalar factor S0(p). It was obtained in the minisuperspace approximation in ref. [46]. The exact form of S0(p) has been known since the unpublished work of the Zamolodchikov brothers [47].4 Returning to the chiral transfer-matrices, let us note that these operators should act in the Hilbert space of the quantum cigar, and therefore their action should commute with the intertwiner S^:5 S^ ( ) = ( ) S^ : In practice, this condition implies that all matrix elements of the (dual) nonlocal IM in the basis of Fock states (2.63) are even functions of p1 (for illustration see eqs. (2.48), (2.49)). The quantum cigar also possesses an in nite set of the so-called local IM acting in Hk;m. To get some feeling for these operators, we need to remind ourselves of an important feature of the model. Namely, it admits an equivalent \dual" description in terms of the so-called sine-Liouville model. The dual Lagrangian is given by [47] (2.66) (2.67) (2.68) (2.69) (2.70) can L (dual) = 1 4 pn' cos p n + 2 # ; with the sine-Liouville elds satisfying the boundary conditions '(t; x + R) = '(t; x) ; #(t; x + R) = #(t; x) + p 2 m n + 2 : Notice that the \coupling" M is a somewhat fake parameter of the Lagrangian | by an additive shift ' 7! ' + const the value of M can be chosen to be any real number. Nevertheless, it is convenient to keep it unspeci ed. To understand the relation between the elds in the NLSM and its dual description, let us take the \zero-mode" of the eld ' '0 = Z R dx 0 R '(x) ; potential term in the action (2.68) can be neglected and p'n and consider the region '0 ! +1 in con guration space. In this asymptotic domain, the # + const, while pn+2 be identi ed with ~ | the eld from the cigar NLSM de ned by the relation J = where J stands for the Noether U(1)-current. 4Although Zamolodchikov's notes have never been published, they were broadly distributed within the scienti c community. 5The intertwiner S^ should not to be confused with the so called \re ection" operator R^ : Fp1; p2 Fp1; p2 7! F(Lp)1; p2 ers ^L : Fp1; p2 7! F (L) Fp1; p2 , and [R^; ( )] = 0. Note that R^ = ^ S^ where ^ = ^L ^R and the chiral intertwinp1; p2 are de ned by the conditions ^Lam = am ^L; ^Lbm = +bm ^L; ^L j p1; p2 i = j p1; p2 i, and similar for ^R (see, e.g., [44]). is 1 = R 2 s 1 Z R dx It turns out that for any even s = 2j there exists a local density (de ned modulo the addition of a total derivative and an overall multiplicative constant) such that i2j 1 is an integral of motion and satis es the commutativity conditions [ i2j 1; ( ) ] = [ i2j 1; i2j0 1 ] = 0 : These operators are referred to as the (chiral) local IM. They were studied in ref. [42], where the explicit form for the rst local IM and their vacuum eigenvalues, i2j 1(p1; p2) for j = 1; 2; 3, can be found. Here we only note that for any j = 1; 2; : : : the numerical coe cients Cl(ms) can be written as where the dots stand for monomials which include higher derivatives of @+' and @+# and Cl(mj) = C2j 1 ( 2)j+1(2j 2)! (n + 2)( 12 (j + 1)! j) l ( n)( 12 l! m! j) m ( n)l 1 (n + 2)m 1 : Here (a)m = Qim=01(a + i) is the Pochhammer symbol. The overall normalization constant C2j 1 is usually set to C2j 1 = 2 3j (j + 1)! n(n + 2) (n + 2)( 12 j) j ( n)( 12 j) j : The twist parameter k has a natural interpretation in the dual description | it can be identi ed with the so-called quasimomentum. The sine-Liouville Lagrangian is invariant 2 under the transformation # 7! # + pn+2 . Due to this periodicity, the space of states of the theory with the boundary conditions (2.69), splits on the orthogonal subspaces Hk;m such that for any state j A i 2 Hk;m, the corresponding wave functional A[ '(x); #(x) ] transforms as 2 A '(x); #(x) + p = e2 ik A[ '(x); #(x) ] : higher derivatives. All such elds are periodic in x, so that one can introduce the integral, HJEP01(28) (2.71) (2.72) (2.73) (2.74) (2.75) (2.76) P2j = X l+m=j 3 Chiral transfer-matrix for Zn parafermions While quantizing the sausage model within the BLZ approach, we have run into the problem of nding the spectrum of the chiral transfer-matrices for the cigar NLSM. As it has been explained, we can consider ( ) as an operator acting in the Fock space Fp with real p = (p1; p2). From the formal point of view, the same spectral problem can be posed for any complex values of p. Notice, that for real p2, 2 and pure imaginary p1, the operator One of the most e ective methods for the calculation of the spectrum of commuting families of operators including the transfer-matrices in integrable quantum eld theory is based on the ODE/IQFT correspondence. From our study of the parafermionic transfermatrix, we proposed the ODE counterpart in the correspondence for the cigar NLSM. It turns out to be identical to that which was introduced earlier in the context of the so-called paperclip model in the works [42, 43]. Based on the results of these papers, we derived non-linear integral equations for determining the vacuum eigenvalues of the chiral transfermatrix which work both for the cigar and the parafermionic regimes. We believe that this might be a good starting point for applying the powerful fermionic methods [92{96] to the sausage/O(3) NLSM. In refs. [27, 58], a conceptual explanation was given of how the ODE/IQFT correspondence for integrable conformal eld theory can be generalized to the massive IQFT. Following this route, we extended the ODE/IQFT correspondence from the cigar to the sausage NLSM. With the correspondence one can uncover the basic integrable structures by studying the properties of the connection coe cients of the ordinary di erential equations. The main result of this paper is the list of properties (i){(x) from section 5.3 for the commuting families of operators in the sausage model which includes the quantum transfer-matrix. The technical result that deserves to be mentioned is the system of NLIE which describes the vacuum eigenvalues of the commuting families of operators. Among other things, it allows one to calculate the k-vacuum energies of the sausage/O(3) NLSM. There are many results in the literature concerning the energy spectrum of the O(3) sigma model in the sector with k = 0 [76, 97{99]. In ref. [14] a system of TBA equations was proposed which allows one to calculate the ground state energy for k = 0 and integer values of the dimensionless coupling n 3 of the sausage model. Recently Ahn, Balog and Ravanini [70] transformed this system of TBA to a system of three non linear integral equations which, it is a rmed, works for any real positive n. Their main assumption is a periodicity condition for the Q-function given by eq. (3.16) from that paper. In our investigations, we did not nd any trace of a Q-function satisfying such a strong periodic condition. Nevertheless, the numerical results presented in gure 2 from that paper seem to be in agreement with the data obtained from the solution of our NLIE (5.30), (5.31) with k = 0 and n = 1. This situation needs to be clari ed. Let us brie y touch on some problems which have not been discussed in the text but are directly related to the subject of this work. We did not make any mention of the sausage model with the topological term equal to which is also expected to be an integrable QFT [14, 100]. Another closely related model is the so-called 3D sausage model introduced by Fateev in [80]. In the work [15], classical integrability was established for a four-parameter family of NLSM with torsion which includes the 3D sausage as a two parameter subfamily. We believe that extending the ODE/IQFT approach to these models will be useful, both as a step in the development of the method, and in terms of applications. Some results in this direction have already been obtained in ref. [81]. There are also the remarkable works [101{103] on toroidal algebras, which are deeply connected to this eld. All the models mentioned above are based on the sl(2)-algebra and its associated integrable structures. Since the work of Klimcik [104] there has been increasing interest 2 3 4 5 0 p 2 48 0:0546105 0:0661040 0:0683646 n 6 7 8 9 10 0:0658731 0:0613178 0:0561029 0:0509101 0:0460445 in \deformed" integrable NLSM associated with higher rank Lie algebras [105{108]. The rst principle quantization of such theories seems to be a very interesting problem. In the recent works [109, 110], an important step in this direction was taken where a one parameter deformation was found of the set of \circular brane" local integrals of motion introduced in ref. [111]. This o ers the possibility for the quantization of the deformed O(N ) NLSM along the lines of this work. Perhaps the main motivation for studying NLSM is based on the fact that certain types of SUSY sigma models are at the heart of the celebrated AdS/CFT correspondence, and integrability is an important possibility. In particular, the NLSM associated with the AdS side of the correspondence for N = 4 SUSY Yang Mills theory was argued to be integrable [ 112, 113 ]. As brie y discussed in the introduction, the study of the rst principles quantization of the NLSM by traditional techniques has proven to be di cult. A similar situation exists with sigma models on supergroups and superspaces, which are expected to provide theoretical descriptions of condensed matter systems with disorder [114]. That is where one is most tempted to try the power of the ODE/IQFT approach. Acknowledgments PHY-1404056. S. L.'s gratitude goes to A. B. Zamolodchikov for sharing his insights and support. The research of G. K. and S. L. is supported by the NSF under grant number NSFA Scaling behaviour of discretized chiral transfer-matrices To investigate the scaling behaviour of T (N)( ) (3.39){(3.44), we conducted numerical work for integer n when the discretized operator is a nite dimensional matrix that can be diagonalized by means of the Bethe ansatz (see appendix B for details). We focused only on the vacuum eigenvalue in the sector Hj(Nm2) and considered the cases with n = 2; 3; : : : ; 6 and all admissible values of j; m (3.14). Let (vac)( ) be the vacuum eigenvalue of the chiral transfer-matrix in the parafermionic subspace Vj the vacuum eigenvalue of T scaling limit of the discretized operator. To estimate numerical values of (vac)( ) we used data obtained for a set of nite N and then performed a certain interpolation procedure to N = 1. The results were compared with predictions coming from the properties of (vac)( ) (m). We expect that it can be obtained from (N)( ) by using the formula (3.50) which explicitly describes the root # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0:4818860 1:4891566 2:4919329 3:4935044 4:4946127 5:4955294 6:4963870 7:4972634 8:4982121 9:4992734 10:500481 11:501864 12:503448 13:505258 14:507318 0:4818829 1:4891424 2:4918863 3:4933890 4:4943769 5:4951073 6:4956974 7:4962107 8:4966857 9:4971480 10:497616 11:498105 12:498625 13:499187 14:499799 0:4818820 1:4891392 2:4918769 3:4933666 4:4943321 5:4950277 6:4955682 7:4960140 8:4964010 9:4967523 10:497084 11:497406 12:497729 13:498060 14:498404 0:4818814 1:4891372 2:4918715 3:4933541 4:4943074 5:4949844 6:4954981 7:4959077 8:4962476 9:4965392 10:496797 11:497031 12:497248 13:497455 14:497655 0:4818809 1:4891359 2:491868 3:493348 4:494295 5:494962 6:495463 7:495854 8:496171 9:496432 10:49665 11:49684 12:49701 13:49715 14:49728 0:47349 1:48725 2:49093 3:49276 4:49387 5:49464 6:49521 7:49564 8:49599 9:49628 10:49652 11:49672 12:49690 13:49705 14:49719 HJEP01(28) g~1(p1; p2) = t~1(p1; p2) 2 cos 2n+p22 ; of the discretized operator T (N)( ) for n = 4, j = m = 0. The column \N = 1" was obtained by interpolating the results for nite N . The entries in the last column were calculated by using the asymptotic formula (A.2) truncated at the rst non-zero term in the series. discussed in section 2.2, specialized to the values p1 = 2i m and p2 = j + 12 . Agreement was found in all cases considered. In this appendix, some of our numerical work is presented. Let fulgl1=1 be the set of zeroes of (vac)( ) considered as a function of 2 . From the numerical data it was found that all the zeroes are simple, real, positive, and accumulate towards 2 = 1 with the leading asymptotic behaviour ul 1 n 2 2 8 < l : l 2j < n2 , this is consistent with the asymptotically exact formula, n ul2 + 1 2 1 X g~m m=1 i 2 which can be easily derived from eqs. (2.53){(2.54). Knowledge of the coe cients g~m allows us to compute systematic corrections to the leading asymptotic behaviour (A.1). As it follows from eq. (2.52), the rst coe cient is with t~1(p1; p2) | vacuum eigenvalue of ~t1 | given by eq. (2.49). Notice that for p2 = j + 12 = n+42 (n-even), the denominator in (A.3) is zero so that (A.2) is no longer valid. (A.1) (A.3) 1.5 large (+ 2) asymptotics N = 1 1.0 asymptotic following from eq. (2.53). On the right panel, ~(vac) = (vac) exp 2 ( 2) 32 is plotted and compared with the large ( 2) asymptotic derived from eqs. (2.43), (2.44). The scaling function was numerically estimated by interpolating to N = 1 the data for N = 500; 1000; 2000; 4000. Also when j = m = 0, g~1 vanishes, but for this case the second term in the sum in (A.2) is known, since g~2 0; 1 2 t~2 0; 12 2 cos n+2 of T and numerical values of t~2(0; 12 ) were calculated in ref. [43] and are reproduced in table 4. Truncating the series in (A.2) at the rst non vanishing term, we calculated the corrections to the leading asymptotic (A.1). This was compared to the zeroes of the vacuum eigenvalue (N)( ) for increasing N . In all cases good agreement was observed. As an example, in table 5 the results for n = 4, j = m = 0 are shown. As 2 1, the asymptotic behaviour of (vac) is dictated by eqs. (2.43), (2.44). Truncating the sum in (2.44) at the rst non-zero term and substituting ~tj by its vacuum eigenvalue, we compared this to the results of the N = 1 interpolation. The agreement was good considering that the interpolation procedure becomes rapidly less e cient for increasing values of ( 2). Figure 9 shows a plot of the estimated scaling function versus the asymptotics for n = 3 and 2j = m = 1. Another check that can be made is to consider the Taylor expansion of (vac)( ) at zero following from formulae (2.42) and (2.48). The coe cient t1(p1; p2) (p1 = 2i m; p2 = j + 12 ) can be compared to the corresponding term in the vacuum eigenvalue of the discretized operator: T (N;vac)( ) = 2 cos + t(1N) 2 + O( 4) : Note that t(1N) is a divergent quantity for large N and must be regularized. According to eq. (3.50), for n > 2, the following limit exists and converges to t1: i 2 t1 n cos( mn ) cos( n ) : (A.4) m n 1 t(N;reg) = 0:54474 0:43807 0:54519 0:44710 0:54542 0:45357 0:54553 0:45818 0:5456440 0:469649 N = 1 0:8630048 0:419808 0:5456445 0:469446 eq. (2.48) 0:8630049 0:419632 to the expression for t1 2i m; j + 12 given by eq. (2.48). The column \N = 1" was obtained by We compared the value of t1 2i m; j + 12 t(N;reg) and found good agreement for n = 3; 4 : : : ; 6 and all the allowed values of j; m. A 1 given by eq. (2.48) to the numerical values of few cases are presented in table 6. Finally, let us mention that for n = 2, analytic expressions exist for both (vac) and the vacuum eigenvalue of T (N). In the case j = m = 0, N Y m=1 T (N;vac)( ) = 2 1 2 cot 2N m 1 2 ; and using the formula (3.50), the scaling limit can be taken explicitly to yield (vac)( ) = e 2 2 2 2 1 2 It is easy to verify that this is consistent with the properties of the chiral transfer-matrix discussed in section 2.2. For n = 2 and 2j = m = 1, the discretized operator turns out to be zero for any N and hence, ( ) = 0. B Scaling behaviour of Bethe roots for the vacuum state space Hj(Nm2) . Recall that Hj(Nm2) denotes the eigenspace of the matrix In this appendix we will consider the vacuum eigenvalue of the matrices Z ( ) in the H(N) (Z) (3.41), (3.45) having eigenvalue !j m2 , where j and m are restricted as in (3.14). Our considerations are entirely based on the properties of Z ( ) (i){(v) listed in section 4.1. Let Z ( )( ) be the eigenvalue corresponding to a common eigenvector j i of the commuting family Z ( ). Using the analytical conditions (iv) and ! symmetry (v), it can be written in the form, Z (n 1)N 2j m Y i=1 1 i (n odd) (B.1) Z+( )( ) = B(N; ) m Z Q type relations (iii), it follows that the overall coe cient B(N; ) (depending on the state j i) is the same for both Z+ is that the roots satisfy the following Bethe ansatz equations: ( ) and Z ( ). Another consequence of this relation Y i=1 n2N j m2 vi Y i=1 (n 22)YN j m2 wi i=1 vi wi i + q 1 l = i + q+1 l q2m q+ 21 l q 21 l 2N q 2 wl = q+2 wl q 2 vl = q+2 vl q2m q2m 1 1 q+1 vl q 1 vl 2N (n even) (B.2) (n odd) (B.3) (n even) (B.4) Similar equations for the Fateev-Zamolodchikov spin chain (3.47) with periodic boundary conditions were previously derived in the works [115] and [116] for odd and even n, respectively. Notice that the constant B(N; ) in (B.1), (B.2) is determined (up to an overall sign) by the quantum Wronskian type relations (ii). The Bethe ansatz equations are valid for all integer n 2 and j; m restricted to (3.14), except for 2j = m = n2 (n even) which requires special attention. In this case, for certain sectors of H0 (N) a signi cant simpli cation occurs; Z ( ) vanishes so that the T Q type relations (iii) become trivial and the quantum Wronskian type relations (ii) can be used to obtain much simpler equations for the roots. For instance, for the vacuum eigenvalue, Z (vac)( ) = 0 and Z+ (vac) is given explicitly by Z+ (vac)( ) = 2 p N n 2 N 1 Y l=1 1 + n cot l 2N n 2 ; ; 2j = m = n even : (B.5) Recall that the vacuum is de ned as the lowest energy state of the Fateev-Zamolodchikov spin chain Hamiltonian (3.47), (3.48), which commutes with both Z+( ) and Z ( ) for any . We studied the solutions to the Bethe ansatz equations corresponding to the low energy states j i of the Fateev-Zamolodchikov spin chain. It was found that the roots accumulate ×× 0.5 ● 1.5 1.0 -1.0 -1.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1.5 ● -0.5 ● -0.5 0.5 functions of 2 are shown for n = 6, 2j = 3; m = 1 and N = 8. (vac) (circles) and Z (vac) (crosses) as HJEP01(28) along the rays given by (see gure 10) arg( ) = arg( 2) = arg( 2) = 2 2 n p ; p ; p ; p = 1; 3; : : : ; n ( i p = 1; 3; : : : ; n 1 (vi p = 2; 4; : : : ; n (wi roots) roots) roots) In the scaling limit most of the roots become densely packed along the rays. However we observed that at the edges of the distribution, the roots exhibit a certain scaling behaviour. In particular, at the edge next to zero of the locus labeled by the integer p, with index i enumerating the roots ordered by increasing absolute value, the following limits exist iN!x1ed lim N n1 i(;Np; ) ; iN!x1ed lim N n2 vi(;Np ; ) ; iN!x1ed lim N n wi;p 2 (N; ) : Here we temporarily exhibit the dependence of the roots on N and the state j i the scaling limit can be de ned for the coe cient B(N; ) in formulae (B.1), (B.2): N!1 B m ( ) = s lim ( =N ) n B(N; ) : Keeping N nite, consider the logarithm of the r.h.s. of eqs. (4.1) and (4.2) for a given eigenvalue. With Z ( ) of the form (B.1), (B.2) it is straightforward to nd their Taylor series at = 0. In the case of odd n, the expansion coe cients are given by M m(N) = Mn(N) = M m(N) = 1 m nN 1 m N X i N m n i n X i i m + n + X i 2 n i m (n 1) log ( 1)m N m cos 2n N e (m < n) (m > n) . Also, (B.6) (B.7) For even n, Vm(N) = W m(N) = V n(N) = W n(N) = 2 2 Vm(N) = 1 1 2 nN 2 nN 1 N N i N 2m 2m 2m X vi m i X w m + i cos N n n X vi 2 + 2 log N e n X w 2 i (n 2) log N e 2 X vi m ; W m(N) = 1 N 2m X w m i It is expected that the following limits exist, M m( ) = slim M m(N) N!1 (n odd) N!1 N!1 Vm( ) = slim Vm(N) ; W m( ) = slim W m(N) (n even) and coincide with the expansion coe cients in e n of the CFT eigenvalues of log : log ( )( ) = log B( ) + X ( 1)m M m( ) e mn log +( )( ) = log B( ) + X Vm( ) e 2mn log ( )( ) = log B( ) + + W m( ) e 2mn : n m m n 2 e e 1 m=1 1 X m=1 (n (n odd) even) Recall that the symbol \slim" stands for the scaling limit which is applied for low energy eigenstates only. For numerical checks, we focused only on the vacuum of the FateevZamolodchikov spin chain (3.47), (3.48). Our numerical work con rmed the existence of the limits (B.9) for n = 3; 4; : : : ; 6 and all admissible values of j and m (3.14). Since a few of the expansion coe cients in (4.35) are available in explicit form, we have the following analytical predictions for some of the limits in (B.9). Let f0;1 = f0;1(p1; p2) be de ned by eq. (4.33) and (x) (x)= (1 x). Then for n > 2 one has (here the superscript \(vac)" in the notation for the coe cients (B.9) is (B.8) HJEP01(28) exact Numerical values of the coe cients (B.7), (B.8) for the vacuum of the FateevZamolodchikov spin chain (3.47), (3.48). The column N = 1 was obtained by interpolating the nite-N data. The last column lists the exact predictions given in (B.10), (B.11). HJEP01(28) W n (0; j) = 4 log E log(n) + 8 (1 + j) 1 + + 2 E The numerical data agreed with these explicit formulae. This is shown, for a few cases, in (m = 1; 3; : : : ; n 2 2m ) m = 0; 1; 2; : : : ; n 2 n 2 n 2 n 2 2 n 2 (B.10) (B.11) n 2 2 1 n 1 n 4 4 1 1 Mm(m; j) = 0 M2(m; j) = n n 1 f0;1 V1 W1 2 V n (0; j) n n 2 2 2 2 2 ; j ; j f0;1 Mn(0; j) = 2 log log(n) + 4 (1 + j) 1 + + E Vm(m; j) = Wm(m; j) m = 1; 2; : : : ; m V1(m; j) = W1(m; j) = f0;1 ; j + m = 0; 1; 2; : : : ; 2 1) ; j + 1 2 + 1 2 n n 2 2 e 2 1 2n ; j + 1 2 E 1 2 i m 2 i(n i(n i(n 4 2 n 4 4 4 n 2) 2) ; j + ; j + i m 2 1 2 1 2 1 2 n n 2 2 + n n 1 2 1 2 1 n 1 n 1.626210 N = 1 N = 1 N = 1 Zamolodchikov spin chain (3.47), (3.48). The column \N = 1" contains the results of numerical interpolation from the nite N data. The analytical expression for Bs(m) is given by (4.34). As was already mentioned, the constant B(N; ) 2 can be found using the quantum Wronskian type relations (ii) from section 4.1. The r.h.s. of these relations is proportional to the lattice shift operator P(N) (3.49) whose eigenvalues are pure phases (3.52). By explicit diagonalization of Z for small N we found that 1 arg B(N; ) 1 nN s) s + n (L L) + s (mod 2) ; (B.12) 12 m and L, L are non-negative integers depending on the state j i. For the vacuum state L = L = 0, and the overall sign of the limit B(vac) (B.6) is ( 1)s. This coincides with the sign factor in Bs(m) (4.34). For large values of N , when direct diagonalization becomes impossible, we veri ed by means of the Bethe ansatz that the m absolute value of ( =N ) n B(N;vac) converges to ( 1)s Bs(m) (see table 8). Recall that 2j = m = n2 with even n is a special case. Using eq. (B.5) the scaling functions can be found explicitly, e 2 1 + 2e ; (vac)( ) = 0 : This formula can be applied for n = 2. For the remaining n = 2 case, j = m = 0, it is easy 1 2 cot 1) Z (vac)( ) = p 2 ; to show that for nite N Z+ (vac)( ) = p 2 N Y m=1 so that the scaling functions are given by 2 12 + 2e 2 e 4N 2e ; C Derivation of NLIE In this appendix we sketch some technical details in the derivation of the system of NLIE (4.94). (vac)( ) = p and are real, then Hermiticity tions (4.12), (4.62), (4.67) and the periodicity (4.60) imply that = e i e e i p2 e+i p2 ( + i ) + + : e i e e i p2 ( ) + e+i p2 +( ) = e+i e e i p2 +( ) e+i p2 ( ) + + ; (C.1) e+i e e i p2 ( + i ) + e+i p2 i n i n + i n 2 + i n 2 i n 2 = i n 2 i n 2 = i n 2 Due to the analyticity of the operators ( ), these relations should be satis ed for any complex . Let us introduce the shortcut notations +( ) ( ) B0 = ; B1 = i 2n ) and = exp 2 i e . Then (C.1) can be rewritten as B0 = U 1 + 1 + B1 = U 1 + 1 + A+ A : Solving these equations w.r.t. A+ and A , one nds 1 2 A+ = 1 B0B1 U 1 1 2 B0 1 2 B1 and similar for A . This formula, combined with the quantum Wronskian relation (4.63) Together with the periodicity condition ( + i ) = i ) the last equation implies A+ 2 A+ i 2 + i2 2i sin 2n+p22 2 U 1 + i 2 As it follows from the quantum Wronskian relation (4.68): U 1 i 2 U 1 i 2 = 2i e 2 ip2 sin 2n+p22 +( + i (n 1) 2 + + i (n+1) : 2 B0 B1 1 B0 ( ) ( + i ) 2i sin 2n+p22 ; +( ) +( + i ) 2i U( ) sin 2 p2 n+2 +( ) ( + i ) + ( ) +( + i ) 4i sin 2n+p22 written in the form leads to A+( ) = cos 2 e i 2 : U 1 : This can be substituted into the previous formula, yielding eq. (4.76) with the subscript \+". Of course the formula is valid for the \ " case also. Let us now take a closer look at the second equation in (C.1) specialized to the eigenvalues corresponding to a common eigenvector j i. Suppose j is a zero of +( )( ). As follows from the quantum Wronskian relation (4.63), ( )( j ) 6= 0, and therefore we conclude that i n 2 e i (e j +p2) +( ) j ei (e j +p2) +( ) j i + i n 2 ; which can be equivalently written in the form (4.81). As was mentioned in the main body of the text, the zeroes of the entire periodic function (vac)( + i ) are simple, located on the lines =m( ) = + (2m + 1); m 2 Z, and accumulate toward <e( ) ! +1. Also, assuming that the parameters p1 and p2 are restricted as in cases (b), (c) from section 4.5, it is expected that the entire function (vac)( ) does not have any zeroes within the strip j=m( )j < 2 (n + 2). Therefore, as + follows from the de nition (4.77), "(vac)( ) is an analytic function for j=m( )j < where it has the leading asymptotic behaviour (4.79) at <e( ) ! +1. Combining this analytic information with the \quantization condition" (4.82) for the zeroes of (vac)( ) and the + asymptotic behaviour see eq. (4.74) log +(vac)( ) = 2 e k + o(1) as <e( ) ! +1 =m( ) < ; with k = n2+p22 , it is a simple exercise (see however appendix D) to derive a dispersion-type 2L(vac)( 0 i ) + i "(vac)( 0 i ) 4 e 0 i + 2 k : relation log (C.2) (C.3) (C.4) (vac) Z 1 d 0 1 2 (vac) 1 1 + e2 2 0+2i i 2 = 2 e 2k Here 2 (0; 2 ) is an arbitrary constant and the notation L(vac)( ) = log 1 + exp i"(vac)( ) The next important property employed in the derivation of the system of integral equations (4.94){(4.96) is that "(vac)( ) can be written in terms of the Fourier integral "(vac)( ) = 4 e 2 k + ei "~( ) : Z d 2 Notice that the existence of the Fourier transform is ensured by the asymptotic behaviour (4.79) at ! +1, and formulae (4.89), (4.90) for 1. One can expect that the function "~( ) decays su ciently fast as 1, so that the integral in (C.4) converges for any in the strip of analyticity j=m( )j < . It is not di cult to see now log (vac) 2k + i i (n + 1) 2 Z R+i0 (vac) i (n + 1) cosh( (n+1) ) sinh( n2 ) 2 2 and also that the imaginary part of the function (C.3) with having in nitesimally small negative imaginary part, can be represented by the convergent integral =m L(vac)( i0) = L~( ) : Z R+i0 Similarly for the function !( ) (4.87) with real, one has Z R+i0 2 !(vac)( ) = 4 e + !~( ) ; log 1 + e !(vac)( ) = M~ ( ) : The remaining part of the derivation of the NLIE consists of straightforward manipulations with the Fourier images "~, L~, !~, M~ . Finally, going back to functions of the variable , one derives the system of integral equations (4.94){(4.96). Knowing the functions "(vac)( ), !(vac)( ) from the solution of the NLIE, and the asymptotic formulae (4.70), (4.74), one can recover the vacuum eigenvalues of the operators +( ) and +( ) from (C.2), (C.5). The corresponding explicit relations are given below, where we drop the superscript \(vac)" like in the NLIE (4.94){(4.96): log +( ) = (C.6) valid for j=m( )j < 2 (n + 2) , and log +( ) = 2 e k + sin( 2n ) e 1 F (CFT)( 3 F (CFT)( k + Z 1 d 0 h F (CFT)( 1 1 2 i 0 i ) L( 0 i ) 1 2 i Z 1 d 0 h F (CFT)( 3 0 i ) L( 0 i ) F4( ) = n + 2 1 1 n + 2 1 n+2 with tanh coth n + 2 n + 2 sin( n+2 ) 2(n + 2) sinh( n++i22 ) sinh( n+22 ) i "~( ) Z R+i0 (C.5) HJEP01(28) 1 1 i Z 1 d 0 F2( sin( n+2 ) 2(n + 2) cosh n++i22 cosh i 2 n+2 1 2 cosh( ) . Here the kernels are given by F (CFT)( ) = F1( ) 1 n+12 , F2( ) = 1 sinh( ) s~m(p1; p2) = Z 1 d 1 2 n + 2 Z 1 d 1 The vacuum eigenvalues of the chiral transfer-matrix can be obtained using the T (vac)(i ) = +( ) +( ) with = e n : Combining (C.6), (C.7) with the general asymptotic expansions at <e( ) ! +1 found in (4.70), (4.74), the expressions for the local and dual nonlocal integrals of motion follow 2 =mhe(2m 1)( i )L( i ) + ( 1)me(2m 1) log 1 + e !( ) (C.9) 2m( i ) =m e n+2 L( i ) sin 2m e n+2 log 1 + e !( ) : D Modi ed NLIE for oscillating asymptotics Here we discuss the modi cations to the integral equations (4.94){(4.96) for the case of 12 , when the asymptotics of the functions "(vac)( ) and !(vac)( ) at real p1 6= 0 and p2 > 1 oscillate (4.88). The rst important di erence in this case is that (vac)( ) has a set of zeroes f m( )gm1=1 + in the strip j=m( )j < 2 (n + 2) whose asymptotic behaviour is given by relation (4.93). Secondly, in the derivation of (4.94) presented in the previous appendix, we implicitly assumed that all values on the real axis, such that "(vac)( ) = (mod 2 ) arise from the quantization condition (4.82), i.e., i for some j = 1; 2; : : : (recall that (vac) j =m( j ) = ). In other words all such are related to the zeroes of +(vac)( ) and, therefore, form an increasing semi-in nite sequence extending towards +1 on the real axis (see (4.83)). For the oscillating asymptotics (4.88) this is no longer true. Indeed, it is easy to check from (4.88) that the condition "(vac)(~m) = 1) with m = 1; 2; : : : is satis ed for an in nite set of values ~ 1 m m=1 which extend towards 1 such that m = n 2p1 1 2 (p1; p2) + o (m=p1) 1 ; valid up to an exponentially small correction. Here (p1; p2) = 4p1 0=n + i log cos( (p2 + ip1))= cos( (p2 ip1) coincides with the scattering phase de ned by eq. (5.11). In the terminology of the Bethe ansatz we have an in nite number of \holes" where the phase passes a resonant value without a corresponding zero j . Therefore the integrals in the r.h.s. of (4.94) contain spurious contributions from non-existent roots. To exclude these unwanted contributions one needs to add extra source terms to the r.h.s. of eqs. (4.94). Introduce the notation J (")( ) = i X log J (!)( ) = 1 m=1 1 X log m=1 S( ~m) t S + i2 (n + 2) t + i2 ~ + i2 + i2 (n + 2) ( ) ( ) where S( ) and t( ) are de ned in (E.1) below. Then the modi ed equations (4.94) can be written as 2 k + J (")( i ) + Z 1 d 0 hG( 1 2 i 0 2i ) L( 0 i ) 0) L( 0 i ) + G1( 0 i ) log 1 + e !( 0) !( ) = 4 e + J (!)( ) + =m G1( (D.2) L( ) = log 1 + e i"( ) : One can check that the leading terms in the asymptotics (4.88) solves these equations at 1, i.e., when the exponential terms proportional to e in the r.h.s. are omitted. E Two-particle S-matrix for the sausage model model [14]. The S-matrix satis es the Yang-Baxter equation and was originally introduced as the Boltzmann weights of the so-called 19-vertex model [117]. S( ) = S++++( ) = S ( ) = T ( ) = S++ ( ) = S ++( ) = S(i ) sinh( in+2 ) sinh( in++2 ) t( ) = S++00( ) = S00++( ) = S 00( ) = S00 ( ) = = S+00 (i ) = S00+(i ) = i ) = S+0+0 ( ) = S0++0( ) = S0 0 ( ) = S0 0( ) = S0+0 (i ) = S00+(i sinh( n+2 ) sinh( in+2 ) sinh( 2ni+2 ) sinh( in++2 ) sin( n2+2 ) sinh( in+2 ) sinh( 2ni+2 ) sinh( in++2 ) (E.1) R( ) = S++ ( ) = S++( ) = ( ) = S0000( ) = S++00( ) + S++ ( ) sinh( 2ni+2 ) sinh( in++2 ) As a 9 9 matrix S(27!2) satis es the conditions S(27!2) y S(27!2) = I(2) for =m( ) = 0 det S(27!2)( ) = sinh2( in+2 ) sinh( 2ni++2 ) 4 sinh2( in++2 ) sinh( 2ni+2 ) : (E.2) Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] L.D. Faddeev, E.K. Sklyanin and L.A. 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Vladimir V. Bazhanov, Gleb A. Kotousov, Sergei L. Lukyanov. Quantum transfer-matrices for the sausage model, Journal of High Energy Physics, 2018, 21, DOI: 10.1007/JHEP01(2018)021