In the last three decades, there have been extraordinary developments in our understanding of topological phases, which challenge traditional condensed matter frameworks in many ways. In contrast to Landau’s symmetry classification of matter, where phases are described by symmetry-breaking order parameters, topological phases are distinguished by nonlocal topological invariants. These phases are fundamentally quantum mechanical with no classical analogue; they are characterized by exotic emergent excitations of the bulk, which are often accompanied by gapless edge degrees of freedoms. Understanding these phases may lead to more energy efficient materials, circuits for spin manipulations, and perhaps fault-tolerant quantum computing.
Simulation of quantum systems
A crucial aspect of numerical modeling is the simulation of quantum systems on a classical computer. For example, such simulator would take in a Hamiltonian as input and produce a phase diagram, thermodynamics properties, or response functions as outputs. For reasonable systems in one-dimensions, the DMRG (density matrix renormalization group) algorithm computes the ground state wavefunction, and subsequently various physical observables. The success of DMRG lies in the ability to efficiently write the quantum ground state wavefunction in terms of classical information, using an ansatz known as a "tensor network state". I am interested extending these ideas and tools to higher dimensions. This will lead to a deeper understanding of phases in higher dimensions, as well tools to simulate new material properties before they are created in the lab.
“Universal topological quantum computation from a superconductor/Abelian quantum Hall heterostructure,” Roger Mong, David J. Clarke, Jason Alicea, Netanel H. Lindner, Paul Fendley, Chetan Nayak, Yuval Oreg, Ady Stern, Erez Berg, Kirill Shtengel, and Matthew P. A. Fisher, Phys. Rev. X 4, 011036 (2014). [arXiv:1307.4403]
“Exact matrix product states for quantum Hall wave functions,” Michael P. Zaletel and Roger Mong, Phys. Rev. B 86, 245305 (2012). [arXiv:1208.4862]
“Quantum Transport and Two-Parameter Scaling at the Surface of a Weak Topological Insulator,” Roger Mong, Jens H. Bardarson, and Joel E. Moore, Phys. Rev. Lett. 108, 076804 (2012). [arXiv:1109.3201]
“Antiferromagnetic topological insulators,” Roger Mong, Andrew M. Essin, and Joel E. Moore, Phys. Rev. B 81, 245209 (2010). [arXiv:1004.1403]