Matrix Product State and X-ray Absorption

The first part of the thesis is dedicated to the study of Matrix Product State (MPS). Initially named "finitely correlated state", MPS is the basis of the Tensor Network State that is widely used to represent quantum many-body wave functions, and is also the essential concept in powerful numerical tools like Density-Matrix Renormalization Group. It is an ansatz that only samples a very small portion of the complete Hilbert space, but can represent ground states of 1-dimensional gapped local Hamiltonians efficiently.

We will first look at the quantum fluctuation of observables in such states. In particular, we consider the Full Counting Statistics of MPS, which is separated into a bulk and boundary term. We identify a central limit theorem like behavior in the limit of large system sizes, and write the corrections of the central limit theorem for a finite size system in terms of the Edgeworth series. We also show that, for special cases of MPSs, like the famous Affleck, Lieb, Kennedy and Tasaki state, because of the topological nature of the state, this description is no longer valid.
Next we look at the time-evolution of an MPS under time-dependent Hamiltonians. We show that in the "injective" case, the Schrodinger equation can be written in terms of the MPS matrices in an interesting way. We show, however, it can never produce an exact time-evolution that changes the entanglement structure of the state.

The second part of the thesis focuses on X-ray absorption and scattering. We are particularly interested in the resonant inelastic X-ray scattering (RIXS), which has been rapidly developing recently. However, there is still an on-going debate about whether the RIXS spectrum of cuprate systems shows collective or quasi-particle physics. We first use a simple perturbation method to get a basic idea about the physics of the process. Then we use a determinant method and the quasi-particle picture to study the RIXS response of a variety of cuprate systems, like CLBLCO, YBCO, and Bi-2201, and find excellent agreement with experiments. We also develop a method to account for the superconducting gap within a meanfield approach and study it in detail for a model of p-wave superconductor. Although the gap is small compared to other band structure parameters, the effect of the gap is rather significant.