The quasiparticles in Quantum Hall liquids carry fractional charge and obey fractional quantum statistics. Of particular recent interest are those with non-Abelian statistics, since their braiding properties could in principle be used for robust coding of quantum information. There is already a good theoretical understanding of quasiholes both in Abelian and non-Abelian QH states. Here we develop conformal field theory methods that allow for an equally precise description of quasielectrons, and explicitly construct two- and four-quasielectron excitations of the non-Abelian Moore-Read state.
In the conformal field theory (CFT) approach to the quantum Hall effect, the multi-electron wave functions are expressed as correlation functions in certain rational CFTs. While this approach has led to a well-understood description of the fractionally charged quasihole excitations, the quasielectrons have turned out to be much harder to handle. In particular, forming quasielectron states requires non-local operators, in sharp contrast to quasiholes that can be created by local chiral vertex operators. In both cases, the operators are strongly constrained by general requirements of symmetry, b…
Energy, interaction, and photoluminescence of spin-reversed quasielectrons in fractional quantum Hall systems
研究了分数量子霍尔区二维电子气的能量谱和光致发光谱。本文计算了反自旋准电子（QE $_{\rm R}$ 's）的单粒子性质，以及它们相互作用和与Laughlin准电子（QE's）、准空穴（QH's）相互作用的赝势。基于$_{\rm R}$ -$_{\rm R}$和$_{\rm R}$ -QE排斥的短程特性，在Halfman的分层方案中，假设了填充因子为$ν={4\over11}$和${5\over13}$的部分非极化不可压态.为了描述光致发光，结合$h($ QE $_{\rm R})_n$状态的价空穴$h$和$n$ QE $_{\rm R}$的家庭预测类似于发现较早的分数带电激子$h$ QE $_n$。比较了两个家族的结合能和光学选择规则。发现$h$ QE $_{\rm R}$与暗的$h$ QE相比具有辐射性，并且发现$h($ QE $_{\rm R})_2$与亮的$h$ QE $_2$相比无辐射性。
From an experimental point of view, quasielectrons and quasiholes play very similar roles in the fractional quantum Hall effect. Nevertheless, the theoretical description of quasielectrons is known to be much harder than the one of quasiholes. The problem is that one obtains a singularity in the wavefunction if one tries to naively construct the quasielectron as the inverse of the quasihole. Here, we demonstrate that the same problem does not arise in lattice fractional quantum Hall models. This result allows us to make detailed investigations of the properties of quasielectrons, including the…
Classical constraints on the reduced density matrix of quantum fluids in a single Landau level, termed as local exclusion conditions (LECs) [B. Yang, arXiv:1901.00047], have recently been shown to characterize the ground state of many FQH phases. In this work, we extend the LEC construction to build the elementary excitations, namely quasiholes and quasielectrons, of these FQH phases. In particular, we elucidate the quasihole counting, categorize various types of quasielectrons, and construct their microscopic wave functions. Our extensive numerical calculations indicate that the undressed qua…
We study fractional quantum Hall states in the cylinder geometry with open boundaries. By truncating the Coulomb interactions between electrons we show that it is possible to construct infinitely many exact eigenstates including the ground state, quasiholes, quasielectrons and the magnetoroton branch of excited states.
The formation and various possible decay processes of neutral and charged excitonic complexes in electronic integral and fractional quantum Hall systems are discussed. The excitonic complexes are bound states of a small number of the relevant negatively and positively charged quasiparticles (e.g., conduction electrons and valence holes, reversed-spin electrons and spin holes, Laughlin quasielectrons and quasiholes, composite fermions) that occur in an electron system under specific conditions (e.g., electron density, well width, electric and magnetic fields, or hydrostatic pressure). The examp…
We develop a formalism to describe quasihole condensates in quantum Hall liquids and thereby extend the conformal field theory approach to the full hierarchy of spin-polarized Abelian states, and to several classes of non-Abelian hierarchical states. Most previously proposed spin-polarized quantum Hall wave functions appear as special cases. In this paper we explain the physical motivations for the approach, and exemplify it by explicitly constructing the level-two quasihole condensate state at filling fraction 2/3, and the two level-three states at 5/13 and 5/7 which are built from combinatio…
Advancing a microscopic framework that rigorously unveils the underlying topological hallmarks of fractional quantum Hall (FQH) fluids is a prerequisite for making progress in the classification of strongly-coupled topological matter. Here we advance a second-quantization framework that helps reveal an exact fusion mechanism for particle fractionalization in FQH fluids, and uncover the fundamental structure behind the condensation of non-local operators characterizing topological order in the lowest-Landau-level (LLL). We show the first exact analytic computation of the quasielectron Berry con…
I demonstrate that the wavefunction for a nu = n+ tilde{nu} quantum Hall state with Landau levels 0,1,...,n-1 filled and a filling fraction tilde{nu} quantum Hall state with 0 < tilde{nu} \leq 1 in the nth Landau level can be obtained hierarchically from the nu = n state by introducing quasielectrons which are then projected into the (conjugate of the) tilde{nu} state. In particular, the tilde{nu}=1 case produces the filled Landau level wavefunctions hierarchically, thus establishing the hierarchical nature of the integer quantum Hall states. It follows that the composite fermion description o…
We develop a theory for the pseudorelativistic fractional quantum Hall effect in graphene, which is based on a multicomponent abelian Chern-Simons theory in the fermionic functional integral approach. Calculations are performed in the Keldysh formalism, directly giving access to real-time correlation functions at finite temperature. We obtain an exact effective action for the Chern-Simons gauge fields, which is expanded to second order in the gauge field fluctations around the mean-field solution. The one-loop fermionic polarization tensor as well as the electromagnetic response tensor in rand…
In this paper, we study a method to obtain non-Abelian FQH state through double-layer FQH states and fractional exciton condensation. In particular, we find that starting with the (330) double-layer state and then increasing the interlayer tunneling strength, we may obtain a single-layer non-Abelian FQH state S(330). We show that the S(330) state is actually the Z_4 parafermion Read-Rezayi state. We also calculate the edge excitation of the S(330) state.
"Wormhole" geometry for entrapping topologically-protected qubits in non-Abelian quantum Hall states and probing them with voltage and noise measurements
We investigate the structure of gapless edge modes propagating at the boundary of some fractional quantum Hall states. We show how to deduce explicit trial wavefunctions from the knowledge of the effective theory governing the edge modes. In general quantum Hall states have many edge states. Here we discuss the case of fractions having only two such modes. The case of spin-polarized and spin-singlet states at filling fraction 2/5 is considered. We give an explicit description of the decoupled charged and neutral modes. Then we discuss the situation involving negative flux acting on the composi…
Non-Abelian states with negative flux: a new series of quantum Hall states
将仿费米子成团的思想应用于具有正负通量的复合玻色子，提出了一系列新的描述分数量子霍尔态的试探波函数。这些非阿贝尔态与其他基态如条纹或复合费米子态竞争填充因子k/（3 k +/- 2）。这两个系列包含了潘等人最近发现的所有状态。90，016801（2003）]，包括偶数分母情况。精确对角化的研究领域和环面点，他们可能的相关性填充因子3/7，3/11和3/8。
Photoluminescence from fractional quantum Hall systems: Role of separation between electron and hole layers
研究了分数量子霍尔区二维电子气（2DEG）的光致发光（PL）谱随电子层和价空穴层间距$d$的变化.在磁长度$λ$的数量级的$d$处，2DEG对光学注入空穴的响应的突然变化导致PL光谱的完全重建。在$d<λ$，空穴结合一个或两个电子形成中性（$X$）或带电（$X^-$）激子，PL光谱探测这些状态的寿命和结合能，而不是2DEG的原始相关性。在$d>2λ$，取决于填充因子$ν$，空穴从2DEG倍增以形成"不相关“状态$h$，或者结合一个或两个劳克林准电子（QE）以形成分数带电激子$h$ QE或$h$ QE $_2$。严格的光学选择规则的束缚态，和唯一的光学活性原来是$h$，$h$ QE*（一个激发态的黑暗$h$ QE），和$h$ QE $_2$。早期研究中提出的“任意子激子”$h$ QE $_3$在任何$d$值下既不稳定也不辐射。不同状态的稳定性对存在的QE的2DEG的关键依赖解释了在PL光谱中观察到的异常在$ν={1\over3}$和${2\over3}$。
In this paper, we report on the study of Abelian and non-Abelian statistics through Fabry-Perot interferometry of fractional quantum Hall (FQH) systems. Our detection of phase slips in quantum interference experiments demonstrates a powerful, new way of detecting braiding of anyons. We confirm the Abelian anyonic braiding statistics in the $ν= 7/3$ FQH state through detection of the predicted statistical phase angle of $2π/3$, consistent with a change of the anyonic particle number by one. The $ν= 5/2$ FQH state is theoretically believed to harbor non-Abelian anyons which are Majorana, meaning…
We study the scaling behavior in the tunneling amplitude when quasiparticles tunnel along a straight path between the two edges of a fractional quantum Hall annulus. Such scaling behavior originates from the propagation and tunneling of charged quasielectrons and quasiholes in an effective field analysis. In the limit when the annulus deforms continuously into a quasi-one-dimensional ring, we conjecture the exact functional form of the tunneling amplitude for several cases, which reproduces the numerical results in finite systems exactly. The results for Abelian quasiparticle tunneling is cons…
Energy spectra of fractional quantum Hall systems in the presence of a valence hole
研究了分数量子霍尔区二维电子气与光注入价带空穴相互作用的能谱随填充因子$ν$和电子层与空穴层间距$d$的变化. 2DEG对空穴的响应在磁长度$λ$量级的$d$处突然改变。在$d<λ$，空穴结合电子形成中性（$X$）或带电（$X^-$）激子，光致发光（PL）光谱探测这些状态的寿命和结合能，而不是2DEG的原始相关性。“缀饰激子”的图片（其中激子和2DEG之间的相互作用被提出来仅仅是为了提高激子质量）受到质疑。相反，低能量状态解释的劳克林相关的费米子（电子和$X^-$的）和形成的两个组件的不可压缩的流体状态的电子-空穴等离子体。在$d>2λ$处，空穴与2DEG的两个Laughlin准电子（QE）结合以形成分数带电激子$h$ QE $_n$。先前发现的“任意子激子”$h$ QE $_3$在$d$的任何值下都是不稳定的。不同$h$ QE $_n$复合物的稳定性对2DEG中QE的存在的临界依赖性导致在$ν={1\over3}$或${2\over3}$处观察到PL光谱的不连续性。
In search of states with non-Abelian statistics, we explore the fractional quantum Hall effect in a system of two-dimensional charge carrier holes. We propose a new method of mapping states of holes confined to a finite width quantum well in a perpendicular magnetic field to states in a spherical shell geometry. This method provides single-particle hole states used in exact diagonalization of systems with a small number of holes in the presence of Coulomb interactions. An incompressible fractional quantum Hall state emerges in a hole liquid at the half-filling of the ground state in a magnetic…
We motivate a close look on the usefulness of the Gaffnian and Haffnian quasihole manifold (null spaces of the respective model Hamiltonians) for well-known gapped fractional quantum Hall (FQH) phases. The conformal invariance of these subspaces are derived explicitly from microscopic many-body states. The resultant CFT description leads to an intriguing emergent primary field with $h=2,c=0$, and we argue the quasihole manifolds are quantum mechanically well-defined and well-behaved. Focusing on the incompressible phases at $ν=1/3$ and $ν=2/5$, we show the low-lying excitations of the Laughlin…
Topological phases of matter are distinguished by topological invariants, such as Chern numbers and topological spins, that quantize their response to electromagnetic currents and changes of ambient geometry. Intriguingly, in the $ν=2/5$ fractional quantum Hall effect, prominent theoretical approaches -- the composite fermion theory and conformal field theory -- have constructed two distinct states, the Jain composite fermion (CF) state and the Gaffnian state, for which many of the topological indices coincide and even the microscopic ground state wave functions have high overlap with each oth…
The quantum Hall phase diagram of the half-filled bilayer system in the second Landau level is studied as a function of tunneling and layer separation using exact diagonalization. We make the striking prediction that bilayer structures would manifest two distinct branches of incompressible fractional quantum Hall effect (FQHE) corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the quantum phase transition between them w…
Partition functions of edge excitations are obtained for non-Abelian Hall states in the second Landau level, such as the anti-Read-Rezayi state, the Bonderson-Slingerland hierarchy and the Wen non-Abelian fluid, as well as for the non-Abelian spin-singlet state. The derivation is straightforward and unique starting from the non-Abelian conformal field theory data and solving the modular invariance conditions. The partition functions provide a complete account of the excitation spectrum and are used to describe experiments of Coulomb blockade and thermopower.
