Twisted bilayer graphene (TBG) is an engineered system with one graphene layer stacked on top of the other and rotated by a certain twisted angle. As a result of the twist, the carbon atoms in the two graphene layers form moire patterns in real space. Around the so called ``magic angles", the low-energy electronic structures of TBG are characterized by four nearly flat bands. These low-energy flat bands are believed to be responsible for the recently observed correlated insulating phase and unusual superconductivity in TBG. These experimental observations have stimulated numerous theoretical studies.
Some recent works suggest the flat bands in TBG exhibits nontrivial "fragile topology". In such a context, my collaborators and I have proposed that the electronic structure of TBG can be understood as Dirac fermions under some pseudo magnetic fields generated by the moir\'e pattern. The two low-energy flat bands from each valley originate from the two zeroth Landau levels of Dirac fermions under such opposite effective magnetic fields and carry opposite Chern numbers +/-1. Such a pseudo Landau-level representation of TBG naturally explains the origin of the magic angles: the magic angles occur when the moire supercell are commensurate with the pseudo-magnetic fluxes such that the pseudo Landau levels are roughly the eigenstates of the systems. The pseudo Landau-level picture also naturally explains the mechanism of the quantum anomalous Hall effect at 3/4 filling observed in experiments: when only one pseudo Landau level is empty, it would be split from the other occupied pseudo Landau levels by exchange Coulomb interactions, opening a gap with nonzero Chern number [1].
We have further proposed that such topological flat bands generically exist in twisted multilayer graphene systems. The total valley Chern numbers in such systems depend on the number of layers and the stacking chiralities, which can be succinctly described by a simple mathematical formula [2]. As each valley is associated with nonzero Chern number, the system would become an orbital ferromagnet once the valley symmetry is broken either spontaneously or by weak magnetic fields [2]. Such orbital ferromagnetic states are not only characterized by (quantum) anomalous Hall effect, but also unusual magneto-optical and nonlinear optical properties [3].
Using a generic unrestricted "moire Hartree-Fock variational method", my collaborator and I have thoroughly studied the correlation effects at almost all integer fillings of the flat bands in TBG. For the first time, we have successfully explained both the correlated insulating states at +/- 1/2 filling and at the charge neutrality point, as well as the quantum anomalous Hall effect (QAHE) at 3/4 and 1/2 fillings within a single theoretical framework. We propose that the correlated insulating states at half an zero fillings are "moire orbital antiferromagnetic states" on an emergent honeycomb lattice with opposite circulating current loops in the moire supercell, which is a unprecedented new phase of matter. We have further predicted a number of unconventional properties of the QAHE in TBG in response to external magnetic fields. For example, the QAHE at electron and hole fillings of TBG are predicted to exhibit hysteresis loops with opposite chiralities, and the QAHE at 1/2 filling can be enhanced by vertical magnetic field, but significantly suppressed by in-plane magnetic field [4].
[1] Jianpeng Liu, Junwei Liu, Xi Dai, Physical Review B 99 (15), 155415 (2019)
[2] Jianpeng Liu, Zhen Ma, Jinhua Gao, Xi Dai, Physical Review X 9 (3), 031021 (2019)
[3] Jianpeng Liu, Xi Dai, arXiv:1907.08932
[4] Jianpeng Liu, Xi Dai, arXiv:1911.03760v2
I have studied the Fermi-surface instability in the surface states of a novel-type of topological semimetal, the line-node semimetal. The line-node semimetal has the so called "drumhead" surface states, the dispersion of which is typically much smaller than that of the bulk bands. This makes the drumhead surface states a perfect platform to study the effects of strong Coulomb correlations. Based on model calculations, I find that the surface exhibits both ferromagnetic and charge-density-wave instabilities. These spin and charge order parameters are exponentially localized at the surface and the bulk remains metallic and non-ordered. The phase transitions from the normal metallic phase to the surface ferromagnetic phase is characterized by an unconventional dynamical critical exponent z=1, which realizes a new universality class for the quantum phase transition at the surface of a 3D topological metal. This work is in collaboration with Leon Balents [1].
I have also studied the possible interaction-driven quantum anomalous Hall insulators (or, Chern insulators) in CrSi(Ge)Te3 systems using advanced first principles calculations. I have proposed that there may emerge the Chern-insulator phase in monolayer CrSi(Ge)Te3 with proper adatoms. Such a topological phase turns out to be purely driven by Coulomb interactions and does not require spin-orbit coupling. In other words, the ground state of such an interaction-driven Chern insulator phase results from a spontaneous time-reversal symmetry breaking in the orbital space, thus carries the current loops, and are dubbed as the flux states. This is the first ab inito proposal of Chern insualtors resulted from interaction-driven spontaneous-time reversal symmetry breaking [2]. This is work is in collaboration with Se Young Park, Kevin Garrity, and David Vanderbilt [2].
[1] Jianpeng Liu and Leon Balents, Phys. Rev. B, 95 075426 (2017)
[2] Jianpeng Liu, Se Young Park, Kevin F. Garrity and David Vanderbilt, Phys. Rev. Lett. 117, 257201(2016)
Magnetic topological semimetals are intriguing to study because it involves the interplay of two interesting subjects: magnetism and topological semimetals. Both subjects contain rich physics, and it is interesting to ask how they are coupled with each other and generate new physics. In this context, I have theoretically studied two noncollinear magnetic topological (semi)metals: Mn3Sn [1] and CeAlGe [2].
In Mn3Sn, each Mn ion has a large magnetic moment ~3 bohr magneton. The Mn ions form layered kagome lattice, with Sn atoms which donate conduction electrons being intercalated between the Mn kagome layers. The magnetic ground state is a in-plane non-collinear antiferromagnetic state with negative vector chirality. In collobration with Leon Balents, we argue that this material possesses a hierarchy of energy scales, which allows a description of the spin structure and spin dynamics using an XY model with Z6 anisotropy. We propose a dynamical equation of motion for the XY order parameter, which implies the presence of Z6 vortex lines, the double-domain pattern in the presence of magnetic fields, and the ability to control domains with charge currents. We also introduce a minimal electronic model that allows efficient calculation of the electronic structure in the antiferromagnetic configuration, and find that there are Fermi arc states localized at the domain walls. These novel bound states associated with the topological defects have their unique transport character: the domain wall Fermi arcs may exhibit its own anomalous Hall effect, and may contribute to a thickness independent domain-wall resistivity. Moreover, we have proposed a possible device based on the domain-dependent anomalous Hall effect [1].
I have studied the magnetotransport in another noncollinear magnetic Weyl semimetal CeAlGe. In experiments, the magnetoresistance of CeAlGe was observed to have a singular angular dependence. Such singular magnetoresistance is attributed to the formation of domains and domain walls in the presence of external magnetic fields along certain high-symmetry directions. I have performed first principles calculations and modeling to study the coupling between the Weyl electrons and the local magnetic moments, and have quantitatively calculated the magnetoresistance across the magnetic domain walls, which is in good agreement with the experimental observations. This work is in collaboration with Takehito Suzuki, Joseph Checkelsky, Lucile Savary, and Leon Balents [2].
[1] Jianpeng Liu and Leon Balents, Phys. Rev. Lett. 119, 087202 (2017)
[2] T. Suzuki, L. Savary, J. Liu, J. W. Lynn, L. Balents, and J. G. Checkelsky, accepted by Science.
A quantity that is closely related to the physics of topological insulators is an isotropic contribution to the orbital magnetoelectric coupling that is known as the “Chern-Simons” or “axion” or “θ coupling” term. This last name comes from the fact that this contribution can be written in the form (e^2/h)*(θ/2π), where θ is a phase angle, i.e., is only gauge-invariant modulo 2π . However, a straightforward finite-difference evaluation of this formula is only practical if a smooth and periodic gauge has been chosen in the entire Brillouin zone. Moreover, previous calculations have shown that for interesting systems expected to exhibit a large θ, such as topological insulators and systems derived from them, it is very difficult to converge the results with respect to k-point sampling. In order to solve this problem, I proposed a new method to compute the Chern Simons coupling. In particular the Brillouin zone is divided into several subvolumes, and the gauge is chosen to be smooth within each subvolume. These subvolumes meet at 2D planes in k-space where there is gauge discontinuity. The total θ response is then divided into contributions of two kinds: 3D integrals of the Chern-Simons 3-form over the subvolumes, and 2D integrals of a planar contribution associated with the gauge discontinuities on the boundary planes. Furthermore, in some cases it is necessary to subdivide the boundary planes into subregions separated by “vortex loops,” which make yet a third contribution in terms of Berry phases defined around the vortex loops. The total θ thus consists of three kinds of terms, expressed as integrals over 3D, 2D and 1D manifolds. We illustrate our method by applying it to the Fu-Kane-Mele model with applied staggered Zeeman field. This work is in collaboration with David Vanderbilt [1].
[1] Jianpeng Liu and David Vanderbilt, Phys. Rev. B 92 (24), 245138 (2015)
Periodically driven quantum systems have received significant attention in recent years. The typical theoretical prescription is to use the Floquet formalism, which allows for the description of a time-periodic system using some effectively time-independent Hamiltonian dubbed as the ``Floquet Hamiltonian". Since the details of the Floquet Hamiltonian are crucially dependent on the frequency, amplitude and polarization of the external drive, it allows one to engineer the physical properties of a quantum system using the laser-light radiations. I have considered driving multi-orbital Mott insulators using laser radiations. I have derived general expressions for periodically driven spin-orbital models using time dependent perturbation theory, and have shown that the effective exchange interactions of the Floquet spin-orbital Hamiltonians are highly tunable by the frequency, amplitude, and polarization of the laser. When the frequency of the laser is within the Mott Hubbard gap, the real doublon-holon pairs would be generated and the system would be heated up. Therefore, the effect of finite bandwidth of doublon-holon excitations are also taken into account in the formalism in order to study the possible heating effects. The formalism is further applied to orthorhombic titanates YTiO3 and LaTiO3 based on first-principles calculations, and find that the “Floquet” spin exchange interactions in these compounds can be engineered to a large extent by tuning the frequency and electric-field amplitude of the laser. Moreover, due to the multi-orbital nature, the antiferromagnetic (LaTiO3) and ferromagnetic (YTiO3) compounds exhibit contrasting responses to lasers at weak electric-field magnitudes, which may be a robust experimental signature. This work is in collaboration with Kasra Hejazi and Leon Balents [1], [2].
[1] Jianpeng Liu, Kasra Hejazi, and Leon Balents, Phys. Rev. Lett. 121, 107201 (2018)
[2] Kasra Hejazi, Jianpeng Liu, and Leon Balents, arXiv preprint arXiv:1809.09800
I have been working on the phase transitions and disorder effects in 3D topological insulators. Based on first principles DFT calculations, we have theoretically studied the phase transitions in In-doped Bi2Se3, and found that the band topology in Bi2Se3 is strongly suppressed by In doping, which leads to a transition from a topological to a normal insulator at very low In concentrations. Moreover, the In impurities tend to segregate and form local clusters. The phase transition in such topological system therefore has to be interpreted from a local percolation picture instead of the usual "band-inversion" picture. This work is in collaboration with David Vanderbilt [1].
I have also worked on the phase transitions in 3D topological insulators with broken inversion symmetry. In 2007 Shuichi Murakami proposed that if inversion symmetry is broken, then the Weyl semimetal phase occurs as an intermediate phase connecting the 3D topological and trivial insulators. However around 2012-2013, some of the theoretical studies on pressurized BiTeI seemed to suggest that there is no such an intermediate Weyl semimetal phase in the topological phase transition of BiTeI (which breaks inversion symmetry); and a different theory was proposed to explain the phase transition behavior. I clarified the general theory of the phase transition in such systems, and insisted that a 3D Weyl seimmetal must show up as an intermediate phase. I also numerically illustrate that an intermediate Weyl semimetal phase indeed shows up in the process of the pressure-driven topological transitions in BiTeI, but just within a very small pressure interval and is hard to be observed even numerically. In the same paper I also made some other predictions as possible candidates of Weyl semimetals with broken inversion symmetry. This work is in collaboration with David Vanderbilt [2].
[1] Jianpeng Liu and David Vanderbilt, Phys. Rev. B 88, 224202 (2013)
[2] Jianpeng Liu and David Vanderbilt, Phys. Rev. B 90, 155316 (2014).