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 ($χ$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.
In this paper, we present a description of Haldane's Luttinger liquid which parallels Laughlin's theory of the Fractional Quantum Hall (FQH) incompressible fluid, both exhibiting similar ground states as well as fractional excitations. These two non-Fermi liquids are instances of a generic structure for low-dimensional quantum liquids which we propose to dub Jastrow-Luttinger Fractional Liquids. An important feature of such liquids is the complete fractionalization of the parent particle. In particular, in both one and two dimensions spin-charge separation can be achieved and is indeed suggest…
Charge fractionalization is the phenomenon where quasi-particle excitations in a many-particle system appear with non-integer values relative to the fundamental charge unit. Examples of such systems are known from field theoretical models and from physical realizations. Recently there has been an interest in charge fractionalization in one-dimensional systems described by Luttinger liquid theory. These are gapless systems and that gives rise to the question whether non-integer charges should be regarded as sharp in the same meaning as in a gapped system. In this talk I first give an introducti…
Helical Luttinger liquids, appearing at the edge of two-dimensional topological insulators, represent a new paradigm of one-dimensional systems, where peculiar quantum phenomena can be investigated. Motivated by recent experiments on charge fractionalization, we propose a setup based on helical Luttinger liquids that allows to time-resolve, in addition to charge fractionalization, also spin-charge separation and pure spin fractionalization. This is due to the combined presence of spin-momentum locking and interactions. We show that electric time-resolved measurements can reveal both charge and…
We report on quantum interference between a pair of counterpropagating quantum Hall edge states that are separated by a high quality tunnel barrier. Observed Aharonov-Bohm oscillations are analyzed in terms of resonant tunneling between coupled Luttinger liquids that creates bound electronic states between pairs of tunnel centers that act like interference slits. We place a lower bound in the range of 20-40 $μ$m for the phase coherence length and directly confirm the extended phase coherence of quantum Hall edge states.
The current noise of a voltage biased interacting quantum wire adiabatically connected to metallic leads is computed in presence of an impurity in the wire. We find that in the weak backscattering limit the Fano factor characterizing the ratio between noise and backscattered current crucially depends on the noise frequency $ω$ relative to the ballistic frequency $v_F/gL$, where $v_F$ is the Fermi velocity, $g$ the Luttinger liquid interaction parameter, and $L$ the length of the wire. In contrast to chiral Luttinger liquids the noise is not only due to the Poissonian backscattering of fraction…
We examine charge fractionalization by chiral separation in a one-dimensional fermion system described by Luttinger liquid theory. The focus is on the question of whether the fractional charges are quantum mechanically sharp, and in the analysis we make a distinction between the global charge, which is restricted by boundary conditions, and the local charge where a background contribution is subtracted. We show, by way of examples, that fractional charges of arbitrary values, all which are quantum mechanically sharp, can be introduced by different initial conditions. Since the system is gaples…
We reconsider the spectrum of the Luttinger liquid (LL) usually understood in terms of phonons (density fluctuations), and within the context of bosonization we give an alternative representation in terms of fractional states. This allows to make contact with Bethe Ansatz which predicts similar fractional states. As an example we study the spinon operator in the absence of spin rotational invariance and derive it from first principles: we find that it is not a semion in general; a trial Jastrow wavefunction is also given for that spinon state. Our construction of the new spectroscopy based on …
We report current transmission data through a split-gate constriction fabricated onto a two-dimensional electron system in the integer quantum Hall (QH) regime. Split-gate biasing drives inter-edge backscattering and is shown to lead to suppressed or enhanced transmission, in marked contrast with the expected linear Fermi-liquid behavior. This evolution is described in terms of particle-hole symmetry and allows us to conclude that an unexpected class of gate-controlled particle-hole-symmetric chiral Luttinger Liquids (CLLs) can exist at the edges of our QH circuit. These results highlight the …
Fractionally charged quasiparticles in edge states, are expected to condense to a chiral Luttinger liquid (CLL). We studied their condensation by measuring the conductance and shot noise due to an artificial backscatterer embedded in their path. At sufficiently low temperatures backscattering events were found to be strongly correlated, producing a highly non-linear current-voltage characteristic and a non-classical shot noise - both are expected in a CLL. When, however, the impinging beam of quasiparticles was made dilute, either artificially via an additional weak backscatterer or by increas…
We analyze the time evolution of spin-polarized electron wave packets injected into the edge states of a two-dimensional topological insulator. In the presence of electron interactions, the system is described as a helical Luttinger liquid and injected electrons fractionalize. However, because of the presence of metallic detectors, no evidences of fractionalization are encoded in dc measurements, and in this regime the system do not show deviations from its non-interacting behavior. Nevertheless, we show that the helical Luttinger liquid nature emerges in the transient dynamics, where signatur…
The effective field theory of the fractional quantum Hall edge is reformulated from microscopic dynamics. Noncommutative Chern-Simons theory is a microscopic description for the quantum Hall fluid. We use it for reference. Considering relabeling symmetry of the electrons and incompressibility of the fluid, we obtain a constraint and derive a chiral Tomonaga-Luttinger liquid theory containing interaction terms. We calculate one-loop corrections to the phonon and electron propagators and get a new tunneling exponent. It agrees with experiments.
We propose a three terminal spin polarized stm setup for probing the helical nature of the Luttinger liquid edge state that appears in the quantum spin Hall system. We show that the three-terminal tunneling conductance strongly depends on the angle ($θ$) between the magnetization direction of the tip and the local orientation of the electron spin on the edge while the two terminal conductance is independent of this angle. We demonstrate that chiral injection of an electron into the helical Luttinger liquid (which occurs when $θ$ is zero or $π$) is associated with fractionalization of the spin …
We study the statistics of charge transfer through an impurity in a chiral Luttinger liquid (realized experimentally as a quantum point contact in a fractional quantum Hall edge state device). Taking advantage of the integrability we present a procedure for obtaining the cumulant generating function of the probability distribution to transfer a fixed amount of charge through the constriction. Using this approach we analyze in detail the behaviour of the third cumulant C_3 as a function of applied voltage, temperature and barrier height. We predict that C_3 can be used to measure the fractional…
We study a simple model of a quantum Hall system with the electrons confined to a linear, narrow channel. The system is mapped to a 1D system which in the low-energy approximation has the form of a Luttinger liquid with different interactions between particles of equal and of opposite chiralities. We study this mapping at the microscopic level, and discuss the relation between the parameters of the 2D system and the corresponding 1D Luttinger liquid parameters. We focus in particular on how the parameters are renormalized by the electron interactions and show that the velocity parameter of the…
We consider the Hall effect in a system of weakly coupled Luttinger chains. We obtain the full conductivity tensor in the absence of dissipation along the chains. We show that while the dependence of the Hall and transverse conductivities on temperature and frequency are affected by the Luttinger interaction very strongly, the Hall resistivity is given by a simple formula corresponding to the noninteracting fermions. We compute the frequency, temperature and field dependence of the transverse conductivity. Consequences for the quasi-one-dimensional organic conductors are discussed.
We consider a Laughlin droplet in a confining potential which is very steep but also weak compared to the ultra-short ranged inter-particle interactions. We find that the eigenstates have a Jack polynomial structure, and have an energy spectrum which is extremely different from the well-known Luttinger liquid edge.
Luttinger's theorem for Fermi liquids equates the volume enclosed by the Fermi surface in momentum space to the electron filling, independent of the strength and nature of interactions. Motivated by recent momentum balance arguments that establish this result in a non-perturbative fashion [M. Oshikawa, Phys. Rev. Lett. {\bf 84}, 3370 (2000)], we present extensions of this momentum balance argument to exotic systems which exhibit quantum number fractionalization focussing on $Z_2$ fractionalized insulators, superfluids and Fermi liquids. These lead to nontrivial relations between the particle f…
We give a microscopic derivation of the chiral Luttinger liquid theory 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 chiral Luttinger liquid theory one needs to go beyond Laughlin's approximation. We show that in the large N limit the I…
We study the electrical conductance in single-mode quantum wires with Rashba spin-orbit interaction subjected to externally applied magnetic fields in the regime in which the ratio of spin-orbit momentum to the Fermi momentum is close to an odd integer, so that a combined effect of multi-electron interaction and applied magnetic field leads to a partial gap in the spectrum. We study how this partial gap manifests itself in the temperature dependence of the fractional conductance of the quantum wire. We use two complementing techniques based on bosonization: refermionization of the model at a p…
We investigate charge fractionalizations in artificial Tomonaga-Luttinger liquids (TLLs) composed of two capacitively coupled quantum Hall edge channels (ECs) in graphene. The interaction strength of the artificial TLLs can be controlled through distance W between the ECs. We show that the fractionalization ratio r and the TLL mode velocity v vary with W. The experimentally obtained relation between v and r follows a unique function predicted by the TLL theory. We also show that charged wavepackets are reflected back and forth multiple times at both ends of the TLL region.
Domain walls in fractional quantum Hall ferromagnets are gapless helical one-dimensional channels formed at the boundaries of topologically distinct quantum Hall (QH) liquids. Naïvely, these helical domain walls (hDWs) constitute two counter-propagating chiral states with opposite spins. Coupled to an s-wave superconductor, helical channels are expected to lead to topological superconductivity with high order non-Abelian excitations. Here we investigate transport properties of hDWs in the $ν=2/3$ fractional QH regime. Experimentally we found that current carried by hDWs is substantially smalle…
We establish a generalization of Luttinger's theorem that applies to fractionalized Fermi liquids, i.e. Fermi liquids coexisting with symmetry enriched topological order. We find that, in the linear relation between the Fermi volume and the density of fermions, the contribution of the density is changed by the filling fraction associated with the topologically ordered sector, which is determined by how the symmetries fractionalize. Conversely, this places constraints on the allowed symmetry enriched topological orders that can manifest in a fractionalized Fermi liquid with a given Fermi volume…
The sliding Luttinger liquids (LL) approach is applied to study fractional topological insulators (FTI). We show that FTI is the low energy fixed point of the theory for realistic spin-orbit and electron-electron interaction. We find that the topological phase pertains in the presence of interaction that breaks the spin invariance and its boundaries are even extended by those terms. Finally we show that one dimensional chiral anomaly in the LL leads to the emergence of topological Chern-Simons terms in the effective gauge theory of the FTI state.
Quantum oscillation phenomena, in conventional 2-dimensional electron systems and in the fractional quantum Hall effect, are usually treated in the Lifshitz-Kosevich formalism. This is justified in three dimensions by Luttinger's expansion, in the parameter $omega_c/μ$. We show that in two dimensions this expansion breaks down, and derive a new expression, exact in the limit where rainbow graphs dominate the self-energy. Application of our results to the fractional quantum Hall effect near half-filling shows very strong deviations from Lifshitz-Kosevich behaviour. We expect that such deviation…
We formulate a theory of non-Abelian fractional quantum Hall states by considering an anisotropic system consisting of coupled, interacting one dimensional wires. We show that Abelian bosonization provides a simple framework for characterizing the Moore Read state, as well as the more general Read Rezayi sequence. This coupled wire construction provides a solvable Hamiltonian formulated in terms of electronic degrees of freedom, and provides a direct route to characterizing the quasiparticles and edge states in terms of conformal field theory. This construction leads to a simple interpretation…
The properties of the strongly interacting edge states of two dimensional topological insulators in the presence of two particle backscattering are investigated. We find an anomalous behavior of the density-density correlation functions, which show oscillations that are neither of Friedel nor of Wigner type: they instead represent a Wigner crystal of fermions of fractional charge e/2, with e the electron charge. By studying the Fermi operator, we show that the state characterized by such fractional oscillations still bears the signatures of spin momentum locking. Finally, we compare the spin-s…
We discuss the signatures of a Kramers pair of Majorana modes formed in a Josephson junction on top of a quantum spin Hall system. We show that, while ignoring interactions on the quantum spin Hall edge allows arbitrary Andreev process in the system, moderate repulsive interactions stabilize Andreev transmission - the hole goes into the opposite lead from where the electron has arrived. We analyze the renormalization group equations and deduce the existence of a non-trivial critical point for sufficiently strong interactions.
It is well established that at low energies one-dimensional (1D) fermionic systems are described by the Luttinger liquid (LL) theory, that predicts phenomena like spin-charge separation, and charge fractionalization into chiral modes. Here we show through the time evolution of an electron injected into a 1D t-J model, obtained with time-dependent density matrix renormalization group, that a further fractionalization of both charge and spin takes place beyond the hydrodynamic limit. Its dynamics can be understood at the supersymmetric point (J=2t) in terms of the excitations of the Bethe-Ansatz…
We study fractional quantum Hall states in the cylinder geometry with open boundaries. We focus on principal fermionic 1/3 and bosonic 1/2 fractions in the case of hard-core interactions. The gap behavior as a function of the cylinder radius is analyzed. By adding enough orbitals to allow for edge modes we show that it is possible to measure the Luttinger parameter of the non-chiral liquid formed by the combination of the two counterpropagating edges when we add a small confining potential. While we measure a Luttinger exponent consistent with the chiral Luttinger theory prediction for the ful…
The Luttinger model of the one-dimensional Fermi gas is the cornerstone of modern understanding of interacting electrons in one dimension. In fact, the enormous class of systems whose universal behavior is adiabatically connected to it are now deemed Luttinger liquids. Recently, it has been shown that metallic single-walled carbon nanotubes are almost perfectly described by the Luttinger Hamiltonian. Indeed, strongly non-Fermi liquid behavior has been observed in a variety of DC transport experiments, in very good agreement with theoretical predictions. Here, we describe how fractional quasipa…
Standard bosonization techniques lead to phonon-like excitations in a Luttinger liquid (LL), reflecting the absence of Landau quasiparticles in these systems. Yet in addition to the above excitations some LL are known to possess solitonic states carrying fractional quantum numbers (e.g. the spin 1/2 Heisenberg chain). We have reconsidered the zero modes in the low-energy spectrum of the gaussian boson LL hamiltonian both for fermionic and bosonic LL: in the spinless case we find that two elementary excitations carrying fractional quantum numbers allow to generate all the charge and current exc…
Coupled-wire constructions use bosonization to analytically tackle the strong interactions underlying fractional topological states of matter. We give an introduction to this technique, discuss its strengths and weaknesses, and provide an overview of the main achievements of coupled-wire constructions.
We argue that a simple way to bypass reflections at the boundaries of a finite Luttinger liquid (LL) connected to electrodes is to match load and drain impedances to the characteristic impedance of the LL viewed as a mesoscopic transmission line. For an impedance matched LL, this implies that the AC and DC shot noise properties of a finite LL are identical to those of an infinite LL. Even for an impedance mismatched LL, we show by a careful analysis of reflections that the intrinsic infinite LL properties can still be extracted yielding possibly irrational charges for the LL elementary excitat…
In this paper, we consider analytically the density evolution of a spinless Fermi liquid with a nonlinear dispersion relation into which one particle is injected. The interaction is point-like and the temperature is zero. We obtain a formula for the evolution of the density and discuss the picture it gives as well as the physics behind it. Compared to the case of a linear spectrum, we find further and more complex fractionalization of the initial density hump: it splits into three humps instead of two, moreover, all three change their shapes in a complicated manner. We analyze the mechanisms o…
We study a simple model of a quantum Hall system with the electrons confined to a linear, narrow channel. The system is mapped to a 1D system which in the low-energy approximation has the form of a Luttinger liquid with different interactions between particles of equal and of opposite chiralities. In a previous paper (Part I) we studied this mapping at the microscopic level, and discussed the relation between the parameters of the 2D system and the corresponding 1D Luttinger liquid parameters. We follow up this study here and derive the low-energy form of the 2D electron correlation function b…
We study spinless electrons in a single channel quantum wire interacting through attractive interaction, and the quantum Hall states that may be constructed by an array of such wires. For a single wire the electrons may form two phases, the Luttinger liquid and the strongly paired phase. The Luttinger liquid is gapless to one- and two-electron excitations, while the strongly paired state is gapped to the former and gapless to the latter. In contrast to the case in which the wire is proximity-coupled to an external superconductor, for an isolated wire there is no separate phase of a topological…
We present the high-temperature (70 K < T < 300 K) resistivity anisotropy and Hall effect measurements of the quasi-one-dimensional (1D) organic conductor (TMTTF)2AsF6. The temperature variations of the resistivity are pronouncedly different for the three different directions, with metallic-like at high temperatures for the a-axis only. Above 220 K the Hall coefficient R_H is constant, positive and strongly enhanced over the expected value; and the corresponding carrier concentration is almost 100 times lower than calculated for one hole/unit cell. Our results give evidence for the existence o…
We consider the edge transport properties of a generic class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model. In particular, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises …
In this paper we study the differential capacitance of a single-level quantum dot attached to quantum wires in the Luttinger liquid phase, or to fractional quantum Hall edges, by both tunneling and interactions. We show that logarithmic or even power-law divergence of the capacitance at low temperatures may result in a substantial region of the parameter space (including the $ν=1/3$ filling). This behavior is a manifestation of generalized two channel Kondo physics, and can be used to extract Luttinger liquid parameters from thermodynamic measurements.
We study theoretically the nonlinear optical response of Tomonaga-Luttinger spin liquid in the context of terahertz (THz) two-dimensional coherent spectroscopy (2DCS). Using the gapless phase of the XXZ-type spin chain as an example, we show that its third-order nonlinear magnetic susceptibilities $χ^{(3)}_{+--+}$ and $χ^{(3)}_{-++-}$ exhibit photon echo, where $\pm$ refers to the left/right-hand circular polarization with respect to the $S^z$ axis. The photon echo arises from a ``lensing'' phenomenon in which the wave packets of fractional excitations move apart and then come back toward each…
The elementary excitations from the conventional magnetic ordered states, such as ferromagnets and antiferromagnets, are magnons. Here, we elaborate a case where the well-defined magnons are absent completely and the spin excitation spectra exhibit an entire continuum in the itinerant edge ferromagnetic state of graphene arising from the flatband edge electronic states. Based on the further studies of the entanglement entropy and finite-size analysis, we show that the continuum other than the Stoner part results from the spin-1/2 spinons deconfined from magnons. The spinon continuum in a magne…
We develop a bosonization approach to study the low temperature properties of one-dimensional gas of particles obeying fractional exclusion statistics (FES). It is shown that such ideal gas reproduces the low-energy excitations and asymptotic exponents of a one-component Luttinger liquid (with no internal degrees of freedom). The bosonized effective theory at low energy (or temperature) is identified to a $c=1$ conformal field theory (CFT) with compactified radius determined by the statistics parameter $λ$. Moreover, this CFT can be put into a form of the harmonic fluid description for Lutting…
We consider the interacting, spin conserving, extended Kane-Mele-Hubbard model, and we rigorously establish the exact quantization of the edge spin conductance and the validity of the Helical Luttinger liquid relations for Drude weights and susceptibilities. Our analysis fully takes into account lattice effects, typically neglected in the Helical Luttinger model approximation, which play an essential role for universality. The analysis is based on exact renormalization group methods and on a combination of lattice and emergent Ward identities, which allow to relate the emergent chiral anomaly …
Power law suppression of local electronic tunneling density of states (TDOS) in the zero-energy limit is a hallmark of Luttinger liquid (LL) phase of the interacting 1-D electron system. We present a theoretical model which hosts LL state with the surprising feature of enhancement rather than suppression in local TDOS originating from non-local and repulsive density-density interactions. Importantly, we find enhancement of TDOS in the manifold of parameter space where the system is stable in the renormalization group (RG) sense. We argue that enhancement of TDOS along with RG stability is poss…
The current noise of a voltage biased interacting quantum wire adiabatically connected to metallic leads is computed in presence of an impurity in the wire. We find that in the weak backscattering limit the Fano factor characterizing the ratio between shot noise and backscattering current crucially depends on the noise frequency relative to the ballistic frequency v_F/gL, where v_F is the Fermi velocity, g the Luttinger liquid interaction parameter, and L the length of the wire. In contrast to chiral Luttinger liquids, the noise is not only due to the Poissonian backscattering of fractionally …
We consider a class of {\em quantum Hall topological insulators}: topologically nontrivial states with zero Chern number at finite magnetic field, in which the counter-propagating edge states are protected by a symmetry (spatial or spin) other than time-reversal. HgTe-type heterostructures and graphene are among the relevant systems. We study the effect of electron interactions on the topological properties of the system. We particularly focus on the vicinity of the topological phase transition, marked by the crossing of two Landau levels, where the system is a strongly interacting quantum Hal…
We present a comprehensive study of the effective Conformal Field Theory (CFT) describing the low energy excitations of a gas of spinless interacting fermions on a circle in the gapless regime (Luttinger liquid). Functional techniques and modular transformation properties are used to compute all correlation functions in a finite size and at finite temperature. Forward scattering disorder is treated exactly. Laughlin experiments on charge transport in a Quantum Hall Fluid on a cylinder are reviewed within this CFT framework. Edge excitations above a given bulk excitation are described by a twis…
Mean-field theory predicts that bilayer quantum Hall systems at odd integer total filling factors can have stripe ground states in which the top Landau level is occupied alternately by electrons in one of the two layers. We report on an analysis of the properties of these states based on a coupled Luttinger liquid description that is able to account for quantum fluctuations of charge-density and position along each stripe edge. The soft modes associated with the broken symmetries of the stripe state lead to an unusual coupled Luttinger liquid system with strongly enhanced low-temperature heat …
We analyze the current noise, generated at a quantum point contact in fractional quantum Hall edge state devices, using the chiral Luttinger liquid model with an impurity and the associated exact field theoretic solution. We demonstrate that an experimentally relevant regime of parameters exists where the noise coincides with the partition noise of independent Laughlin quasiparticles. However, outside of this regime, this independent particle picture breaks down and the inclusion of interaction effects is essential to understand the shot noise.
We consider a resonant level coupled to a chiral Luttinger liquid which can be realized, e.g., at a fractional quantum Hall edge. We study the dependence of the occupation probability n of the level on its energy εfor various values of the Luttinger-liquid parameter g. At g<1/2 a weakly coupled level shows a sharp jump in n(ε) at the Fermi level. As the coupling is increased, the magnitude of the jump decreases until \sqrt{2g}, and then the discontinuity in n(ε) disappears. We show that n(ε) can be expressed in terms of the magnetization of a Kondo impurity as a function of magnetic field.
The gapless edge modes of the Quantum Spin Hall insulator form a helical liquid in which the direction of motion along the edge is determined by the spin orientation of the electrons. In order to probe the Luttinger liquid physics of these edge states and their interaction with a magnetic (Kondo) impurity, we consider a setup where the helical liquid is tunnel-coupled to a semiconductor quantum dot which is excited by optical absorption, thereby inducing an effective quantum quench of the tunneling. At low energy, the absorption spectrum is dominated by a power-law singularity. The correspondi…
We investigate the Hall effect in a quasi one-dimensional system made of weakly coupled Luttinger Liquids at half filling. Using a memory function approach, we compute the Hall coefficient as a function of temperature and frequency in the presence of umklapp scattering. We find a power-law correction to the free-fermion value (band value), with an exponent depending on the Luttinger parameter $K_ρ$. At high enough temperature or frequency the Hall coefficient approaches the band value.
Among the predicted properties of fractional quantum Hall states are fractionally charged quasiparticles and conducting edge-states described as chiral Luttinger liquids. In a system with a narrow constriction, tunneling of quasi-particles 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 quasi-particle, which has been confirmed in several experiments. However, the non-linearity of the resistance predicted by the chiral Luttinger liquid theory…
We calculate a topological invariant, whose value would coincide with the Chern number in case of integer quantum Hall effect, for fractional quantum Hall states. In case of Abelian fractional quantum Hall states, this invariant is shown to be equal to the trace of the K-matrix. In case of non-Abelian fractional quantum Hall states, this invariant can be calculated on a case by case basis from the conformal field theory describing these states. This invariant can be used, for example, to distinguish between different fractional Hall states numerically even though, as a single number, it cannot…
