# New Perspectives of Topological Phases of Matter

The topological phase of matter is one of the center topics of modern condensed matter. The classic example is the fractional quantum Hall effect, which instead of being frozen at low temperature, the complete quench of kinetic energies paves the way for strong intrinsic quantum fluctuations and electron-electron correlations, leading to the formation of “incompressible quantum fluid” that exhibit fractionalized quasi-particle, anyon statistics, protected chiral edge modes etc.

My interests in new perspectives of topological quantum matter include:

(1) **New phenomena in conventional materials.** There have been unabated interests in developing new low temperature techniques and transport measurements in studying this phase of matter. For instance, the recent observed unexpected half-quantized Hall conductance points to the possibility of “particle-hole symmetric Pfaffian” state which was previously not appreciated in theoretical and numerical studies. Understanding the stabilities of the many-body states in the Landau level is one of my interests.

(2) **Light-matter coupled system and topology.** When the collective motion of electrons are coupled to quantum light, mixed light-matter excitations called polariton can occur. The physics is particularly interesting in the strong coupling regime where the interplay between light and matter is possible to alter the properties of quantum materials at a fundamental level. For instance, the recent observed modification of Hall conductivity in cavity indicated the possible of modifying the topological quantization of quantum Hall systems in a cavity [*Science* 375, 1030–1034 (2022)]. Its origin, as well as its practical implications, are within my research interest [2].

(3) **High dimensional generalization.** The fractional quantum Hall effect in new systems also include its high dimensional generalization [1], where on the four dimensional CP2 space we found multiple ways of generalizing the famous Laughlin wavefunction, and anomalous counting of quasi-hole excitations which we argue has close connection to the mathematical subject called "commutative algebra in the plane".

(4) **Local correlations and new numerical techniques.** My last specific aim along this direction is about advancing the understandings of local correlations in fractional quantum Hall effect, which may lead to new concepts and new numerical techniques in the future. My postdoctoral experience on correlated materials at Flatiron institute and my graduate school training on topological matter at Princeton make me realize that there perhaps is a “gap”, both at the conceptual and methodological level, in the condensed matter community of dealing with strong correlation in topological trivial and non-trivial systems.

More specifically, correlation is naturally compressed in a real space local form such as “Hubbard interaction” in topological trivial materials, and numerical techniques based on this local picture such as dynamical mean field theory have been found successful. On the other side, in the field of topological physics, due to the intrinsic topological obstruction to electron localization in real space, people tend to use a reciprocal space picture where the topological winding of wavefunction is clearer.

I am interested in filling this void by studying the real space correlation of fractional quantum Hall effect as the start point. Future interesting direction include extending embedding numerical methods such as dynamical mean field theory and dynamical matrix embedding theory to topological materials, as complementary methods in providing their ground state and excitation information at close to zero or finite temperatures.

[1] **Jie Wang**, Semyon Klevtsov and Michael R. Douglas (2021). Fractional Quantum Hall States on CP2 Space. arXiv:2109.11522

[2] Vasil Rokaj*, Jie Wang* and others [in preparation; * equal contribution]