Factorized S-matrices Research Articles

On the classification of UV completions of integrable Toverline{T} deformations of CFT

It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant Toverline{T} perturbation of dimension 4 has universal properties. In particular, for the most interesting cases, the theory develops a singularity in the ultra-violet (UV), signifying a shortest possible distance, with a Hagedorn transition in applications to string theory. We show that by adding an infinite number of higher [ Toverline{T} ]s>1 irrelevant operators of positive integer scaling dimension 2(s+1) with tuned couplings, this singularity can be resolved and the theory becomes UV complete with a Virasoro central charge cUV> cIR consistent with the c-theorem. We propose an approach to classifying the possible UV completions of a given CFT perturbed by [ Toverline{T} ]s that are integrable. The main tool utilized is the thermodynamic Bethe ansatz. We study this classification for theories with scalar (diagonal) factorizable S-matrices. For the Ising model with cIR = frac{1}{2} we find 3 UV completions based on a single massless Majorana fermion description with cUV = frac{7}{10} and frac{3}{2} , which both have mathcal{N} = 1 SUSY and were previously known, and we argue that these are the only solutions to our classification problem based on this spectrum of particles. We find 3 additional ones with a spectrum of 8 massless particles related to the Lie group E8 appropriate to a magnetic perturbation with cUV = frac{21}{22},frac{15}{12} , and frac{31}{2} . We argue that it is likely there are more cases for this E8 spectrum. We also study simpler cases based on su(3) and su(4) where we can propose complete classifications. For su(3) the infrared (IR) theory is the 3-state Potts model with cIR = frac{4}{5} and we find 3 completions with frac{4}{5} < cUV≤ frac{16}{5} . For the su(4) case, which has 3 particles and cIR = 1, and we find 11 UV completions with 1 < cUV ≤ 5, most of which were previously unknown.

Toverline{T} -deformation and long range spin chains

We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the Toverline{T} -deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of mathcal{N} = 4 super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefore share many features. Both deformations preserve integrability and lead to non-local deformed theories, resulting in a change of the corresponding factorized S-matrices. We also prove a factorisation formula for the expectation value of the operators which trigger the deformation on the lattice; similar results in quantum field theory play an essential role in the solvability of such deformations. We point out that the long range deformation is a natural counterpart of the Toverline{T} -deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.

Yang Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices

Integrable deformation of SU(2) sigma and lambda models are considered at the classical and quantum levels. These are the Yang-Baxter and XXZ-type anisotropic deformations. The XXZ type deformations are UV safe in one regime, while in another regime, like the Yang-Baxter deformations, they exhibit cyclic RG behaviour. The associ-ated affine quantum group symmetry, realized classically at the Poisson bracket level, has q a complex phase in the UV safe regime and q real in the cyclic RG regime, where q is an RG invariant. Based on the symmetries and RG flow we propose exact factorizable S-matrices to describe the scattering of states in the lambda models, from which the sigma models follow by taking a limit and non-abelian T-duality. In the cyclic RG regimes, the S-matrices are periodic functions of rapidity, at large rapidity, and in the Yang-Baxter case violate parity.

The importance of being integrable: Out of the paper, into the lab

The scattering matrix (S-matrix), relating the initial and final states of a physical system undergoing a scattering process, is a fundamental object in quantum mechanics and quantum field theory. The study of factorized S-matrices, in which many-body scattering factorizes into a product of two-body terms to yield an integrable model, has long been considered the domain of mathematical physics. Many beautiful results have been obtained over several decades for integrable models of this kind, with far reaching implications in both mathematics and theoretical physics. The viewpoint that these were only toy models changed dramatically with brilliant experimental advances in realizing low-dimensional quantum many-body systems in the lab. These recent experiments involve both the traditional setting of condensed matter physics and the trapping and cooling of atoms in optical lattices to engineer and study quasi-one-dimensional systems. In some cases the quantum physics of one-dimensional systems is arguably more interesting than their three-dimensional counterparts, because the effect of interactions is more pronounced when atoms are confined to one dimension. This article provides a brief overview of these ongoing developments, which highlight the fundamental importance of integrability.

The structure of non-abelian kinks

We consider a class of integrable quantum field theories in 1+1 dimensions whose classical equations have kink solutions with internal collective coordinates that transform under a non-abelian symmetry group. These generalised sine-Gordon theories have been shown to be related to the world sheet theory of the string in the AdS/CFT correspondence. We provide a careful analysis of the boundary conditions at spatial infinity complicated by the fact that they are defined by actions with a WZ term. We go on to describe the local and non-local charges carried by the kinks and end by showing that their structure is perfectly consistent with the exact factorizable S-matrices that have been proposed to describe these theories.

Nested sequence of projectors. II. Multiparameter multistate statistical models, Hamiltonians, S-matrices

Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors. Statistical models associated to such N2×N2 matrices for odd N are studied here. Presence of 12(N+3)(N−1) free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are N possible states at each site. The trace of the transfer matrix is shown to depend on 12(N−1) parameters. For order r, N eigenvalues constitute the trace and the remaining (Nr−N) eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the rth roots of unity, or lower dimensional multiplets corresponding to factors of the order r when r is not a prime number. The modulus of any eigenvalue is of the form eμθ, where μ is a linear combination of the free parameters, θ being the spectral parameter. For r a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians and potentials corresponding to factorizable S-matrices are constructed starting from our braid matrices. Perspectives are discussed.

ExactS-matrices for supersymmetric sigma models and the Potts model

We study the algebraic formulation of exact factorizable S-matrices for integrable two-dimensional field theories. We show that different formulations of the S-matrices for the Potts field theory are essentially equivalent, in the sense that they can be expressed in the same way as elements of the Temperley-Lieb algebra, in various representations. This enables us to construct the S-matrices for certain nonlinear sigma models that are invariant under the Lie ``supersymmetry'' algebras sl(m+n|n) (m=1,2; n>0), both for the bulk and for the boundary, simply by using another representation of the same algebra. These S-matrices represent the perturbation of the conformal theory at theta=pi by a small change in the topological angle theta. The m=1, n=1 theory has applications to the spin quantum Hall transition in disordered fermion systems. We also find S-matrices describing the flow from weak to strong coupling, both for theta=0 and theta=pi, in certain other supersymmetric sigma models.

Exact S-matrices with affine quantum group symmetry

We show how to construct the exact factorized S-matrices of 1+1 dimensional quantum field theories whose symmetry charges generate a quantum affine algebra. Quantum affine Toda theories are examples of such theories. We take into account that the Lorentz spins of the symmetry charges determine the gradation of the quantum affine algebras. This gives the S-matrices a non-rigid pole structure. It depends on a kind of “quantum” dual Coxeter number which will therefore also determine the quantum mass ratios in these theories. As an example we explicitly construct S-matrices with U q ( c n (1)) symmetry.

Quantum conserved currents in supersymmetric Toda theories

We consider N = 1 supersymmetric Toda theories which admit a fermionic untwisted affine extension, i.e. the systems based on the A( n, n), D( n + 1, n) and B( n, n) superalgebras. We construct the superspace Miura transformations which allow us to determine the W-supercurrents of the conformal theories and we compute their renormalized expressions. The analysis of the renormalization and conservation of higher-spin currents in then performed for the corresponding supersymmetric massive theories. We establish the quantum integrability of these models and show that although their lagrangian is not hermitian, the masses of the fundamental particles are real, a property which is maintained by one-loop corrections. The spectrum is actually much richer, since the theories admit solitons. The existence of quantum conserved higher-spin charges implies that elastic, factorized S-matrices can be constructed.

The full set of cn-invariant factorized S-matrices

We use the method of the tensor product graph to construct rational (Yangian invariant) solutions of the Yang-Baxter equation in fundamental representations of $c_n$ and thence the full set of $c_n$-invariant factorized $S$-matrices. Brief comments are made on their bootstrap structure and on Belavin's scalar Yangian conserved charges.

A new family of diagonal ADE-related scattering theories

We propose the factorizable S-matrices of the massive excitations of the non-unitary minimal model M 2,11 perturbed by the operator φ 1,4. The massive excitations and the whole set of two particle S-matrices of the theory is simply related to the E 8 unitary minimal scattering theory. The counting argument and the thermodynamic Bethe ansatz (TBA) are applied to this scattering theory in order to support this interpretation. Generalizing this result, we describe a new family of non-unitary and diagonal ADE- related scattering theories. A further generalization suggests the magnonic TBA for a large class of non-unitary G ⊗ G/G coset models ( G= A odd , D n, E 6,7,8) perturbed by φ id, id, adj , described by non-diagonal S-matrices.

Quantum conserved currents in affine Toda theories

We study the renormalization and conservation at the quantum level of higher-spin currents in affine Toda theories with particular emphasis on the nonsimply laced cases. For specific examples, namely the spin-3 current for the a 3 (2) and c 2 (1) theories, we prove conservation to all-loop order, thus establishing the existence of factorized S-matrices. For these theories, as well as the simply laced a 2 (1) theory, we compute one-loop corrections to the correspondinh higher-spin charges and study charge conservation for the three-particle vertex function. For the a 3 (2) theory we show that although the current is conserved, anomalous threshold singularities spoil the conservation of the corresponding charge for the on-shell vertex function, implying a breakdown of some of the bootstrap procedures commonly used in determining the exact S-matrix.

Factorizable S-matrices from nonlocal ZNcharges

Conserved nonlocal ZN charges Q and Q are defined through their action on asymptotic particle states. This action entails a nontrivial coproduct. S-matrices invariant under these ZN charges are explicitly found. They enjoy in addition P, C ,T, crossing invariance and unitarity. The unitarization factors are calculated. Finally, the authors identify perturbed conformal field theories described by these factorized S-matrices.

On symmetries of some massless 2D field theories

We describe a few aspects of the quantum symmetries of some massless two-dimensional field theories. We discuss their relations with recent proposals for the factorized scattering theories of the massless PCM 1 and O(3) ϑ= π sigma models. We use these symmetries to propose massless factorized S-matrices for the su(2) sigma models with topological terms at any level, alias the PCM k models, and for the su(2)-coset massless flows.

Scattering matrices for φ1,2 perturbed conformal minimal models in absence of kink states

We determine the spectrum and the factorizable S-matrices of the massive excitations of the non-unitary minimal models M 2,2n + 1 perturbed by the operator ø 1,2. These models present no kinks as asymptotic states, as follows from the reduction of the Zhiber-Mikhailov-Shabat model with respect to the quantum group SL(2) q found by Smirnov. We also give the whole set of S-matrices of the non-unitary minimal model M 2,9 perturbed by the operator ø 1,4, which is related to a RSOS reduction for the ø 1,2 operator of the unitary model M 8,9. The thermodynamical Bethe ansatz and the truncated conformal space approach are applied to these scattering theories in order to support their interpretation.

φ 1,2 deformation of the [formula omitted]2,2 n+1 conformal minimal models

The spectrum and the factorizable S-matrices of the massive excitations of the φ 1,2 deformation of the nonunitary minimal models M 2,2 n+1 is given. These models present no kinks as asymptotic states, as follows from the reduction of the Zhiber-Mikhailov-Shabat model with respect to the quantum group SL(2) q , found by Smirnov. An interesting situation of zeros and poles overlapping in the physical amplitudes is also discussed.

New factorized S-matrices associated with SO( N)

A method is presented for applying the fusion procedure for solutions of the rational Yang-Baxter equation to R-matrices acting in the vector representation of SO( N). As an application, new R-matrices acting in symmetric representations of SO( N) are calculated. R-matrices acting in reducible representations containing the antisymmetric representations of SO( N) are calculated. These are used to construct factorized S-matrices giving the scattering of particles in the principal chiral model. Connections with affine Toda field theory are discussed.

Finite-size effects in theories with factorizable S-matrices

We study the energy spectrum of (1 + 1)-dimensional perturbed conformal field theories defined on the cylinder. The finite-size dependence of the two-particle levels allows a direct numerical measurement of the elastic S-matrix which we compare with the conjectured minimal S-matrix for several perturbations of minimal models. We discuss the simplifications that integrability imposes on the spectrum above threshold. In particular, the ultraviolet limit of the elastic phase shift of two lightest particles is related in a simple way to scaling dimensions of the conformal field theory.

Towards the classification of completely integrable quantum field theories (the Bethe-Ansatz associated with dynkin diagrams and their automorphisms)

The eigenvalues of the unitarizable and factorized S-matrices and their transfer matrices are presented under the form of the Bethe-ansatz equations expressed in terms of the simple root system and of the automorphisms of the corresponding simple Lie algebra.

Factorized S-matrix and the Bethe ansatz for simple lie groups

The factorized S-matrices as well as the eigenvalues of the transfer matrices for symmetrical degrees of fundamental representations of all Lie groups are presented in the form of the Bethe ansatz in terms of roots systems. As an application the explicit expressions of the S-matrices for the lowest dimension representations of some exceptional Lie groups and the solution of the principal chiral model are calculated.