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We propose a general framework that leads to one-dimensional XX and Hubbard models in full generality, based on the decomposition of an arbitrary vector space (possibly infinite dimensional) into a direct sum of two subspaces, the two corresponding orthogonal projectors allowing one to define a R-matrix of a universal XX model, and then of a Hubbard model using a Shastry type construction. The QISM approach ensures integrability of the models, the properties of the obtained R-matrices leading to local Hubbard-like Hamiltonians. In all cases, the energies, the symmetry algebras and the scattering matrices are explicitly determined. The computation of the Bethe Ansatz equations for some subsectors of the universal Hubbard theories are determined, while they are fully computed in the XX case. A perturbative calculation in the large coupling regime is also done for the universal Hubbard models.
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We propose a general formula for the group of invertible topological phases on a space $Y$, possibly equipped with the action of a group $G$. Our formula applies to arbitrary symmetry types. When $Y$ is Euclidean space and $G$ a crystallographic group, the term `topological crystalline phases is sometimes used for these phases of matter.
The quantum $tau_2$-model with generic site-dependent inhomogeneity and arbitrary boundary fields is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix are given in terms of an inhomogeneous T-Q relation, which is based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices. Moreover, the associated Bethe Ansatz equations are also obtained.
Based on the inhomogeneous T-Q relation constructed via the off-diagonal Bethe Ansatz, the Bethe-type eigenstates of the XXZ spin-1/2 chain with arbitrary boundary fields are constructed. It is found that by employing two sets of gauge transformations, proper generators and reference state for constructing Bethe vectors can be obtained respectively. Given an inhomogeneous T-Q relation for an eigenvalue, it is proven that the resulting Bethe state is an eigenstate of the transfer matrix, provided that the parameters of the generators satisfy the associated Bethe Ansatz equations.
The Hayden-Preskill protocol is a quantum information theoretic model of the black hole information paradox. Based on the protocol, it was revealed that information scrambling and entanglement lead to an instant leakage of information. In this paper, we study the information paradox with symmetry in the framework of the Hayden-Preskill protocol. Symmetry is an important feature of black holes that induces yet more conceptual puzzles in the regime of quantum gravity. We especially consider an axial symmetry and clarify its consequences in the information leakage problem. Using a partial decoupling approach, we first show that symmetry induces a emph{delay} of information leakage and an emph{information remnant}, both of which can be macroscopically large for certain initial conditions. We then clarify the physics behind the delay and the information remnant. By introducing the concept of emph{clipping of entanglement}, we show that the delay is characterized by thermodynamic properties of the black hole associated with the symmetry. We also show that the information remnant is closely related to the symmetry-breaking of the black hole. These relations indicate the existence of non-trivial microscopic-macroscopic correspondences in the information leakage problem.
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.
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