## Research

#### Fractional topological insulators

In condensed matter systems, although the building blocks (electrons and ions) carry integer charges in the unit of electron charge e, the charge of quasi-particle excitations may take a fractional value. The most famous example of this type is the quasi-particles and quasi-holes in fractional quantum Hall effects, which is a topological state of matter whose low-energy physics is governed by a topological gauge field theory: the Chern-Simon's gauge theory. These fractional excitations are neither fermions nor bosons. They are known as anyons and carry fractional statistics. Some of the anyons carry non-abelian statistics and can be used to perform topological quantum computation.Recently, it is discovered (by my coworkers and I, as well as other independent groups) that, similar fractional topological states also exit in lattice systems with zero net magnetic field using topological flatbands. To distinguish these new fractional states from the fractional quantum Hall effects, they are referred to as fractional Chern insulators.

Numerical studies indicate that the fractional Chern insulators share the same topological properties with the corresponding fractional quantum Hall states, suggesting that these two types of states may be closely related. This connection is eventually revealed in one of our recent work, showing that these two types of states are adiabatically connected and thus are different regimes of the same phase.

The discovery of fractional Chern insulator not only enlarges the family of fractional topological insulators, it also helps us to understand the nature of fractional topological states.

**[Click the images for larger figures]**

**References:**

- Ying-Hai Wu, J. K. Jain and Kai Sun,
*Adiabatic Continuity Between Hofstadter and Chern Insulators*, Physical Review B 86, 165129 (2012). - Shuo Yang, Zheng-Cheng Gu, Kai Sun and S. Das Sarma,
*Topological flat band models with arbitrary Chern numbers*, arxiv preprint arXiv:1205.5792 (2012). - Shuo Yang, Kai Sun, and S. Das Sarma,
*Quantum Phases of Disordered Flatband Lattice Fractional Quantum Hall Systems*, Physical Review B**85**, 205124 (2012). - D. N. Sheng, Zheng-Cheng Gu, Kai Sun, and L. Sheng,
*Fractional Quantum Hall Effect in the Absence of Landau Levels*, Nature Communications**2**, 389 (2011). - Kai Sun, Zhengcheng Gu, Hosho Katsura, and S. Das Sarma,
*Nearly Flatbands with Nontrivial Topology*, Physical Review Letters**106**, 236803 (2011). - Accompanying viewpoint:
*Fractional quantum Hall effect without Landau levels,*by Rahul Roy and Shivaji L. Sondhi, Physics**4**, 46 (2011).

#### Interaction induced topological insulators

Topological band insulators (without fractional quasi-particles) are band insulators with topologically protected edge states. In topological band insulators, interaction effect is irrelevant as far as the topological nature is concerned. These nontrivial band insulators can be stabilized by introducing some external magnetic field and/or spin-orbital couplings regardless of the interactions, as long as it is not too strong to cause a phase transition. My focus on topological band insulators is to understand the role of interactions and strong correlation effects, e.g. using interactions and many-body effect to stable a topological insulator, via quantum phase transition, in the absence of external magnetic field and/or spin-orbital couplings.**[Click the image for a larger figure]**

**References:**

- Kai Sun, W. Vincent Liu, Andreas Hemmerich, and S. Das Sarma,
*Topological Semimetal in a Fermionic Optical Lattice*, Nature Physics**8**, 67 (2012). - Kai Sun, Hong Yao, Eduardo Fradkin, and Steven A. Kivelson,
*Topological Insulators and Nematic Phases from Spontaneous Symmetry Breaking in 2d Fermi Systems with a Quadratic Band Crossing*, Physical Review Letters**103**, 046811 (2009).

#### Topological Kondo insulators

Topological insulators have been observed in various semiconductors, in which electron-electron interactions are weak. Are there topological insulators in strongly correlated systems? Our recent work suggests a positive answer. There is a family of strongly correlated systems, which is known as heavy fermion compounds. In these systems, the effective masses of electrons and holes are typically orders of magnitudes larger than the mass of an electron. Heavy fermion compounds show a rich variety of phenomena, including superconductivity, quantum criticality and quantum phase transitions, etc. Among these interesting materials, some of them are insulators, which are known as Kondo insulators, and these insulators can be topologically nontrivial.**[Click the image for a larger figure. Figure on the left from JQI news release: An Ideal Material]**

**References:**

- Mengxing Ye, J. W. Allen and Kai Sun,
*Topological crystalline Kondo insulators and universal topological surface states of SmB*, Arxiv preprint arXiv:1307.7191 (2013)._{6} - G. Li, Z. Xiang, F. Yu, T. Asaba, B. Lawson, P. Cai, C. Tinsman, A. Berkley, S. Wolgast, Y. S. Eo, Dae-Jeong Kim, C. Kurdak, J. W. Allen, Kai Sun, X. H. Chen, Y. Y. Wang, Z. Fisk, And Lu Li,
*Quantum oscillations in Kondo Insulator SmB6*, Arxiv preprint arXiv:1306.5221 (2013). - Steven Wolgast, Cagliyan Kurdak, Kai Sun, J. W. Allen, Dae-Jeong Kim
and Zachary Fisk,
*Discovery of the First Topological Kondo Insulator: Samarium Hexaboride*. Arxiv preprint arXiv:1211.5104 (2012) - Maxim Dzero, Kai Sun, Piers Coleman, and Victor Galitski,
*Theory of Topological Kondo Insulators*, Physical Review B**85**, 045130 (2012). - Maxim Dzero, Kai Sun, Victor Galitski, and Piers Coleman,
*Topological Kondo Insulators*, Physical Review Letters**104**, 106408 (2010). - News Report on Nature: Eugenie Samuel Reich,
*Hopes surface for exotic insulator*, Nature**492**, 165, (2012). - JQI News release: An Ideal Material.

#### Isostaticity and Elastic Holography

Topological states of matter, including the fractional topological insulators and topological band insulators discussed above, are quantum systems whose ground state wavefunctions have some nontrivial topological structure. By definition, a topological state must live in the quantum world. However, in our recently study, it is discovered that classical systems can support some of the phenomena observed in quantum topological states of matter as well, including zero-energy edge states and holographic property. However, the theoretical reason behind these phenomena is not due to a nontrivial topology, but caused by an emergent conformal symmetry.**[Click the image for a larger figure]**

**References:**

- Kai Sun, Anton Souslov, Xiaoming Mao and T. C. Lubensky,
*Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices*, Proceedings of the National Academy of Sciences of the United States of America,**109**, 12369 (2012). - See accompanying Commentary:
*Topological soft matter: Kagome lattices with a twist*by Vincenzo Vitelli, PNAS**109**, 12266 (2012)

#### Liquid Crystal Phases in strongly correlated electronic systems

**Quantum melting**: It is well-known that by increasing

*thermal fluctuations*(i.e. temperature), a crystal melts into a liquid. In electronic systems, electrons can also form a crystal (e.g. a Wigner crystal), which will also melt into an electron liquid (e.g. a Fermi liquid) when

*quantum fluctuations*become strong enough. This melting is similar to the melting of a conventional crystal, but there is a key difference. Here it is the quantum fluctuations that drive the melting, instead of thermal fluctuations, and thus it is known as a

*quantum melting*. The quantum melting of an electron crystal is believed to be the key to understand many exotic physics in strongly correlated systems, including High temperature superconductors.

**Classical Liquid Crystal:**It has been known long time ago that some solids don't melt directly into a liquid state. Instead, they first turn into some intermediate states before turning into a liquid. These intermediate states are known as liquid crystals, which can be found in any liquid crystal display (LCD) of your computers, smart phones, etc. From the symmetry point of the view, the phase transition from a liquid to a crystal breaks spontaneously the rotational and translational symmetries. For liquid crystal states, they break only part of these symmetries. For example, the

*nematic*state breaks only rotational symmetries, while the translational symmetries are preserved, and the

*smectic*state breaks the rotational symmetries and translational symmetry in one direction only.

**Quantum Liquid Crystals:**Similarly, when we melt a crystal of electrons, there may be intermediate "liquid crystal" states too, which are known as

*quantum liquid crystal phases*. These quantum phases share the same symmetry properties with their classical counterparts. Therefore, in principle, one can make a quantum liquid crystal display out of them for a quantum computer, and then sell it over the quantum Internet to get quantum money.

Of course, the motivation to study these quantum phases is NOT to create a new display, but to understand strongly correlated systems. These quantum liquid states have been observed in various materials, including high temperature superconductivity and a key question to answer is whether they are directly related to the phenomenon of high temperature superconductivity itself. Although this question still needs further investigations to answer, in a recent paper, we provide theoretical and experimental evidences showing that the nematic order does enhance the superconductivity in two component superconductors (for example the p+ip topological superconductors).

**[Click the image for a larger figure]**

**References:**

- Y.A. Ying, N.E. Staley, Y. Xin, K. Sun, X. Cai, D. Fobes, T. Liu, Z.Q. Mao, and Y. Liu,
*Dislocations and the Enhancement of Superconductivity in Odd-Parity Superconductor Sr*, arXiv preprint arXiv:1205.3250 (2012)._{2}Ruo_{4} - Kai Sun, Congjun Wu, and S. Das Sarma,
*Spontaneous Inhomogeneous Phases in Ultracold Dipolar Fermi Gases*, Physical Review B**82**, 075105 (2010). - Kai Sun, Michael J. Lawler, and Eun-Ah Kim,
*Spin-Charge Interplay in Electronic Liquid Crystals: Fluctuating Spin Stripe Driven by Charge Nematic Ordering*, Physical Review Letters**104**, 106405 (2010). - M. Fregoso Benjamin, Sun Kai, Fradkin Eduardo, and L. Lev Benjamin,
*Biaxial Nematic Phases in Ultracold Dipolar Fermi Gases*, New Journal of Physics**11**, 103003 (2009). - Kai Sun and Eduardo Fradkin,
*Time-Reversal Symmetry Breaking and Spontaneous Anomalous Hall Effect in Fermi Fluids*, Physical Review B**78**, 245122 (2008). - Kai Sun, Benjamin M. Fregoso, Michael J. Lawler, and Eduardo Fradkin,
*Fluctuating Stripes in Strongly Correlated Electron Systems and the Nematic-Smectic Quantum Phase Transition*, Physical Review B**78**, 085124 (2008). - Congjun Wu, Kai Sun, Eduardo Fradkin, and Shou-Cheng Zhang,
*Fermi Liquid Instabilities in the Spin Channel*, Physical Review B**75**, 115103 (2007).