We investigate the magnetic response in the quantized spin Hall (SH) phase of layered-honeycomb lattice system with intrinsic spin-orbit coupling lambda_SO and on-site Hubbard U. The response is characterized by a parameter g= 4 U a^2 d / 3, where a and d are the lattice constant and interlayer distance, respectively. When g< (sigma_{xy}^{s2} mu)^{-1}, where sigma_{xy}^{s} is the quantized spin Hall conductivity and mu is the magnetic permeability, the magnetic field inside the sample oscillates spatially. The oscillation vanishes in the non-interacting limit U -> 0. When g > (sigma_{xy}^{s2} mu)^{-1}, the system shows perfect diamagnetism, i.e., the Meissner effect occurs. We find that superlattice structure with large lattice constant is favorable to see these phenomena. We also point out that, as a result of Zeeman coupling, the topologically-protected helical edge states shows weak diamagnetism which is independent of the parameter g.
We derive the low-energy theory of semi-quantized quantum Hall states, a recently observed class of gapless bilayer fractional quantum Hall states. Our theory shows these states to feature gapless quasiparticles of fractional charge coupled to an emergent Chern-Simons gauge field. These gapless quasiparticles can be understood as composites of electrons and Laughlin-like quasiparticles. We show that semi-quantized quantum Hall states exhibit perfect interlayer drag, host non-Fermi liquid physics, and serve as versatile parent states for fully gapped topological phases hosting anyonic excitatio…
A finite-temperature effective free energy of the boundary of a quantized thermal Hall system is derived microscopically from the bulk two-dimensional Dirac fermion coupled with a gravitational field. In two spatial dimensions, the thermal Hall conductivity of fully gapped insulators and superconductors is quantized and given by the bulk Chern number, in analogy to the quantized electric Hall conductivity in quantum Hall systems. From the perspective of effective action functionals, two distinct types of the field theory have been proposed to describe the quantized thermal Hall effect. One of …
The frictional drag between parallel two-dimensional electron systems has been measured in a regime of strong interlayer correlations. When the bilayer system enters the excitonic quantized Hall state at total Landau level filling factor ν_T=1 the longitudinal component of the drag vanishes but a strong Hall component develops. The Hall drag resistance is observed to be accurately quantized at h/e^2.
We propose a method for studying the strong interaction regimes in twisted bilayer graphene using hybrid Wannier functions, that are Wannier-like in one direction and Bloch-like in the other. We focus on the active bands as given by the continuum model proposed by Bistritzer and MacDonald, and discuss the properties of corresponding hybrid Wannier functions. We then employ the method for a study of the fillings of $\pm 3$ electrons per moiré cell using the Hartree-Fock method. We discuss at length different regimes under which a quantized anomalous Hall effect is seen in these two fillings.
We extend Laughlin's magnetic-flux-threading argument to the quantized thermal Hall effect. A proper analogue of Laughlin's adiabatic magnetic-flux threading process for the case of the thermal Hall effect is given in terms of an external gravitational field. From the perspective of the edge theories of quantum Hall systems, the quantized thermal Hall effect is closely tied to the breakdown of large diffeomorphism invariance, that is, a global gravitational anomaly. In addition, we also give an argument from the bulk perspective in which a free energy, decomposed into its Fourier modes, is adi…
We perform numerical studies to determine if the fractional quantum Hall state observed at filling nu=5/2 is the Moore-Read wavefunction or its particle hole conjugate, the so-called AntiPfaffian. Using a truncated Hilbert space approach we find that for realistic interactions, including Landau-level mixing, the ground state remains fully polarized and the AntiPfaffian is strongly favored.
A theory of collective states in a magnetically quantized two-dimensional electron gas (2DEG) with half-filled Landau level (quantized Hall ferromagnet) in the presence of magnetic 3d impurities is developed. The spectrum of bound and delocalized spin-excitons as well as the renormalization of Zeeman splitting of the impurity 3d levels due to the indirect exchange interaction with the 2DEG are studied for the specific case of n-type GaAs doped with Mn where the Lande` g-factors of impurity and 2DEG have opposite signs. If the sign of the 2DEG g-factor is changed due to external influences, the…
Bi2Se3 is an important semiconductor thermoelectric material and a prototype topological insulator. Here we report observation of Shubnikov-de Hass (SdH) oscillations accompanied by quantized Hall resistances (Rxy) in highly-doped n-type Bi2Se3 with bulk carrier concentrations of few 10^19 cm^-3. Measurements under tilted magnetic fields show that the magnetotransport is 2D-like, where only the c-axis component of the magnetic field controls the Landau level formation. The quantized step size in 1/Rxy is found to scale with the sample thickness, and average ~e2/h per quintuple layer (QL). We s…
We study topological phases in Iridium (Ir) oxide superlattices of orthorhombic perovskite-type grown along the [001] crystallographic axis. Due to strong spin-orbit coupling of Ir 5d-orbitals and electronic correlation effects, Ir oxide bilayer superlattices display topological magnetic insulators exhibiting quantized anomalous Hall effects. Depending on stacking of two layers, we also found a valley Hall insulator with counter-propagating edge currents from two different valleys and a topological crystalline insulator with edge states protected by the crystal lattice symmetry. In a single la…
Motivated by the experimental observation of a quantized 5/2 thermal conductance at filling $ν=5/2$, a result incompatible with both the Pfaffian and the Antipfaffian states, we have pushed the expansion of the effective Hamiltonian of the $5/2$ quantized Hall state to third-order in the parameter $κ=E_c/\hbar ω_c \propto 1/\sqrt{B}$ controlling the Landau level mixing , where $E_c$ is the Coulomb energy and $ω_c$ the cyclotron frequency. Exact diagonalizations of this effective Hamiltonian show that the difference in overlap with the Pfaffian and the AntiPfaffian induced at second-order is re…
Second-quantized approach to the study of Halperin state in fractional quantum Hall effect
College of Physics and Electronic Science, Hubei Normal University（湖北师范大学物理与电子科学学院）；Hubei Key Laboratory of Photoelectric Materials and Devices, Hubei Normal University（湖北师范大学光电材料与器件湖北省重点实验室）
We present results of extensive numerical calculations on the ground state of electrons in the first excited (n=1) Landau level with Coulomb interactions, and including non-zero thickness effects, for filling factors 12/5 and 13/5 in the torus geometry. In a region that includes these experimentally-relevant values, we find that the energy spectrum and the overlaps with the trial states support the previous hypothesis that the system is in the non-Abelian k = 3 liquid phase we introduced in a previous paper.
The quantized thermal Hall effect is an important probe for detecting chiral topological order and revealing the nature of chiral gapless edge states. The standard Kubo formula approach for the thermal Hall conductance $κ_{xy}$ based on the linear-response theory faces difficulties in practical application due to the lack of a reliable numerical method for calculating dynamical quantities in microscopic models at finite temperature. In this work, we propose an approach for calculating $κ_{xy}$ in two-dimensional lattice models displaying chiral topological order. Our approach targets at the ed…
We study the fractional boundary charges (FBCs) occurring in nanowires in the presence of periodically modulated chemical potentials and connect them to the FBCs occurring in a two-dimensional electron gas in the presence of a perpendicular magnetic field in the integer quantum Hall effect (QHE) regime. First, we show that in nanowires the FBCs take fractional values and change linearly as a function of phase offset of the modulated chemical potential. This linear slope takes quantized values determined by the period of the modulation and depends only on the number of the filled bands. Next, w…
We have investigated the sample dependence of the half-integer thermal Hall effect in $α$-RuCl$_3$ under a magnetic field tilted 45 degree from the $c$ axis to the $a$ axis. We find that the sample with the largest longitudinal thermal conductivity ($κ_{xx}$) shows the half-integer quantized thermal Hall effect expected in the Kitaev model. On the other hand, the quantized thermal Hall effect was not observed in the samples with smaller $κ_{xx}$. We suggest that suppressing the magnetic scattering effects on the phonon thermal conduction, which broaden the field-induced gap protecting the chir…
Although numerical studies modeling the quantum hall effect at filling fraction $5/2$ predict either the Pfaffian (Pf) or its particle hole conjugate, the anti-Pfaffian (aPf) state, recent experiments appear to favor a quantized thermal hall conductivity with quantum number $K=5/2$ , rather than the value $K=7/2$ or $K=3/2$ expected for the Pf or aPF state, respectively. While a particle hole symmetric topological order (the PH-Pfaffian) would be consistent with the experiments, this state is believed to be energetically unfavorable in a homogenous system. Here we study the effects of disorder…
The recent observation of a half-integer quantized thermal Hall effect in $α$-RuCl$_3$ is interpreted as a unique signature of a chiral spin liquid with a Majorana edge mode. A similar quantized thermal Hall effect is expected in chiral topological superconductors. The unavoidable presence of gapless acoustic phonons, however, implies that, in contrast to the quantized electrical conductivity, the thermal Hall conductivity $κ_xy$ is never exactly quantized in real materials. Here, we investigate how phonons affect the quantization of the thermal conductivity focusing on the edge theory. As an …
The observation of new insulating phases of two-dimensional electrons in the first excited Landau level is reported. These states, which are manifested as re-entrant integer quantized Hall effects, exist alongside well-developed even-denominator fractional quantized Hall states at nu=7/2 and 5/2 and new odd-denominator states at nu=3+1/5 and 3+4/5.
We suggest a theory for a deformable and sliding charge density wave (CDW) in the Hall bar geometry for the quantum limit when the carriers in remnant small pockets are concentrated at lowest Landau levels (LL) forming a fractionally ($ν<1$) filled quantum Hall state. The gigantic polarizability of the CDW allows for a strong redistribution of electronic densities up to a complete charge segregation when all carriers occupy, with the maximum filling, a fraction $ν$ of the chain length - thus forming the integer quantum Hall state, while leaving the fraction $(1-ν)$ of the chain length unoccupi…
The quintessential two-dimensional lattice model that describes the competition between the kinetic energy of electrons and their short-range repulsive interactions is the repulsive Hubbard model. We study a time-reversal symmetric variant of the repulsive Hubbard model defined on a planar lattice: Whereas the interaction is unchanged, any fully occupied band supports a quantized spin Hall effect. We show that at 1/2 filling of this band, the ground state develops spontaneously and simultaneously Ising ferromagnetic long-range order and a quantized charge Hall effect when the interaction is su…
We report the observation of a quantum anomalous Hall effect in twisted bilayer graphene showing Hall resistance quantized to within .1\% of the von Klitzing constant $h/e^2$ at zero magnetic field.The effect is driven by intrinsic strong correlations, which polarize the electron system into a single spin and valley resolved moiré miniband with Chern number $C=1$. In contrast to extrinsic, magnetically doped systems, the measured transport energy gap $Δ/k_B\approx 27$~K is larger than the Curie temperature for magnetic ordering $T_C\approx 9$~K, and Hall quantization persists to temperatures o…
We consider a honeycomb network built of quantum wires, with each node of the network having a Y-junction of three wires with a ring through which flux can be inserted. The junctions are the basic circuit elements for the network, and they are characterized by 3 x 3 conductance tensors. The low energy stable fixed point tensor conductances result from quantum effects, and are determined by the strength of the interactions in each wire and the magnetic flux through the ring. We consider the limit where there is decoherence in the wires between any two nodes, and study the array as a network of …
A weak superconducting proximity effect in the vicinity of the topological transition of a quantum anomalous Hall system has been proposed as a venue to realize a topological superconductor (TSC) with chiral Majorana edge modes (CMEMs). A recent experiment [Science 357, 294 (2017)] claimed to have observed such CMEMs in the form of a half-integer quantized conductance plateau in the two-terminal transport measurement of a quantum anomalous Hall-superconductor junction. Although the presence of a superconducting proximity effect generically splits the quantum Hall transition into two phase tran…
We present a theory of the magneto-optical Faraday and Kerr effects of topological insulator (TI) films. For film thicknesses short compared to wavelength, we find that the low-frequency Faraday effect in ideal systems is quantized at integer multiples of the fine structure constant, and that the Kerr effect exhibits a giant $π/2$ rotation for either normal or oblique incidence. For thick films that contain an integer number of half wavelengths, we find that the Faraday and Kerr effects are both quantized at integer multiples of the fine structure constant. For TI films with bulk parallel cond…
We present a phase diagram for a double quantum well bilayer electron gas in the quantum Hall regime at total filling factor $ν=1$, based on exact numerical calculations of the topological Chern number matrix and the (inter-layer) superfluid density. We find three phases: a quantized Hall state with pseudo-spin superfluidity, a quantized Hall state with pseudo-spin ``gauge-glass'' order, and a decoupled composite Fermi liquid. Comparison with experiments provides a consistent explanation of the observed quantum Hall plateau, Hall drag plateau and vanishing Hall drag resistance, as well as the …
We present a Landau-Ginzburg theory for a fractional quantized Hall nematic state and the transition to it from an isotropic fractional quantum Hall state. This justifies Lifshitz-Chern-Simons theory -- which is shown to be its dual -- on a more microscopic basis and enables us to compute a ground state wave function in the symmetry-broken phase. In such a state of matter, the Hall resistance remains quantized while the longitudinal DC resistivity due to thermally-excited quasiparticles is anisotropic. We interpret recent experiments at Landau level filling factor ν=7/3 in terms of our theory.
Recent experimental work on locally gated graphene layers resulting in p-n junctions have revealed quantum Hall effect in their transport behavior. We explain the observed conductance quantization which is fractional in the bipolar regime and integer in the unipolar regime in terms of quantum Hall edge modes propagating along and across the p-n interface. In the bipolar regime the electron and hole modes can mix at the p-n boundary, leading to current partition and quantized shot noise plateaus similar to those of conductance, while in the unipolar regime transport is noiseless. These quantum …
Heat transport mediated by Majorana edge modes in a magnetic insulator leads to a half-integer thermal quantum Hall conductance, which has recently been reported for the two-dimensional honeycomb material $α$-RuCl$_3$. While the conventional electronic Hall effect requires a perpendicular magnetic field, we find that this is not the case in $α$-RuCl$_3$. Strikingly, the thermal Hall plateau appears even for a magnetic field with no out-of-plane components. The field-angular variation of the quantized thermal Hall conductance has the same sign structure of the topological Chern number, which is…
We discuss quantum Hall effects in a gapped insulator on a periodic two-dimensional lattice. We derive a universal relation among the the quantized Hall conductivity, and charge and flux densities per physical unit cell. This follows from the magnetic translation symmetry and the large gauge invariance, and holds for a very general class of interacting many-body systems. It can be understood as a combination of Laughlin's gauge invariance argument and Lieb-Schultz-Mattis-type theorem. A variety of complementary arguments, based on a cut-and-glue procedure, the many-body electric polarization, …
Symmetry protected topological (SPT) states are short-range entangled states with symmetry. Nontrivial SPT states have symmetry protected gapless edge excitations. In 2-dimension (2D), there are infinite number of nontrivial SPT phases with SU(2) or SO(3) symmetry. These phases can be described by SU(2)/SO(3) nonlinear-sigma models with a quantized topological θ-term. At open boundary, the θ-term becomes the Wess-Zumino-Witten term and consequently the boundary excitations are decoupled gapless left movers and right movers. Only the left movers (if θ>0) carry the SU(2)/SO(3) quantum numbers. A…
By ungauging a recently discovered lattice rotor model for Chern-Simons theory, we create an exactly soluble path integral on spacetime lattice for $U^κ(1)$ Symmetry Protected Topological (SPT) phases in $2+1$ dimensions with a non-zero Hall conductance. We then convert the path integral on a $2+1$d spacetime lattice into a $2$d Hamiltonian lattice model, and show that the Hamiltonian consists of mutually commuting local projectors. We confirm the non-zero Hall conductance by calculating the Chern number of the exact ground state. It has recently been suggested that no commuting projector mode…
We discuss the construction of a ground state wavefunction on a finite ring given the ground state on a long chain. In the presence of local symmetries, we can obtain the ground state with arbitrary flux inserted through the ring. A key ingredient are the quantum numbers of the entanglement spectrum. This method allows us to characterize phases by measuring quantized responses, such as the Hall conductance, using data contained in the entanglement spectrum. As concrete examples, we show how the Berry phase allows us to map out the phase diagram of a spin-1 model and calculate the Hall conducti…
Quantized Transport of $ν= 2/3$ Fractional Quantum Hall Edge with Disordered Superconducting Proximity
Princeton Center for Theoretical Science, Princeton University（普林斯顿大学理论科学中心）；Department of Physics, Princeton University（普林斯顿大学物理系）；Department of Electrical and Computer Engineering, Princeton University（普林斯顿大学电气与计算机工程系）
Despite extensive investigations, a topological state that hosts Majorana edge modes in the magnetic field-induced quantum disordered state of the Kitaev candidate material $α$-RuCl$_3$ has been hotly debated. To gain more insight into this issue, we measured the thermal Hall conductivity $κ_{xy}$ of various samples grown by the Bridgman method. The results show that the half-integer quantum thermal Hall effect is intimately related to the magnitude of longitudinal thermal conductivity and the Néel temperature at zero field, both of which are sample dependent. Samples exhibiting the half-integ…
Topological insulators are bulk electronic insulators which possess symmetry protected gapless modes on their surfaces. Breaking the symmetries that underlie the gapless nature of the surface modes is predicted to give rise to exotic new states of matter. In particular, it has recently been predicted and shown that breaking of time reversal symmetry in the form of ferromagnetism can give rise to a gapped state characterized by a zero magnetic field quantized Hall response and dissipationless longitudinal transport known as the Quantum Anomalous Hall (QAH) state. A key question that has thus fa…
