About Me
Hello! I am Nan Cheng, currently a Ph.D candidate in physics at University of Michigan, Ann Arbor. I received my Bachelor's degree (with highest honor) in physics from Fudan University in 2019.
My research interests include condensed matter physics and statistical physics. I enjoy collaborating on projects related to systems with novel symmetries, self-assembly problems, elasticity, and non-Hermitian physics.
Tel: +1-7348340799
Email: nancheng@umich.edu
Google Scholar: My google scholar site
Office: 2247 Randall Laboratory, 450 Church Street, Ann Arbor, MI 48109
CV
You can view or download my CV here: View CV (PDF).
Projects
Here is a brief overview of the projects I've worked on:
- Band Theory and Boundary Modes of High-Dimensional Representations of Infinite Hyperbolic Lattices: Periodic lattices in hyperbolic space are characterized by symmetries beyond Euclidean crystallographic groups, offering a new platform for classical and quantum waves, demonstrating great potentials for a new class of topological metamaterials. One important feature of hyperbolic lattices is that their translation group is nonabelian, permitting high-dimensional irreducible representations (irreps), in contrast to abelian translation groups in Euclidean lattices. In this work, I generalized the concept of Bravais lattice to hyperbolic space using an algebraic correspondence between translation group and unit cells. Then I introduced a new hyperbolic Bloch's theorem to hyperbolic Bravais lattices with a focus on high dimensional irreps. I also discussed its implications on unusual mode-counting and degeneracy, as well as bulk-edge correspondence in hyperbolic lattices.
- Universal Spectral Moment Theorem and Its Applications in Non-Hermitian Systems: The high sensitivity of the spectrum and wavefunctions to boundary conditions, termed the non-Hermitian skin effect, represents a fundamental aspect of non-Hermitian systems. While it endows non-Hermitian systems with unprecedented physical properties, it presents notable obstacles in grasping universal properties that are robust against microscopic details and boundary conditions. In this work, I introduced a pivotal theorem: in the thermodynamic limit, for any non-Hermitian systems with finite-range interactions, all spectral moments are invariant quantities, independent of boundary conditions, posing strong constraints on the spectrum. Based on this theorem, I introduced new phases termed bulk dispersive phase and bulk proliferative phase in non-Hermitian band systems.
- Geometrically frustrated self-assembly of hyperbolic crystals from icosahedral nanoparticles: Geometric frustration is a fundamental concept in various areas of physics, and its role in self-assembly processes has recently been recognized as a source of intricate self-limited structures. In this work, I presented an analytic theory of the geometrically frustrated self-assembly of regular icosahedral nanoparticle based on the non-Euclidean crystal {3,5,3} formed by icosahedra in hyperbolic space. I found a possible ground state of these nanoparticles in the {3,5,3} crystal by an argument based on the geometrical interpretation of the gaussian curvature and numerically embedded that ground state in hyperbolic space H3 into Euclidean space E3.
- Backscattering-free edge states below all bands in two-dimensional auxetic media: Unidirectional and backscattering-free propagation of sound waves is of fundamental interest in physics and highly sought-after in engineering. Current strategies utilize topologically protected chiral edge modes in bandgaps, or complex mechanisms involving active constituents or nonlinearity. In this work, my collaborators and I proposed a new class of passive, linear, one-way edge states based on spin-momentum locking of Rayleigh waves in two-dimensional media in the limit of vanishing bulk modulus, which provides perfect unidirectional and backscattering-free edge propagation immune to any edge roughness at a broad range of frequencies instead of residing in gaps between bulk bands. I further showed that such modes are characterized by a new topological winding number that protects the linear momentum of the wave along the edge.
My Notes
Here are the notes I wrote during my Ph.D:
- The geometric interpretation of Berry Phase: This note introduces Berry phase from a geometric point of view. There are serval advantage of this point of view. First, the fact that the total Chern number of all bands must be zero is trivial from this point of view. Second, the gauge invariance of Berry phase is also trivial from this point of view. Third, from this point of view, the formula for computing non-abelian Berry phase is a natural extension for the abelian one. In this note, we first briefly introduce tangent spaces, differential forms and vector bundles. Then we explain geometrically parallel transport, covariant derivatives, sectional curvature and holonomy angles. We show that Berry connection is an induced connection of some natural connection on some natural vector bundles, Berry curvature is the curvature of Berry connection, and Berry phase is the holonomy angle. We further recover the formula computing (non-abelian) Berry phase and Chern numbers using differential forms.
- Linear Elasticity: This note shows step by step how the assumptions of homogeneity and isotropy together with the fact that translation and rotations don't cost energy constrain on the Lagrangian for linear elasticity to the form we are familiar with. We further derive constitute relation between strain and stress from the Lagrangian.
- Symmetries in Quantum Mechanics: This note is originally written for my math friends introducing to them symmetries in quantum mechanics. Mathematically speaking, symmetrics are operations keeping certain object invariant. In this note, we define symmetries in quantum mechanics as operations preserving the solution of Schrödinger equation plus two other technical conditions. From this new definition, we show that symmetries are in one to one correspondence with unitary or anti-unitary operators commuting with the Hamiltonian. Then we introduce spatial symmetry, time reversal symmetry, and some basic representation theory. Finally, we introduce spin and explain why in classical mechanics we use SO(3) for angular momentum but in quantum mechiancs we use SU(2) for spin angular momentum.
- ADM Mass: This note introduces ADM mass, momentum and angular momentum for asymptotically flat spacetime.