Study On The Lax Pair Operator Method For Special Open Boundary Integrable Spin Chains

The integrable model is essential branch in the field of physics.Its exact solution can clarify some critical physical concepts and provide references for experimental physics.In general,the Yang-Baxter equation guarantees the integrability of models with periodic boundary conditions.For the model with open boundary conditions,the Yang-Baxter equation and the reflection equation proposed by Sklyanin(boundary Yang-Baxter equation)guarantee the integrability.On the other hand,if one can find a Lax pair such that the corresponding Lax equation is equivalent to the equation of motion of the system,then the system is completely integrable in the sense of Lax.For the model with periodic boundary conditions,Izergin and Korepin first gave the Lax pair of the model and express its equation of motion as the equivalent Lax equation.For the model with open boundary conditions,Guan and other physicists proposed a method to prove the integrability of the model using Lax pair of boundary conditions.Furthermore,the left and right boundary K-matrices compatible with the integrability of the model are obtained.They are proved to be the same as the boundary K-matrices obtained by using Sklyanin reflection equation,the feasibility of this method is verified.The Shor-Movassagh spin chain model is obtained by mapping the Motzkin path of quantum random walk to a spin chain.This model has strong quantum entanglement.The fact makes the model have great potential application value in quantum information.We use the Yang-Baxter equation to prove the integrability of the Shor-Movassagh spin chain with periodic boundary conditions when partial interactions are “lost”.We call this integrable model the free Shor-Movassagh spin chain.When proving the integrability of the model with open boundary conditions,it is found that the crossing unitarity of the model is breaking due to the partial transpose of the R-matrix is not invertible.Sklyanin’s reflection equation cannot be used to construct the integrability condition of the model.Therefore,we use the Lax pair operator method to prove its integrability.The Lax pair in the bulk of free Shor-Movassagh open chain is constructed,so the integrability of the model with periodic boundary conditions is proved in the Lax sense.Then the Lax pair on the boundary of the model is obtained,so the integrability of the free Shor-Movassagh open chain is proved.The double-row monodromy matrix and transfer matrix of the model are constructed.The left and right boundary K-matrices compatible with the integrability of the model on the open interval are also obtained.Finally,from the open boundary integrable condition,we derive the open boundary Hamiltonian of the model,and it is more general than the original Hamiltonian.