Chiral Luttinger Liquid Theory Research Articles

Quantum Nonlinear Optics on the Edge of a Few-Particle Fractional Quantum Hall Fluid in a Small Lattice.

We study the quantum dynamics in response to time-dependent external potentials of the edge modes of a small fractional quantum Hall fluid composed of few particles on a lattice in a bosonic Laughlin-like state at filling ν=1/2. We show that the nonlinear chiral Luttinger liquid theory provides a quantitatively accurate description even for the small lattices that are available in state-of-the-art experiments, away from the continuum limit. Experimentally accessible data related to the quantized value of the bulk transverse Hall conductivity are identified both in the linear and the non-linear response to an external excitation. The strong nonlinearity induced by the open boundaries is responsible for sizable quantum blockade effects, leading to the generation of nonclassical states of the edge modes.

Linear and nonlinear edge dynamics of trapped fractional quantum Hall droplets

We report numerical studies of the linear and nonlinear edge dynamics of a non-harmonically-confined macroscopic fractional quantum Hall fluid. In the long-wavelength and weak excitation limit, observable consequences of the fractional transverse conductivity are recovered. The first nonuniversal corrections to the chiral Luttinger liquid theory are then characterized: for a weak excitation in the linear response regime, cubic corrections to the linear wave dispersion and a broadening of the dynamical structure factor of the edge excitations are identified; for stronger excitations, sizable nonlinear effects are found in the dynamics. The numerically observed features are quantitatively captured by a nonlinear chiral Luttinger liquid quantum Hamiltonian that reduces to a driven Korteweg--de Vries equation in the semiclassical limit. Experimental observability of our predictions is finally discussed.

Nonlocal transport of heat in equilibrium drift-diffusion systems

The amount of heat an integer quantum Hall edge state can carry in equilibrium is quantized in universal units of the heat flux quantum ${J}_{q}=\frac{\ensuremath{\pi}{k}_{B}^{2}}{12\ensuremath{\hbar}}{T}^{2}$ per edge state. We address the question of how heat transport in realistic one-dimensional devices can differ from the usual chiral Luttinger liquid theory. We show that a local measurement can reveal a nonquantized amount of heat carried by the edge states, despite a globally equilibrium situation. More specifically, we report a heat enhancement effect in edge states interacting with Ohmic reservoirs in the presence of nonlocal interactions or chirality-breaking diffusive currents. In contrast to a nonequilibrium, nonlinear drag effect, we report an equilibrium, linear phenomenon. The chirality of the edge states creates additional correlations between the reservoirs, reflected in a higher-than-quantum heat flux in the chiral channel. We show that for different types of coupling the enhancement can be understood as static or dynamical back action of the reservoirs on the chiral channel. We show that our results qualitatively hold by replacing the dissipative Ohmic reservoirs by an energy-conserving mesoscopic capacitor and consider the respective transmission lines for different types of interaction.

Entanglement spectrum edge reconstruction and correlation hole of fractional quantum Hall liquids

The edge of the electronic fractional quantum Hall (FQH) system obeys the law of the chiral Luttinger liquid theory due to its intrinsic topological properties and the relation of bulk-edge correspondence. However, in a realistic experimental system, such as the usual Hall bar setup, the soften of the background confinement potential can induce the reconstruction of the edge spectrum which breaks the chirality and universality of the FQH edge. The entanglement spectrum (ES) of the FQH ground state has the same counting structure as that in the energy spectrum indicating the topological characters of the quantum state. In this work, we report that the ES can also have an edge reconstruction while sweeping the area of the sub-system in real space cut. Moreover, we found the critical area of the sub-system matches accurately with the intrinsic building block of the fractional quantum Hall liquids, namely the correlation hole of the FQH liquids. The above results seem like be universal after our studying a series of typical FQH states, such as two Laughlin states at $\nu = 1/3$ and $\nu = 1/5$, and the Moore-Read state for $\nu = 5/2$.

Universal Edge Transport in Interacting Hall Systems

We study the edge transport properties of $2d$ interacting Hall systems, displaying single-mode chiral edge currents. For this class of many-body lattice models, including for instance the interacting Haldane model, we prove the quantization of the edge charge conductance and the bulk-edge correspondence. Instead, the edge Drude weight and the edge susceptibility are interaction-dependent; nevertheless, they satisfy exact universal scaling relations, in agreement with the chiral Luttinger liquid theory. Moreover, charge and spin excitations differ in their velocities, giving rise to the spin-charge separation phenomenon. The analysis is based on exact renormalization group methods, and on a combination of lattice and emergent Ward identities. The invariance of the emergent chiral anomaly under the renormalization group flow plays a crucial role in the proof.

Photoassisted shot noise spectroscopy at fractional filling factor

We study the photoassisted shot noise generated by a periodic voltage in the fractional quantum Hall regime. Fluctuations of the current are due to the presence of a quantum point contact operating in the weak backscattering regime. We show how to reconstruct the photoassisted absorption and emission probabilities by varying independently the dc and ac contributions to the voltage drive. This is made possible by the peculiar power-law behavior of the tunneling rates in the chiral Luttinger liquid theory, which allow to approximate the typical infinite sums of the photoassisted transport formalism in a simple and particularly convenient way.

Wave functions for fractional Chern insulators in disk geometry

Recently, fractional Chern insulators (FCIs), also called fractional quantum anomalous Hall (FQAH) states, have been theoretically established in lattice systems with topological flat bands. These systems exhibit similar fractionalization phenomena to the conventional fractional quantum Hall (FQH) systems. Using the mapping relationship between the FQH states and the FCI/FQAH states, we construct the many-body wave functions of the fermionic FCI/FQAH states in disk geometry with the aid of the generalized Pauli principle (GPP) and Jack polynomials. Compared with the ground state by the exact diagonalization method, the wave-function overlap is higher than 0.97, even when the Hilbert space dimension is as large as 3 × 106. We also use the GPP and the Jack polynomials to construct edge excitations for the fermionic FCI/FQAH states. The quasi-degeneracy sequences of fermionic FCI/FQAH systems reproduce the prediction of the chiral Luttinger liquid theory, complementing the exact diagonalization results with larger lattice sizes and more particles.

Nonequilibrium noise in transport across a tunneling contact betweenν=23fractional quantum Hall edges

In a recent experimental paper [1] a qualitative confirmation of the existence of upstream neutral modes at $\nu = 2/3$ quantum Hall edge was reported. Using the chiral Luttinger liquid theory of quantum Hall edge we develop a quantitative model of the experiment [1]. A good quantitative agreement of our theory with the experimental data reinforces the conclusion of existence of the upstream neutral mode. Our model also enables us to extract important quantitative information about non-equilibrium processes in Ohmic and tunneling contacts from the experimental data. In particular, for $\nu = 2/3$, we find a power-law dependence of the neutral mode temperature on the charge current injected from the Ohmic contact.

Edge excitations in fractional Chern insulators

Recent theoretical papers have demonstrated the realization of fractional quantum anomalous Hall states (also called fractional Chern insulators) in topological flat band lattice models without an external magnetic field. Such newly proposed lattice systems play a vital role in obtaining a large class of fractional topological phases. Here we report the exact numerical studies of edge excitations for such systems in a disk geometry loaded with hard-core bosons, which will serve as a more viable experimental probe for such topologically ordered states. We find convincing numerical evidence of a series of edge excitations characterized by the chiral Luttinger liquid theory for the bosonic fractional Chern insulators in both the honeycomb disk Haldane model and the kagome-lattice disk model. We further verify these current-carrying chiral edge states by inserting a central flux to test their compressibility.

Chiral Luttinger liquids and a generalized Luttinger theorem in fractional quantum Hall edges via finite-entanglement scaling

We use bosonic field theories and the infinite system density matrix renormalization group (iDMRG) method to study infinite strips of fractional quantum Hall (FQH) states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge mode exponents and momenta without finite-size errors. We analyze states in the first and second level of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid ($\chi$LL) theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the non-chiral case. We prove a generalized Luttinger's theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in 1D.

The Universal Edge Physics in Fractional Quantum Hall Liquids

The chiral Luttinger liquid theory for fractional quantum Hall edge transport predicts universal power-law behavior in the current-voltage (I-V) characteristics for electrons tunneling into the edge. However, it has not been unambiguously observed in experiments in two-dimensional electron gases based on GaAs/GaAlAs heterostructures or quantum wells. One plausible cause is the fractional quantum Hall edge reconstruction, which introduces non-chiral edge modes. The coupling between counterpropagating edge modes can modify the exponent of the I-V characteristics. By comparing the v = 1/3 fractional quantum Hall states in modulation-doped semiconductor devices and in graphene devices, we show that the graphene-based systems have an experimental accessible parameter region to avoid the edge reconstruction, which is suitable for the exploration of the universal edge tunneling exponent predicted by the chiral Luttinger liquid theory.

Edge excitations of bosonic fractional quantum Hall phases in optical lattices

The rapid development of artificial gauge fields in ultracold gases suggests that atomic realization of fractional quantum Hall physics will become experimentally practical in the near future. While it is known that bosons on lattices can support quantum Hall states, the universal edge excitations that provide the most likely experimental probe of the topological order have not been obtained. We find that the edge excitations of an interacting boson lattice model are surprisingly sensitive to interedge hybridization and edge-bulk mixing for some confining potentials. With properly chosen potentials and fluxes, the edge spectrum is surprisingly clear even for small systems with strong lattice effects such as bandwidth. Various fractional quantum Hall phases for bosons can be obtained, and the phases $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=2/3$ have the edge spectra predicted by the chiral Luttinger liquid theory.

Effect of the boundary shape in the effective theory of fractional quantum Hall edges

Starting from a microscopic description of a system of strongly interacting electrons in a strong magnetic field in a finite geometry, we construct the boundary low energy effective theory for a fractional quantum Hall droplet taking into account the effects of a smooth edge. The effective theory obtained is the standard chiral boson theory (chiral Luttinger theory) with an additional self-interacting term which is induced by the boundary. As an example of the consequences of this model, we show that such modification leads to a nonuniversal reduction in the tunneling exponent which is independent of the filling fraction. This is in qualitative agreement with experiments, which systematically found exponents smaller than those predicted by the ordinary chiral Luttinger liquid theory.

Edge excitations and topological order in a rotating Bose gas

The edge excitations and related topological orders of correlated states of a fast rotating Bose gas are studied. Using exact diagonalization of small systems, we compute the energies and number of edge excitations, as well as the boson occupancy near the edge for various states. The chiral Luttinger-liquid theory of Wen is found to be a good description of the edges of the bosonic Laughlin and other states identified as members of the principal Jain sequence for bosons. However, we find that in a harmonic trap the edge of the state identified as the Moore-Read (Pfaffian) state shows a number of anomalies. An experimental way of detecting these correlated states is also discussed.

Microscopic construction of the chiral Luttinger liquid theory of the quantum Hall edge

We give a microscopic derivation of the chiral Luttinger liquid theory $(\ensuremath{\chi}LL)$ for the Laughlin states. Starting from the wave function describing an arbitrary incompressibly deformed Laughlin state (IDLS) we quantize these deformations. In this way we obtain the low-energy projections of local microscopic operators and derive the quantum field theory of edge excitations directly from quantum mechanics of electrons. This shows that to describe experimental and numeric deviations from $\ensuremath{\chi}LL$ one needs to go beyond Laughlin's approximation. We show that in the large $N$ limit the IDLS is described by the dispersionless Toda hierarchy.

Field theoretical description of quantum Hall edge reconstruction.

We propose a generalization of the chiral Luttinger liquid theory to allow for a unified description of quantum Hall edges with or without edge reconstruction. Within this description edge reconstruction is found to be a quantum phase transition in the universality class of a one-dimensional dilute Bose gas transition, whose critical behavior can be obtained exactly. At principal filling factors nu=1/m, we show the additional edge mode due to edge reconstruction modifies the point contact tunneling exponent in the low-energy limit, by a small and nonuniversal amount.

Nonuniversal behavior of scattering between fractional quantum Hall edges.

In a fractional quantum Hall system with a narrow constriction, tunneling of quasiparticles between states at different edges can lead to resistance and to shot noise. The ratio of the shot noise to the backscattered current, in the weak scattering regime, measures the fractional charge of the quasiparticle, which has been confirmed in several experiments. However, the predicted nonlinearity of the resistance was apparently not observed in some of these cases. As a possible explanation, we consider a model where coupling between the current carrying edge mode and additional phononlike edge modes can lead to nonuniversal exponents in the current-voltage characteristic.

Effective action of a compressible quantum Hall state edge: Application to tunneling

The electrodynamical response of the edge of a compressible quantum Hall system affects tunneling into the edge. Using composite Fermi liquid theory, we derive an effective action for the edge modes interacting with tunneling charge. This action generalizes the chiral Luttinger liquid theory of the quantum Hall edge to compressible systems in which transport is characterized by a finite Hall angle. In addition to the standard terms, the action contains a dissipative term. The tunneling exponent is calculated as a function of the filling fraction for several models, including screened and unscreened long-range Coulomb interaction, as well as a short-range interaction. We find that tunneling exponents are robust and to a large extent insensitive to the particular model. We discuss recent tunneling measurements in overgrown cleaved edge systems, and demonstrate that the profile of charge density near the edge is very sensitive to the parameters of the system. In general, the density is nonmonotonic, and can deviate from the bulk value by up to 30%. Implications for the correspondence to chiral Luttinger edge theories are discussed.

Coulomb blockade in the fractional quantum Hall effect regime

We use chiral Luttinger liquid theory to study transport through a quantum dot in the fractional quantum Hall effect regime and find rich non-Fermi-liquid tunneling characteristics. In particular, we predict a remarkable Coulomb-blockade-type energy gap that is quantized in units of the noninteracting level spacing, new power-law tunneling exponents for voltages beyond threshold, and a line shape as a function of gate voltage that is dramatically different than that for a Fermi liquid. We propose experiments to use these unique spectral properties as a new probe of the fractional quantum Hall effect.

Evolution of quasiparticle charge in the fractional quantum hall regime

The charge of quasiparticles in a fractional quantum Hall (FQH) liquid, tunneling through a partly reflecting constriction with transmission t, was determined via shot noise measurements. In the nu = 1/3 FQH state, a charge smoothly evolving from e(*) = e/3 for t(1/3) congruent with 1 to e(*) = e for t(1/3)<<1 was determined, agreeing with chiral Luttinger liquid theory. In the nu = 2/5 FQH state the quasiparticle charge evolves smoothly from e(*) = e/5 at t(2/5) congruent with 1 to a maximum charge less than e(*) = e/3 at t(2/5)<<1. Thus it appears that quasiparticles with an approximate charge e/5 pass a barrier they see as almost opaque.