Thermodynamics And Bethe States Of The Quantum Integrable Models Without U(1) Symmetry

Off-diagonal Bethe ansatz method is a recently developed method to solve quantum integrable models.Thanks to this new method,quantum integrable models without U(1)symmetry can be solved exactly.Based on the off-diagonal Bethe ansatz method,further properties about some U(1)symmetry broken quantum systems are discussed as follows:First,one dimensional Hubbard model with arbitrary boundary magnetic fields is solved exactly.In chapter 2,by using off-diagonal Bethe ansatz method together with coordinate Bethe ansatz method,we solve the one dimensional Hubbard model with arbitrary magnetic boundary fields exactly.With the coordinate Bethe ansatz method,the second eigenvalue problem is constructed.Then the second eigenvalue eigenvalue problem is transformed to that of XXX spin chain with arbitrary boundary fields which can be solved via the off-diagonal Bethe ansatz method.Second,a method to approach the thermodynamic limit of quantum integrable models that can be exactly solved via off-diagonal Bethe ansatz method is introduced.In chapter 3,as an example,the XXZ spin chain with arbitrary boundary fields is discussed in the thermodynamic limit N ? ?,and the surface energy of this model is derived.For imaginary crossing parameter ? case,it is found that at a sequence of degenerate points of ?,the off-diagonal Bethe ansatz equations reduce to the conventional ones.This kind of reduction allows us to extrapolate the formulae derived from the reduced BAEs to arbitrary ? case with O(N-2)corrections.While for real crossing parameter ? case,for any given one boundary field,say the left one,we can always choose a proper right boundary field to match the constraint c = 0,under which the off-diagonal Bethe ansatz equations also reduce to the conventional ones.And then the surface energy can be exactly solved.Third,Bethe-type states of the XXX spin-12 chain with arbitrary boundary fields is retrieved based on the off-diagonal Bethe ansatz solutions of the model.In chapter 4,a set of orthogonal complete basis in Hilbert space is constructed with the separation of variables(SoV)method,and thus any eigenstate of the model can be decomposed as a linear combination of the basis.Finally,the Bethe state of the model is constructed with the help of off-diagonal Bethe ansatz solutions.