The algebraic Bethe Ansatz is carried out for models (dimer models on a hexagonal lattice) whose matrix of local Boltzmann weights fails to satisfy the Yang–Baxter equation. The operator algebra is constructed instead by a generalizable alternative process using only the coproduct decomposition of the operators. The new technique makes it possible to solve many lattice N-body models whose local Boltzmann weights fail to meet the algebraic requirements of the Yang–Baxter equation.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y
The algebraic Bethe Ansatz is carried out for models (dimer models on a hexagonal lattice) whose matrix of local Boltzmann weights fails to satisfy the Yang–Baxter equation. The operator algebra is constructed instead by a generalizable alternative process using only the coproduct decomposition of the operators. The new technique makes it possible to solve many lattice N-body models whose local Boltzmann weights fail to meet the algebraic requirements of the Yang–Baxter equation.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y
A crash course on the Yang–Baxter equation and its applications in the statistical mechanics of lattice vertex models is presented. Using the simple example of a one-dimensional lattice gas, the basic terminology and standard mathematical procedures of statistical mechanics are illustrated. The coproduct notation is introduced via a discussion of the two-dimensional dimer model, and the transfer matrix formulation is elaborated further. The algebraic Bethe Ansatz (ABA) is introduced in the context of the ice model, leading to the Yang–Baxter equations. The motivation for introducing the ABA is developed, emphasizing the similarity to the quantum oscillator problem, in features such as the generation of “excited states”, and the determination of the eigenvalue spectrum. Finally, these methods from the theory of lattice models are applied to the case of a quantum many-body problem, the Heisenberg chain. From the condition that the monodromy matrices satisfy the Yang–Baxter equation, a complete set of mutually commuting operators is derived, and the Heisenberg chain problem is completely solved.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y
The algebraic Bethe ansatz is one of the main approaches to solving integrable models in statistical mechanics. This thesis is a detailed introduction to aspects of this technique. In chapter 1, following Takhtajan1, we fully explain all preliminaries that are required by this technique in the context of the 8-vertex model, including the solution of the Yang-Baxter equations in terms of Boltzmann weights that are parameterized by elliptic function. In chapter 2, also following Takhtajan, we restrict our attention to the special case of the 6-vertex model, with Boltzmann weights parameterized by hyperbolic trigonometric functions, formulate the model in detail on a finite lattice with periodic boundary conditions, and show how the algebraic Bethe ansatz can be used to characterize the eigenvectors and eigenvalues, in the homogeneous case (all vertical rapidities are equal, and all horizontal rapidities are also equal). In chapter 3, following Bogoliubov et al2, we consider the 6-vertex model in inhomogeneous case (all rapidities are now independenti variables) but with domain wall boundary conditions. Using the algebraic Bethe ansatz, and Izergin’s result for the partition function, we calculate boundary 1-point functions of the six-vertex model on a finite lattice and represent them in determinant form. Takhatajan, L. J. Introduction to Agebraic Bethe Ansatz, Lecture Notes in Physics, Exactly solvable problems in condensed matter and relativistic field theory 2 Boundary correlation functions of the six-vertex model J. Phys. A: Math. Gen. 35, 5525-5541 (2002)
We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model. Specifying the parameters appropriately, we obtain the Bethe ansatz equations of the supersymmetric t-J model, the Hubbard model, or of Yang’s model of electrons with delta interaction. This means that the Bethe ansatz equations of these (and many other) models can be obtained from a common algebraic source, namely from the Yang-Baxter algebra generated by the gl(1|2) invariant R-matrix.
The formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide quantum group representations even when q is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact SU(2)q invariance on the lattice. The algebra of the Yang-Baxter (YB) and SU(2)q generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the SU(2)q generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of SU(2)q. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS(p) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of SU(2)q, at q=eiπ/p, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J
The Yang-Baxter algebras (YBA) are introduced in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Bethe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. This approach permit to define and solve rigorously massive QFT as an appropiate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underly the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter. To conclude, some recent developments in the domain of integrable theories are summarized.
The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA into unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary R matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method on the spin- 12 XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models on 4 sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.