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Quantum Entanglement and Topological Phases of Matter

Final Report Summary - QETPM (Quantum Entanglement and Topological Phases of Matter)

Summary of Objectives

This project has as its main objective the investigation of how to generate and classsify new topological phases of matter in lattice systems, and as a secondary pbjective to further the identification of opological orer using numerical methods e.g. based on entanglement entropy.

Description of Work

We have focused our efforts on systems of particles hopping on a lattice in the presence of a magnetic field. Such systems are realized both in ultrcold atomic gases and are also of great interest to the theoretical desciption of a class of materials called topological insulators. We studied multiparticle systems with various types of interactions in this setting using both analytical and numerical methods. Analytical work focused on the application of conformal field theory to the contruction of trial wave functions and the description of the excitation spectra of these systems as well as on the identification and study of exactly solvable Hamiltonians which can describe the various phases. Numerical work focuse on exact diagonalization of the Hamiltonian of the system away from the solvable points and calculating overlaps between variational and exact wave functions for small systems, as well as various characteristic features of the excitation spectrum (notably its angular momentum structure).

Main Results
1 – Establishing a rigorous relation between the non-interacting Hofstadter problem (particles hopping on a lattice in a magnetic field) and topological insulators with energy bands of higher Chern numbers. E.g. we can identify a correspondence between basis states for these systems. This has been completed for the case of Chern number C = 2 and have taken some steps towards proving it for general C.

2 – Identification and classification of multi-component topological phases in the interacting system based on Chern number and the type of interaction. In a previous work, together with other collaborators the researcher had identified a bi-layer fractional quantum Hall state for interacting particles in a band of Chern number C = 2. We examined the possibility of generalizing this state to particles in bands with higher Chern numbers, leading to multi-layer generalizations of the fractional quantum Hall states. We constructed Hamiltonians for such multi-layer states, taking inspiration from ultracold bose gases on optical lattices, with interactions and umklapp scattering. We have produced predictions for topological orders of these multi-layer states, starting from the order of a system with non-interacting layers, using the mechanism of topological phase transitions known as topological Bose condensation.

3 – Construction of exactly solvable short range interaction Hamiltonians for two points in the phase diagram of the two-layer system and exact determination of the zero energy eigenstates and order at these points. One of the points is representative of a topologically ordered phase, the other is gapless but nevertheless displays very interesting structure in its spectrum, related to conformal field theory.

4 – Confirmation that the topological phase in point 3 can be described either in terms of stong interlayer 3-body interactions with weak interlayer tunneling or in terms of 2-body intra-layer interactions with strong interlayer pair tunneling.

5 – Numerical confirmation that the phase referred to in points 3 and 4 is stable against various perturbations.

6 – Establishing a conformal field theory description of the class of states mentioned above, which allows one to determine the properties of the excitations of these states.

Expected Final Results and their potential Impact and Use

We expect the final scientific results of this project to be an important contribution to the understanding of the phases of strongly interacting many particle systems on a lattice in the presence of a strong magnetic field, or in bands with nontrivial Chern numbers. This will be of great interest to scientist working on ultracold bose systems on optical lattices, interacting topological insulators and on a host of two dimensional materials, notably graphene.

Knowledge of for example the conformal field theories describing the various proposed phases is essential in devising methods that can be used to identify these states in an experimental set up.
A number of the materials to which our work is relevant have many promising (future) applications. E.g. graphene is mentioned a.o. in connection with bioengineering, optical electronic, energy production (photovoltaics) and storage (batteries, capacitors).
Further impact will be achieved through Dr. Hormozi's association with NUI Maynooth and through the many contacts she has made in Europe during the fellowship, as she progresses into permanent academic employment.

Website for information related to the project:

Further enquiries can be directed in the first instance to
Dr. J.K.Slingerland
Department of Mathematical Physics
National University of Ireland Maynooth