Deep inelastic scattering (DIS) samples a part of the wave function of a hadron in the vicinity of the light cone. Lipatov constructed a spin chain which describes the amplitude of DIS in leading logarithmic approximation. Kharzeev and Levin proposed the entanglement entropy as an observable in DIS [ Phys. Rev. D 95, 114008 (2017)], and suggested a relation between the entanglement entropy and parton distributions. Here we represent the DIS process as a local quench in Lipatov’s spin chain and study the time evolution of the produced entanglement entropy. We show that the resulting entanglement entropy depends on time logarithmically, $\mathcal{S}(t)=1/3\ln (t/\tau )$ with $\tau =1/m$ for $1/m\le t\le (mx{)}^{-1}$, where $m$ is the proton mass and $x$ is the Bjorken $x$. The central charge $c$ of Lipatov’s spin chain is determined here to be $c=1$; using the proposed relation between the entanglement entropy and parton distributions, this corresponds to the gluon structure function growing at small $x$ as $xG(x)\sim 1/x^{1/3}$.

The Izergin-Korepin model with general non-diagonal boundary terms, a typical integrable model beyond A-type and without U(1)-symmetry, is studied via the offdiagonal Bethe ansatz method. Based on some intrinsic properties of the R-matrix and the K-matrices, certain operator product identities of the transfer matrix are obtained at some special points of the spectral parameter. These identities and the asymptotic behaviors of the transfer matrix together allow us to construct the inhomogeneous T − Q relation and the associated Bethe ansatz equations. In the diagonal boundary limit, the reduced results coincide exactly with those obtained via other methods.

The ground state of the Shor–Movassagh chain can be analytically described by the Motzkin paths. There is no analytical description of the excited states. The model is not solvable. We prove the integrability of the model without interacting part in this paper (free Shor–Movassagh). The Lax pair for the free Shor–Movassagh open chain is explicitly constructed. We further obtain the boundary K -matrices compatible with the integrability of the model on the open interval. Our construction provides a direct demonstration for the quantum integrability of the model, described by the Yang–Baxter algebra. Because the partial transpose of the R matrix is not invertible, the model does not have crossing unitarity and the integrable open chain cannot be constructed by the reflection equation (boundary Yang–Baxter equation).

The off-diagonal Bethe ansatz method is generalized to the integrable model associated with thesp(4) (orC₂) Lie algebra. By using the fusion technique, we obtain the complete operator product identities among the fused transfer matrices. These relations, together with some asymptotic behaviors and values of the transfer matrices at certain points, enable us to determine the eigenvalues of the transfer matrices completely. For the periodic boundary condition case, we recover the sameT−Qrelations obtained via conventional Bethe ansatz methods previously, while for the off-diagonal boundary condition case, the eigenvalues are given in terms of inhomogeneousT−Qrelations, which could not be obtained by the conventional Bethe ansatz methods. The method developed in this paper can be directly generalized to genericsp(2n) (i.e.,C_n) integrable model.

We investigate the thermodynamic limit of the exact solution, which is given by an inhomogeneous T − Q relation, of the one-dimensional supersymmetric t − J model with unparallel boundary magnetic fields. It is shown that the contribution of the inhomogeneous term at the ground state satisfies the L⁻¹ scaling law, where L is the system-size. This fact enables us to calculate the surface (or boundary) energy of the system. The method used in this paper can be generalized to study the thermodynamic limit and surface energy of other models related to rational R-matrices.