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 remained an open problem. Here we develop a geometric approach for constructing pseudopotential Hamiltonians in a universal manner that naturally applies to all geometries. Our method generalizes to the multicomponent SU(n) cases with a combination of spin or pseudospin (layer, subband, valley) degrees of freedom. We demonstrate the utility of the approach through several examples, including certain non-Abelian multicomponent states whose parent Hamiltonians were previously unknown, and verify the method by numerically computing their entanglement properties.
We construct a class of lattice Hamiltonians that exhibit fractional Hall conductivity. These Hamiltonians, while not being exactly solvable, can be controllably solved in their low energy sectors, through a combination of perturbative and exact techniques. Our construction demonstrates a systematic way to circumvent the Kapustin-Fidkowski no-go theorem and is generalizable.
The Kapustin-Fidkowski no-go theorem forbids $U(1)$ symmetric topological orders with non-trivial Hall conductivity in (2+1)d from admitting commuting projector Hamiltonians, where the latter is the paradigmatic method to construct exactly solvable lattice models for topological orders. Even if a topological order would intrinsically have admitted commuting projector Hamiltonians, the theorem forbids so once its interplay with $U(1)$ global symmetry which generates Hall conductivity is taken into consideration. Nonetheless, in this work, we show that for all (2+1)d $U(1)$ symmetric abelian top…
We study frustration-free Hamiltonians of fractional quantum Hall (FQH) states from the point of view of the matrix product state (MPS) representation of their ground and excited states. There is a wealth of solvable models relating to FQH physics, which, however, is mostly derived and analyzed from the vantage point of first-quantized "analytic clustering properties". In contrast, one obtains long-ranged frustration-free lattice models when these Hamiltonians are studied in an orbital basis, which is the natural basis for the MPS representation of FQH states. The connection between MPS-like s…
We prove that neither Integer nor Fractional Quantum Hall Effects with nonzero Hall conductivity are possible in gapped systems described by Local Commuting Projector Hamiltonians.
We use the Hamiltonian theory developed by Shankar and Murthy to study a quantum Hall system in a tilted magnetic field. With a finite width of the system in the $z$ direction, the parallel component of the magnetic field introduces anisotropy into the effective two-dimensional interactions. The effects of such anisotropy can be effectively captured by the recently proposed generalized pseudo-potentials. We find that the off-diagonal components of the pseudo-potentials lead to mixing of composite fermions Landau levels, which is a perturbation to the picture of $p$ filled Landau levels in comp…
We discuss anomalous fractional quantum Hall effect that exists without external magnetic field. We propose that excitations in such systems may be described effectively by non-interacting particles with the Hamiltonians defined on the Brillouin zone with a branch cut. Hall conductivity of such a system is expressed through the one-particle Green function. We demonstrate that for the Hamiltonians of the proposed type this expression takes fractional values times Klitzing constant. Possible relation of the proposed construction with degeneracy of ground state is discussed as well.
We consider the process of flux insertion for ground states of almost local commuting projector Hamiltonians in two spatial dimensions. In the case of finite dimensional local Hilbert spaces, we prove that this process cannot pump any charge and we conclude that the Hall conductance must vanish.
Zero-Clustering Geometry in Realistic Fractional Quantum Hall Wave Functions
Zhejiang Institute of Modern Physics and Zhejiang Key Laboratory of Micro-Nano Quantum Chips and Quantum Control, Zhejiang University（浙江大学现代物理研究所和浙江省微纳量子芯片与量子控制重点实验室）；Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University（重庆大学物理学院和重庆市强耦合物理重点实验室）
We investigate a model of hard-core bosons with correlated hopping on the honeycomb lattice in an external magnetic field by means of a coupled-wire approach. It has been numerically shown that this model exhibits at half filling the bosonic integer quantum Hall (BIQH) state, which is a symmetry-protected topological phase protected by the $U(1)$ particle conservation [Y.-C. He et al., Phys. Rev. Lett. 115, 116803 (2015)]. By combining the bosonization approach and a coupled-wire construction, we analytically confirm this finding and show that it even holds in the strongly anisotropic (quasi-o…
Conformal field theory has recently been applied to derive few-body Hamiltonians whose ground states are lattice versions of fractional quantum Hall states. The exact lattice models involve interactions over long distances, which is difficult to realize in experiments. It seems, however, that such long-range interactions should not be necessary, as the correlations decay exponentially in the bulk. This poses the question, whether the Hamiltonians can be truncated to contain only local interactions without changing the physics of the ground state. Previous studies have in a couple of cases with…
We study the transition of $ν=1/3$ and $2/5$ fractional quantum Hall states of the honeycomb Hofstadter model as we tune to a two-orbital moiré superlattice Hamiltonian, motivated by the flat bands of twisted bilayer graphene in a perpendicular magnetic field. In doing so, we address the extent to which these states survive in moiré systems and analyze the nature of the transition. Through the use of a Peierls substitution, we determine the Landau-level splitting for the moiré Hamiltonian, and study the structure of the Chern bands for a range of magnetic flux per plaquette. We identify topolo…
Considering lattice Hamiltonians designed using conformal field theory to have fractional quantum Hall states as ground states, we study the dynamics of one or two particles on such lattices. Examining the eigenspectrum and dynamics of the single-particle sector, we demonstrate that these Hamiltonians cannot be regarded as describing interacting particles placed on a Chern band, so that the physics is fundamentally different from fractional Chern insulators. The single-particle spectrum is shown to consist of eigenstates localized in shells which have larger radius for larger eigenenergies. Th…
We show that there is an emergent lattice description for the continuous fractional quantum Hall (FQH) systems, with a generalised set of few-body coherent states. In particular, model Hamiltonians of the FQH effect are equivalent to the real space von Neumann lattice of local projection operators imposed on a continuous system in the thermodynamic limit. It can be analytically derived that tuning local one-body potentials in such lattices amounts to the tuning of individual two or few-body pseudopotentials. For some cases, we can realise pure few-body pseudopotentials important for stabilisin…
Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase. In this work, we derived a set of analytical results for the energy gap of the graviton modes with two-body and three-body Hamiltonians in both the long-wavelength and thermodynamic limit. These allow us to construct model Hamiltonians for the graviton modes in different FQH phases, and to elucidate a hierarchical structure of conformal Hilbert spaces (nullspaces of model Hamiltonians) with respect to the graviton modes and their corresponding ground states. Using the analytical …
Proposed effective Hamiltonians from the literature for the material $α$-RuCl${}_3$ are used to compute the magnon thermal Hall conductivity, $κ_{xy}$, using linear spin wave theory for the magnetically ordered state. No model previously proposed that was tested explains published experimental data. Models with Kitaev interaction $K>0$ are seen to predict $κ_{xy}\gtrsim 0$, which is inconsistent with the data. Fluctuations toward a Kitaev-type spin liquid would have the wrong sign to explain the data. However, a slight variant of a previously proposed model predicts a large $κ_{xy}$, demonstra…
Lattice construction of pseudopotential Hamiltonians for Fractional Chern Insulators
分数Chern绝缘体是分数量子霍尔态在无轨道磁场晶格系统中的新实现。这些状态可以通过Wannier状态表示（Phys. Rev. Lett. 107，126803（2011））。本文利用Wannier态表象构造了分数阶Chern绝缘体的赝势哈密顿量，它是以某些理想模型波函数作为精确基态的相互作用哈密顿量。我们表明，这些赝势哈密顿可以近似的短程相互作用的分数陈省身绝缘体，和他们的范围将被最小化的最佳规范选择的Wannier状态。作为说明性的例子，我们明确地写出了几个分数陈绝缘体模型的最低赝势的形式，包括晶格狄拉克模型和棋盘模型与陈数1，和d波模型和三角晶格模型与陈数2。当Chern数C=1时，赝势哈密顿算符的基态为1/3 Laughlin态;当C=2时，赝势哈密顿算符的基态为拓扑的330态.还包括d-波模型和两个解耦层的晶格狄拉克模型之间的插值的结果，这明确地证明了C=2分数陈绝缘体和双层分数量子霍尔态之间的关系。所提出的状态可以通过未来的数值工作进行验证，特别是提供了一个模型哈密顿量的拓扑结构的状态，还没有实现数字。
A formalism is developed for the rigorous study of solvable fractional quantum Hall parent Hamiltonians with Landau level mixing. The idea of organization through "generalized Pauli principles" is expanded to allow for root level entanglement, giving rise to "entangled Pauli principles". Through the latter, aspects of the effective field theory description become ingrained in exact microscopic solutions for a great wealth of phases for which no similar single Landau level description is known. We discuss in detail braiding statistic, edge theory, and rigorous zero mode counting for the Jain-22…
We introduce and prove the "root theorem", which establishes a condition for families of operators to annihilate all root states associated with zero modes of a given positive semi-definite $k$-body Hamiltonian chosen from a large class. This class is motivated by fractional quantum Hall and related problems, and features generally long-ranged, one-dimensional, dipole-conserving terms. Our theorem streamlines analysis of zero-modes in contexts where "generalized" or "entangled" Pauli principles apply. One major application of the theorem is to parent Hamiltonians for mixed Landau-level wave fu…
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…
The Fibonacci topological order is the simplest platform for a universal topological quantum computer, consisting of a single type of non-Abelian anyon, $τ$, with fusion rule $τ\timesτ=1+τ$. While it has been proposed that the anyon spectrum of the $ν=12/5$ fractional quantum Hall state includes a Fibonacci sector, a dynamical picture of how a pure Fibonacci state may emerge in a quantum Hall system has been lacking. Here we use recently proposed non-Abelian dualities to construct a Fibonacci state of bosons at filling $ν=2$ starting from a trilayer of integer quantum Hall states. Our parent t…
Inter-Landau-level transitions break particle hole symmetry and will choose either the Pfaffian or the anti-Pfaffian state as the absolute ground state at 5/2 filling of the fractional quantum Hall effect. An approach based on truncating the Hilbert space has favored the anti-Pfaffian. A second approach based on an effective Hamiltonian produced the Pfaffian. In this letter perturbation theory is applied to finite sizes without bias to any specific pseudo-potential component. This method also singles out the anti-Pfaffian. A critical piece of the effective Hamiltonian, which was absent in prev…
Model Fractional Quantum Hall States on Lattices: Exact Parent Hamiltonians and Routes to Realization
International Center for Quantum Materials, Peking University（北京大学量子材料中心）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学量子器件北京市重点实验室）
We show that every uniform state on the sphere is essentially a superposition of regular graphs. In addition, we develop a graph-based ansatz to construct trial FHQ ground states sharing the local properties of Jack polynomials. In particular, our graphic states have the $(k,r)$ clustering property. Moreover, a subclass of the construction is realizable as the densest zero-energy state of a model that modifies the projection Hamiltonian.
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 study the $ν={\pm}1/2$ fractional quantum Hall states in graphene observed by Zibrov et al. [Nat. Phys. 14, 930 (2018)]. The parton construction is employed to provide a valley unpolarized trial wave function for these states. The lattice scale valley anisotropic terms in the Hamiltonian soften the repulsion between electrons in different valleys to favor the valley unpolarized state. The validity of our proposal is corroborated by numerical calculations.
We study the quantum self-organization of interacting particles in one-dimensional(1D) many-body systems, modeled via Hubbard chains with short-range interactions between the particles. We show the emergence of 1D states with density-wave and clustering order, related to topology, at odd denominator fillings that appear also in the fractional-quantum-Hall-effect (FQHE), which is a 2D electronic system with Coulomb interactions between the electrons and a perpendicular magnetic field. For our analysis we use an effective topological measure applied on the real space wavefunction of the system, …
We analyse various microscopic properties of the nematic fractional quantum Hall effect (FQHN) in the thermodynamic limit, and present necessary conditions required of the microscopic Hamiltonians for the nematic FQHE to be robust. Analytical expressions for the degenerate ground state manifold, ground state energies, and gapless nematic modes are given in compact forms with the input interaction and the corresponding ground state structure factors. We relate the long wavelength limit of the neutral excitations to the guiding center metric deformation, and show explicitly the family of trial w…
In this letter, we analyze the topological response of a fermionic model defined on the Lieb lattice in presence of an electromagnetic field. The tight-binding model is built in terms of three species of spinless fermions and supports a topological Varma phase due to the spontaneous breaking of time-reversal symmetry. In the low-energy regime, the emergent effective Hamiltonian coincides with the so-called Duffin-Kemmer-Petiau (DKP) Hamiltonian, which describes relativistic pseudospin-0 quasiparticles. By considering a minimal coupling between the DKP quasiparticles and an external Abelian gau…
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…
