Spin chains with boundary inhomogeneities

We investigate the effect of introducing a boundary inhomogeneity in the transfer matrix of an integrable open quantum spin chain. We find that it is possible to construct a local Hamiltonian, and to have quantum group symmetry. The boundary inhomogeneity has a profound effect on the Bethe ansatz solution.

Thermodynamic limit of the spin-$$ \frac{1}{2} $$ XYZ spin chain with the antiperiodic boundary condition

Based on its off-diagonal Bethe ansatz solution, we study the thermodynamic limit of the spin-$$ \frac{1}{2} $$ 1 2 XYZ spin chain with the antiperiodic boundary condition. The key point of our method is that there exist some degenerate points of the crossing parameter ηm,l, at which the associated inhomogeneous T − Q relation becomes a homogeneous one. This makes extrapolating the formulae deriving from the homogeneous one to an arbitrary η with O(N−2) corrections for a large N possible. The ground state energy and elementary excitations of the system are obtained. By taking the trigonometric limit, we also give the results of antiperiodic XXZ spin chain within the gapless region in the thermodynamic limit, which does not have any degenerate points.

Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line

We consider the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height h(x,t)h(x,t) on the positive half line with boundary condition \partial_x h(x,t)|_{x=0}=A∂xh(x,t)|x=0=A. It is equivalent to a continuum directed polymer (DP) in a random potential in half-space with a wall at x=0x=0 either repulsive A>0A>0, or attractive A<0A<0. We provide an exact solution, using replica Bethe ansatz methods, to two problems which were recently proved to be equivalent [Parekh, arXiv:1901.09449]: the droplet initial condition for arbitrary A \geqslant -1/2A≥−1/2, and the Brownian initial condition with a drift for A=+\inftyA=+∞ (infinite hard wall). We study the height at x=0x=0 and obtain (i) at all time the Laplace transform of the distribution of its exponential (ii) at infinite time, its exact probability distribution function (PDF). These are expressed in two equivalent forms, either as a Fredholm Pfaffian with a matrix valued kernel, or as a Fredholm determinant with a scalar kernel. For droplet initial conditions and A> - \frac{1}{2}A>−12 the large time PDF is the GSE Tracy-Widom distribution. For A= \frac{1}{2}A=12, the critical point at which the DP binds to the wall, we obtain the GOE Tracy-Widom distribution. In the critical region, A+\frac{1}{2} = \epsilon t^{-1/3} \to 0A+12=ϵt−1/3→0 with fixed \epsilon = \mathcal{O}(1)ϵ=𝒪(1), we obtain a transition kernel continuously depending on \epsilonϵ. Our work extends the results obtained previously for A=+\inftyA=+∞, A=0A=0 and A=- \frac{1}{2}A=−12.

Thermodynamics of the Repulsive Lieb–Liniger Model

This chapter discusses how the Bethe ansatz solution of the one-dimensional Bose gas with repulsive δ‎-function interaction is extended to finite temperatures, the thermody- namic Bethe ansatz. The excitations of this system consist of particle and hole excitations, which can be described by the corresponding densities of Bethe ansatz roots. It shows how these Bethe ansatz root densities are used to define an appropriate expression for the entropy of the system of Bose particles, which is the main ingredient for the extension of the Bethe ansatz method to finite temperature.

The propagator of the finite XXZ spin-$\tfrac{1}{2}$ chain

We derive contour integral formulas for the real space propagator of the spin-\tfrac1212 XXZ chain. The exact results are valid in any finite volume with periodic boundary conditions, and for any value of the anisotropy parameter. The integrals are on fixed contours, that are independent of the Bethe Ansatz solution of the model and the string hypothesis. The propagator is obtained by two different methods. First we compute it through the spectral sum of a deformed model, and as a by-product we also compute the propagator of the XXZ chain perturbed by a Dzyaloshinskii-Moriya interaction term. As a second way we also compute the propagator through a lattice path integral, which is evaluated exactly utilizing the so-called FF-basis in the mirror (or quantum) channel. The final expressions are similar to the Yudson representation of the infinite volume propagator, with the volume entering as a parameter. As an application of the propagator we compute the Loschmidt amplitude for the quantum quench from a domain wall state.

Mobile impurities in integrable models

We use a mobile impurity or depleton model to study elementary excitations in one-dimensional integrable systems. For Lieb-Liniger and bosonic Yang-Gaudin models we express two phenomenological parameters characterising renormalised interactions of mobile impurities with superfluid background: the number of depleted particles, NN and the superfluid phase drop \pi JπJ in terms of the corresponding Bethe Ansatz solution and demonstrate, in the leading order, the absence of two-phonon scattering resulting in vanishing rates of inelastic processes such as viscosity experienced by the mobile impurities.

Completeness of the Bethe states for the rational, spin-1/2 Richardson-Gaudin system

Establishing the completeness of a Bethe Ansatz solution for an exactly solved model is a perennial challenge, which is typically approached on a case by case basis. For the rational, spin-1/2 Richardson–Gaudin system it will be argued that, for generic values of the system’s coupling parameters, the Bethe states are complete. This method does not depend on knowledge of the distribution of Bethe roots, such as a string hypothesis, and is generalisable to a wider class of systems.

Lie Symmetry and the Bethe Ansatz Solution of a New Quasi-Exactly Solvable Double-Well Potential

We study the Schrödinger equation with a new quasi-exactly solvable double-well potential. Exact expressions for the energies, the corresponding wave functions, and the allowed values of the potential parameters are obtained using two different methods, the Bethe ansatz method and the Lie algebraic approach. Some numerical results are reported and it is shown that the results are in good agreement with each other and with those obtained previously via a different method.