Yang-Baxter and other relations for free-fermion and Ising models

Gao-Yuan Chiang, Chin-Rong Lee, Hsiang-nan Li

Masahiro Shiroishi, Miki Wadati

We present a new example of the tetrahedral Zamolodchikov algebra which is related to the six-vertex free-fermion model. As an application of the tetrahedral Zamolodchikov algebra, we obtain a new solution of the Yang-Baxter equation, which is a generalization of the ] R -matrix of the one-dimensional Hubbard model.

Bo-Yu Hou, Wen‐Li Yang

Akash Sinha, Somnath Maity, Pramod Padmanabhan +1 more

We introduce a one dimensional spin $\frac{1}{2}$ Hamiltonian with multi-site interactions, but still local. The algebra of its Hamiltonian densities resembles that of the transverse field Ising model. Using this fact we show that its spectrum is free-fermionic but with a huge degeneracy for each level. The source of the degeneracy is a set of local conserved quantities that act like a classical background field for the quantum system. The thermodynamics of this system is contrasted with the standard Ising model. At the gapless points in the energy spectrum, we show that this system can be derived from the quantum inverse scattering method adapted to a multi-site generalization of the Yang-Baxter equation as introduced by E. Rowell and Z. Wang. The $R$-matrix is constructed using generators of extraspecial 2-groups. This helps us extract all the conserved charges and lay the framework for a general mechanism to generate such multi-site interaction spin systems that are transverse field Ising models under the hood. A remark on how to obtain P. Fendley's free-fermion in disguise models in this formalism is also included.

S. Dotsenko, Alexander M. Polyakov

Daniel Bump, Slava Naprienko

A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid $\mathfrak{G}$ together with a map $π:\mathfrak{G}\to\operatorname{End}(V\otimes V)$ for some vector space $V$ such that the Yang-Baxter commutator $[[ π(u),π(w),π(v)]]=0$ if $u,v\in\mathfrak{G}$ are such that the groupoid composition $w=u\star v$ is defined. An important role is played by an object map $Δ:\mathfrak{G}\to M$ for some set $M$ such that $Δ(u)=Δ(v')$, $Δ(w)=Δ(v)$ and $Δ(w')=Δ(u')$, where $v\mapsto v'$ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group $\operatorname{GL}(2)\times\operatorname{GL}(1)$. For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.