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Register a new userGround state solutions of inhomogeneous Bethe equations
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The distribution of Bethe roots, solution of the inhomogeneous Bethe equations, which characterize the ground state of the periodic XXX Heisenberg spin-$frac{1}{2}$ chain is investigated. Numerical calculations shows that, for this state, the new inhomogeneous term does not contribute to the Baxter T-Q equation in the thermodynamic limit. Different families of Bethe roots are identified and their large N behaviour are conjectured and validated.
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The transfer-matrix eigenvalues of the isotropic open Heisenberg quantum spin-1/2 chain with non-diagonal boundary magnetic fields are known to satisfy a TQ-equation with an inhomogeneous term. We derive here a discrete Wronskian-type formula relating a solution of this inhomogeneous TQ-equation to the corresponding solution of a dual inhomogeneous TQ-equation.
For a given polynomial $V(x)in mathbb C[x]$, a random matrix eigenvalues measure is a measure $prod_{1leq i<jleq N}(x_i-x_j)^2 prod_{i=1}^N e^{-V(x_i)}dx_i$ on $gamma^N$. Hermitian matrices have real eigenvalues $gamma=mathbb R$, which generalize to $gamma$ a complex Jordan arc, or actually a linear combination of homotopy classes of Jordan arcs, chosen such that integrals are absolutely convergent. Polynomial moments of such measure satisfy a set of linear equations called loop equations. We prove that every solution of loop equations are necessarily polynomial moments of some random matrix measure for some choice of arcs. There is an isomorphism between the homology space of integrable arcs and the set of solutions of loop equations. We also generalize this to a 2-matrix model and to the chain of matrices, and to cases where $V$ is not a polynomial but $V(x)in mathbb C(x)$.
We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in defining (minimal fundamental) and adjoint representations. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T^{otimes 2} in two cases, when T is the defining and the adjoint representation. In the case when T is the defining representation, these projectors and the split Casimir operator are used to explicitly write down invariant solutions of the Yang-Baxter equations. In the case when T is the adjoint representation, these projectors and characteristic identities are considered from the viewpoint of the universal description of the simple Lie algebras in terms of the Vogel parameters.
We consider closed XXX spin chains with broken total spin $U(1)$ symmetry within the framework of the modified algebraic Bethe ansatz. We study multiple actions of the modified monodromy matrix entries on the modified Bethe vectors. The obtained formulas of the multiple actions allow us to calculate the scalar products of the modified Bethe vectors. We find an analog of Izergin-Korepin formula for the scalar products. This formula involves modified Izergin determinants and can be expressed as sums over partitions of the Bethe parameters.
We consider $mathfrak{gl}_2$-invariant quantum integrable models solvable by the algebraic Bethe ansatz. We show that the form of on-shell Bethe vectors is preserved under certain twist transformations of the monodromy matrix. We also derive the actions of the twisted monodromy matrix entries onto twisted off-shell Bethe vectors.
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