Multilayer fractional quantum Hall wave functions can be used to construct the non-Abelian states of the $\mathbb{Z}_k$ Read-Rezayi series upon symmetrization over the layer index. Unfortunately, this construction does not yield the complete set of $\mathbb{Z}_k$ ground states on the torus. We develop an alternative projective construction of $\mathbb{Z}_k$ Read-Rezayi states that complements the existing one. On the multi-layer torus geometry, our construction consists of introducing twisted boundary conditions connecting the layers before performing the symmetrization. We give a comprehensive account of this construction for bosonic states, and numerically show that the full ground state and quasihole manifolds are recovered for all computationally accessible system sizes. Furthermore, we analyze the neutral excitation modes above the Moore-Read on the torus through an extensive exact diagonalization study. We show numerically that our construction can be used to obtain excellent approximations to these modes. Finally, we extend the new symmetrization scheme to the plane and sphere geometries.
We present a systematic perturbative approach to study excitations in the thin cylinder/torus limit of the quantum Hall states. The approach is applied to the Haldane-Rezayi and Gaffnian quantum Hall states, which are both expected to have gapless excitations in the usual two-dimensional thermodynamic limit. For the Haldane-Rezayi state, we confirm that gapless excitations are present also in the "one-dimensional" thermodynamic limit of an infinite thin cylinder, in agreement with earlier considerations based on the wave functions alone. In contrast, we identify the lowest excitations of the G…
We derive an exact matrix product state representation of the Haldane-Rezayi state on both the cylinder and torus geometry. Our derivation is based on the description of the Haldane-Rezayi state as a correlator in a non-unitary logarithmic conformal field theory. This construction faithfully captures the ten degenerate ground states of this model state on the torus. Using the cylinder geometry, we probe the gapless nature of the phase by extracting the correlation length, which diverges in the thermodynamic limit. The numerically extracted topological entanglement entropies seem to only probe …
Read-Rezayi fractional Chern insulators in modulated Bernal graphene
Massachusetts Institute of Technology（麻省理工学院）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；Stockholm University（斯德哥尔摩大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）；Uppsala University（乌普萨拉大学）
We study the possibility of fractional quantum Hall effects in HgTe quantum wells using exact diagonalization. Our results show that Laughlin states, the Moore-Read state, and the Read-Rezayi $Z_3$ state can all be supported. However, near the level crossing point (of the single-particle spectrum), the gap can be destroyed by Landau level mixing, and the Moore-Read state and the Read-Rezayi state dominate over their respective competing states only for wide wells. For smaller well widths the Moore-Read state crosses over to the composite fermion Fermi sea, while the Read-Rezayi state loses its…
We provide a full derivation of Laughlin's Jastrow-type wave functions for quantized Hall states subject to periodic boundary conditions using an operator formalism. The construction includes the quasi-hole and the technically more challenging quasi-electron excitation, which was left as an open problem in the classic paper by Haldane and Rezayi [Phys. Rev. B 31, 2529 (1985)].
Stability of the k=3 Read-Rezayi state in chiral two-dimensional systems with tunable interactions
k=3的Read-Rezayi（RR）仿费米子量子Hall态具有非阿贝尔激发，这为普适拓扑量子计算提供了一个平台.虽然在GaAs基二维电子系统中，当填充因子ν=12/5时，可以实现RR态，但相应的量子霍尔态很弱，目前几乎不可能在实验上研究。在这里，我们认为RR态可以在一类具有无质量和有质量的类Dirac能带结构的手性材料中实现。这类材料包括单层和双层石墨烯，以及拓扑绝缘体。我们表明，相比GaAs，这些系统提供了几个重要的优势，在实现和研究RR状态。最重要的是，可以通过改变外部磁场和通过设计样品的介电环境来调整{ \it in situ}有效相互作用。这种可调谐性使得在不同的朗道能级上实现具有可控能隙的RR态成为可能。它还允许人们探测到其他可压缩和不可压缩相的量子相变。
Many fractional quantum Hall wave functions are known to be unique and highest-density zero modes of certain "pseudopotential" Hamiltonians. Examples include the Read-Rezayi series (in particular, the Laughlin, Moore-Read and Read-Rezayi Z_3 states), and more exotic non-unitary (Haldane-Rezayi, Gaffnian states) or irrational states (Haffnian state). While a systematic method to construct such Hamiltonians is available for the infinite plane or sphere geometry, its generalization to manifolds such as the cylinder or torus, where relative angular momentum is not an exact quantum number, has rema…
The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling $ν= k/(k+2)$. Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted "$\bar{2}^{k}1^{k+1}$," and show that they lie in the same universality classes as the particle-hole-conjugate RR ("anti-RR") states. Our analytical results imply that a $(U(1)_{k+1} \times U(2k)_{-1})/(SU(k)_{-2} \times U(1)_{-1})$ coset conformal field theory describes the edge excitations of the $\b…
Quasiholes in certain fractional quantum Hall states are promising candidates for the experimental realization of non-Abelian anyons. They are assumed to be localized excitations, and to display non-Abelian statistics when sufficiently separated, but these properties have not been explicitly demonstrated except for the Moore-Read state. In this work, we apply the newly developed matrix product state technique to examine these exotic excitations. For the Moore-Read and the $\mathbb{Z}_3$ Read-Rezayi states, we estimate the quasihole radii, and determine the correlation lengths associated with t…
The $ν=12/5$ fractional quantum Hall plateau observed in $\mathrm{GaAs}$ semiconductor wells is a suspect in the search for non-Abelian Fibonacci anyons. Using the infinite density matrix renormalization group, we find clear evidence that fillings $ν= 12/5$ and $ν=13/5$ are in the $k = 3$ Read-Rezayi phase in the absence of particle-hole symmetry-breaking effects. The lowest energy charged excitation is identified as a non-Abelian Fibonacci anyon, distinguished from its Abelian counterpart by its local quadrupole moment. However, several experiments at $ν= 13/5$ observe a re-entrant integer qu…
Recently it was shown that the topological properties of $2D$ and $3D$ topological insulators are captured by a $\mathbb{Z}_2$ chiral anomaly in the boundary field theory. It remained, however, unclear whether the anomaly survives electron-electron interactions. We show that this is indeed the case, thereby providing an alternative formalism for treating topological insulators in the interacting regime. We apply this formalism to fractional topological insulators (FTI) via projective/parton constructions and use it to test the robustness of all fractional topological insulators which can be de…
We analyse the inner products of edge state wavefunctions in the fractional quantum Hall effect, specifically for the Laughlin and Moore-Read states. We use an effective description for these inner products given by a large-$N$ expansion ansatz proposed in recent work by J. Dubail, N. Read and E. Rezayi, PRB 86, 245310 (2012). As noted by these authors, the terms in this ansatz can be constrained using symmetry, a procedure we perform to high orders. We then check this conjecture by calculating the overlaps exactly for small system sizes and compare the numerics with our high-order expansion. …
The problem of bounding the (abelian) many-anyon ground-state energy from above, with a dependence on the statistics parameter which matches that of currently available lower bounds, is reduced to studying the correlation functions of Moore-Read (Pfaffian) and Read-Rezayi type clustering states.
Supersymmetry (SUSY) is a symmetry transforming bosons to fermions and vice versa. Indications of its existence have been extensively sought after in high-energy experiments. However, signatures of SUSY have yet to be detected. In this manuscript we propose a condensed matter realization of SUSY on the edge of a Read-Rezayi quantum Hall state, given by filling factors of the form $ν=\frac{k}{k+2}$, where $k$ is an integer. As we show, this strongly interacting state exhibits an $\mathcal{N}=2$ SUSY. This allows us to use a topological invariant - the Witten index - defined specifically for sup…
Using truncated conformal field theory (CFT), we present the formalism necessary to obtain exact matrix product state (MPS) representations for any fractional quantum hall model state which can be written as an expectation value of primary fields in some conformal field theory. The auxiliary bond space is the Hilbert space of the descendants of primary fields and matrix elements can be calculated using well-known CFT commutation relations. We obtain a site-independent MPS representation on the thin annulus and cylinder, while a site-dependent representation is possible on the disk and sphere g…
We propose a matrix quantum mechanics for a class of non-Abelian quantum Hall states. The model describes electrons which carry an internal SU(p) spin. The ground states of the matrix model include spin-singlet generalisations of the Moore-Read and Read-Rezayi states and, in general, lie in a class previously introduced by Blok and Wen. The effective action for these states is a U(p) Chern-Simons theory. We show how the matrix model can be derived from quantisation of the vortices in this Chern-Simons theory and how the matrix model ground states can be reconstructed as correlation functions i…
Many bosonic (fermionic) fractional quantum Hall states, such as Laughlin, Moore-Read and Read-Rezayi wavefunctions, belong to a special class of orthogonal polynomials: the Jack polynomials (times a Vandermonde determinant). This fundamental observation allows to point out two different recurrence relations for the coefficients of the permanent (Slater) decomposition of the bosonic (fermionic) states. Here we provide an explicit Fock space representation for these wavefunctions by introducing a two-body squeezing operator which represents them as a Jastrow operator applied to reference states…
It is an important open problem to understand the landscape of non-Abelian fractional quantum Hall phases which can be obtained starting from physically motivated theories of Abelian composite particles. We show that progress on this problem can be made using recently proposed non-Abelian bosonization dualities in 2+1 dimensions, which morally relate $U(N)_k$ and $SU(k)_{-N}$ Chern-Simons-matter theories. The advantage of these dualities is that regions of the phase diagram which may be obscure on one side of the duality can be accessed by condensing local operators on the other side. Starting…
We study the edge-mode excitations of a fractional quantum Hall droplet by expressing the edge state wavefunctions as linear combinations of Jack polynomials with a negative parameter. We show that the exact diagonalization within subspace of Jack polynomials can be used to generate the chiral edge-mode excitation spectrum in the Laughlin phase and the Moore-Read phase with realistic Coulomb interaction. The truncation technique for the edge excitations simplifies the procedure to extract reliably the edge-mode velocities, which avoids the otherwise complicated analysis of the full spectrum th…
We introduce a broad class of simple models for quantum Hall states based on the expansion of their parent Hamiltonians near the one-dimensional limit of "thin cylinders", i.e. when one dimension $L_y$ of the Hall surface becomes comparable to the magnetic length $\ell_B$. Formally, the models can be viewed as topological generalizations of the 1D Hubbard model with center-of-mass-preserving hopping of multiparticle clusters. In some cases, we show that the models can be exactly solved using elementary techniques, and yield simple wave functions for the ground states as well as the entire neut…
Certain fractional quantum Hall wavefunctions -- particularly including the Laughlin, Moore-Read, and Read-Rezayi wavefunctions -- have special structure that makes them amenable to analysis using an exeptionally wide range of techniques including conformal field theory (CFT), thin cylinder or torus limit, study of symmetric polynomials and Jack polynomials, and so-called ``special" parent Hamiltonians. This review discusses these techniques as well as explaining to what degree some other quantum Hall wavefunctions share this special structure. Along the way we will explore the physics of quan…
A fractional quantum Hall factory on quantum processors: constant-depth preparation of clustered non-Abelian states
University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校）；Max Planck Institute for Chemical Physics of Solids（马克斯·普朗克固体化学物理研究所）；National University of Singapore（新加坡国立大学）；Ludwig-Maximilians-Universität München（慕尼黑路德维希-马克西米利安大学）
We introduce a coupled wire model for a sequence of non-Abelian quantum Hall states that generalize the Z4 parafermion Read Rezayi state. The Z4 orbifold quantum Hall states occur at filling factors ν= 2/(2m-p) for odd integers $m$ and $p$, and have a topological order with a neutral sector characterized by the orbifold conformal field theory with central charge $c=1$ at radius $R=\sqrt{p/2}$. When $p=1$ the state is Abelian. The state with $p=3$ is the $Z_4$ Read Rezayi state, and the series of $p\ge 3$ defines a sequence of non-Abelian states that resembles the Laughlin sequence. Our model i…
The neutral fermionic edge mode is essential to the non-Abelian topological property and its experimental detection in $Z_k$ fractional quantum Hall (FQH) state for $k > 1$. Usually, the identification of the edge modes in a finite size system is difficult, especially near the region of the edge reconstruction, due to mixing with the bosonic edge mode and the bulk states as well. We study the edge-mode excitations of the Moore-Read (MR) and Read-Rezayi (RR) states by using Jack polynomials in the truncated subspace. It is found that the electron density, as a detector, has marked different beh…
The construction of fractional quantum Hall (FQH) states from the two-dimensional array of quantum wires provides a useful way to control strong interactions in microscopic models and has been successfully applied to the Laughlin, Moore-Read, and Read-Rezayi states. We extend this construction to the Abelian and non-Abelian $SU(N-1)$-singlet FQH states at filling fraction $ν=k(N-1)/[N+k(N-1)m]$ labeled by integers $k$ and $m$, which are potentially realized in multi-component quantum Hall systems or $SU(N)$ spin systems. Utilizing the bosonization approach and conformal field theory (CFT), we …
