The collective behaviour of a large number of particles with strong interactions can be drastically different from that of a single free particle or those interacting in a few-body system as in particle colliders. Such a striking phenomenon called emergence is accountable for the richness and colourfulness of the macroscopic physical world and life, despite consisting of a handful of elementary particles obeying simple interaction rules. Exactly solvable models of statistical mechanics and quantum many-body physics provide analytically accessible benchmarks for more physically realistic descriptions of condensed matter systems, and are usually valuable complements to our understanding obtained from more generic approaches that rely on perturbative methods or numerical simulation.
One particularly successful such model is the Affleck-Kennedy-Lieb-Tasaki (AKLT) spin chain, which has lead to far-reaching discoveries such as topological order and tensor network. There has been exciting develop in recent years when (more) exact excited states are solved, which forms a spectrum generating algebra that unifies similar structures of other models with exact excited states equidistant in energy. Such a structure can have dynamical implications as certain initial state will periodically revive in time evolution, as recently observed in Rydberg blockade experiments, which is a novel type of ergodicity breaking named quantum many-body scars.
Another related but different type of models is integrable models, the most famous one being the Heisenberg spin chain, solved with Bethe ansatz. Unlike the AKLT chain, which has a frustration-free Hamiltonian, meaning the ground state is given by the common lowest energy eigenstate of each local interaction term, the Bethe Ansatz solution is believed to cover the complete spectrum of spin-1/2 Heisenberg model, or the anisotropic XXZ chain. In the higher spin or SU(N) counterpart, however, breaking the symmetry violates the Yang-Baxter equation of scattering matrix and the resulting model is only partially integrable, which also has potential dynamical implications of slow thermalisation, and provides yet another example of violation of the eigenstate thermalisation hypothesis.
The aim of this project is to construct new partially solvable or integrable one-dimensional lattice models that "interpolate" between the two types of models described above: one with sub-spectrum identified from a bottom-up approach by acting spectrum generating operators on top of the solvable ground state, the other from a top-down approach by breaking integrability and examine which integrable eigenstates survive. A key reference (arXiv: 2309.13911) of pioneering work in this direction generalised the frustration-free condition and combined the matrix product state (MPS) solution with Bethe ansatz into a matrix product ansatz solution. We will try to further generalise these results to obtain a wider class of partially solvable models, potentially apply the developed technique to the Motzkin spin chains, and better understand the big picture landscape of solvability and integrability.
The student will have the opportunity to acquire first-hand knowledge of matrix product states, tensor network, Bethe ansatz, and q-deformation, develop analytic skills of solving complicated physics problems with simple mathematical tools, and with the help of Mathematica. Depending on the preparation and ambition of the student, the outcome of the thesis project can either be the characterisation of physical properties of a generalised or deformed known model, such as spectral gap, correlation function, order parameter, and entanglement entropy, or the discovery of new types of (partially) solvable models of his or her own.