The nonlinear integral equations describing the spectra of the left and right (continuous) quantu... more The nonlinear integral equations describing the spectra of the left and right (continuous) quantum KdV equations on the cylinder are derived from integrable lattice field theories, which turn out to allow the Bethe Ansatz equations of a twisted "spin −1/2" chain. A very useful mapping to the more common nonlinear integral equation of the twisted continuous spin +1/2 chain is found. The diagonalization of the transfer matrix is performed. The vacua sector is analysed in detail detecting the primary states of the minimal conformal models and giving integral expressions for the eigenvalues of the transfer matrix. Contact with the seminal papers [1, 2] by Bazhanov, Lukyanov and Zamolodchikov is realised. General expressions for the eigenvalues of the infinite-dimensional abelian algebra of local integrals of motion are given and explicitly calculated at the free fermion point.
Journal of Physics A: Mathematical and General, 2002
In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix... more In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of U q (sl 2) over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator. His approach was to start with the 6-vertex model Bethe ansatz, and to derive certain functional relations between the transfer matrix T (v) and a matrix Q(v)-the elements of both matrices being entire functions. He went on to show that the reverse argument could be used in order to start from the functional relations (and some other properties of T (v) and Q(v)) and derive the Bethe equations. He then considered the 8-vertex model, constructed a Q(v) operator that obeyed the correct requirements, and used the reverse argument to derive Bethe equations. The approach is described clearly in Baxter's book [9]. Later on in the 70s, the quantum inverse scattering method (QISM) was developed and used to produce a rather simpler derivation of the same Bethe equations for the 8-vertex model (the algebraic Bethe ansatz approach) [10]. Baxter also invented his corner transfer matrix technique for the 8-vertex model [9]. So, remarkably successful though it was, the Q-operator approach perhaps came to be considered by many as a historical curiosity. However, in the last few years there has been something of a revival of interest in Q. The reasons for this include the following: • Some understanding has been obtained into how Q fits into the QISM/quantum-groups picture of solvable lattice models [11, 12, 13, 14, 15, 16, 17, 18]. • The discovery of the mysterious ODE/IM models correspondence-relating functional relations obeyed by the solutions and spectral determinants of certain ODEs to Bethe ansatz functional relations [19, 20, 21]. • The role of Q in classical integrable systems as a generator of Backlünd transformation has been understood in certain cases (see [22] and references therein). In this paper, we are concerned with the first point. The key to the QISM approach to solvable lattice models is to understand them in terms of an underlying algebra A. The generators of A are matrix elements L ij (z), where i, j ∈ {0, 1} (in the simplest case) and z is a spectral parameter. The set of relations amongst the generators are given by the matrix relation R(z/z ′)L 1 (z)L 2 (z ′) = L 2 (z ′)L 1 (z)R(z/z ′), (1.1) where L 1 (z) = L(z) ⊗ 1, L 2 (z) = 1 ⊗ L(z), and R(z) is a 4 × 4 matrix. This QISM description was later refined in terms of quantum groups. In this picture A is recognised as a quasi-triangular Hopf algebra (aka a quantum group). For the vertex models of the title, the algebra A is U q (sl 2). Families of R-matrices and L-operators are then all given in terms of representations of a universal R-matrix R ∈ U q (b +) ⊗ U q (b −), where U q (b ±) are two Borel subalgebras of U q (sl 2). The relevant U q (sl 2) representations are the spin-n/2 evaluation representations (π (n) z , V (n) z) defined in Section 3 (in this paper, a representation of an algebra A is specified by a pair (π, V), consisting of an A module V and the associated map π : A → End(V)). Then we have R(z/z ′) ≡ (π (1) z ⊗ π (1) z ′)R, L(z) ≡ (π (1) z ⊗ 1)R,
Journal of Physics A: Mathematical and General, 2001
Quantum monodromy matrices coming from a theory of two coupled (m)KdV equations are modified in o... more Quantum monodromy matrices coming from a theory of two coupled (m)KdV equations are modified in order to satisfy the usual Yang-Baxter relation. As a consequence, a general connection between braided and unbraided (usual) Yang-Baxter algebras is derived and also analysed.
Journal of Physics A: Mathematical and General, 2002
A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m... more A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in involution which form the Cartan sub-algebra of the braided quantum group. Representations diagonalizing these operators are described through relying on an easy generalization of Algebraic Bethe Ansatz techniques. The conjecture that this monodromy matrix algebra leads, in the cylinder continuum limit, to a Perturbed Minimal Conformal Field Theory description is analysed and supported.
Journal of Physics a Mathematical and General, 2002
In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix... more In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of U q (sl 2) over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator. His approach was to start with the 6-vertex model Bethe ansatz, and to derive certain functional relations between the transfer matrix T (v) and a matrix Q(v)-the elements of both matrices being entire functions. He went on to show that the reverse argument could be used in order to start from the functional relations (and some other properties of T (v) and Q(v)) and derive the Bethe equations. He then considered the 8-vertex model, constructed a Q(v) operator that obeyed the correct requirements, and used the reverse argument to derive Bethe equations. The approach is described clearly in Baxter's book [9]. Later on in the 70s, the quantum inverse scattering method (QISM) was developed and used to produce a rather simpler derivation of the same Bethe equations for the 8-vertex model (the algebraic Bethe ansatz approach) [10]. Baxter also invented his corner transfer matrix technique for the 8-vertex model [9]. So, remarkably successful though it was, the Q-operator approach perhaps came to be considered by many as a historical curiosity. However, in the last few years there has been something of a revival of interest in Q. The reasons for this include the following: • Some understanding has been obtained into how Q fits into the QISM/quantum-groups picture of solvable lattice models [11, 12, 13, 14, 15, 16, 17, 18]. • The discovery of the mysterious ODE/IM models correspondence-relating functional relations obeyed by the solutions and spectral determinants of certain ODEs to Bethe ansatz functional relations [19, 20, 21]. • The role of Q in classical integrable systems as a generator of Backlünd transformation has been understood in certain cases (see [22] and references therein). In this paper, we are concerned with the first point. The key to the QISM approach to solvable lattice models is to understand them in terms of an underlying algebra A. The generators of A are matrix elements L ij (z), where i, j ∈ {0, 1} (in the simplest case) and z is a spectral parameter. The set of relations amongst the generators are given by the matrix relation R(z/z ′)L 1 (z)L 2 (z ′) = L 2 (z ′)L 1 (z)R(z/z ′), (1.1) where L 1 (z) = L(z) ⊗ 1, L 2 (z) = 1 ⊗ L(z), and R(z) is a 4 × 4 matrix. This QISM description was later refined in terms of quantum groups. In this picture A is recognised as a quasi-triangular Hopf algebra (aka a quantum group). For the vertex models of the title, the algebra A is U q (sl 2). Families of R-matrices and L-operators are then all given in terms of representations of a universal R-matrix R ∈ U q (b +) ⊗ U q (b −), where U q (b ±) are two Borel subalgebras of U q (sl 2). The relevant U q (sl 2) representations are the spin-n/2 evaluation representations (π (n) z , V (n) z) defined in Section 3 (in this paper, a representation of an algebra A is specified by a pair (π, V), consisting of an A module V and the associated map π : A → End(V)). Then we have R(z/z ′) ≡ (π (1) z ⊗ π (1) z ′)R, L(z) ≡ (π (1) z ⊗ 1)R,
We construct anyon fields in a (2 + 1)-dimensional relativistic gauge theory without a Chern-Simo... more We construct anyon fields in a (2 + 1)-dimensional relativistic gauge theory without a Chern-Simons term. We also investigate the basic characteristic properties of these fields.
With the aim of exploring a massive family of models, the nonlinear integral equation for a quant... more With the aim of exploring a massive family of models, the nonlinear integral equation for a quantum system consisting of left and right KdV equations coupled on the cylinder is derived from an integrable lattice field theory. The eigenvalues of the energy and of the transfer matrix (and of all the other local integrals of motion) are expressed in terms of the corresponding solutions of the nonlinear integral equation. The family of models turns out to correspond to the Φ (1,3) perturbation of Conformal Field Theories. The analytic and asymptotic behaviours of the transfer matrix are studied and given.
We construct Drinfel'd twists that define deformed Hopf structures. In particular, we obtain defo... more We construct Drinfel'd twists that define deformed Hopf structures. In particular, we obtain deformed double Yangians and dynamical double Yangians.
We study the first sub-leading correction O((ln s) 0) to the cusp anomalous dimension in the high... more We study the first sub-leading correction O((ln s) 0) to the cusp anomalous dimension in the high spin expansion of finite twist operators in N = 4 SYM theory. Since this approximation is still governed by a linear integral equation (derived already from the Bethe Ansatz equations in a previous paper), we finalise it better in order to study the weak and strong coupling regimes. In fact, we emphasise how easily the weak coupling expansion can be obtained, confirms the known four loop result and predicts the higher orders. Eventually, we pay particular attention to the strong coupling regime showing agreement and predictions in comparison with string expansion; speculations on the 'universal' part (upon subtracting the collinear anomalous dimension) are brought forward.
We propose a scheme for determining a generalised scaling function, namely the Sudakov factor in ... more We propose a scheme for determining a generalised scaling function, namely the Sudakov factor in a peculiar double scaling limit for high spin and large twist operators belonging to the sl(2) sector of planar N = 4 SYM. In particular, we perform explicitly the all-order computation at strong 't Hooft coupling regarding the first (contribution to the) generalised scaling function. Moreover, we compare our asymptotic results with the numerical solutions finding a very good agreement and evaluate numerically the non-asymptotic contributions. Eventually, we illustrate the agreement and prediction on the string side.
Journal of Statistical Mechanics: Theory and Experiment, 2007
As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the a... more As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the all loop expansion of the dimensions in the SU (2) sector of the planar N = 4 SYM, we compute an exact non-perturbative expression for it. For this aim, we use the effective and well-known idea in 2D statistical field theory to convert the Bethe Ansatz equations into two coupled non-linear integral equations (NLIEs). We focus our attention on the highest anomalous dimension for fixed bare dimension or length, L, analysing the many advantages of this method for extracting exact behaviours varying the length and the 't Hooft coupling, λ. For instance, we will show that the large L (asymptotic) expansion is exactly reproduced by its analogue in the BDS Bethe Ansatz, though the exact expression clearly differs from the BDS one (by non-analytic terms). Performing the limits on L and λ in different orders is also under strict control. Eventually, the precision of numerical integration of the NLIEs is as much impressive as in other easierlooking theories.
Bound state excitations of the spin 1/2-XYZ model are considered inside the Bethe Ansatz framewor... more Bound state excitations of the spin 1/2-XYZ model are considered inside the Bethe Ansatz framework by exploiting the equivalent Non-Linear Integral Equations. Of course, these bound states go to the sine-Gordon breathers in the suitable limit and therefore the scattering factors between them are explicitly computed by inspecting the corresponding Non-Linear Integral Equations. As a consequence, abstracting from the physical model the Zamolodchikov-Faddeev algebra of two n-th elliptic breathers defines a tower of n-order Deformed Virasoro Algebras, reproducing the n = 1 case the usual well-known algebra of Shiraishi-Kubo-Awata-Odake [1].
Initially, we derive a nonlinear integral equation for the vacuum counting function of the spin 1... more Initially, we derive a nonlinear integral equation for the vacuum counting function of the spin 1/2-XYZ chain in the disordered regime, thus paralleling similar results by Klümper [1], achieved through a different technique in the antiferroelectric regime. In terms of the counting function we obtain the usual physical quantities, like the energy and the transfer matrix (eigenvalues). Then, we introduce a double scaling limit which appears to describe the sine-Gordon theory on cylindrical geometry, so generalising famous results in the plane by Luther [2] and Johnson et al. [3]. Furthermore, after extending the nonlinear integral equation to excitations, we derive scattering amplitudes involving solitons/antisolitons first, and bound states later. The latter case comes out as manifestly related to the Deformed Virasoro Algebra of Shiraishi et al. [4]. Although this nonlinear integral equations framework was contrived to deal with finite geometries, we prove it to be effective for discovering or rediscovering S-matrices. As a particular example, we prove that this unique model furnishes explicitly two S-matrices, proposed respectively by Zamolodchikov [5] and Lukyanov-Mussardo-Penati [6, 7] as plausible scattering description of unknown integrable field theories.
The nonlinear integral equations describing the spectra of the left and right (continuous) quantu... more The nonlinear integral equations describing the spectra of the left and right (continuous) quantum KdV equations on the cylinder are derived from integrable lattice field theories, which turn out to allow the Bethe Ansatz equations of a twisted "spin −1/2" chain. A very useful mapping to the more common nonlinear integral equation of the twisted continuous spin +1/2 chain is found. The diagonalization of the transfer matrix is performed. The vacua sector is analysed in detail detecting the primary states of the minimal conformal models and giving integral expressions for the eigenvalues of the transfer matrix. Contact with the seminal papers [1, 2] by Bazhanov, Lukyanov and Zamolodchikov is realised. General expressions for the eigenvalues of the infinite-dimensional abelian algebra of local integrals of motion are given and explicitly calculated at the free fermion point.
Journal of Physics A: Mathematical and General, 2002
In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix... more In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of U q (sl 2) over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator. His approach was to start with the 6-vertex model Bethe ansatz, and to derive certain functional relations between the transfer matrix T (v) and a matrix Q(v)-the elements of both matrices being entire functions. He went on to show that the reverse argument could be used in order to start from the functional relations (and some other properties of T (v) and Q(v)) and derive the Bethe equations. He then considered the 8-vertex model, constructed a Q(v) operator that obeyed the correct requirements, and used the reverse argument to derive Bethe equations. The approach is described clearly in Baxter's book [9]. Later on in the 70s, the quantum inverse scattering method (QISM) was developed and used to produce a rather simpler derivation of the same Bethe equations for the 8-vertex model (the algebraic Bethe ansatz approach) [10]. Baxter also invented his corner transfer matrix technique for the 8-vertex model [9]. So, remarkably successful though it was, the Q-operator approach perhaps came to be considered by many as a historical curiosity. However, in the last few years there has been something of a revival of interest in Q. The reasons for this include the following: • Some understanding has been obtained into how Q fits into the QISM/quantum-groups picture of solvable lattice models [11, 12, 13, 14, 15, 16, 17, 18]. • The discovery of the mysterious ODE/IM models correspondence-relating functional relations obeyed by the solutions and spectral determinants of certain ODEs to Bethe ansatz functional relations [19, 20, 21]. • The role of Q in classical integrable systems as a generator of Backlünd transformation has been understood in certain cases (see [22] and references therein). In this paper, we are concerned with the first point. The key to the QISM approach to solvable lattice models is to understand them in terms of an underlying algebra A. The generators of A are matrix elements L ij (z), where i, j ∈ {0, 1} (in the simplest case) and z is a spectral parameter. The set of relations amongst the generators are given by the matrix relation R(z/z ′)L 1 (z)L 2 (z ′) = L 2 (z ′)L 1 (z)R(z/z ′), (1.1) where L 1 (z) = L(z) ⊗ 1, L 2 (z) = 1 ⊗ L(z), and R(z) is a 4 × 4 matrix. This QISM description was later refined in terms of quantum groups. In this picture A is recognised as a quasi-triangular Hopf algebra (aka a quantum group). For the vertex models of the title, the algebra A is U q (sl 2). Families of R-matrices and L-operators are then all given in terms of representations of a universal R-matrix R ∈ U q (b +) ⊗ U q (b −), where U q (b ±) are two Borel subalgebras of U q (sl 2). The relevant U q (sl 2) representations are the spin-n/2 evaluation representations (π (n) z , V (n) z) defined in Section 3 (in this paper, a representation of an algebra A is specified by a pair (π, V), consisting of an A module V and the associated map π : A → End(V)). Then we have R(z/z ′) ≡ (π (1) z ⊗ π (1) z ′)R, L(z) ≡ (π (1) z ⊗ 1)R,
Journal of Physics A: Mathematical and General, 2001
Quantum monodromy matrices coming from a theory of two coupled (m)KdV equations are modified in o... more Quantum monodromy matrices coming from a theory of two coupled (m)KdV equations are modified in order to satisfy the usual Yang-Baxter relation. As a consequence, a general connection between braided and unbraided (usual) Yang-Baxter algebras is derived and also analysed.
Journal of Physics A: Mathematical and General, 2002
A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m... more A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in involution which form the Cartan sub-algebra of the braided quantum group. Representations diagonalizing these operators are described through relying on an easy generalization of Algebraic Bethe Ansatz techniques. The conjecture that this monodromy matrix algebra leads, in the cylinder continuum limit, to a Perturbed Minimal Conformal Field Theory description is analysed and supported.
Journal of Physics a Mathematical and General, 2002
In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix... more In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of U q (sl 2) over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator. His approach was to start with the 6-vertex model Bethe ansatz, and to derive certain functional relations between the transfer matrix T (v) and a matrix Q(v)-the elements of both matrices being entire functions. He went on to show that the reverse argument could be used in order to start from the functional relations (and some other properties of T (v) and Q(v)) and derive the Bethe equations. He then considered the 8-vertex model, constructed a Q(v) operator that obeyed the correct requirements, and used the reverse argument to derive Bethe equations. The approach is described clearly in Baxter's book [9]. Later on in the 70s, the quantum inverse scattering method (QISM) was developed and used to produce a rather simpler derivation of the same Bethe equations for the 8-vertex model (the algebraic Bethe ansatz approach) [10]. Baxter also invented his corner transfer matrix technique for the 8-vertex model [9]. So, remarkably successful though it was, the Q-operator approach perhaps came to be considered by many as a historical curiosity. However, in the last few years there has been something of a revival of interest in Q. The reasons for this include the following: • Some understanding has been obtained into how Q fits into the QISM/quantum-groups picture of solvable lattice models [11, 12, 13, 14, 15, 16, 17, 18]. • The discovery of the mysterious ODE/IM models correspondence-relating functional relations obeyed by the solutions and spectral determinants of certain ODEs to Bethe ansatz functional relations [19, 20, 21]. • The role of Q in classical integrable systems as a generator of Backlünd transformation has been understood in certain cases (see [22] and references therein). In this paper, we are concerned with the first point. The key to the QISM approach to solvable lattice models is to understand them in terms of an underlying algebra A. The generators of A are matrix elements L ij (z), where i, j ∈ {0, 1} (in the simplest case) and z is a spectral parameter. The set of relations amongst the generators are given by the matrix relation R(z/z ′)L 1 (z)L 2 (z ′) = L 2 (z ′)L 1 (z)R(z/z ′), (1.1) where L 1 (z) = L(z) ⊗ 1, L 2 (z) = 1 ⊗ L(z), and R(z) is a 4 × 4 matrix. This QISM description was later refined in terms of quantum groups. In this picture A is recognised as a quasi-triangular Hopf algebra (aka a quantum group). For the vertex models of the title, the algebra A is U q (sl 2). Families of R-matrices and L-operators are then all given in terms of representations of a universal R-matrix R ∈ U q (b +) ⊗ U q (b −), where U q (b ±) are two Borel subalgebras of U q (sl 2). The relevant U q (sl 2) representations are the spin-n/2 evaluation representations (π (n) z , V (n) z) defined in Section 3 (in this paper, a representation of an algebra A is specified by a pair (π, V), consisting of an A module V and the associated map π : A → End(V)). Then we have R(z/z ′) ≡ (π (1) z ⊗ π (1) z ′)R, L(z) ≡ (π (1) z ⊗ 1)R,
We construct anyon fields in a (2 + 1)-dimensional relativistic gauge theory without a Chern-Simo... more We construct anyon fields in a (2 + 1)-dimensional relativistic gauge theory without a Chern-Simons term. We also investigate the basic characteristic properties of these fields.
With the aim of exploring a massive family of models, the nonlinear integral equation for a quant... more With the aim of exploring a massive family of models, the nonlinear integral equation for a quantum system consisting of left and right KdV equations coupled on the cylinder is derived from an integrable lattice field theory. The eigenvalues of the energy and of the transfer matrix (and of all the other local integrals of motion) are expressed in terms of the corresponding solutions of the nonlinear integral equation. The family of models turns out to correspond to the Φ (1,3) perturbation of Conformal Field Theories. The analytic and asymptotic behaviours of the transfer matrix are studied and given.
We construct Drinfel'd twists that define deformed Hopf structures. In particular, we obtain defo... more We construct Drinfel'd twists that define deformed Hopf structures. In particular, we obtain deformed double Yangians and dynamical double Yangians.
We study the first sub-leading correction O((ln s) 0) to the cusp anomalous dimension in the high... more We study the first sub-leading correction O((ln s) 0) to the cusp anomalous dimension in the high spin expansion of finite twist operators in N = 4 SYM theory. Since this approximation is still governed by a linear integral equation (derived already from the Bethe Ansatz equations in a previous paper), we finalise it better in order to study the weak and strong coupling regimes. In fact, we emphasise how easily the weak coupling expansion can be obtained, confirms the known four loop result and predicts the higher orders. Eventually, we pay particular attention to the strong coupling regime showing agreement and predictions in comparison with string expansion; speculations on the 'universal' part (upon subtracting the collinear anomalous dimension) are brought forward.
We propose a scheme for determining a generalised scaling function, namely the Sudakov factor in ... more We propose a scheme for determining a generalised scaling function, namely the Sudakov factor in a peculiar double scaling limit for high spin and large twist operators belonging to the sl(2) sector of planar N = 4 SYM. In particular, we perform explicitly the all-order computation at strong 't Hooft coupling regarding the first (contribution to the) generalised scaling function. Moreover, we compare our asymptotic results with the numerical solutions finding a very good agreement and evaluate numerically the non-asymptotic contributions. Eventually, we illustrate the agreement and prediction on the string side.
Journal of Statistical Mechanics: Theory and Experiment, 2007
As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the a... more As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the all loop expansion of the dimensions in the SU (2) sector of the planar N = 4 SYM, we compute an exact non-perturbative expression for it. For this aim, we use the effective and well-known idea in 2D statistical field theory to convert the Bethe Ansatz equations into two coupled non-linear integral equations (NLIEs). We focus our attention on the highest anomalous dimension for fixed bare dimension or length, L, analysing the many advantages of this method for extracting exact behaviours varying the length and the 't Hooft coupling, λ. For instance, we will show that the large L (asymptotic) expansion is exactly reproduced by its analogue in the BDS Bethe Ansatz, though the exact expression clearly differs from the BDS one (by non-analytic terms). Performing the limits on L and λ in different orders is also under strict control. Eventually, the precision of numerical integration of the NLIEs is as much impressive as in other easierlooking theories.
Bound state excitations of the spin 1/2-XYZ model are considered inside the Bethe Ansatz framewor... more Bound state excitations of the spin 1/2-XYZ model are considered inside the Bethe Ansatz framework by exploiting the equivalent Non-Linear Integral Equations. Of course, these bound states go to the sine-Gordon breathers in the suitable limit and therefore the scattering factors between them are explicitly computed by inspecting the corresponding Non-Linear Integral Equations. As a consequence, abstracting from the physical model the Zamolodchikov-Faddeev algebra of two n-th elliptic breathers defines a tower of n-order Deformed Virasoro Algebras, reproducing the n = 1 case the usual well-known algebra of Shiraishi-Kubo-Awata-Odake [1].
Initially, we derive a nonlinear integral equation for the vacuum counting function of the spin 1... more Initially, we derive a nonlinear integral equation for the vacuum counting function of the spin 1/2-XYZ chain in the disordered regime, thus paralleling similar results by Klümper [1], achieved through a different technique in the antiferroelectric regime. In terms of the counting function we obtain the usual physical quantities, like the energy and the transfer matrix (eigenvalues). Then, we introduce a double scaling limit which appears to describe the sine-Gordon theory on cylindrical geometry, so generalising famous results in the plane by Luther [2] and Johnson et al. [3]. Furthermore, after extending the nonlinear integral equation to excitations, we derive scattering amplitudes involving solitons/antisolitons first, and bound states later. The latter case comes out as manifestly related to the Deformed Virasoro Algebra of Shiraishi et al. [4]. Although this nonlinear integral equations framework was contrived to deal with finite geometries, we prove it to be effective for discovering or rediscovering S-matrices. As a particular example, we prove that this unique model furnishes explicitly two S-matrices, proposed respectively by Zamolodchikov [5] and Lukyanov-Mussardo-Penati [6, 7] as plausible scattering description of unknown integrable field theories.
Uploads
Papers by Marco Rossi