## The Bethe ansatz for the six-vertex and XXZ models: an exposition.(English)Zbl 1430.60080

Summary: In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $$\psi$$ and energy $$\Lambda$$, which satisfy $$V\psi =\Lambda\psi$$, where $$V$$ is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $$a=b=1$$ and $$c>0$$. We also show that the same vector $$\psi$$ satisfies $$H\psi =E\psi$$, where $$H$$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $$E$$ computed explicitly.
Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [“Discontinuity of the phase transition for the planar random-cluster and Potts models with $$q>4$$”, Preprint, arXiv:1611.09877] that amounts to proving that the random-cluster model on $$\mathbb{Z}^2$$ with cluster weight $$q>4$$ exhibits a first-order phase transition.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B23 Exactly solvable models; Bethe ansatz 82B26 Phase transitions (general) in equilibrium statistical mechanics

### Keywords:

transfer matrix; Bethe ansatz; six vertex model; XXZ model
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### References:

  R. J. Baxter. Partition function of the eight vertex lattice model. Annals Phys., 70:193-228, 1972. · Zbl 0236.60070  R. J. Baxter. Exactly solved models in statistical mechanics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1989. Reprint of the 1982 original. · Zbl 0723.60120  H. Bethe. Zur Theorie der Metalle I. Eigenwerte und Eigenfunktionen der Hnearen Atomkette. Zeitschrift für Physik, 71(3):205-226, 1931. · JFM 57.1587.01  N. M. Bogoliubov, A. G. Izergin, and V. E. Korepin. Quantum inverse scattering method and correlation functions. Cambridge university press, 1997. · Zbl 0787.47006  H. Duminil-Copin, M. Gagnebin, M. Harel, I. Manolescu, and V. Tassion. Discontinuity of the phase transition for the planar random-cluster and Potts models with $$q>4$$. Preprint, https://arxiv.org/abs/1611.09877, 2016. · Zbl 1430.60080  M. Jimbo and T. Miwa. Algebraic Analysis of Solvable Lattice Models. American Mathematical Soc., 1994. · Zbl 0828.17018  M. Karbach, K. Hu, and G. Müller. Introduction to the Bethe ansatz II. Computers in Physics, 12:565-573, 1998.  M. Karbach and G. Müller. Introduction to the Bethe ansatz I. Computers in Physics, 11:36-43, 1997.  E. H. Lieb. Residual entropy of square ice. Physical Review, 162(1):162, 1967. · Zbl 1352.82008  N. Reshetikhin. Lectures on the integrability of the 6-vertex model. https://arxiv.org/abs/1010.5031, 2010. · Zbl 1202.82022  C. N. Yang and C. P. Yang. One-dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe’s hypothesis for ground state in a finite system. Phys. Rev., 150:321-327, 1966. · Zbl 1202.82022  C. N. Yang and C. P. Yang. One-dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe’s hypothesis for ground state in a finite system. Phys. Rev., 150:321-327, 1966.
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