Pattern-of-Zeroes Models of Quantum Phases of Matter Adam Dai, Alex Meiburg Dos Pueblos High School Mentor: Zhenghan Wang, Microsoft Q Station & UCSB July 12, 2013 Abstract Fractional quantum Hall states can be modeled by symmetric or anti-symmetric polynomials of infinite variables. The pattern-of-zeros approach is a useful tool in analyzing these polynomials. One salient feature of the pattern-of-zeros of the fractional quantum Hall wave functions is their quadratic growth rate. We construct a family of polynomials in which the pattern-of-zeros grows cubically, and exam- ine its behavior. Our numerical simulation indicates that our wave functions achieve maximum values when the electrons align along a certain direction in the plane. We believe that these new polynomial wave functions potentially model crystal states of electrons, and may be useful in understanding the transition between liquid and crystal states of electrons. 1 Introduction We are all familiar with classical states of matter such as solid, liquid, and gas. But quantum states or phases of matter are much more mysterious. Physicists recently discovered an exciting new category of quantum phases of matter—topological insulators. Together with fractional quantum Hall liquids, they form the class of topological phases of matter with potential application to topological quantum computing. It is an interesting question to find theoretical models for these quantum phases of matter, and thus 1