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Liang Fu

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cond-mat.mes-hall cond-mat.str-el cond-mat.mtrl-sci cond-mat.supr-con cond-mat.stat-mech physics.optics quant-ph cond-mat.quant-gas hep-th Artificial Intelligence cond-mat.dis-nn hep-lat Machine Learning physics.app-ph physics.comp-ph

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preprint2026arXiv

QERNEL: a Scalable Large Electron Model

We introduce QERNEL, a foundational neural wavefunction that variationally solves families of parameterized many-electron Hamiltonians and captures their ground states throughout parameter space within a single model. QERNEL combines FiLM-based parameter conditioning with scale-efficient architectural elements -- mixture of experts and grouped-query attention, substantially improving expressivity at low computational cost. We apply QERNEL to interacting electrons in semiconductor moiré heterobilayers, training a single weight-shared model for systems of up to 150 electrons. By solving the many-electron Schrödinger equation conditioned on moiré potential depth, QERNEL captures both quantum liquid and crystal states and discovers the sharp phase transition between them, marked by abrupt changes in interaction energy and charge density. Our work establishes a foundation model for moiré quantum materials and a scalable architecture toward a Large Electron Model for solids.

preprint2022arXiv

Dirac electron under periodic magnetic field: Platform for fractional Chern insulator and generalized Wigner crystal

We propose a platform for flat Chern band by subjecting two-dimensional Dirac materials -- such as graphene and topological insulator thin films -- to a periodic magnetic field, which can be created by the vortex lattice of a type-II superconductor. As a generalization of the $n=0$ Landau level, the flat band of Dirac fermion under a nonuniform magnetic field remains at zero energy, exactly dispersionless and topologically protected, while its local density of states is spatially modulated due to the magnetic field variation. In the presence of short-range repulsion, we find fractional Chern insulators emerge at filling factors $ν=1/m$, whose ground states are generalized Laughlin wavefunctions. We further argue that generalized Wigner crystals may emerge at certain commensurate fillings under a highly nonuniform magnetic field in the form of a flux line lattice.

preprint2022arXiv

Moiré Landau fans and magic zeros

We study the energy spectrum of moiré systems under a uniform magnetic field. The superlattice potential generally broadens Landau levels into Chern bands with finite bandwidth. However, we find that these Chern bands become flat at a discrete set of magnetic fields which we dub "magic zeros". The flat band subspace is generally different from the Landau level subspace in the absence of the moiré superlattice. By developing a semiclassical quantization method and taking account of superlattice induced Bragg reflection, we prove that magic zeros arise from the simultaneous quantization of two distinct $k$-space orbits. The flat bands at magic zeros provide a new setting for exploring crystalline fractional quantum Hall physics.

preprint2022arXiv

Non-Abelian nonsymmorphic chiral symmetries

The Hofstadter model exemplifies a large class of physical systems characterized by particles hopping on a lattice immersed in a gauge field. Recent advancements on various synthetic platforms have enabled highly-controllable simulations of such systems with tailored gauge fields featuring complex spatial textures. These synthetic gauge fields could introduce synthetic symmetries that do not appear in electronic materials. Here, in an SU(2) non-Abelian Hofstadter model, we theoretically show the emergence of multiple nonsymmorphic chiral symmetries, which combine an internal unitary anti-symmetry with fractional spatial translation. Depending on the values of the gauge fields, the nonsymmorphic chiral symmetries can exhibit non-Abelian algebra and protect Kramer quartet states in the bulk band structure, creating general four-fold degeneracy at all momenta. These nonsymmorphic chiral symmetries protect double Dirac semimetals at zero energy, which become gapped into quantum confined insulating phases upon introducing a boundary. Moreover, the parity of the system size can determine whether the resulting insulating phase is trivial or topological. Our work indicates a pathway for creating topology via synthetic symmetries emergent from synthetic gauge fields.

preprint2022arXiv

Pseudogap metal and magnetization plateau from doping moiré Mott insulator

The problem of doping Mott insulators is of fundamental importance and long-standing interest in the study of strongly correlated electron systems. The advent of semiconductor based moiré materials opens a new ground for simulating the Hubbard model on the triangular lattice and exploring the rich phase diagram of doped Mott insulators as a function of doping and external magnetic field. Based on our recent identification of spin polaron quasiparticle in Mott insulator, in this work we predict a new metallic state emerges at small doping and intermediate field range, a pseudogap metal that exhibits a single-particle gap and a doping-dependent magnetization plateau.

preprint2022arXiv

Quantum anomalous Hall effect from inverted charge transfer gap

A general mechanism is presented by which topological physics arises in strongly correlated systems without flat bands. Starting from a charge transfer insulator, topology emerges when the charge transfer energy between the cation and anion is reduced to invert the lower Hubbard band and the spin-degenerate charge transfer band. A universal low-energy theory is developed for the inversion of charge transfer gap in a quantum antiferromagnet. The inverted state is found to be a quantum anomalous Hall (QAH) insulator with non-coplanar magnetism. Interactions play two essential roles in this mechanism: producing the insulating gap and quasiparticle bands prior to the band inversion, and causing the change of magnetic order necessary for the QAH effect after inversion. Our theory explains the electric field induced transition from correlated insulator to QAH state in AB-stacked TMD bilayer MoTe$_2$/WSe$_2$.

preprint2022arXiv

Supercurrent diode effect and finite momentum superconductivity

When both inversion and time-reversal symmetries are broken, the critical current of a superconductor can be nonreciprocal. In this work we show that in certain classes of two-dimensional superconductors with antisymmetric spin-orbit coupling, Cooper pairs acquire a finite momentum upon the application of an in-plane magnetic field, and as a result, critical currents in the direction parallel and antiparallel to the Cooper pair momentum become unequal. This supercurrent diode effect is also manifested in the polarity-dependence of in-plane critical fields induced by a supercurrent. These nonreciprocal effects may be found in polar SrTiO$_3$ film, few-layer MoTe$_2$ in the $T_d$ phase, and twisted bilayer graphene in which the valley degrees of freedom plays the role analogous to spin.

preprint2022arXiv

Supercurrent induced resonant optical response

The optical conductivity encodes the current response to a time dependent electric field. We develop a theory of the optical conductivity $σ(ω)$ in presence of a dc supercurrent. Current induced optical response is prohibited by Galilean invariance from occurring in systems with a single parabolic band. However, we show that lattice effects give rise to a pronounced current dependent peak in $σ(ω)$ at the gap edge $ω= 2 Δ$, which diverges in the clean limit. We demonstrate this in a model of a multi-band superconductor. Our theory explains the recent observation of a current induced peak in the optical conductivity in NbN by Nakamura et al. (2019), and provides a new mechanism for direct activation of the Higgs mode with light.

preprint2021arXiv

Ferromagnetic helical nodal line and Kane-Mele spin-orbit coupling in kagome metal Fe3Sn2

The two-dimensional kagome lattice hosts Dirac fermions at its Brillouin zone corners K and K', analogous to the honeycomb lattice. In the density functional theory electronic structure of ferromagnetic kagome metal Fe$_3$Sn$_2$, without spin-orbit coupling we identify two energetically split helical nodal lines winding along $z$ in the vicinity of K and K' resulting from the trigonal stacking of the kagome layers. We find that hopping across A-A stacking introduces a layer splitting in energy while that across A-B stacking controls the momentum space amplitude of the helical nodal lines. The effect of spin-orbit coupling is found to resemble that of a Kane-Mele term, where the nodal lines can either be fully gapped to quasi-two-dimensional massive Dirac fermions, or remain gapless at discrete Weyl points depending on the ferromagnetic moment orientation. Aside from numerically establishing Fe$_3$Sn$_2$ as a model Dirac kagome metal, our results provide insights into materials design of topological phases from the lattice point of view, where paradigmatic low dimensional lattice models often find realizations in crystalline materials with three-dimensional stacking.

preprint2021arXiv

Interfacial ferroelectricity in rhombohedral-stacked bilayer transition metal dichalcogenides

Van der Waals (vdW) materials have greatly expanded our design space of heterostructures by allowing individual layers to be stacked at non-equilibrium configurations, for example via control of the twist angle. Such heterostructures not only combine characteristics of the individual building blocks, but can also exhibit emergent physical properties absent in the parent compounds through interlayer interactions. Here we report on a new family of emergent, nanometer-thick, semiconductor 2D ferroelectrics, where the individual constituents are well-studied non-ferroelectric monolayer transition metal dichalcogenides (TMDs), namely WSe2, MoSe2, WS2, and MoS2. By stacking two identical monolayer TMDs in parallel, we obtain electrically switchable rhombohedral-stacking configurations, with out-of-plane polarization that is flipped by in-plane sliding motion. Fabricating nearly-parallel stacked bilayers enables the visualization of moiré ferroelectric domains as well as electric-field-induced domain wall motion with piezoelectric force microscopy (PFM). Furthermore, by using a nearby graphene electronic sensor in a ferroelectric field transistor geometry, we quantify the ferroelectric built-in interlayer potential, in good agreement with first-principles calculations. The novel semiconducting ferroelectric properties of these four new TMDs opens up the possibility of studying the interplay between ferroelectricity and their rich electric and optical properties.

preprint2021arXiv

New mechanism and exact theory of superconductivity from strong repulsive interaction

We introduce a new and general mechanism for superconductivity in Fermi systems with strong repulsive interaction. Because kinetic terms are small compared to the bare repulsion, the dynamic of charge carriers is constrained by the the presence of other nearby carriers. By treating kinetic terms as a perturbation around the atomic limit, we show that pairing can be induced by correlated multi-particle tunneling processes that favor two itinerant carriers to be close together. Our analytically-controlled theory provides a quantitative formula relating $T_c$ to microscopic parameters, with maximum $T_c$ reaching about 10% of the Fermi temperature. Our work demonstrates a powerful method for studying strong coupling superconductivity with unconventional pairing symmetry. It also offers a realistic new route to realizing finite angular momentum superfluidity of spin-polarized fermions in optical lattice.

preprint2021arXiv

Spin-triplet superconductivity from interband effect in doped insulators

Despite being of fundamental importance and potential interest for topological quantum computing, spin-triplet superconductors remain rare in solid state materials after decades of research. In this work, we present a general mechanism for spin-triplet superconductivity in multi-band systems, where a non-retarded pairing interaction between conduction electrons is produced by their electronic repulsion to a third-electron undergoing a virtual interband transition. Our theory is analytically controlled by an interband hybridization parameter, and explicitly demonstrated in doped band insulators with the example of an extended Hubbard model. In light of this theory, we propose that recently discovered dilute superconductors such as ZrNCl and WTe$_2$ are spin-triplet, and compare the expected consequences of our theory with experimental data.

preprint2021arXiv

Unconventional superconductivity due to interband polarization

We analyze in detail the superconductivity that arises in an extended Hubbard model describing a multiband system with repulsive interactions. We show that virtual interband processes induce an effective attractive interaction for small momentum transfers, a situation not found in most models of superconductivity from repulsion. This attraction can be traced back, in real space, to the presence of correlated hopping terms induced by interband polarization. We reveal this physics with both strong-coupling expansion and many-body perturbation theory, supplemented by numerical calculations. Finally, we point out interesting similarities with the problem of interacting electrons in twisted bilayer graphene, suggesting the importance of interband contribution to superconductivity.

preprint2020arXiv

Charge Transfer Excitations, Pair Density Waves, and Superconductivity in Moiré Materials

Transition metal dichalcogenide (TMD) bilayers are a new class of tunable moiré systems attracting interest as quantum simulators of strongly-interacting electrons in two dimensions. In particular, recent theory predicts that the correlated insulator observed in WSe$_2$/WS$_2$ at half filling is a charge-transfer insulator similar to cuprates and, upon further hole doping, exhibits a transfer of charge from anion-like to cation-like orbitals at different locations in the moiré unit cell. In this work, we demonstrate that in this doped charge-transfer insulator, tightly-bound charge-2e excitations can form to lower the total electrostatic repulsion. This composite excitation, which we dub a trimer, consists of a pair of holes bound to a charge-transfer exciton. When the bandwidth of doped holes is small, trimers crystallize into insulating pair density waves at a sequence of commensurate doping levels. When the bandwidth becomes comparable to the pair binding energy, itinerant holes and charge-2e trimers interact resonantly, leading to unconventional superconductivity similar to superfluidity in an ultracold Fermi gas near Feshbach resonance. Our theory is broadly applicable to strongly-interacting charge-transfer insulators, such as WSe$_2$/WS$_2$ or TMD homobilayers under an applied electric field.

preprint2020arXiv

Density functional approach to correlated moire states: itinerant magnetism

Two-dimensional moire superlattices have recently emerged as a fertile ground for creating novel electronic phases of matter with unprecedented control. Despite intensive efforts, theoretical investigation of correlated moire systems has been challenged by the large number of atoms in a superlattice unit cell and the inherent difficulty of treating electron correlation. The physics of correlated moire systems is governed by low-energy electrons in a coarse-grained long-wavelength potential, unlike the singular Coulomb potential of atomically-spaced ions in natural solids. Motivated by the separation between moire and atomic length scales, in this work we apply density functional theory to study directly the continuum model of interacting electrons in the periodic moire potential. Using this quantitatively accurate method, we predict itinerant spin-valley ferromagnetism in transition metal dichalchogenide heterobilayers, which originates from the constructive interplay between moire potential and Coulomb interaction in a two-dimensional electron system.

preprint2020arXiv

Discovery of segmented Fermi surface induced by Cooper pair momentum

Since the early days of Bardeen-Cooper-Schrieffer theory, it has been predicted that a sufficiently large supercurrent can close the energy gap in a superconductor and creates gapless Bogoliubov quasiparticles through the Doppler shift of quasiparticle energy due to the Cooper pair momentum. In this gapless superconducting state, zero-energy quasiparticles reside on a segment of the normal state Fermi surface, while its remaining part is still gapped. The finite density of states of field-induced quasiparticles, known as the Volovik effect, has been observed in tunneling and specific heat measurements on d- and s-wave superconductors. However, the segmented Fermi surface of a finite-momentum state carrying a supercurrent has never been detected directly. Here we use quasiparticle interference (QPI) technique to image field-controlled Fermi surface of Bi$_2$Te$_3$ thin films proximitized by the superconductor NbSe$_2$. By applying a small in-plane magnetic field, a screening supercurrent is induced which leads to finite-momentum pairing on topological surface states of Bi$_2$Te$_3$. Our measurements and analysis reveal the strong impact of finite Cooper pair momentum on the quasiparticle spectrum, and thus pave the way for STM study of pair density wave and FFLO states in unconventional superconductors.

preprint2020arXiv

Enhanced anomalous Nernst effect in disordered Dirac and Weyl materials

We analyze the thermoelectric response of Dirac and Weyl semimetals using the semiclassical approach, focusing on the extrinsic contributions due to skew-scattering and side jump. Our results apply to linear response Nernst effect in ferromagnetic Dirac materials such as Fe$_3$Sn$_2$, Weyl semimetal Co$_3$Sn$_2$S$_2$ and to second order response of monolayer graphene on hBN with trigonal warping. Our analysis indicates that the extrinsic contributions can be a significant component of anomalous Nernst response and used to explain an enhanced thermoelectric response.

preprint2020arXiv

Enhanced Superconductivity in Monolayer $T_d$-MoTe$_2$ with Tilted Ising Spin Texture

Crystalline two-dimensional (2D) superconductors with low carrier density are an exciting new class of materials in which superconductivity coexists with strong interactions, the effects of complex topology are not obscured by disorder, and electronic properties can be strongly tuned by electrostatic gating. Very recently, two such materials, 'magic-angle' twisted bilayer graphene and monolayer $T_d$-WTe$_2$, have been reported to show superconductivity at temperatures near 1 K. Here we report superconductivity in semimetallic monolayer $T_d$-MoTe$_2$. The critical temperature $T_\textrm{c}$ reaches 8 K, a sixty-fold enhancement as compared to the bulk. This anomalous increase in $T_\textrm{c}$ is only observed in monolayers, and may be indicative of electronically mediated pairing. Reflecting the low carrier density, the critical temperature, magnetic field, and current density are all tunable by an applied gate voltage, revealing a superconducting dome that extends across both hole and electron pockets. The temperature dependence of the in-plane upper critical field is distinct from that of $2H$ transition metal dichalcogenides (TMDs), consistent with a tilted spin texture as predicted by \textit{ab initio} theory.

preprint2020arXiv

High-frequency rectification via chiral Bloch electrons

Rectification is a process that converts electromagnetic fields into a direct current. Such a process underlies a wide range of technologies such as wireless communication, wireless charging, energy harvesting, and infrared detection. Existing rectifiers are mostly based on semiconductor diodes, with limited applicability to small voltage or high frequency inputs. Here, we present an alternative approach to current rectification that uses the intrinsic electronic properties of quantum crystals without using semiconductor junctions. We identify a previously unknown mechanism for rectification from skew scattering due to the inherent chirality of itinerant electrons in time-reversal-invariant but inversion-breaking materials. Our calculations reveal large, tunable rectification effects in graphene multilayers and transition metal dichalcogenides. Our work demonstrates the possibility of realizing high-frequency rectifiers by rational material design and quantum wavefunction engineering.

preprint2020arXiv

Local Probes for Quantum Hall Ferroelectrics and Nematics

Two-dimensional multi-valley electronic systems in which the dispersion of individual pockets has low symmetry give rise to quantum Hall ferroelectric and nematic states in the presence of strong quantising magnetic fields. We investigate local signatures of these states arising near impurities that can be probed via Scanning Tunnelling Microscopy (STM) spectroscopy. For quantum Hall ferroelectrics, we demonstrate a direct relation between the dipole moment measured at impurity bound states and the ideal bulk dipole moment obtained from the modern theory of polarisation. We also study the many-body problem with a single impurity via exact diagonalization and find that near strong impurities non-trivial excitonic state can form with specific features that can be easily identified via STM spectroscopy.

preprint2020arXiv

Nearly quantized conductance plateau of vortex zero mode in an iron-based superconductor

Majorana zero-modes (MZMs) are spatially-localized zero-energy fractional quasiparticles with non-Abelian braiding statistics that hold a great promise for topological quantum computing. Due to its particle-antiparticle equivalence, an MZM exhibits robust resonant Andreev reflection and 2e2/h quantized conductance at low temperature. By utilizing variable-tunnel-coupled scanning tunneling spectroscopy, we study tunneling conductance of vortex bound states on FeTe0.55Se0.45 superconductors. We report observations of conductance plateaus as a function of tunnel coupling for zero-energy vortex bound states with values close to or even reaching the 2e2/h quantum conductance. In contrast, no such plateau behaviors were observed on either finite energy Caroli-de Genne-Matricon bound states or in the continuum of electronic states outside the superconducting gap. This unique behavior of the zero-mode conductance reaching a plateau strongly supports the existence of MZMs in this iron-based superconductor, which serves as a promising single-material platform for Majorana braiding at a relatively high temperature.

preprint2020arXiv

Observation of a thermoelectric Hall plateau in the extreme quantum limit

The thermoelectric Hall effect is the generation of a transverse heat current upon applying an electric field in the presence of a magnetic field. Here we demonstrate that the thermoelectric Hall conductivity $α_{xy}$ in the three-dimensional Dirac semimetal ZrTe$_5$ acquires a robust plateau in the extreme quantum limit of magnetic field. The plateau value is independent of the field strength, disorder strength, carrier concentration, or carrier sign. We explain this plateau theoretically and show that it is a unique signature of three-dimensional Dirac or Weyl electrons in the extreme quantum limit. We further find that other thermoelectric coefficients, such as the thermopower and Nernst coefficient, are greatly enhanced over their zero-field values even at relatively low fields.

preprint2019arXiv

Classification of Critical Points in Energy Bands Based on Topology, Scaling and Symmetry

A critical point of the energy dispersion is the momentum where electron velocity vanishes. At the corresponding energy, the density of states (DOS) exhibits non-analyticity such as divergence. Critical points can be first classified as ordinary and high-order ones, and the ordinary critical points have been studied thoroughly by Léon van Hove. In this work, we describe and classify high-order critical points based on topology, scaling and symmetry, which are beyond Léon van Hove's framework. We show that high-order critical points can have power-law divergent DOS with particle-hole asymmetry, and can be realized at generic or symmetric momenta by tuning a few parameters such as twist angle, strain, pressure and/or external fields.

preprint2019arXiv

Magic of high order van Hove singularity

We introduce a new type of van Hove singularity in two dimensions, where a saddle point in momentum space is changed from second-order to high-order. Correspondingly, the density of states near such ``high-order van Hove singularity'' is significantly enhanced from logarithmic to power-law divergence, which promises stronger electron correlation effects. High-order van Hove singularity can be generally achieved by tuning the band structure with a single parameter in moiré superlattices, such as twisted bilayer graphene by tuning twist angle or applying pressure, and trilayer graphene by applying vertical electric field.

preprint2019arXiv

Supermetal

We study the effect of electron interaction in an electronic system with a high-order Van Hove singularity, where the density of states shows a power-law divergence. Owing to scale invariance, we perform a renormalization group (RG) analysis to find a nontrivial metallic behavior where various divergent susceptibilities coexist but no long-range order appears. We term such a metallic state as a supermetal. Our RG analysis reveals noninteracting and interacting fixed points, which draws an analogy to the $ϕ^4$ theory. We further present a finite anomalous dimension at the interacting fixed point by a controlled RG analysis, thus establishing an interacting supermetal as a non-Fermi liquid.

preprint2019arXiv

Thermoelectric response and entropy of fractional quantum Hall systems

We study thermoelectric transport properties of fractional quantum Hall systems based on exact diagonalization calculation. Based on the relation between thermoelectric response and thermal entropy, we demonstrate that thermoelectric Hall conductivity $α_{xy}$ has powerlaw scaling $α_{xy} \propto T^η$ for gapless composite Fermi-liquid states at filling number $ν=1/2$ and $1/4$ at low temperature ($T$), with exponent $η\sim 0.5$ distinctly different from Fermi liquids. The powerlaw scaling remains unchanged for different forms of interaction including Coulomb and short-range ones, demonstrating the robustness of non-Fermi-liquid behavior at low $T$. In contrast, for $1/3$ fractional quantum Hall state, $α_{xy}$ vanishes at low $T$ with an activation gap associated with neutral collective modes rather than charged quasiparticles. Our results establish a new manifestation of the non-Fermi-liquid nature of quantum Hall fluids at finite temperature.

preprint2017arXiv

Rotation Anomaly and Topological Crystalline Insulators

We show that in the presence of $n$-fold rotation symmetries and time-reversal symmetry, the number of fermion flavors must be a multiple of $2n$ ($n=2,3,4,6$) on two-dimensional lattices, a stronger version of the well-known fermion doubling theorem in the presence of only time-reversal symmetry. The violation of the multiplication theorems indicates anomalies, and may only occur on the surface of new classes of topological crystalline insulators. Put on a cylinder, these states have $n$ Dirac cones on the top and on the bottom surfaces, connected by $n$ helical edge modes on the side surface.

preprint2016arXiv

A new form of (unexpected) Dirac fermions in the strongly-correlated cerium monopnictides

Discovering Dirac fermions with novel properties has become an important front in condensed matter and materials sciences. Here, we report the observation of unusual Dirac fermion states in a strongly-correlated electron setting, which are uniquely distinct from those of graphene and conventional topological insulators. In strongly-correlated cerium monopnictides, we find two sets of highly anisotropic Dirac fermions that interpenetrate each other with negligible hybridization, and show a peculiar four-fold degeneracy where their Dirac nodes overlap. Despite the lack of protection by crystalline or time-reversal symmetries, this four-fold degeneracy is robust across magnetic phase transitions. Comparison of these experimental findings with our theoretical calculations suggests that the observed surface Dirac fermions arise from bulk band inversions at an odd number of high-symmetry points, which is analogous to the band topology which describes a $\mathbb{Z}_{2}$-topological phase. Our findings open up an unprecedented and long-sought-for platform for exploring novel Dirac fermion physics in a strongly-correlated semimetal.

preprint2016arXiv

Braiding without Braiding: Teleportation-Based Quantum Information Processing with Majorana Zero Modes

We present a new measurement-based scheme for performing braiding operations on Majorana zero modes and for detecting their non-Abelian statistics without moving or hybridizing them. In our scheme, the topological qubit encoded in any pair of well-separated Majorana zero modes in a mesoscopic superconductor island is read out from the transmission phase shift in electron teleportation through the island in the Coulomb blockade regime. We propose experimental setups to measure the teleportation phase shift via conductance in an electron interferometer or persistent current in a closed loop.

preprint2016arXiv

Conetronics in 2D Metal-Organic Frameworks: Double Dirac Cones, Magnetic Half Dirac Cones and Quantum Anomalous Hall Effect

Based on recently synthesized Ni3C12S12 class 2D metal-organic frameworks, we predict electronic properties of M3C12S12 and M3C12O12, where M is Zn, Cd, Hg, Be, or Mg with no M orbital contributions to bands near Fermi level. For M3C12S12, their band structures exhibit double Dirac cones with different Fermi velocities that are n and p type, respectively, which are switchable by few-percent strain. The crossing of two cones are symmetry-protected to be non-hybridizing, leading to two independent channels in 2D node-line semimetals at the same k-point akin to spin-channels in spintronics, rendering conetronics device possible. The node line rings right at their crossing, which are both electron and hole pockets at the Fermi level, can give rise to magnetoresistance that will not saturate when the magnetic field is infinitely large, due to perfect n-p compensation. For M3C12O12, together with conjugated metal-tricatecholate polymers M3(HHTP)2, the spin-polarized slow Dirac cone center is pinned precisely at the Fermi level, making the systems conducting in only one spin or cone channel. Quantum anomalous Hall effect can arise in MOFs with non-negligible spin-orbit coupling like Cu3C12O12. Compounds of M3C12S12 and M3C12O12 with different M, can be used to build spintronic and cone-selecting heterostructure devices, tunable by strain or electrostatic gating.

preprint2016arXiv

Fracton Topological Order, Generalized Lattice Gauge Theory and Duality

We introduce a generalization of conventional lattice gauge theory to describe fracton topological phases, which are characterized by immobile, point-like topological excitations, and sub-extensive topological degeneracy. We demonstrate a duality between fracton topological order and interacting spin systems with symmetries along extensive, lower-dimensional subsystems, which may be used to systematically search for and characterize fracton topological phases. Commutative algebra and elementary algebraic geometry provide an effective mathematical toolset for our results. Our work paves the way for identifying possible material realizations of fracton topological phases.

preprint2016arXiv

Identification of nematic superconductivity from the upper critical field

Recent nuclear magnetic resonance and specific heat measurements have provided concurring evidence of spontaneously broken rotational symmetry in the superconducting state of the doped topological insulator Cu$_x$Bi$_2$Se$_3$. This suggests that the pairing symmetry corresponds to a two-dimensional representation of the $D_{3d}$ crystal point group, and that Cu$_x$Bi$_2$Se$_3$ is a nematic superconductor. In this work, we present a comprehensive study of the upper critical field $H_{c2}$ of nematic superconductors within Ginzburg-Landau (GL) theory. Contrary to typical GL theories which have an emergent U(1) rotational symmetry obscuring the discrete symmetry of the crystal, the theory of two-component superconductors in trigonal crystals reflects the true $D_{3d}$ crystal symmetry. This has direct implications for the upper critical field. First, $H_{c2}$ of trigonal superconductors exhibits a sixfold anisotropy in the basal plane. Second, when the degeneracy of the two components is lifted by, e.g., uniaxial strain, $H_{c2}$ exhibits a twofold anisotropy with characteristic angle and temperature dependence. Our thorough study shows that measurement of the upper critical field is a direct method of detecting nematic superconductivity, which is directly applicable to recently-discovered trigonal superconductors Cu$_x$Bi$_2$Se$_3$, Sr$_x$Bi$_2$Se$_3$, Nb$_x$Bi$_2$Se$_3$, and Tl$_x$Bi$_2$Se$_3$.

preprint2016arXiv

Majorana fermions in a superconducting Mobius strip

Recently, much attention has been paid to search for Majorana fermions in solid-state systems. Among various proposals there is one based on radio-frequency superconducting quantum interference devices (rf-SQUIDs), in which the appearance of 4$π$-period energy-phase relations is regarded as smoking-gun evidence of Majorana fermion states. Here we report the observation of truncated 4$π$-period (i.e., 2$π$-period but fully skewed) oscillatory patterns of contact resistance on rf-SQUIDs constructed on the surface of three-dimensional topological insulator Bi$_2$Te$_3$. The results reveal the existence of 1/2 fractional modes of Cooper pairs and the occurrence of parity switchings, both of which are necessary signatures accompanied with the formation of Majorana fermion states.

preprint2016arXiv

New quantum critical points of $j=3/2$ Dirac electrons in antiperovskite topological crystalline insulators

We study the effect of the long-range Coulomb interaction in $j=3/2$ Dirac electrons in cubic crystals with the $O_h$ symmetry, which serves as an effective model for antiperovskite topological crystalline insulators. The renormalization group analysis reveals three fixed points that are Lorentz invariant, rotationally invariant, and $O_h$ invariant. Among them, the Lorentz- and $O_h$-invariant fixed points are stable in the low-energy limit while the rotationally invariant fixed point is unstable. The existence of a stable $O_h$-invariant fixed point of Dirac fermions with finite velocity anisotropy presents an interesting counterexample to emergent Lorentz invariance in solids.

preprint2016arXiv

Observation of Majorana fermions with spin selective Andreev reflection in the vortex of topological superconductor

Majorana fermion (MF) whose antiparticle is itself has been predicted in condensed matter systems. Signatures of the MFs have been reported as zero energy modes in various systems. More definitive evidences are highly desired to verify the existence of the MF. Very recently, theory has predicted MFs to induce spin selective Andreev reflection (SSAR), a novel magnetic property which can be used to detect the MFs. Here we report the first observation of the SSAR from MFs inside vortices in Bi2Te3/NbSe2 hetero-structure, in which topological superconductivity was previously established. By using spin-polarized scanning tunneling microscopy/spectroscopy (STM/STS), we show that the zero-bias peak of the tunneling differential conductance at the vortex center is substantially higher when the tip polarization and the external magnetic field are parallel than anti-parallel to each other. Such strong spin dependence of the tunneling is absent away from the vortex center, or in a conventional superconductor. The observed spin dependent tunneling effect is a direct evidence for the SSAR from MFs, fully consistent with theoretical analyses. Our work provides definitive evidences of MFs and will stimulate the MFs research on their novel physical properties, hence a step towards their statistics and application in quantum computing.

preprint2016arXiv

Odd-parity superconductors with two-component order parameters: nematic and chiral, full gap and Majorana node

Motivated by the recent experiment indicating that superconductivity in the doped topological insulator Cu$_x$Bi$_2$Se$_3$ has an odd-parity pairing symmetry with rotational symmetry breaking, we study the general class of odd-parity superconductors with two-component order parameters in trigonal and hexagonal crystal systems. In the presence of strong spin-orbit interaction, we find two possible superconducting phases below $T_c$, a time-reversal-breaking (i.e., chiral) phase and an anisotropic (i.e., nematic) phase, and determine their relative energetics from the gap function in momentum space. The nematic superconductor generally has a full quasi-particle gap, whereas the chiral superconductor with a three-dimensional (3D) Fermi surface has point nodes with lifted spin degeneracy, resulting in itinerant Majorana fermions in the bulk and topological Majorana arcs on the surface.

preprint2016arXiv

Tunable Weyl fermions and Fermi arcs in magnetized topological crystalline insulators

Based on $k\cdot p$ analysis and realistic tight-binding calculations, we find that time-reversal-breaking Weyl semimetals can be realized in magnetically-doped (Mn, Eu, Cr etc.) Sn$_{1-x}$Pb$_x$(Te,Se) class of topological crystalline insulators. All the Weyl points are well separated in momentum space and possess nearly the same energy due to high crystalline symmetry. Moreover, both the Weyl points and Fermi arcs are highly tunable by varying Pb/Sn composition, pressure, magnetization, temperature, surface potential etc., opening up the possibility of manipulating Weyl points and rewiring the Fermi arcs.

preprint2015arXiv

A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations

We introduce exactly solvable models of interacting (Majorana) fermions in $d \ge 3$ spatial dimensions that realize a new kind of topological quantum order, building on a model presented in ref. [1]. These models have extensive topological ground-state degeneracy and a hierarchy of point-like, topological excitations that are only free to move within sub-manifolds of the lattice. In particular, one of our models has fundamental excitations that are completely stationary. To demonstrate these results, we introduce a powerful polynomial representation of commuting Majorana Hamiltonians. Remarkably, the physical properties of the topologically-ordered state are encoded in an algebraic variety, defined by the common zeros of a set of polynomials over a finite field. This provides a "geometric" framework for the emergence of topological order.

preprint2015arXiv

Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators

The surface of a three-dimensional topological electron system often hosts symmetry-protected gapless surface states. With the effect of electron interactions, these surface states can be gapped out without symmetry breaking by a surface topological order, in which the anyon excitations carry anomalous symmetry fractionalization that cannot be realized in a genuine two-dimensional system. We show that for a mirror-symmetry-protected topological crystalline insulator with mirror Chern number $n=4$, its surface can be gapped out by an anomalous $\mathbb Z_2$ topological order, where all anyons carry mirror-symmetry fractionalization $M^2=-1$. The identification of such anomalous crystalline symmetry fractionalization implies that in a two-dimensional $\mathbb Z_2$ spin liquid the vison excitation cannot carry $M^2=-1$ if the spinon carries $M^2=-1$ or a half-integer spin.

preprint2015arXiv

Crystal Field Effect Induced Topological Crystalline Insulators In Monolayer IV-VI Semiconductors

Two-dimensional (2D) topological crystalline insulators (TCIs) were recently predicted in thin films of the SnTe class of IV-VI semiconductors, which can host metallic edge states protected by mirror symmetry. As thickness decreases, quantum confinement effect will increase and surpass the inverted gap below a critical thickness, turning TCIs into normal insulators. Surprisingly, based on first-principles calculations, here we demonstrate that (001) monolayers of rocksalt IV-VI semiconductors XY (X=Ge, Sn, Pb and Y= S, Se, Te) are 2D TCIs with the fundamental band gap as large as 260 meV in monolayer PbTe, providing a materials platform for realizing two-dimensional Dirac fermion systems with tunable band gap. This unexpected nontrivial topological phase stems from the strong {\it crystal field effect} in the monolayer, which lifts the degeneracy between $p_{x,y}$ and $p_z$ orbitals and leads to band inversion between cation $p_z$ and anion $p_{x,y}$ orbitals. This crystal field effect induced topological phase offers a new strategy to find and design other atomically thin 2D topological materials.

preprint2015arXiv

Detecting crystal symmetry fractionalization from the ground state: Application to $\mathbb Z_2$ spin liquids on the kagome lattice

In quantum spin liquid states, the fractionalized spinon excitations can carry fractional crystal symmetry quantum numbers, and this symmetry fractionalization distinguishes different topologically ordered spin liquid states. In this work we propose a simple way to detect signatures of such crystal symmetry fractionalizations from the crystal symmetry representations of the ground state wave function. We demonstrate our method on projected $\mathbb Z_2$ spin liquid wave functions on the kagome lattice, and show that it can be used to classify generic wave functions. Particularly our method can be used to distinguish several proposed candidates of $\mathbb Z_2$ spin liquid states on the kagome lattice.

preprint2015arXiv

Electrically Tunable Quantum Spin Hall State in Topological Crystalline Insulator Thin films

Based on a combination of $k \cdot p$ theory, band topology analysis and electronic structure calculations, we predict the (111) thin films of the SnTe class of three-dimensional (3D) topological crystalline insulators realize the quantum spin Hall phase in a wide range of thickness. The nontrivial topology originates from the inter-surface coupling of the topological surface states of TCI in the 3D limit. The inter-surface coupling changes sign and gives rise to topological phase transitions as a function of film thickness. Furthermore, this coupling can be strongly affected by an external electric field, hence the quantum spin Hall phase can be effectively tuned under experimentally accessible the electric field. Our results show that (111) thin films of SnTe-class TCI can be an ideal platform to realize the novel applications of quantum spin Hall insulators.

preprint2015arXiv

Experimental observation of Weyl points

In 1929, Hermann Weyl derived the massless solutions from the Dirac equation - the relativistic wave equation for electrons. Neutrinos were thought, for decades, to be Weyl fermions until the discovery of the neutrino mass. Moreover, it has been suggested that low energy excitations in condensed matter can be the solutions to the Weyl Hamiltonian. Recently, photons have also been proposed to emerge as Weyl particles inside photonic crystals. In all cases, two linear dispersion bands in the three-dimensional (3D) momentum space intersect at a single degenerate point - the Weyl point. Remarkably, these Weyl points are monopoles of Berry flux with topological charges defined by the Chern numbers. These topological invariants enable materials containing Weyl points to exhibit a wide variety of novel phenomena including surface Fermi arcs, chiral anomaly, negative magnetoresistance, nonlocal transport, quantum anomalous Hall effect, unconventional superconductivity[15] and others [16, 17]. Nevertheless, Weyl points are yet to be experimentally observed in nature. In this work, we report on precisely such an observation in an inversion-breaking 3D double-gyroid photonic crystal without breaking time-reversal symmetry.

preprint2015arXiv

Interacting Dirac fermions under spatially alternating pseudo-magnetic field: Realization of spontaneous quantum Hall effect

Both topological crystalline insulators surfaces and graphene host multi-valley massless Dirac fermions which are not pinned to a high-symmetry point of the Brillouin zone. Strain couples to the low-energy electrons as a time-reversal invariant gauge field, leading to the formation of pseudo-Landau levels (PLL). Here we study periodic pseudo-magnetic fields originating from strain superlattices. We study the low-energy Dirac PLL spectrum induced by the strain superlattice and analyze the effect of various polarized states. Through self-consistent Hartree-Fock calculations we establish that, due to the strain superlattice and PLL electronic structure, a valley-ordered state spontaneously breaking time-reversal and realizing a quantum Hall phase is favored, while others are suppressed.

preprint2015arXiv

Majorana Fermion Surface Code for Universal Quantum Computation

We introduce an exactly solvable model of interacting Majorana fermions realizing $Z_{2}$ topological order with a $Z_{2}$ fermion parity grading and lattice symmetries permuting the three fundamental anyon types. We propose a concrete physical realization by utilizing quantum phase slips in an array of Josephson-coupled mesoscopic topological superconductors, which can be implemented in a wide range of solid state systems, including topological insulators, nanowires or two-dimensional electron gases, proximitized by $s$-wave superconductors. Our model finds a natural application as a Majorana fermion surface code for universal quantum computation, with a single-step stabilizer measurement requiring no physical ancilla qubits, increased error tolerance, and simpler logical gates than a surface code with bosonic physical qubits. We thoroughly discuss protocols for stabilizer measurements, encoding and manipulating logical qubits, and gate implementations.

preprint2015arXiv

New classes of topological crystalline insulators with unpinned surface Dirac cones

We theoretically predict two new classes of three-dimensional topological crystalline insulators (TCIs), which have an odd number of unpinned surface Dirac cones protected by crystal symmetries. The first class is protected by a single glide plane symmetry; the second class is protected by a composition of a twofold rotation and time-reversal symmetry. Both classes of TCIs are characterized by a quantized $π$ Berry phase associated with surface states and a $Z_2$ topological invariant associated with the bulk bands. In the presence of disorder, these TCI surface states are protected against localization by the average crystal symmetries, and exhibit critical conductivity in the universality class of the quantum Hall plateau transition. These new TCIs exist in time-reversal-breaking systems with or without spin-orbital coupling, and their material realizations are discussed.

preprint2015arXiv

Odd-parity superconductivity in the vicinity of inversion symmetry breaking in spin-orbit-coupled systems

We study superconductivity in spin-orbit-coupled systems in the vicinity of inversion symmetry breaking. We find that due to the presence of spin-orbit coupling, fluctuations of the incipient parity-breaking order generate an attractive pairing interaction in an odd-parity pairing channel, which competes with the s-wave pairing. We show that applying a Zeeman field suppresses the s-wave pairing and promotes the odd-parity superconducting state. Our work provides a new mechanism for odd-parity pairing and opens a route to novel topological superconductivity.

preprint2015arXiv

Parity-breaking phases of spin-orbit-coupled metals with gyrotropic, ferroelectric and multipolar orders

We study Fermi liquid instabilities in spin-orbit-coupled metals with inversion symmetry. By introducing a canonical basis for the doubly degenerate Bloch bands in momentum space, we derive the general form of interaction functions. A variety of time-reversal-invariant, parity-breaking phases is found, whose Fermi surface is spontaneously deformed and spin-split. In terms of symmetry, these phases possess gyrotropic, ferroelectric and multipolar orders. The ferroelectric and multipolar phases are accompanied by structural distortions, from which the electronic orders can be identified. The gyrotropic phase exhibits a unique nonlinear optical property. Based on recent experiments, we identify several interesting quantum materials including pyrochlore oxides, which show evidence of these parity breaking orders.

preprint2015arXiv

Physical Implementation of a Majorana Fermion Surface Code for Fault-Tolerant Quantum Computation

We propose a physical realization of a commuting Hamiltonian of interacting Majorana fermions realizing $Z_{2}$ topological order, using an array of Josephson-coupled topological superconductor islands. The required multi-body interaction Hamiltonian is naturally generated by a combination of charging energy induced quantum phase-slips on the superconducting islands and electron tunneling. Our setup improves on a recent proposal for implementing a Majorana fermion surface code [1], a 'hybrid' approach to fault-tolerant quantum computation that combines (1) the engineering of a stabilizer Hamiltonian with a topologically ordered ground state with (2) projective stabilizer measurements to implement error correction and a universal set of logical gates. Our hybrid strategy has advantages over the traditional surface code architecture in error suppression and single-step stabilizer measurements, and is widely applicable to implementing stabilizer codes for quantum computation.

preprint2015arXiv

Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials

It is well-known that a non-vanishing Hall conductivity requires time-reversal symmetry breaking. However, in this work, we demonstrate that a Hall-like transverse current can occur in second-order response to an external electric field in a wide class of time-reversal invariant and inversion breaking materials, at both zero and twice the optical frequency. This nonlinear Hall effect has a quantum origin arising from the dipole moment of the Berry curvature in momentum space, which generates a net anomalous velocity when the system is in a current-carrying state. We show that the nonlinear Hall coefficient is a rank-two pseudo-tensor, whose form is determined by point group symmetry. We discus optimal conditions to observe this effect and propose candidate two- and three-dimensional materials, including topological crystalline insulators, transition metal dichalcogenides and Weyl semimetals.

preprint2015arXiv

Theory of interacting topological crystalline insulators

We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multi-valley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group $Z_8$. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap 8 flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U(1)\times Z_2$ symmetry-protected topological (SPT) phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between 3D topological phases protected by spatial symmetries and 2D topological phases protected by internal symmetries.

preprint2015arXiv

Three-dimensional topological photonic crystal with a single surface Dirac cone

A single Dirac cone on the surface is the hallmark of three-dimensional (3D) topological insulators, where the double degeneracy at the Dirac point is protected by time-reversal symmetry and the spin-splitting away from the point is provided by the spin-orbital coupling. Here we predict a single Dirac-cone surface state in a 3D photonic crystal, where the degeneracy at the Dirac point is protected by a nonsymmorphic glide reflection and the linear splitting away from it is enabled by breaking time-reversal symmetry. Such a gapless surface state is fully robust against random disorder of any type. This bosonic topological band structure is achieved by applying alternating magnetization to gap out the 3D "generalized Dirac points" discovered in the bulk of our crystal. The $Z_2$ bulk invariant is characterized through the evolution of Wannier centers. Our proposal--readily realizable using ferrimagnetic materials at microwave frequencies--can also be regarded as the photonic analog of topological crystalline insulators, providing the first 3D bosonic symmetry-protected topological system.

preprint2015arXiv

Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials

In this review, we discuss recent progress in the explorations of topological materials beyond topological insulators; specifically, we focus on topological crystalline insulators and bulk topological superconductors. The basic concepts, model Hamiltonians, and novel electronic properties of these new topological materials are explained. The key role of symmetries that underlie their topological properties is elucidated. Key issues in their materials realizations are also discussed.

preprint2015arXiv

Topological nodal line semimetals with and without spin-orbital coupling

We theoretically study three-dimensional topological semimetals (TSMs) with nodal lines protected by crystalline symmetries. Compared with TSMs with point nodes, e.g., Weyl semimetals and Dirac semimetals, where the conduction and the valence bands touch at discrete points, in these new TSMs the two bands cross at closed lines in the Brillouin zone. We propose two new classes of symmetry protected nodal lines in the absence and in the presence of spin-orbital coupling (SOC), respectively. In the former, we discuss nodal lines that are protected by the combination of inversion symmetry and time-reversal symmetry; yet unlike any previously studied nodal lines in the same symmetry class, each nodal line has a $Z_2$ monopole charge and can only be created (annihilated) in pairs. In the second class, with SOC, we show that a nonsymmorphic symmetry (screw axis) protects a four-band crossing nodal line in systems having both inversion and time-reversal symmetries.

preprint2015arXiv

Topological semimetals and topological insulators in rare earth monopnictides

We use first principles calculations to study the electronic properties of rock salt rare earth monopnictides La$X$ ($X=$N, P, As, Sb, Bi). A new type of topological band crossing termed `linked nodal rings' is found in LaN when the small spin-orbital coupling (SOC) on nitrogen orbitals is neglected. Turning on SOC gaps the nodal rings at all but two points, which remain gapless due to $C_4$-symmetry and leads to a 3D Dirac semimetal. Interestingly, unlike LaN, compounds with other elements in the pnictogen group are found to be topological insulators (TIs), as a result of band reordering due to the increased lattice constant as well as the enhanced SOC on the pnictogen atom. These TI compounds exhibit multi-valley surface Dirac cones at three $\bar{M}$-points on the $(111)$-surface.

preprint2014arXiv

Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

A quantum phase transition is usually achieved by tuning physical parameters in a Hamiltonian at zero temperature. Here, we demonstrate that the ground state of a topological phase itself encodes critical properties of its transition to a trivial phase. To extract this information, we introduce a partition of the system into two subsystems both of which extend throughout the bulk in all directions. The resulting bulk entanglement spectrum has a low-lying part that resembles the excitation spectrum of a bulk Hamiltonian, which allows us to probe a topological phase transition from a single wavefunction by tuning either the geometry of the partition or the entanglement temperature. As an example, this remarkable correspondence between topological phase transition and entanglement criticality is rigorously established for integer quantum Hall states.

preprint2014arXiv

Dirac mass generation from crystal symmetry breaking on the surfaces of topological crystalline insulators

The tunability of topological surface states (SS) and controllable opening of the Dirac gap are of fundamental and practical interest in the field of topological materials. In topological crystalline insulators (TCIs), a spontaneously generated Dirac gap was recently observed, which was ascribed to broken cubic crystal symmetry. However, this structural distortion has not been directly observed so far, and the microscopic mechanism of Dirac gap opening via crystal symmetry breaking remains elusive. In this work, we present scanning tunneling microscopy (STM) measurements of a TCI Pb$_{1-x}$Sn$_x$Se for a wide range of alloy compositions spanning the topological and non-topological regimes. STM topographies directly reveal a symmetry-breaking distortion on the surface, which imparts mass to the otherwise massless Dirac electrons - a mechanism analogous to the long sought-after Higgs mechanism in particle physics. Remarkably, our measurements show that the Dirac gap scales with alloy composition, while the magnitude of the distortion remains nearly constant. Based on theoretical calculations, we find the Dirac mass is controlled by the composition-dependent SS penetration depth, which determines the weight of SS in the distorted region that is confined to the surface. Finally, we discover the existence of SS in the non-topological regime, which have the characteristics of gapped, double-branched Dirac fermions.

preprint2014arXiv

Entanglement Spectrum of a Random Partition: Connection with the Localization Transition

We study the entanglement spectrum of a translationally-invariant lattice system under a random partition, implemented by choosing each site to be in one subsystem with probability $p\in[0, 1]$. We apply this random partitioning to a translationally-invariant (i.e., clean) topological state, and argue on general grounds that the corresponding entanglement spectrum captures the universal behavior about its disorder-driven transition to a trivial localized phase. Specifically, as a function of the partitioning probability $p$, the entanglement Hamiltonian $H_{A}$ must go through a topological phase transition driven by the percolation of a random network of edge-states. As an example, we analytically derive the entanglement Hamiltonian for a one-dimensional topological superconductor under a random partition, and demonstrate that its phase diagram includes transitions between Griffiths phases.

preprint2014arXiv

Experimental observation of Dirac-like surface states and topological phase transition in Pb$_{1-x}$Sn$_x$Te(111) films

The surface of a topological crystalline insulator (TCI) carries an even number of Dirac cones protected by crystalline symmetry. We epitaxially grew high quality Pb$_{1-x}$Sn$_x$Te(111) films and investigated the TCI phase by in-situ angle-resolved photoemission spectroscopy. Pb$_{1-x}$Sn$_x$Te(111) films undergo a topological phase transition from trivial insulator to TCI via increasing the Sn/Pb ratio, accompanied by a crossover from n-type to p-type doping. In addition, a hybridization gap is opened in the surface states when the thickness of film is reduced to the two-dimensional limit. The work demonstrates an approach to manipulating the topological properties of TCI, which is of importance for future fundamental research and applications based on TCI.

preprint2014arXiv

Loops, sign structures and emergent Fermi statistics in three-dimensional quantum dimer models

We introduce and study three-dimensional quantum dimer models with positive resonance terms. We demonstrate that their ground state wave functions exhibit a nonlocal sign structure that can be exactly formulated in terms of loops, and as a direct consequence, monomer excitations obey Fermi statistics. The sign structure and Fermi statistics in these "signful" quantum dimer models can be naturally described by a parton construction, which becomes exact at the solvable point.

preprint2014arXiv

Odd-parity topological superconductor with nematic order: Application to CuxBi2Se3

CuxBi2Se3 was recently proposed as a promising candidate for time-reversal-invariant topological superconductors[1]. In this work, we argue that the unusual anisotropy of the Knight shift observed by Zheng[2], taken together with specific heat measurements, provides strong support for an unconventional odd-parity pairing in the two-dimensional E_u representation of the D3d crystal point group[1], which spontaneously breaks the three-fold rotational symmetry of the crystal, leading to a subsidiary nematic order. We predict that the spin-orbit interaction associated with hexagonal warping plays a crucial role in pinning the two-component order parameter and makes the superconducting state fully-gapped, leading to a topological superconductor. Experimental signatures of the E_u pairing related to the nematic order are discussed.

preprint2014arXiv

Quantum Spin Hall Effect and Topological Field Effect Transistor in Two-Dimensional Transition Metal Dichalcogenides

We report a new class of large-gap quantum spin Hall insulators in two-dimensional transition metal dichalcogenides, namely, MX$_2$ with M=(Mo, W) and X=(S, Se, and Te), whose topological electronic properties are highly tunable by external electric field. We propose a novel topological field effect transistor made of these atomic layer materials and their van der Waals heterostructures. Our device exhibits parametrically enhanced charge-spin conductance through topologically protected transport channels, and can be rapidly switched off by electric field through topological phase transition instead of carrier depletion. Our work provides a practical material platform and device architecture for topological quantum electronics.

preprint2014arXiv

Strain-induced partially flat band, helical snake states, and interface superconductivity in topological crystalline insulators

Topological crystalline insulators in IV-VI compounds host novel topological surface states consisting of multi-valley massless Dirac fermions at low energy. Here we show that strain generically acts as an effective gauge field on these Dirac fermions and creates pseudo-Landau orbitals without breaking time-reversal symmetry. We predict the realization of this phenomenon in IV-VI semiconductor heterostructures, due to a naturally occurring misfit dislocation array at the interface that produces a periodically varying strain field. Remarkably, the zero-energy Landau orbitals form a flat band in the vicinity of the Dirac point, and coexist with a network of snake states at higher energy. We propose that the high density of states of this flat band gives rise to interface superconductivity observed in IV-VI semiconductor multilayers at unusually high temperatures, with non-BCS behavior. Our work demonstrates a new route to altering macroscopic electronic properties to achieve a partially flat band, and paves the way for realizing novel correlated states of matter.

preprint2014arXiv

Symmetry breaking and Landau quantization in topological crystalline insulators

In the recently discovered topological crystalline insulators SnTe and Pb_{1-x}Sn_{x}(Te,Se), crystal symmetry and electronic topology intertwine to create topological surface states with many interesting features including Lifshitz transition, Van-Hove singularity and fermion mass generation. These surface states are protected by mirror symmetry with respect to the (110) plane. In this work we present a comprehensive study of the effects of different mirror-symmetry-breaking perturbations on the (001) surface band structure. Pristine (001) surface states have four branches of Dirac fermions at low-energy. We show that ferroelectric-type structural distortion generates a mass and gaps out some or all of these Dirac points, while strain shifts Dirac points in the Brillouin zone. An in-plane magnetic field leaves surface state gapless, but introduces asymmetry between Dirac points. Finally, an out-of-plane magnetic field leads to discrete Landau levels. We show that the Landau level spectrum has an unusual pattern of degeneracy and interesting features due to the unique underlying band structure. This suggests that Landau level spectroscopy can detect and distinguish between different mechanisms of symmetry breaking in topological crystalline insulators.

preprint2014arXiv

Tensor Network Implementation of Bulk Entanglement Spectrum

Many topologically nontrivial states of matter possess gapless degrees of freedom on the boundary, and when these boundary states delocalize into the bulk, a phase transition occurs and the system becomes topologically trivial. We show that tensor networks provide a natural framework for analyzing such topological phase transitions in terms of the boundary degrees of freedom which mediate it. To do so, we make use of a correspondence between a topologically nontrivial ground state and its phase transition to a trivial phase established in [1]. This involved computing the bulk entanglement spectrum (BES) of the ground state, upon tracing out an extensive subsystem. This work implements BES via tensor network representations of ground states. In this framework, the universality class of the quantum critical entanglement Hamiltonian in d spatial dimensions is either derived analytically or mapped to a classical statistical model in d + 1 dimensions, which can be studied using Monte Carlo or tensor renormalization group methods. As an example, we analytically derive the universality classes of topological phase transitions from the spin-1 chain Haldane phase and demonstrate that the AKLT wavefunction (and its generalizations) remarkably contains critical six-vertex (and in general eight-vertex) models within it.

preprint2014arXiv

Topological Crystalline Insulator Nanomembrane with Strain-Tunable Band Gap

The ability to fine-tune band gap and band inversion in topological materials is highly desirable for the development of novel functional devices. Here we propose that the electronic properties of a free-standing nanomembrane of topological crystalline insulator (TCI) SnTe and Pb$_{1-x}$Sn$_x$(Se,Te) are highly tunable by engineering elastic strain and controlling membrane thickness, resulting in tunable band gap and giant piezoconductivity. Membrane thickness governs the hybridization of topological electronic states on opposite surfaces, while elastic strain can further modulate the hybridization strength by controlling the penetration length of surface states. We propose a frequency-resolved infrared photodetector using force-concentration induced inhomogeneous elastic strain in TCI nanomembrane with spatially varying width. The predicted tunable band gap accompanied by strong spin-textured electronic states will open up new avenues for fabricating piezoresistive devices, thermoelectrics, infrared detectors and energy-efficient electronic and optoelectronic devices based on TCI nanomembrane.

preprint2014arXiv

Topological Crystalline Insulators and Dirac Octets in Anti-perovskites

We predict a new class of topological crystalline insulators (TCI) in the anti-perovskite material family with the chemical formula A$_3$BX. Here the nontrivial topology arises from band inversion between two $J=3/2$ quartets, which is described by a generalized Dirac equation for a "Dirac octet". Our work suggests that anti-perovskites are a promising new venue for exploring the cooperative interplay between band topology, crystal symmetry and electron correlation.

preprint2014arXiv

Valley-selective optical Stark effect in monolayer WS2

Breaking space-time symmetries in two-dimensional crystals (2D) can dramatically influence their macroscopic electronic properties. Monolayer transition-metal dichalcogenides (TMDs) are prime examples where the intrinsically broken crystal inversion symmetry permits the generation of valley-selective electron populations, even though the two valleys are energetically degenerate, locked by time-reversal symmetry. Lifting the valley degeneracy in these materials is of great interest because it would allow for valley-specific band engineering and offer additional control in valleytronic applications. While applying a magnetic field should in principle accomplish this task, experiments to date have observed no valley-selective energy level shifts in fields accessible in the laboratory. Here we show the first direct evidence of lifted valley degeneracy in the monolayer TMD WS2. By applying intense circularly polarized light, which breaks time-reversal symmetry, we demonstrate that the exciton level in each valley can be selectively tuned by as much as 18 meV via the optical Stark effect. These results offer a novel way to control valley degree of freedom, and may provide a means to realize new valley-selective Floquet topological phases in 2D TMDs.

preprint2013arXiv

Anomalous Supercurrent from Majorana States in Topological Insulator Josephson Junctions

We propose a Josephson junction setup based on a topological insulator (TI) thin film to detect Majorana states, which exploits the unique helical and extended nature of the TI surface state. When the magnetic flux through the junction is close to an integer number of flux quanta, Majorana states, present on both surfaces of the film, give rise to a narrow peak-dip structure in the current- phase relation by hybridizing at the edge of the junction. Moreover, the maximal Majorana-state contribution to Josephson current takes a (nearly) universal value, approximately equal to the supercurrent capacity of a single quantum-channel. These features provide a characteristic signature of Majorana states based entirely on supercurrent.

preprint2013arXiv

Inducing time reversal invariant topological superconductivity and fermion parity pumping in quantum wires

We propose a setup to realize time-reversal invariant topological superconductors in quantum wires, proximity coupled to conventional superconductors. We consider a model of quantum wire with strong spin-orbit coupling and proximity coupling to two s-wave superconductors. When the relative phase between the two superconductors is $ϕ=π$ a Kramers' pair of Majorana zero modes appears at each edge of the wire. We study the robustness of the phase in presence of both time-reversal invariant and time-reversal breaking perturbations. In addition, we show that the system forms a natural realization of a fermion parity pump, switching the local fermion parity of both edges when the relative phase between the superconductors is changed adiabatically by $2π$.

preprint2013arXiv

Mapping the unconventional orbital texture in topological crystalline insulators

The newly discovered topological crystalline insulators (TCIs) harbor a complex band structure involving multiple Dirac cones. These materials are potentially highly tunable by external electric field, temperature or strain and could find future applications in field-effect transistors, photodetectors, and nano-mechanical systems. Theoretically, it has been predicted that different Dirac cones, offset in energy and momentum-space, might harbor vastly different orbital character, a unique property which if experimentally realized, would present an ideal platform for accomplishing new spintronic devices. However, the orbital texture of the Dirac cones, which is of immense importance in determining a variety of materials properties, still remains elusive in TCIs. Here, we unveil the orbital texture in a prototypical TCI Pb$_{1-x}$Sn$_x$Se. By using Fourier-transform (FT) scanning tunneling spectroscopy (STS) we measure the interference patterns produced by the scattering of surface state electrons. We discover that the intensity and energy dependences of FTs show distinct characteristics, which can directly be attributed to orbital effects. Our experiments reveal the complex band topology involving two Lifshitz transitions and establish the orbital nature of the Dirac bands in this new class of topological materials, which could provide a different pathway towards future quantum applications.

preprint2013arXiv

Observation of Dirac node formation and mass acquisition in a topological crystalline insulator

In the recently discovered topological crystalline insulators (TCIs), topology and crystal symmetry intertwine to create surface states with a unique set of characteristics. Among the theoretical predictions for TCIs is the possibility of imparting mass to the massless Dirac fermions by breaking crystal symmetry, as well as a Lifshitz transition with a change of Fermi surface topology. Here we report high resolution scanning tunneling microscopy studies of a TCI, Pb1-xSnxSe. We demonstrate the formation of zero mass Dirac fermions protected by crystal symmetry and the mechanism of mass generation via symmetry breaking, which constitute the defining characteristics of TCIs. In addition, we show two distinct regimes of fermiology separated by a Van-Hove singularity at the Lifshitz transition point. Our work paves the way for engineering the Dirac band gap and realizing interaction-driven topological quantum phenomena in TCIs.

preprint2013arXiv

Spin-filtered Edge States with an Electrically Tunable Gap in a Two-Dimensional Topological Crystalline Insulator

Three-dimensional topological crystalline insulators were recently predicted and observed in the SnTe class of IV-VI semiconductors, which host metallic surface states protected by crystal symmetries. In this work, we study thin films of these materials and expose their potential for device applications. We demonstrate that thin films of SnTe and Pb(1-x)Sn(x)Se(Te) grown along the (001) direction are topologically nontrivial in a wide range of film thickness and carry conducting spin-filtered edge states that are protected by the (001) mirror symmetry via a topological invariant. Application of an electric field perpendicular to the film will break the mirror symmetry and generate a band gap in these edge states. This functionality motivates us to propose a novel topological transistor device, in which charge and spin transport are maximally entangled and simultaneously controlled by an electric field. The high on/off operation speed and coupling of spin and charge in such a device may lead to electronic and spintronic applications for topological crystalline insulators.

preprint2013arXiv

Surface States of Topological Crystalline Insulators in IV-VI Semiconductors

Topological crystalline insulators (TCI) are new topological phases of matter protected by crystal symmetry of solids. Recently, the first realization of TCI has been predicted and observed in IV-VI semiconductor SnTe and related alloys Pb_{1-x}Sn_{x}(Te, Se). By combining k.p theory and band structure calculation, we present a unified approach to study topological surface states on various crystal surfaces of TCI in IV-VI semiconductors. We explicitly derive k.p Hamiltonian for topological surface states from electronic structure of the bulk, thereby providing a microscopic understanding of bulk-boundary correspondence in TCI. Depending on the surface orientation, we find two types of surface states with qualitatively different properties. In particular, we predict that (111) surface states consist of four Dirac cones centered at time-reversal-invariant momenta Γ and M, while (110) surface states consist of Dirac cones at non-time-reversal-invariant momenta, similar to (001). Moreover, both (001) and (110) surface states exhibit a Lifshitz transition as a function of Fermi energy, which is accompanied by a Van-Hove singularity in density of states arising from saddle points in the band structure.

preprint2013arXiv

Weak Topological Insulators in PbTe/SnTe Superlattices

It is desirable to realize topological phases in artificial structures by engineering electronic band structures. In this paper, we investigate $(PbTe)_m(SnTe)_{2n-m}$ superlattices along [001] direction and find a robust weak topological insulator phase for a large variety of layer numbers m and 2n-m. We confirm this topologically non-trivial phase by calculating Z2 topological invariants and topological surface states based on the first-principles calculations. We show that the folding of Brillouin zone due to the superlattice structure plays an essential role in inducing topologically non-trivial phases in this system. This mechanism can be generalized to other systems in which band inversion occurs at multiple momenta, and gives us a brand-new way to engineer topological materials in artificial structures.

preprint2012arXiv

Majorana Fermions and Exotic Surface Andreev Bound States in Topological Superconductors: Application to CuxBi2Se3

The recently discovered superconductor CuxBi2Se3 is a candidate for three-dimensional time- reversal-invariant topological superconductors, which are predicted to have robust surface Andreev bound states hosting massless Majorana fermions. In this work, we analytically and numerically find the linearly dispersing Majorana fermions at k = 0, which smoothly evolve into a new branch of gapless surface Andreev bound states near the Fermi momentum. The latter is a new type of Andreev bound states resulting from both the nontrivial band structure and the odd-parity pairing symmetry. The tunneling spectra of these surface Andreev bound states agree well with a recent point-contact spectroscopy experiment[1] and yield additional predictions for low temperature tunneling and photoemission experiments.

preprint2012arXiv

Odd-Parity Pairing and Topological Superconductivity in a Strongly Spin-Orbit Coupled Semiconductor

The existence of topological superconductors preserving time-reversal symmetry was recently predicted, and they are expected to provide a solid-state realization of itinerant massless Majorana fermions and a route to topological quantum computation. Their first concrete example, CuxBi2Se3, was discovered last year, but the search for new materials has so far been hindered by the lack of guiding principle. Here, we report point-contact spectroscopy experiments showing that the low-carrier-density superconductor Sn_{1-x}In_{x}Te is accompanied with surface Andreev bound states which, with the help of theoretical analysis, give evidence for odd-parity pairing and topological superconductivity. The present and previous finding of topological superconductivity in Sn_{1-x}In_{x}Te and CuxBi2Se3 demonstrates that odd-parity pairing favored by strong spin-orbit coupling is a common underlying mechanism for materializing topological superconductivity.

preprint2012arXiv

Pair Superfluid and Supersolid of Correlated Hard-Core Bosons on a Triangular Lattice

We have systematically studied the hard-core Bose-Hubbard model with correlated hopping on a triangular lattice using density-matrix renormalization group method. A rich ground state phase diagram is determined. In this phase diagram there is a supersolid phase and a pair superfluid phase due to the interplay between the ordinary frustrated boson hopping and an unusual correlated hopping. In particular, we find that the quantum phase transition between the supersolid phase and the pair superfluid phase is continuous.

preprint2012arXiv

Proximity-effect-induced superconductivity in topological insulator-related material Bi2Se3

We have studied the electron transport properties of topological insulator-related material Bi2Se3 near the superconducting Pb-Bi2Se3 interface, and found that a superconducting state is induced over an extended volume in Bi2Se3. This state can carry a Josephson supercurrent, and demonstrates a gap-like structure in the conductance spectra as probed by a normal-metal electrode. The establishment of the gap is not by confining the electrons into a narrow space close to the superconductor-normal metal interface, as previously observed in other systems, but presumably via electron-electron attractive interaction in Bi2Se3.

preprint2012arXiv

Robustness of odd-parity superconductivity with spin-orbit locking against disorder

Unconventional superconductors host a plethora of interesting physical phenomena. However, the standard theory of superconductivity suggests that unconventional pairing is highly sensitive to disorder, and hence can only be observed in ultraclean systems. We find that due to an emergent chiral symmetry, spin-orbital locking can parametrically suppress pair decoherence introduced by impurity scattering in odd-parity superconductors. Our work demonstrates that disorder is not an obstacle to realize odd-parity superconductivity in materials with strong spin-orbit coupling.

preprint2012arXiv

Topological Crystalline Insulators in the SnTe Material Class

Topological crystalline insulators are new states of matter in which the topological nature of electronic structures arises from crystal symmetries. Here we predict the first material realization of topological crystalline insulator in the semiconductor SnTe, by identifying its nonzero topological index. We predict that as a manifestation of this nontrivial topology, SnTe has metallic surface states with an even number of Dirac cones on high-symmetry crystal surfaces such as {001}, {110} and {111}. These surface states form a new type of high-mobility chiral electron gas, which is robust against disorder and topologically protected by reflection symmetry of the crystal with respect to {110} mirror plane. Breaking this mirror symmetry via elastic strain engineering or applying an in-plane magnetic field can open up a continuously tunable band gap on the surface, which may lead to wide-ranging applications in thermoelectrics, infrared detection, and tunable electronics. Closely related semiconductors PbTe and PbSe also become topological crystalline insulators after band inversion by pressure, strain and alloying.

preprint2012arXiv

Topology, Delocalization via Average Symmetry and the Symplectic Anderson Transition

A field theory of the Anderson transition in two dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortex-like topological defects is essential for localization. The sign of vortex fugacity determines the $Z_2$ topological class of the localized phase. There are two distinct, but equivalent transitions between the metallic phase and the two insulating phases. The critical conductivity and correlation length exponent of these transitions are computed in a $N=1-ε$ expansion in the number of replicas, where for small $ε$ the critical points are perturbatively connected to the Kosterlitz Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.

preprint2012arXiv

Weyl points and line nodes in gapless gyroid photonic crystals

Weyl points and line nodes are three-dimensional linear point- and line-degeneracies between two bands. In contrast to Dirac points, which are their two-dimensional analogues, Weyl points are stable in the momentum space and the associated surface states are predicted to be topologically non-trivial. However, Weyl points are yet to be discovered in nature. Here, we report photonic crystals, based on the double-gyroid structures, exhibiting frequency-isolated Weyl points with intricate phase diagrams. The surface states associated with the non-zero Chern numbers are demonstrated. Line nodes are also found in similar geometries; the associated surface states are shown to be flat bands. Our results are readily experimentally realizable at both microwave and optical frequencies.

preprint2011arXiv

Spin-orbital groundstates of Superconducting doped topological insulators (A Majorana Platform)

The Bi$_2$Se$_3$ class of topological insulators has recently been shown to undergo a superconducting transition upon hole or electron doping (Cu$_x$-Bi$_2$Se$_3$ with T$_C$=3.8$^o$K and Pd$_x$-Bi$_2$Te$_3$ with T$_C$=5$^o$K), raising the possibilities that these are the first known "topological superconductors" or realizes a superconducting state that can be potentially used as Majorana platforms (L.A. Wray \textit{et.al.}, Nature Phys. \textbf{6}, 855-859 (2010)). We use angle resolved photoemission spectroscopy to examine the full details of the spin-orbital groundstates of these materials including Bi$_2$Te$_3$, observing that the spin-momentum locked topological surface states remain well defined and non-degenerate with respect to bulk electronic states at the Fermi level in the optimally doped superconductor and obtaining their experimental Fermi energies. The implications of this unconventional surface (that undergoes superconducting at lower temperatures) topology are discussed, and we also explore the possibility of realizing the same topology in superconducting variants of Bi$_2$Te$_3$ (with T$_C$ $\sim$ 5$^o$K). Characteristics of the experimentally measured three dimensional bulk states are examined in detail for these materials with respect to the superconducting state and topological properties, showing that a single Majorana fermion zero mode is expected to be bound at each superconducting vortex on the surface. Systematic measurements also reveal intriguing renormalization and charge correlation instabilities of the surface-localized electronic modes.

preprint2011arXiv

SU(2)-invariant spin liquids on the triangular lattice with spinful Majorana excitations

We describe a new class of spin liquids with global SU(2) spin rotation symmetry in spin 1/2 systems on the triangular lattice, which have real Majorana fermion excitations carrying spin S = 1. The simplest translationally-invariant mean-field state on the triangular lattice breaks time-reversal symmetry and is stable to fluctuations. It generically possesses gapless excitations along 3 Fermi lines in the Brillouin zone. These intersect at a single point where the excitations scale with a dynamic exponent z = 3. An external magnetic field has no orbital coupling to the SU(2) spin rotation-invariant fermion bilinears that can give rise to a transverse thermal conductivity, thus leading to the absence of a thermal Hall effect. The Zeeman coupling is found to gap out two-thirds of the z = 3 excitations near the intersection point and this leads to a suppression of the low temperature specific heat, the spin susceptibility and the Wilson ratio. We also compute physical properties in the presence of weak disorder and discuss possible connections to recent experiments on organic insulators.

preprint2011arXiv

Topological Crystalline Insulators

The recent discovery of topological insulators has revived interest in the topological properties of insulating band structures. In this work, we extend the topological classification of insulating band structures to include certain point group symmetry of crystals. We find a class of three-dimensional "topological crystalline insulators" which have metallic surface states on certain high symmetry crystal surfaces. These topological crystalline insulators can be viewed as the counterpart of topological insulators in materials without spin-orbit coupling. Their surface states have quadratic band degeneracy instead of linear Dirac dispersion. Their band structures are characterized by new Z2 invariants. We hope this work will enlarge the family of topological phases in band insulators and stimulate the search for them in real materials.

preprint2011arXiv

Topological Superconductivity and Surface Andreev Bound States in Doped Semiconductors: Application to CuxBi2Se3

The recently discovered superconductor CuxBi2Se3[1] is a candidate for three-dimensional time-reversal-invariant topological superconductors[2], which are predicted to have robust surface Andreev bound states hosting massless Majorana fermions. In this work, we present an analytical and numerical study of the surface Andreev bound state wavefunction and dispersion. We find the topologically protected Majorana fermions at k = 0, as well as a new type of surface Andreev bound states at finite k. We relate our results to a recent point-contact spectroscopy experiment[3].

preprint2011arXiv

Transport properties of non-equilibrium systems under the application of light: Photo-induced quantum Hall insulators without Landau levels

In this paper, we study transport properties of non-equilibrium systems under the application of light in many-terminal measurements, using the Floquet picture. We propose and demonstrate that the quantum transport properties can be controlled in materials such as graphene and topological insulators, via the application of light. Remarkably, under the application of off-resonant light, topological transport properties can be induced; these systems exhibits quantum Hall effects in the absence of a magnetic field with a near quantization of the Hall conductance, realizing so-called quantum Hall systems without Landau levels first proposed by Haldane.

preprint2010arXiv

Electron Teleportation via Majorana Bound States in a Mesoscopic Superconductor

Zero-energy Majorana bound states in superconductors have been proposed to be potential building blocks of a topological quantum computer, because quantum information can be encoded in the fermion occupation of a pair of {\it spatially separated} Majorana bound states. However, despite intensive theoretical efforts, non-local signatures of Majorana bound states have not been found in charge transport. In this work, we predict a striking non-local phase-coherent electron transfer process by virtue of tunneling in and out of a pair of Majorana bound states. This teleportation phenomena only exists in a mesoscopic superconductor because of an all-important but previously over-looked charging energy. We propose an experimental setup to detect this phenomena in a superconductor/quantum spin Hall insulator/magnetic insulator hybrid system.

preprint2010arXiv

Fractionalization in Superconductor Josephson Junction Arrays Hinged by Quantum Spin Hall edges

In this paper we study a novel superconductor-ferromagnet-superconductor (SC-FM-SC) Josephson junction array deposited on top of a two-dimensional quantum spin Hall (QSH) insulator. The existence of Majorana bound states at the interface between SC and FM gives rise to charge-e tunneling, in addition to the usual charge-2e Cooper pair tunneling, between neighboring superconductor islands. Moreover, because Majorana fermions encode the information of charge number parity, an exact Z_2 gauge structure naturally emerges and leads to many new insulating phases, including a deconfined phase where electrons fractionalize into charge-e bosons and topological defects. A new superconductor-insulator transition has also been found.

preprint2010arXiv

Geometric phases and competing orders in two dimensions

We discuss the problem of characterizing "quantum disordered" ground states, obtained upon loss of antiferromagnetic order on general lattices in two spatial dimensions, with arbitrary electronic band structure. A key result is the response in electron bilinears to the skyrmion density in the local antiferromagnetic order, induced by geometric phases. We also discuss the connection to topological terms obtained under situations where the electronic spectrum has a Dirac form.

preprint2010arXiv

Symmetry fractional quantization in two dimensions

We introduce a solvable spin-rotational and time-reversal invariant spin-1 model in two dimensions. Depending on parameters, the ground state is an equal-weight superposition of all valence loops called "resonating valence loop" (RVL) or an equal-weight superposition of all valence bonds known as "resonating valence bond" (RVB). In RVL, ends of open loops are deconfined spinons of spin-1/2 that cannot be obtained by simple combinations of spin-1 -- a phenomenon of fractionalization; while in RVB, all quasiparticles carry an integer spin, hence no fractionalization. RVL and RVB are spin liquids with identical topological order but different spin-rotational and time-reversal symmetry quantum numbers of quasiparticles. We propose that quantized symmetry quantum number gives a systematic way to (partially) classify phases with identical topological order in dimensions greater than one.

preprint2009arXiv

Hexagonal Warping Effects in the Surface States of Topological Insulator Bi$_2$Te$_3$

A single two-dimensinoal Dirac fermion state has been recently observed on the surface of topological insulator Bi$_2$Te$_3$ by angle-resolved photoemission spectroscopy (ARPES). We study the surface band structure using $k \cdot p$ theory and find an unconventional hexagonal warping term, which is the counterpart of cubic Dresselhaus spin-orbit coupling in rhombohedral structures. We show that this hexagonal warping term naturally explains the observed hexagonal snow-flake Fermi surface. The strength of hexagonal warping is characterized by a single parameter, which is extracted from the size of the Fermi surface. We predict a number of testable signatures of hexagonal warping in spectroscopy experiments on Bi$_2$Te$_3$. We also explore the possibility of a spin-density wave due to strong nesting of the Fermi surface.

preprint2009arXiv

Odd-Parity Topological Superconductors: Theory and Application to Cu$_x$Bi$_2$Se$_3$

Topological superconductors have been theoretically predicted as a new class of time-reversal-invariant superconductors which are fully gapped in the bulk but have protected gapless surface Andreev bound states. In this work, we provide a simple criterion that directly identifies this topological phase in \textit{odd-parity} superconductors. We next propose a two-orbital $U-V$ pairing model for the newly discovered superconductor Cu$_x$Bi$_2$Se$_3$%. Due to its peculiar three-dimensional Dirac band structure, we find that an inter-orbital triplet pairing with odd-parity is favored in a significant part of the phase diagram, and therefore gives rise to a topological superconductor phase. Finally we propose sharp experimental tests of such a pairing symmetry.

preprint2009arXiv

Probing Neutral Majorana Fermion Edge Modes with Charge Transport

We propose two experiments to probe the Majorana fermion edge states that occur at a junction between a superconductor and a magnet deposited on the surface of a topological insulator. Combining two Majorana fermions into a single Dirac fermion on a magnetic domain wall allows the neutral Majorana fermions to be probed with charge transport. We will discuss a novel interferometer for Majorana fermions, which probes their Z_2 phase. This setup also allows the transmission of neutral Majorana fermions through a point contact to be measured. We introduce a point contact formed by a superconducting junction and show that its transmission can be controlled by the phase difference across the junction. We discuss the feasibility of these experiments using the recently discovered topological insulator Bi_2 Se_3.

preprint2008arXiv

Josephson Current and Noise at a Superconductor-Quantum Spin Hall Insulator-Superconductor Junction

We study junctions between superconductors mediated by the edge states of a quantum spin Hall insulator. We show that such junctions exhibit a fractional Josephson effect, in which the current phase relation has a 4π, rather than a 2πperiodicity. This effect is a consequence of the conservation of fermion parity - the number of electrons modulo 2 - in a superconducting junction, and is closely related to the Z_2 topological structure of the quantum spin Hall insulator. Inelastic processes, which violate the conservation of fermion parity, lead to telegraph noise in the equilibrium supercurrent. We predict that the low frequency noise due these processes diverges exponentially with temperature T as T -> 0. Possible experiments on HgCdTe quantum wells will be discussed.

preprint2007arXiv

Topological Insulators in Three Dimensions

We study three dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where the QSH effect is distinguished by a single $Z_2$ topological invariant, in three dimensions there are 4 invariants distinguishing 16 "topological insulator" phases. There are two general classes: weak (WTI) and strong (STI) topological insulators. The WTI states are equivalent to layered 2D QSH states, but are fragile because disorder continuously connects them to band insulators. The STI states are robust and have surface states that realize the 2+1 dimensional parity anomaly without fermion doubling, giving rise to a novel "topological metal" surface phase. We introduce a tight binding model which realizes both the WTI and STI phases, and we discuss the relevance of this model to real three dimensional materials, including bismuth.

Liang Fu

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97 published item(s)

QERNEL: a Scalable Large Electron Model

Dirac electron under periodic magnetic field: Platform for fractional Chern insulator and generalized Wigner crystal

Moiré Landau fans and magic zeros

Non-Abelian nonsymmorphic chiral symmetries

Pseudogap metal and magnetization plateau from doping moiré Mott insulator

Quantum anomalous Hall effect from inverted charge transfer gap

Supercurrent diode effect and finite momentum superconductivity

Supercurrent induced resonant optical response

Ferromagnetic helical nodal line and Kane-Mele spin-orbit coupling in kagome metal Fe3Sn2

Interfacial ferroelectricity in rhombohedral-stacked bilayer transition metal dichalcogenides

New mechanism and exact theory of superconductivity from strong repulsive interaction

Spin-triplet superconductivity from interband effect in doped insulators

Unconventional superconductivity due to interband polarization

Charge Transfer Excitations, Pair Density Waves, and Superconductivity in Moiré Materials

Density functional approach to correlated moire states: itinerant magnetism

Discovery of segmented Fermi surface induced by Cooper pair momentum

Enhanced anomalous Nernst effect in disordered Dirac and Weyl materials

Enhanced Superconductivity in Monolayer $T_d$-MoTe$_2$ with Tilted Ising Spin Texture

High-frequency rectification via chiral Bloch electrons

Local Probes for Quantum Hall Ferroelectrics and Nematics

Nearly quantized conductance plateau of vortex zero mode in an iron-based superconductor

Observation of a thermoelectric Hall plateau in the extreme quantum limit

Classification of Critical Points in Energy Bands Based on Topology, Scaling and Symmetry

Magic of high order van Hove singularity

Supermetal

Thermoelectric response and entropy of fractional quantum Hall systems

Rotation Anomaly and Topological Crystalline Insulators

A new form of (unexpected) Dirac fermions in the strongly-correlated cerium monopnictides

Braiding without Braiding: Teleportation-Based Quantum Information Processing with Majorana Zero Modes

Conetronics in 2D Metal-Organic Frameworks: Double Dirac Cones, Magnetic Half Dirac Cones and Quantum Anomalous Hall Effect

Fracton Topological Order, Generalized Lattice Gauge Theory and Duality

Identification of nematic superconductivity from the upper critical field

Majorana fermions in a superconducting Mobius strip

New quantum critical points of $j=3/2$ Dirac electrons in antiperovskite topological crystalline insulators

Observation of Majorana fermions with spin selective Andreev reflection in the vortex of topological superconductor

Odd-parity superconductors with two-component order parameters: nematic and chiral, full gap and Majorana node

Tunable Weyl fermions and Fermi arcs in magnetized topological crystalline insulators

A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations

Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators

Crystal Field Effect Induced Topological Crystalline Insulators In Monolayer IV-VI Semiconductors

Detecting crystal symmetry fractionalization from the ground state: Application to $\mathbb Z_2$ spin liquids on the kagome lattice

Electrically Tunable Quantum Spin Hall State in Topological Crystalline Insulator Thin films

Experimental observation of Weyl points

Interacting Dirac fermions under spatially alternating pseudo-magnetic field: Realization of spontaneous quantum Hall effect

Majorana Fermion Surface Code for Universal Quantum Computation

New classes of topological crystalline insulators with unpinned surface Dirac cones

Odd-parity superconductivity in the vicinity of inversion symmetry breaking in spin-orbit-coupled systems

Parity-breaking phases of spin-orbit-coupled metals with gyrotropic, ferroelectric and multipolar orders

Physical Implementation of a Majorana Fermion Surface Code for Fault-Tolerant Quantum Computation

Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials

Theory of interacting topological crystalline insulators

Three-dimensional topological photonic crystal with a single surface Dirac cone

Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials

Topological nodal line semimetals with and without spin-orbital coupling

Topological semimetals and topological insulators in rare earth monopnictides

Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

Dirac mass generation from crystal symmetry breaking on the surfaces of topological crystalline insulators

Entanglement Spectrum of a Random Partition: Connection with the Localization Transition

Experimental observation of Dirac-like surface states and topological phase transition in Pb$_{1-x}$Sn$_x$Te(111) films

Loops, sign structures and emergent Fermi statistics in three-dimensional quantum dimer models

Odd-parity topological superconductor with nematic order: Application to CuxBi2Se3

Quantum Spin Hall Effect and Topological Field Effect Transistor in Two-Dimensional Transition Metal Dichalcogenides

Strain-induced partially flat band, helical snake states, and interface superconductivity in topological crystalline insulators

Symmetry breaking and Landau quantization in topological crystalline insulators

Tensor Network Implementation of Bulk Entanglement Spectrum

Topological Crystalline Insulator Nanomembrane with Strain-Tunable Band Gap

Topological Crystalline Insulators and Dirac Octets in Anti-perovskites

Valley-selective optical Stark effect in monolayer WS2

Anomalous Supercurrent from Majorana States in Topological Insulator Josephson Junctions

Inducing time reversal invariant topological superconductivity and fermion parity pumping in quantum wires

Mapping the unconventional orbital texture in topological crystalline insulators

Observation of Dirac node formation and mass acquisition in a topological crystalline insulator

Spin-filtered Edge States with an Electrically Tunable Gap in a Two-Dimensional Topological Crystalline Insulator

Surface States of Topological Crystalline Insulators in IV-VI Semiconductors