In this note, inspired partly by the works on the analytic trans-series of free-energies in 2D integrable QFTs in the UV limit, we study the corresponding asymptotic expansions in the IR limit of the integrable XXX spin chains, at the anti-ferromagnetic critical point. Starting from the linear TBA integral equation, we generate the perturbative series in the minimal-scheme coupling constants, up to the first order where the $ζ_3$ appears. To all perturbative orders, at spin one-half, the perturbative series relates to that of the $β^2=8π^-$ massive sine-Gordon by a sign flip of the coupling constant, and this relation generalizes for an arbitrary spin, to certain massive deformation of $su(2)_k$ WZW. This relation partially supports the conjectured field-theoretical interpretations of such critical points, at the leading-power $\&\&$ all-logarithmic accuracy. We also compute the perturbative coefficients attached to the first exponentially-small corrections up to the order where $ζ_3$ appears, in case $s>1$. An interesting feature is that the largest exponentially-small corrections at the locations $h^{2+\frac{2k}{s}}$ are non-vanishing, but the corresponding Borel singularities in the perturbative series are canceled for generic $s>1$, not necessarily integers or half-integers. This becomes most dramatic in the a 't Hooft type large $s$ limit, in which the perturbative series has a finite convergence radius, while these exponentially small corrections still survive.
Using the linearized TBA equations recently obtained in [arXiv:1002.1711] we show analytically that the 5-loop anomalous dimension of the Konishi operator agrees with the result obtained previously from the generalized Luscher formulae. The proof is based on the relation between this linear system and the XXX model TBA equations.
We consider integrable boundary states in the XXX spin-1/2 spin chain. We begin by briefly reviewing the algebraic Bethe Ansatz as well as integrable boundary states in spin chains. Then a recently discovered class of integrable states known as crosscap states is described and expanded. In these states each spin is entangled with its antipodal spin. We present a novel proof of the integrability of both a crosscap state that is known in the literature and one that has not previously been studied. We then use the machinery of the algebraic Bethe Ansatz to derive the overlaps between the crosscap…
We present a general method of folding an integrable spin chain, defined on a line, to obtain an integrable open spin chain, defined on a half-line. We illustrate our method through two fundamental models with sl(2) Lie algebra symmetry: the Heisenberg XXX and the Inozemtsev hyperbolic spin chains. We obtain new long-range boundary Hamiltonians and demonstrate that they exhibit Yangian symmetries, thus ensuring integrability of the models we obtain. The method presented provides a "bottom-up" approach for constructing integrable boundaries and can be applied to any spin chain model.
We derive the fusion hierarchy of functional equations for critical A-D-E lattice models related to, the sl(2) unitary minimal models, the parafermionic models and the supersymmetric models of conformal field theory, and deduce the related TBA functional equations. The derivation uses fusion projectors and applies in the presence of all known integrable boundary conditions on the torus and cylinder. The resulting TBA functional equations are_universal_ in the sense that they depend only on the Coxeter number of the A-D-E graph and are independent of the particular integrable boundary condition…
We study the spectrum of the integrable open XXX Heisenberg spin chain subject to non-diagonal boundary magnetic fields. The spectral problem for this model can be formulated in terms of functional equations obtained by separation of variables or, equivalently, from the fusion of transfer matrices. For generic boundary conditions the eigenvalues cannot be obtained from the solution of finitely many algebraic Bethe equations. Based on careful finite size studies of the analytic properties of the underlying hierarchy of transfer matrices we devise two approaches to analyze the functional equatio…
For the integrable higher-spin XXX and XXZ spin chains we present multiple-integral representations for the correlation function of an arbitrary product of Hermitian elementary matrices in the massless ground state. We give a formula expressing it by a single term of multiple integrals. In particular, we explicitly derive the emptiness formation probability (EFP). We assume $2s$-strings for the ground-state solution of the Bethe ansatz equations for the spin-$s$ XXZ chain, and solve the integral equations for the spin-$s$ Gaudin matrix. In terms of the XXZ coupling $Δ$ we define $ζ$ by $Δ=\cos…
The thermodynamic Bethe ansatz (TBA) and the excited state TBA equations for an integrable spin chain related to the Lie superalgebra osp(1|2) are proposed by the quantum transfer matrix (QTM) method. We introduce the fusion hierarchy of the QTM and derive the functional relations among them (T-system) and their certain combinations (Y-system). Their analytical property leads to the non-linear integral equations which describe the free energy and the correlation length at any finite temperatures. With regard to the free energy, they coincide with the TBA equation (math-ph/9911010, Mod. Phys. L…
The XXX Gaudin model with generic integrable boundaries specified by the most general non-diagonal K-matrices is studied by the off-diagonal Bethe ansatz method. The eigenvalues of the associated Gaudin operators and the corresponding Bethe ansatz equations are obtained.
We compute the tree-level bosonic S matrix in light-cone gauge for superstrings on pure-NSNS $\text{AdS}_{3}\times\text{S}^3\times\text{S}^3\times\text{S}^1$. We show that it is proportional to the identity and that it takes the same form as for $\text{AdS}_{3} \times \text{S}^3\times\text{T}^4$ and for flat space. Based on this, we make a conjecture for the exact worldsheet S matrix and derive the mirror thermodynamic Bethe ansatz (TBA) equations describing the spectrum. Despite a non-trivial vacuum energy, they can be solved in closed form and coincide with a simple set of Bethe ansatz equat…
In arXiv:0908.4052, Nekrasov and Shatashvili pointed out that the N=2 instanton partition function in a special limit of the Omega-deformation parameters is characterized by certain thermodynamic Bethe ansatz (TBA) like equations. In this work we present an explicit derivation of this fact as well as generalizations to quiver gauge theories. To do so we combine various techniques like the iterated Mayer expansion, the method of expansion by regions, and the path integral tricks for non-perturbative summation. The TBA equations derived entirely within gauge theory have been proposed to encode t…
We give a direct derivation of a proposal of Nekrasov-Shatashvili concerning the quantization conditions of the Toda chain. The quantization conditions are formulated in terms of solutions to a nonlinear integral equation similar to the ones coming from the thermodynamic Bethe ansatz. This is equivalent to extremizing a certain function called Yang's potential. It is shown that the Nekrasov-Shatashvili formulation of the quantization conditions follows from the solution theory of the Baxter equation, suggesting that this way of formulating the quantization conditions should indeed be applicabl…
We put forward an idea that the boundary entropy associated with integrable massless flow of thermodynamic Bethe ansatz (TBA) is identified with tachyon action of boundary string field theory. We show that the temperature parameterizing a massless flow in the TBA formalism can be identified with tachyon energy for the classical action at least near the ultraviolet fixed point, i.e. the open string vacuum.
We analyze the Thermodynamic Bethe Ansatz (TBA) for various integrable S-matrices in the context of generalized $T\bar T$ deformations. We focus on the sinh-Gordon model and its elliptic deformation in both its fermionic and bosonic realizations. We confirm that the determining factor for a turning point in the TBA, interpreted as a finite Hagedorn temperature, is the difference between the number of bound states and resonances in the theory. Implementing the numerical pseudo-arclength continuation method, we are able to follow the solutions to the TBA equations past the turning point all the …
We consider integrable deformations of the XXZ spin chain for periodic and open boundary conditions. In particular, we classify all long-range deformations and study their impact on the spectrum. As compared to the XXX case, we have the z-spin at our disposal, which induces two additional deformations: the short-range magnetic twist and a new long-range momentum-dependent twist.
By considering the continuum scaling limit of the $A_{4}$ RSOS lattice model of Andrews-Baxter-Forrester with integrable boundaries, we derive excited state TBA equations describing the boundary flows of the tricritical Ising model. Fixing the bulk weights to their critical values, the integrable boundary weights admit a parameter $ξ$ which plays the role of the perturbing boundary field $ϕ_{1,3}$ and induces the renormalization group flow between boundary fixed points. The boundary TBA equations determining the RG flows are derived in the $\mathcal{B}_{(1,2)}\to \mathcal{B}_{(2,1)}$ example. …
Using the Algebraic Bethe Ansatz we consider the correlation functions of the integrable higher spin chains. We apply a method recently developed for the spin $\frac 12$ Heisenberg chain, based on the solution of the quantum inverse problem. We construct a representation for the correlation functions on a finite chain for arbitrary spin. Then we show how the string solutions of the Bethe equations can be considered in the framework of this approach in the thermodynamic limit. Finally, a multiple integral representation for the spin 1 zero temperature correlation functions is obtained in the th…
We propose an exact summation method to compute thermodynamic observables in integrable quantum field theories. The key idea is to use the matrix-tree theorem to write the Gaudin determinants that appear in the cluster expansion as a sum over graphs. For theories with a diagonal S-matrix, this method is more powerful than the standard Thermodynamic Bethe Ansatz (TBA) technique as it is exact to all orders of powers in inverse volume. We have obtained using this method the TBA equation, the excited state energies in finite volume, the Leclair-Mussardo formula for one point functions, the finite…
The off-diagonal Bethe Ansatz method [1] is used to revisit the periodic XXX Heisenberg spin-1/2 chain. It is found that the spectrum of the transfer matrix can be characterized by an inhomogeneous T-Q relation, a natural but nontrivial extension of Baxter's T-Q relation [2].
We show how a lattice approach can be used to derive Thermodynamic Bethe Ansatz (TBA) equations describing all excitations for boundary flows. The method is illustrated for a prototypical flow of the tricritical Ising model by considering the continuum scaling limit of the A4 lattice model with integrable boundaries. Fixing the bulk weights to their critical values, the integrable boundary weights admit two boundary fields $ξ$ and $η$ which play the role of the perturbing boundary fields $ϕ_{1,3}$ and $ϕ_{1,2}$ inducing the renormalization group flow between boundary fixed points. The excitati…
Thermodynamics of the Heisenberg XXX chain with negative spin
Institute of Modern Physics, Northwest University（西北大学现代物理研究所）；C.N. Yang Institute for Theoretical Physics, Stony Brook University（纽约州立大学石溪分校杨振宁理论物理研究所）
We consider the non-unitary Lee-Yang minimal model ${\cal M}(2,5)$ in three different finite geometries: (i) on the interval with integrable boundary conditions labelled by the Kac labels $(r,s)=(1,1),(1,2)$, (ii) on the circle with periodic boundary conditions and (iii) on the periodic circle including an integrable purely transmitting defect. We apply $φ_{1,3}$ integrable perturbations on the boundary and on the defect and describe the flow of the spectrum. Adding a $Φ_{1,3}$ integrable perturbation to move off-criticality in the bulk, we determine the finite size spectrum of the massive sca…
We study a series of $N\!=\!1$ supersymmetric integrable particle theories in $d=1+1$ dimensions. These theories are represented as integrable perturbations of specific $N\!=\!1$ superconformal field theories. Starting from the conjectured $S$-matrices for these theories, we develop the Thermodynamic Bethe Ansatz (TBA), where we use that the 2-particle $S$-matrices satisfy a free fermion condition. Our analysis proves a conjecture by E.~Melzer, who proposed that these $N\!=\!1$ supersymmetric TBA systems are ``folded'' versions of $N\!=\!2$ supersymmetric TBA systems that were first studied by…
The general solutions for the factorization equations of the reflection matrices $K^{\pm}(θ)$ for the eight vertex and six vertex models (XYZ, XXZ and XXX chains) are found. The associated integrable magnetic Hamiltonians are explicitly derived, finding families dependig on several continuous as well as discrete parameters.
The integrals of motion of the tricritical Ising model are obtained by Thermodynamic Bethe Ansatz (TBA) equations derived from the A_4 integrable lattice model. They are compared with those given by the conformal field theory leading to a unique one-to-one lattice-conformal correspondence. They can also be followed along the renormalization group flows generated by the action of the boundary field ϕ_{1,3} on conformal boundary conditions in close analogy to the usual TBA description of energies.
Recently, an exact description of instanton corrections to the moduli spaces of 4d N=2 supersymmetric gauge theories compactified on a circle and Calabi-Yau compactifications of Type II superstring theories was found. The equations determining the instanton contributions turn out to have the form of Thermodynamic Bethe Ansatz. We explore further this relation and, in particular, we identify the contact potential of quaternionic string moduli space with the free energy of the integrable system and the Kahler potential of the gauge theory moduli space with the Yang-Yang functional. We also show …
We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of particles which dynamically interacts via the scattering matrix of affine Toda field theory and whose statistical interaction is of a general Haldane type. Up to the first leading order, we provide general approximated analytical expressions for the solutions of these equations from which we derive general formulae for the ultraviolet scaling functions for theories in which the underlying Lie algebra is simply laced. For several explicit models we compare the quality of the approximated analytical solutions against t…
New boundary conditions for integrable nonlinear lattices of the XXX type, such as the Heisenberg chain and the Toda lattice are presented. These integrable extensions are formulated in terms of a generic XXX Heisenberg magnet interacting with two additional spins at each end of the chain. The construction uses the most general rank 1 ansatz for the 2x2 L-operator satisfying the reflection equation algebra with rational r-matrix. The associated quadratic algebra is shown to be the one of dynamical symmetry for the A1 and BC2 Calogero-Moser problems. Other physical realizations of our quadratic…
Based on the inhomogeneous T-Q relation constructed via the off-diagonal Bethe Ansatz, a systematic method for retrieving the Bethe-type eigenstates of integrable models without obvious reference state is developed by employing certain orthogonal basis of the Hilbert space. With the XXZ spin torus model and the open XXX spin-1/2 chain as examples, we show that for a given inhomogeneous T-Q relation and the associated Bethe Ansatz equations, the constructed Bethe-type eigenstate has a well-defined homogeneous limit.
We examine certain classical continuum long wave-length limits of prototype integrable quantum spin chains. We define the corresponding construction of classical continuum Lax operators. Our discussion starts with the XXX chain, the anisotropic Heisenberg model and their generalizations and extends to the generic isotropic and anisotropic gl_n magnets. Certain classical and quantum integrable models emerging from special "dualities" of quantum spin chains, parametrized by c-number matrices, are also presented.
The structure of a new family of factorised $S$-matrix theories with resonance poles is reviewed. They are conjectured to correspond to the Homogeneous sine-Gordon theories associated with simply laced compact Lie groups. Two of their more remarkable properties are, first, that some of the resonance poles can be traced to the presence of unstable particles in the spectrum, and, second, that they involve several independent mass scales. The conjectured relationship with the simply laced HSG theories has been checked by means of the Thermodynamic Bethe ansatz (TBA) and, more recently, through th…
In this paper we study the boundary effects for off-critical integrable field theories which have close analogs with integrable lattice models. Our models are the $SU(2)_{k}\otimes SU(2)_{l}/SU(2)_{k+l}$ coset conformal field theories perturbed by integrable boundary and bulk operators. The boundary interactions are encoded into the boundary reflection matrix. Using the TBA method, we verify the flows of the conformal BCs by computing the boundary entropies. These flows of the BCs have direct interpretations for the fusion RSOS lattice models. For super CFTs ($k=2$) we show that these flows ar…
We investigate the exact overlaps between eigenstates of integrable spin chains and a special class of states called "integrable initial/final states". These states satisfy a special integrability constraint, and they are closely related to integrable boundary conditions. We derive new algebraic relations for the integrable states, which lead to a set of recursion relations for the exact overlaps. We solve these recursion relations and thus we derive new overlap formulas, valid in the XXX Heisenberg chain and its integrable higher spin generalizations. Afterwards we generalize the integrabilit…
We study the thermodynamic behaviour of Inozemtsev's long-range elliptic spin chain using the Bethe ansatz equations describing the spectrum of the model in the infinite-length limit. We classify all solutions of these equations in that limit and argue which of these solutions determine the spectrum in the thermodynamic limit. Interestingly, some of the solutions are not selfconjugate, which puts the model in sharp contrast to one of the model's limiting cases, the Heisenberg xxx spin chain. Invoking the string hypothesis we derive the thermodynamic Bethe ansatz equations (TBA-equations) from …
We show some symmetry relations among the correlation functions of the integrable higher-spin XXX and XXZ spin chains, where we explicitly evaluate the multiple integrals representing the one-point functions in the spin-1 case. We review the multiple-integral representations of correlation functions for the integrable higher-spin XXZ chains derived in a region of the massless regime including the anti-ferromagnetic point. Here we make use of the gauge transformations between the symmetric and asymmetric R-matrices, which correspond to the principal and homogeneous gradings, respectively, and w…
Type-I quantum impurities are investigated in the context of the integrable Heisenberg model. This type of defects is associated to the (q)-harmonic oscillator algebra. The transmission matrices associated to this particular type of defects are computed via the Bethe ansatz methodology for the XXX model, as well as for the critical and non-critical XXZ spin chain. In the attractive regime of the critical XXZ spin chain the transmission amplitudes for the breathers are also identified.
The integrable XXX spin s quantum chain and the alternating $s^{1}$, $s^{2}$ ($s^{1}-s^{2}={1\over 2}$) chain with boundaries are considered. The scattering of their excitations with the boundaries via the Bethe ansatz method is studied, and the exact boundary S matrices are computed in the limit $s, s^{1, 2} \to \infty$. Moreover, the connection of these models with the SU(2) Principal Chiral, WZW and the RSOS models is discussed.
We present determinant formulae for the form factors of spin operators of general integrable XXX Heisenberg spin chains for arbitrary (finite dimensional) spin representations. The results apply to any "mixed" spin chains, such as alternating spin chains, or to spin chains with magnetic impurities.
We consider the massless tricritical Ising model M(4,5) perturbed by the thermal operator in a cylindrical geometry and apply integrable boundary conditions, labelled by the Kac labels (r,s), that are natural off-critical perturbations of known conformal boundary conditions. We derive massless thermodynamic Bethe ansatz (TBA) equations for all excitations by solving, in the continuum scaling limit, the TBA functional equation satisfied by the double-row transfer matrices of the A_4 lattice model of Andrews, Baxter and Forrester (ABF) in Regime IV. The resulting TBA equations describe the massl…
The ground-state energy of integrably-twisted theories is analyzed in finite volume. We derive the leading and next-to-leading order (NLO) Lüscher-type corrections for large volumes of the vacuum energy for integrable theories with twisted boundary conditions and twisted S-matrix. We then derive the twisted thermodynamic Bethe ansatz (TBA) equations to describe exactly the ground state, from which we obtain an untwisted Y-system. The two approaches are compared by expanding the TBA equations to NLO, and exact agreement is found. We give explicit results for the O(4) model and for the three-par…
We examine some excited state energies in the non-unitary integrable quantum field theory obtained from the perturbation of the minimal conformal field theory model $M_{3,5}$ by its operator $ϕ_{2,1}$. Using the correspondence of this IQFT to the scaling limit of the dilute $A_2$ lattice model (in a particular regime) we derive the functional equations for the QFT commuting transfer matrices. These functional equations can be transformed to a closed set of TBA-like integral equations which determine the excited state energies in the finite-size system. In particular, we explicitly construct th…
In presence of a static pair of sources, the spectrum of low-lying states of whatever confining gauge theory in D space-time dimensions is described, at large source separations, by an effective string theory. In the far infrared the latter flows, in the static gauge, to a two-dimensional massless free-field theory. It is known that the Lorentz invariance of the gauge theory fixes uniquely the first few subleading corrections of this free-field limit. We point out that the first allowed correction - a quartic polynomial in the field derivatives - is exactly the composite field $T\bar{T}$, buil…
The g-function was introduced by Affleck and Ludwig in the context of critical quantum systems with boundaries. In the framework of the thermodynamic Bethe ansatz (TBA) method for relativistic scattering theories, all attempts to write an exact integral equation for the off-critical version of this quantity have, up to now, been unsuccesful. We tackle this problem by using an n-particle cluster expansion, close in spirit to form-factor calculations of correlators on the plane. The leading contribution already disagrees with all previous proposals, but a study of this and subsequent terms allow…
The thermodynamic Bethe Ansatz equations arising in the context of the $AdS_5/CFT_4$ correspondence exhibit an important difference with respect to their analogues in relativistic integrable quantum field theories: their solutions (the Y functions) are not meromorphic functions of the rapidity, but live on a complicated Riemann surface with an infinite number of branch points and therefore enjoy a new kind of extended Y-system. In this paper we review the analytic properties of the TBA solutions, and present new information coming from their numerical study. An identity allowing to simplify th…
We suggest an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the Thermodynamic Bethe Ansatz equations for its ground state. The idea relies on analytic continuation through complex values of the coupling constant, and an analysis of the monodromies that the equations and their solutions undergo. For the scaling Lee-Yang model, we find equations in this way for the one- and two- particle states in the spin-zero sector, and suggest various generalisations. Numerical results show excellent agreement with the truncated conformal s…
We use relativistic invariance to investigate two aspects of integrable AdS${}_3$ string theory. Firstly, we write down the all-loop TBA equations for the massless sector of the theory with R-R flux, using the recently discovered hidden relativistic symmetry. Secondly, for the low-energy relativistic limit of the theory with NS-NS flux we write down the S matrix, dressing factors and TBA. We find that the integrable system coincides with a restriction to AdS${}_3$ of the relativistic q-deformed AdS${}_5$ theory. We also comment on the relativistic limit of the the small-$k$ NS-NS theory.
We investigate low-energy string excitations in AdS$_3\times$S$^3\times$T$^4$. When the worldsheet is decompactified, the theory has gapless modes whose spectrum at low energies is determined by massless relativistic integrable S matrices of the type introduced by Al. B. Zamolodchikov. The S matrices are non-trivial only for excitations with identical worldsheet chirality, indicating that the low-energy theory is a CFT$_2$. We construct a Thermodynamic Bethe Ansatz (TBA) for these excitations and show how the massless modes' wrapping effects may be incorporated into the AdS$_3$ spectral proble…
In this paper we compute the scaling functions of the effective central charges for various quantum integrable models in a deep ultraviolet region $R\to 0$ using two independent methods. One is based on the ``reflection amplitudes'' of the (super-)Liouville field theory where the scaling functions are given by the conjugate momentum to the zero-modes. The conjugate momentum is quantized for the sinh-Gordon, the Bullough-Dodd, and the super sinh-Gordon models where the quantization conditions depend on the size $R$ of the system and the reflection amplitudes. The other method is to solve the st…
We develop a new method to compute the exact overlaps between integrable boundary states and on-shell Bethe states for integrable spin chains. Our method is based on the coordinate Bethe Ansatz and does not rely on the "rotation trick" of the corresponding lattice model. It leads to a rigorous proof of the factorized overlap formulas in a number of cases, some of which were inaccessible to earlier methods. As concrete examples, we consider the compact XXX and XXZ Heisenberg spin chains, and the non-compact $SL(2,\mathbb{R})$ spin chain.
We consider the continuum limit of 4d planar fishnet diagrams using integrable spin chain methods borrowed from the $\mathcal{N}=4$ Super-Yang-Mills theory. These techniques give us control on the scaling dimensions of single-trace operators for all values of the coupling constant in the fishnet theory. We use them to study the thermodynamical limit of the BMN operator corresponding to the spin chain ferromagnetic vacuum. We find that its scaling dimension exhibits a critical behaviour when the coupling constant approaches Zamolodchikov's critical coupling. Analysis close to that point suggest…
We consider the critical non-unitary minimal model {\cal M}(3,5) with integrable boundaries. We analyze the patterns of zeros of the eigenvalues of the transfer matrix and then determine the spectrum of the critical theory through the Thermodynamic Bethe Ansatz (TBA) equations. By solving the TBA functional equation satisfied by the transfer matrices of the associated A_{4} RSOS lattice model of Forrester and Baxter in Regime III in the continuum scaling limit, we derive the integral TBA equations for all excitations in the (r=1,s=1) sector then determine their corresponding energies. The exci…
I study the technique of Algebraic Bethe Ansatz for solving integrable models and show how it works in detail on the simplest example of spin 1/2 XXX magnetic chain. Several other models are treated more superficially, only the specific details are given. Several parameters, appearing in these generalizations: spin $s$, anisotropy parameter $\ga$, shift $\om$ in the alternating chain, allow to include in our treatment most known examples of soliton theory, including relativistic model of Quantum Field Theory.
We present the first known integrable relativistic field theories with interacting massive and massless sectors. And we demonstrate that knowledge of the massless sector is essential for understanding of the spectrum of the massive sector. Terms in this spectrum polynomial in the spatial volume (the accuracy for which the Bethe ansatz would suffice in a massive theory) require not just Lüscher-like corrections (usually exponentially small) but the full TBA integral equations. We are motivated by the implications of these ideas for AdS/CFT, but present here only field-theory results.
The relation between the dilatation operator of N=4 Yang-Mills theory and integrable spin chains makes it possible to compute the one-loop anomalous dimensions of all operators in the theory. In this paper we show how to apply the technology of integrable spin chains to the calculation of Yang-Mills correlation functions by expressing them in terms of matrix elements of spin operators on the corresponding spin chain. We illustrate this method with several examples in the SU(2) sector described by the XXX_1/2 chain.
We solve the longstanding problem to define a functional characterization of the spectrum of the transfer matrix associated to the most general spin-1/2 representations of the 6-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-d…
Baxter's Q-operator is generally believed to be the most powerful tool for the exact diagonalization of integrable models. Curiously, it has hitherto not yet been properly constructed in the simplest such system, the compact spin-1/2 Heisenberg-Bethe XXX spin chain. Here we attempt to fill this gap and show how two linearly independent operatorial solutions to Baxter's TQ equation may be constructed as commuting transfer matrices if a twist field is present. The latter are obtained by tracing over infinitely many oscillator states living in the auxiliary channel of an associated monodromy matr…
We study on-shell and off-shell properties of the supersymmetric sinh-Gordon and perturbed SUSY Yang-Lee models using the thermodynamic Bethe ansatz and form factors. Identifying the supersymmetric models with the Eight Vertex Free Fermion Model, we derive inversion relation for inhomogeneous transfer matrix and TBA equations and get correct UV results. We obtain two-point form factors of the trace of energy-momentum tensor using the Watson equations and their SUSY transformations. As an application, we compute the UV central charge using these form factors and spectral representation of the $…
We present a detailed analysis of the structure of the conservation laws in quantum integrable chains of the XYZ-type and in the Hubbard model. With the use of the boost operator, we establish the general form of the XYZ conserved charges in terms of simple polynomials in spin variables and derive recursion relations for the relative coefficients of these polynomials. For two submodels of the XYZ chain - namely the XXX and XY cases, all the charges can be calculated in closed form. For the XXX case, a simple description of conserved charges is found in terms of a Catalan tree. This constructio…
