Vertex Operator Algebra in the XXZ Chain with a Boundary

均 今野, Michio Jimbo, Rinat Kedem +2 more

Takeo Kojima

Tetsuo Deguchi

We discuss a family of operators which commute or anti-commute with the twisted transfer matrix of the six-vertex model at q being roots of unity, q 2 N = 1. The operators commute with the Hamiltonian of the XXZ spin chain under the twisted boundary conditions, and they are also valid for the inhomogeneous case. For the case of the anti-periodic boundary conditions, we show explicitly that the operators generate the sl 2 loop algebra in the sector of the total spin operator S Z ≡ N /2(mod N ). The infinite-dimensional symmetry leads to exponentially-large spectral degeneracies, as shown for the periodic boundary conditions (Deguchi T, Fabricius K and McCoy B M 2001 J. Stat. Phys. 102 701). Furthermore, we derive explicitly the sl 2 loop algebra symmetry for the periodic XXZ spin chain with an odd number of sites in the sector S Z ≡ N /2(mod N ) when q is a primitive N th root of unity with N odd. Interestingly, in the case of N = 3, various conjectures of combinatorial formulae for the XXZ spin chain with odd sites have been given by Stroganov and other authors. We also note a connection to the spectral degeneracies of the eight-vertex model.

Yasuhiro Fujii, Miki Wadati

The bounded XXZ spin chain is studied. The operator appeared in the q -difference equation discovered by Cherednik is modified into the transfer matrix of the model. Its eigenstate is composed by the vertex operators of the quantum affine algebra. By use of a geometric symmetry called the “turning symmetry” that the spin chain without boundary magnetic fields satisfies, the asymptotic form of the eigenstate is found and the spontaneous magnetization at a boundary is calculated. It is roughly twice as large as that in the half-infinite XXZ spin chain.

Ze Tao, Fujun Liu

We formulate a boundary-residual-closure mechanism to systematically construct finite-parameter open XXZ Hamiltonians featuring non-diagonal boundary data. By retaining the standard trigonometric XXZ R -matrix, we promote the auxiliary boundary entries to operator-valued generators and establish the sixteen reflection-residual components as the defining relations of a boundary extension algebra. Reducing this algebraic system to its regular Hermitian scalar sector endogenously generates a non-diagonal six-vertex boundary representative, ensuring that the required amplitude and phase emerge prior to the formulation of the double-row transfer object. Subsequently, we embed this residual-generated representative into a mixed detuned double-row seed. By extracting the regularity-point logarithmic derivative of this composite structure, we derive an explicit Hermitian open XXZ Hamiltonian with a transverse left boundary field. Ultimately, this exact mathematical framework ensures the resulting system strictly recovers the standard diagonal open XXZ chain as its zero-amplitude anchor.