## Research

Currently, I am working with Natan Andrei on the nonequilibrium dynamics of exactly solvable 1+1 dimensional problems. These are brief summaries of the problems I worked on or currently working on. Feel free to contact me for the notes and/or discussion. I will be in the March meeting 2017 and hopefully will send a note on arXiv for the Kondo problem dynamics (second paragraph below).

**Massive Thirring Dynamics:**There is an approach to find the exact dynamics of integrable models for open systems. See this and references therein. The main idea is to extend the sum over bound state solutions (strings) to continuous complex plane and try to define contours in a clever way to capture the poles due to the bound states. I am following this approach to solve the open system dynamics for massive Thirring model or Interacting Resonant Level (IRL) model. IRL can be mapped to a special case of massive Thirring model. XYZ is the spin model one gets from massive Thirring and sine-Gordon model is the bosonized version of the massive Thirring. One interesting physical problem one can solve is the Mott insulator to superfluid quench problem which can be realized in cold atom systems. In other words, what happens if ones starts with the ground state in Mott insulator phase and evolve it with the Hamiltonian with parameters in the superfluid phase.**Kondo Dynamics:**One can also set up the Kondo Hamiltonian using the Yudson approach above. One simplification that happens for the Kondo model is the fact that the "bare" S-matrix is independent of the "bare" momenta. So one can write the time evolution of an arbitrary initial state as a symmetric sum over orderings. Now using this formalism we are trying to calculate quench dynamics observables. One interesting observable is the spin impurity as a function of time. Consider the impurity spin was decoupled and set to spin up and at time zero we turn on the Kondo interaction. Also one can prepare other initial conditions using magnetic field. Now the main difficulty is that the spectrum is set to be linear and one needs to fill the Fermi sea. This means we need an antisymmetric sum over permutations and the symmetric sum over orderings that we got, which is about e(N!)^2 terms. At this moment, we are trying to see how one can group these sums into determinants (and elementary symmetric sums). This can prove that problem is computable and then we can study the universality limit computationally.**Lieb-Liniger Gas in Harmonic Potential:**Usually studying the Lieb-Liniger (LL) gas experimentally involves including a harmonic potential. Classically speaking, or using mean field, the nonlinear Schrodinger (NS) equation must be switched with the more general Gross-Pitaevski (GP) equation. We know that exact solutions exist for the LL model. The question is how to solve the LL model in the presence of a harmonic potential. Classically, using Painleve integrability and gauge transformations, one can map the solutions of NS to the solutions of some special forms of GP equation. We want to understand if similar transformations can be understood in Bethe ansatz language. This article gives a quick introduction to how general NS equation can get, preserving either Painleve integrability or "complete" integrability conditions. Another related question is the correspondence between the classical bright and dark soliton solutions to the repulsive and interacting LL solutions. For bright solitons, this is a somewhat resolved problem. But overall one needs more details so to understand the corresponding extensions of LL model.**Integrability of Spin-Orbital Kondo:**Motivated by the study of the Hund's metals, one can consider an impurity model like Kondo, but considering both the spin and orbital degrees of freedom. See this article for the Hamiltonian and an RG analysis. The question is if SU(N)xSU(M) model is exactly solvable by Bethe ansatz. By checking the solutions for the Yang-Baxter equation (YBE), seems like there is an easy proof that there is no solutions beyond the known rational solutions for SU(MN) or separable orbital SU(M) and spin part SU(N). This is also proven for the Heisenberg magnet model (bulk). A direction to follow might be the application of colored YBE and possible solutions in this setting. There are some big work to be done in this direction though. Formulating an extended algebraic Bethe ansatz to systematically write the Bethe ansatz equations being one of the challenges.