A one-dimensional Bose-Hubbard model with unidirectional hopping is shown to be exactly solvable. Applying the algebraic Bethe ansatz method, we prove the integrability of the model and derive the Bethe ansatz equations. The exact eigenvalue spectrum can be obtained by solving these equations. The distribution of Bethe roots reveals the presence of a superfluid-Mott insulator transition at the ground state, and the critical point is determined. By adjusting the boundary parameter, we demonstrate the existence of a non-Hermitian skin effect even in the presence of interaction, but it is completely suppressed for the Mott insulator state in the thermodynamical limit. Our result represents a new class of exactly solvable non-Hermitian many-body systems, which has no Hermitian correspondence and can be used as a benchmark for various numerical techniques developed for non-Hermitian many-body systems.

An analytic method is proposed to compute the surface energy and elementary excitations of the XXZ spin chain with generic nondiagonal boundary fields. For the gapped case, in some boundary parameter regimes the contributions of the two boundary fields to the surface energy are nonadditive. Such a correlation effect between the two boundaries also depends on the parity of the site number $N$ even in the thermodynamic limit $N\to \infty$. For the gapless case, contributions of the two boundary fields to the surface energy are additive due to the absence of long-range correlation in the bulk. Although the $U\left(1\right)$ symmetry of the system is broken, exact spinonlike excitations, which obviously do not carry spin-$\frac{1}{2}$, are observed. The present method provides a universal procedure to deal with quantum integrable systems either with or without $U\left(1\right)$ symmetry.

A novel Bethe ansatz scheme is proposed to calculate physical properties of quantum integrable systems without $U\left(1\right)$ symmetry. As an example, the antiperiodic XXZ spin chain, a typical correlated many-body system embedded in a topological manifold, is examined. Conserved “momentum” and “charge” operators are constructed despite the absence of translational invariance and $U\left(1\right)$ symmetry. The ground state energy and elementary excitations are derived exactly. It is found that two intrinsic fractional (one half) zero modes accounting for the double degeneracy exist in the eigenstates. The elementary excitations show quite a different picture from that of a periodic chain. This method can be applied to other quantum integrable models either with or without $U\left(1\right)$ symmetry.

We study the topological properties of the one-dimensional non-Hermitian Kitaev model with complex either periodic or quasiperiodic potentials. We obtain the energy spectrum and the phase diagrams of the system by using the transfer matrix method as well as the topological invariant. The phase transition points are given analytically. The Majorana zero modes in the topological nontrivial regimes are obtained. Focusing on the quasiperiodic potential, we obtain the phase transition from the topological superconducting phase to the Anderson localization, which is accompanied with the Anderson localization–delocalization transition in this non-Hermitian system. We also find that the topological regime can be reduced by increasing the non-Hermiticity.

Finding out root patterns of quantum integrable models is an important step to study their physical properties in the thermodynamic limit. Especially for models without U(1) symmetry, their spectra are usually given by inhomogeneous T − Q relations and the Bethe root patterns are still unclear. In this paper with the antiperiodic XXZ spin chain as an example, an analytic method to derive both the Bethe root patterns and the transfer-matrix root patterns in the thermodynamic limit is proposed. Based on them the ground state energy and elementary excitations in the gapped regime are derived. The present method provides an universal procedure to compute physical properties of quantum integrable models in the thermodynamic limit.

We investigate the thermodynamic limit of the one-dimensional ferromagnetic XXZ model with twisted (or antiperiodic ) boundary condition. It is shown that the distribution of the Bethe roots of the inhomogeneous Bethe Ansatz equations (BAEs) for the ground state as well as for the low-lying excited states satisfy the string hypothesis, although the inhomogeneous BAEs are not in the standard product form which has made the study of the corresponding thermodynamic limit nontrivial. We also obtain the twisted boundary energy induced by the non-trivial twisted boundary conditions in the thermodynamic limit.