Integrable Lattice Models Research Articles

Multicoloured Temperley–Lieb lattice models. The ground state

Using the inversion relation method, we calculate the ground-state energy for the lattice integrable models, based on baxterization of multicoloured generalization of Temperley–Lieb algebras. The simplest vertex model is analysed and found to have a mass gap.

Low-temperature asymptotics of integrable systems in an external field

An asymptotic low-temperature expansion is performed for an integrable bosonic lattice model and for the critical spin-1/2 Heisenberg chain in a magnetic field. The results apply to the integrable Bose gas as well. We also comment on a high-temperature expansion of the bosonic lattice model.

Logarithmic minimal models

Working in the dense loop representation, we use the planar Temperley–Liebalgebra to build integrable lattice models called logarithmic minimal models . Specifically, we construct Yang–Baxter integrable Temperley–Lieb models on the stripacting on link states and consider their associated Hamiltonian limits. These models andtheir associated representations of the Temperley–Lieb algebra are inherentlynon-local and not (time-reversal) symmetric. We argue that, in the continuumscaling limit, they yield logarithmic conformal field theories with central chargesc = 1−(6(p−p′)2/pp′),where p, p′ = 1, 2, ... are coprime. The first few members of the principal series are critical dense polymers (m = 1, c = −2), criticalpercolation (m = 2,c = 0) and the logarithmicIsing model (m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformalboundary conditions organized in an extended Kac table with conformal weightsΔr,s = (((m+1)r−ms)2−1)/4m(m+1),r, s = 1, 2, .... The associated conformal partition functions are given in terms of Virasoro characters ofhighest-weight representations. Individually, these characters decompose into a finitenumber of characters of irreducible representations. We show with examples howindecomposable representations arise from fusion.

Quantization of models with non-compact quantum group symmetry: modular XXZ magnet and lattice sinh-Gordon model

We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of U_q(sl_2)) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R-matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called Q-operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous space-time. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures.

Diagonalization of an Integrable Discretization of the Repulsive Delta Bose Gas on the Circle

We introduce an integrable lattice discretization of the quantum system of n bosonic particles on a ring interacting pairwise via repulsive delta potentials. The corresponding (finite-dimensional) spectral problem of the integrable lattice model is solved by means of the Bethe Ansatz method. The resulting eigenfunctions turn out to be given by specializations of the Hall-Littlewood polynomials. In the continuum limit the solution of the repulsive delta Bose gas due to Lieb and Liniger is recovered, including the orthogonality of the Bethe wave functions first proved by Dorlas (extending previous work of C.N. Yang and C.P. Yang).

Zamolodchikov's tetrahedron equation and hidden structure of quantum groups

The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra $U_q(\hat{sl}(n))$, where the rank $n$ coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the $sl(n)$-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2+1 dimensions.

Positive and negative hierarchies of nonlinear integrable lattice models and three integrable coupling systems associated with a discrete spectral problem

Positive and negative hierarchies of nonlinear integrable lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct three integrable coupling systems of the positive hierarchy through enlarging Lax pair method.

Integrals of motion from TBA and lattice-conformal dictionary

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.

An integrable model with a non-reducible three particleR-matrix

We define an integrable lattice model which, in the notation of Yang, in addition to the conventional 2-particle $R$-matrices also contains non-reducible 3-particle $R$-matrices. The corresponding modified Yang-Baxter equations are solved and an expression for the transfer matrix is found as a normal ordered exponential of a (non-local) Hamiltonian.

Entanglement entropy and quantum field theory

We carry out a systematic study of entanglement entropy in relativisticquantum field theory. This is defined as the von Neumann entropySA = −Tr ρAlogρA corresponding to the reduced density matrixρA of a subsystemA. For thecase of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central chargec, we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and whenA consists of an arbitrary number of disjoint intervals. For such asystem away from its critical point, when the correlation lengthξ is large but finite, we show that , where is the number of boundary points ofA.These results are verified for a free massive field theory, which is also used to confirm a scaling ansatzfor the case of finite size off-critical systems, and for integrable lattice models, such as the Ising andXXZ models, which are solvable by corner transfer matrix methods. Finally the free fieldresults are extended to higher dimensions, and used to motivate a scaling form forthe singular part of the entanglement entropy near a quantum phase transition.

Phase diagram and quantum critical behavior of an integrable Kondo lattice model

An integrable Kondo lattice model describing a strongly correlated electron host interacting with a spin-$\frac{1}{2}$ lattice is proposed. It is found that with the variations of the Kondo coupling J, the hole concentration ${n}_{h},$ and the magnetic field H, the system may fall into a variety of phases. The phase boundaries of the ground state are determined exactly. The marginal excitations and the quantum critical behavior at the phase boundaries are discussed.

Integrable lattice models for conjugate A(1)n

A new class of $A^{(1)}_n$ integrable lattice models is presented. These are interaction-round-a-face models based on fundamental nimrep graphs associated with the $A^{(1)}_n$ conjugate modular invariants, there being a model for each value of the rank and level. The Boltzmann weights are parameterized by elliptic theta functions and satisfy the Yang-Baxter equation for any fixed value of the elliptic nome q. At q=0, the models provide representations of the Hecke algebra and are expected to lead in the continuum limit to coset conformal field theories related to the $A^{(1)}_n$ conjugate modular invariants.

Positive and Negative Hierarchies of Integrable Lattice Models Associated with a Hamiltonian Pair

A difference Hamiltonian operator involving two arbitrary constants is presented, and it is used to construct a pair of nondegenerate Hamiltonian operators. The resulting Hamiltonian pair yields two difference hereditary operators, and the associated positive and negative hierarchies of nonlinear integrable lattice models are derived through the bi-Hamiltonian formulation. Moreover, the two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. The use of zero curvature equation leads us to conclude that all resulting integrable lattice models are local and that the integrable lattice models in the positive hierarchy are of polynomial type and the integrable lattice models in the negative hierarchy are of rational type.

Quantum projectors and local operators in lattice integrable models

In the framework of the quantum inverse scattering method, we consider a problem of constructing local operators for one-dimensional quantum integrable models, especially for the lattice versions of the nonlinear Schrodinger and sine-Gordon models. We show that a certain class of local operators can be constructed from the matrix elements of the monodromy matrix in a simple way. They are closely related to the quantum projectors and have nice commutation relations with half of the matrix elements of the elementary monodromy matrix. The form factors of these operators can be calculated by using the standard algebraic Bethe ansatz techniques.

Fused integrable lattice models with quantum impurities and open boundaries

The alternating integrable spin chain and the RSOS( q 1, q 2; p) model in the presence of a quantum impurity are investigated. The boundary free energy due to the impurity is derived, the ratios of the corresponding g functions at low and high temperature are specified and their relevance to boundary flows in unitary minimal and generalized coset models is discussed. Finally, the alternating spin chain with diagonal and non-diagonal integrable boundaries is studied, and the corresponding boundary free energy and g functions are derived.

Classicalr-matrices and the Feigin–Odesskii algebra via Hamiltonian and Poisson reductions

We present a formula for a classical $r$-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the respective gauge transformations are Lie group ones. Our formula is in terms of Dirac brackets, and some new observations on these brackets are made. We apply our method to derive a classical $r$-matrix for the elliptic Calogero-Moser system with spin starting from the Higgs bundle over an elliptic curve with marked points. In the paper we also derive a classical Feigin-Odesskii algebra by a Poisson reduction of some modification of the Higgs bundle over an elliptic curve. This allows us to include integrable lattice models in a Hitchin type construction.

Explicit free parametrization of the modified tetrahedron equation

The modified tetrahedron equation (MTE) with affine Weyl quantum variables at the Nth root of unity is solved by a rational mapping operator which is obtained from the solution of a linear problem. We show that the solutions can be parametrized in terms of eight free parameters and 16 discrete phase choices, thus providing a broad starting point for the construction of three-dimensional integrable lattice models. The Fermat-curve points parametrizing the representation of the mapping operator in terms of cyclic functions are expressed in terms of the independent parameters. An explicit formula for the density factor of the MTE is derived. For the example N = 2 we write the MTE in full detail.

Incommensurate and superconducting phases in an exactly solvable model

An integrable lattice model of strongly interacted electrons with nearest-neighbor and next-nearest-neighbor hoppings is proposed and investigated. The exact ground-state phase diagram is calculated as a function of an on-site interaction, a value of the hopping integral between next-nearest-neighbor lattice sites and a band filling (the nearest-neighbor hopping integral is chosen equal to unity). The model describes incommensurate and superconducting phases which are realized simultaneously in a definite region of interaction parameters and a band filling. The long-distance asymptotics of the density-density and one-particle correlation functions are calculated in the incommensurate phase.

On the quantum inverse scattering problem

A general method for solving the so-called quantum inverse scattering problem (namely the reconstruction of local quantum (field) operators in term of the quantum monodromy matrix satisfying a Yang–Baxter quadratic algebra governed by an R-matrix) for a large class of lattice quantum integrable models is given. The principal requirement being the initial condition ( R(0)= P, the permutation operator) for the quantum R-matrix solving the Yang–Baxter equation, it applies not only to most known integrable fundamental lattice models (such as Heisenberg spin chains) but also to lattice models with arbitrary number of impurities and to the so-called fused lattice models (including integrable higher spin generalizations of Heisenberg chains). Our method is then applied to several important examples like the sl n XXZ model, the XYZ spin- 1 2 chain and also to the spin- s Heisenberg chains.

Renormalization of twist-three operators and integrable lattice models

We address the problem of solution of the QCD three-particle evolution equations which govern the Q2-dependence of the chiral-even quark–gluon–quark and three-gluon correlators contributing to a number of asymmetries at leading order and the transversely polarized structure function g2(xBj). The quark–gluon–quark case is completely integrable in multicolour limit and corresponds to a spin chain with non-periodic boundary conditions, while the pure gluonic sector contains, apart from a piece in the Hamiltonian equivalent to XXX Heisenberg magnet of spin s=−32, a non-integrable addendum which can be treated perturbatively for a bulk of the spectrum except for a few lowest energy levels. We construct a quasiclassical expansion with respect to the total conformal spin of the system and describe fairly well the energy spectra of quark–gluon–quark and three-gluon systems.