We conduct theoretical research on quantum condensed matter physics. Below are some background information for non-experts and some examples of current or past projects.

#### 1. Ultracold Quantum Gases

**Background:** The term “quantum gases” refers to quantum degenerate gases of neutral atoms (such as Rubidium, Potassium or Lithium) or molecules confined in vacuum by laser beams and chilled to temperatures as low as 0.000000001 degrees above absolute zero. They are literally the “coolest” stuff ever created by mankind. These systems provide well-controlled settings to advance our fundamental understanding of the many-body physics of strongly interacting quantum particles. Pioneering experimental works leading to these achievements were awarded Nobel Prize in 1997 (for “development of methods to cool and trap atoms with laser light”) and 2001 (for “achievement of Bose-Einstein condensation in dilute gases of alkali atoms”). We are interested in the quantum phases of Fermi gases in new parameter regimes being realized in ongoing cold atoms/molecules experiments. Here are some examples.

A. Dipolar Fermi gases. Interacting fermions undergo spontaneous symmetry breaking as the temperature is lowered. This gives rise to quantum phases with long range order such as superconductivity/superfluidity, charge density waves, and magnetism. Motivated by quantum gas experiments on magnetic atoms and dipolar molecules, we studied the competing instabilities of fermions with dipole-dipole interactions, which are anisotropic and long-ranged, in two dimensions using functional renormalization group. We discovered bond-ordered solid phases, or p-wave density waves, of dipolar fermions on lattice [PRL 12] and for fermions with quadrupolar moments [PRL 13]. Unconventional (p-wave) spin density waves also emerge for spin 1/2 fermions [PRA 13]. We have also worked out the functional renormalization group phase diagram of continuous dipolar Fermi gases in 2D [PRA 16]. Recently, we have systematically studied the phase diagram of spin models describing localized dipolar molecules in optical lattice, leading to the discovery of quantum paramagnetic phases (spin liquids) [PRL 17; PRL 18].

B. Interacting Fermi gases on higher orbital bands. Electrons of d- or f-orbitals play a central role in correlated electronic materials such as higher temperature superconductors. What interesting things can p-orbitals do? We have systematically studied strongly correlated p-orbital fermions on optical lattices. The low energy effective model turns out to be the quantum 120-degree model, a geometrically frustrated compass spin model, on the honeycomb lattice [PRL 08]. We also derived the spin-orbital exchange of spin 1/2 p-band fermions [PRL 15]. Interestingly, on an optical ladder, p-band fermions form a topological insulating phase, in the absence of spin-orbit coupling or gauge field [Nat. Comm. 13]. Other examples include the orbital order near an optical Feshbach resonance [PRA(R) 11] and the Mott transition of p-band bosons [PRA 11]. Recently, we unified the 120 degree model, the Kitaev model, and the Ising model into a single continuum model of quantum spins, the tripod model, on the honeycomb lattice and obtained its phase diagram using tensor networks [NJP, 16].

C. Floquet (periodically driven) quantum matter. Repeatedly “kicking” or “shaking” a quantum many-body system may fundamentally alter its topological properties and give rise to new excitations. Such phenomena may have no static analogs and lie outside the well known “periodical table of topological insulators and superconductors.” We discovered counter-propagating edge modes in a periodically kicked Hofstadter model of quantum Hall system [PRL 14] and showed it is rather generic for other driving protocols such as sinusoidal driving [PRB 14]. In a recent study, we find that periodic driving can turn coupled Kitaev chains (1D p-wave superconductors) into 2D p-wave topological superconductors with chiral edge modes or completely flat edge states [ZNA, 16].

D. One dimensional Fermi gases. Strongly interacting quantum many-body systems are notoriously hard to solve. Exact solutions are not only rare but also prohibitively complicated. From the Bethe ansatz exact solutions, we constructed the low energy effective field theory for 1D Fermi gas with arbitrary spin imbalance and contact attractive interaction, known as the Gaudin-Yang model. This enable us to compute all the correlation functions and work out the phase diagram for quasi-1D imbalanced Fermi gases [PRA, 08]. We further simplified the thermodynamic Bethe ansatz equation for 1D imbalanced Fermi gas down to 4 algebraic equation in the experimental regime [PRL 09]. This makes it much easier to compute the density profiles of the trapped gas and determine its temperature.

#### 2. Quantum Materials

**Background:** Quantum mechanics with all its weirdness is often said to dictate the behaviors of small objects such as electrons and atoms. But it can also manifest itself on the macroscopic scale. A hall mark example is superconductivity. At low temperatures, many materials become superconductors which have zero electrical resistance and expel weak magnetic field. A superconductor can be thought as a “perfect” quantum fluid of pairs of electrons, all sharing the same quantum state. It has become a leading competitor in building the next generation quantum devices such as qubits, the building blocks for a quantum computer. We are interested in understanding the quantum phases (magnetism, density waves, superconductivity, topological insulators) in new materials, and the general schemes to classify them using symmetry and topology.

E. Topological Superconductivity. Piecing a superconductor and another material together may give birth to brand new particles or collective excitations. We found that Weyl fermions arise in a periodic stacking (superlattice) of superconductor and magnetic material with spin-orbit coupling [arXiv:1506.05166]. We systematically studied the the Andreev bound states at the interface between an s-wave superconductor and a topological insulator [PRB 11], the current-phase relation for Josephson effect through helical metal [PRB 12], and the evolution from Majorana fermions for a pi junction fabricated on the surface of a topological insulator [PRB(R) 12]. We are currently working on how the helical and chiral Majorana fermions in p-wave superconductors can be “braided” for topological quantum computing [arXiv 08].

F. Quantum Magnetism. We develop numerical many-body algorithms for frustrated quantum spin models. Two examples are functional renormalization group and tensor network ansatz. We have applied these methods to solve compass models that includes the Kitaev and the 120-degree model as special limits [NJP 16], the dipolar Heisenberg model on the triangular [PRL 18], square [PRB 18] and zigzag lattice [arXiv 18], and found strong evidence for quantum paramagnetic phases (likely quantum spin liquids) and symmetry protected phases of spin 1/2 systems.

G. Quantum Transport. Josephson junctions are the building blocks of superconducting qubits. Josephson’s Nobel prize winning work elucidated how dc and ac charge current flow in these devices. But how about heat and spin currents? We established a new formulation to describe nonequilibrium superconductivity near an arbitrary (e.g. spin-active) interface using the quasiclassical Keldysh Green functions [*PRB 04*]. This technique enabled us to formulate the theory of heat conduction [*PRL 03*] and spin transport [*PRL 07, PRB 08*] in Josephson junctions. In collaboration with Dr. Qiliang Li (ECE department, George Mason) and NIST scientists, we provide theoretical support in making the first field-effect transistor based on nanowires of topological insulator Bi2Se3 [*SR 13*], and back-gate field-effect transistors of topological crystalline insulator SnTe[*APL 14*].