Bloch Model Wave Functions and Pseudopotentials for All Fractional Chern Insulators

Wu, Yang-Le; Regnault, Nicolas; Bernevig, B. Andrei

doi:10.1103/physrevlett.110.106802

Cited by 108 publications

(152 citation statements)

References 48 publications

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“…This approach relies on the relative angular momentum, which, in this case, is an exact quantum number. As such, it cannot be applied to the cylinder, torus, or any quantum Hall lattice model such as fractional Chern insulators, for which the analogue of pseudopotentials has been developed recently [44][45][46][47].…”

Section: Introductionmentioning

confidence: 99%

Geometric Construction of Quantum Hall Clustering Hamiltonians

2015

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Many fractional quantum Hall wave functions are known to be unique highest-density zero modes of certain "pseudopotential" Hamiltonians. While a systematic method to construct such parent Hamiltonians has been available for the infinite plane and sphere geometries, the generalization to manifolds where relative angular momentum is not an exact quantum number, i.e., the cylinder or torus, remains an open problem. This is particularly true for non-Abelian states, such as the Read-Rezayi series (in particular, the Moore-Read and Read-Rezayi Z 3 states) and more exotic nonunitary (Haldane-Rezayi and Gaffnian) or irrational (Haffnian) states, whose parent Hamiltonians involve complicated many-body interactions. Here, we develop a universal geometric approach for constructing pseudopotential Hamiltonians that is applicable to all geometries. Our method straightforwardly generalizes to the multicomponent SUðnÞ cases with a combination of spin or pseudospin (layer, subband, or valley) degrees of freedom. We demonstrate the utility of our approach through several examples, some of which involve non-Abelian multicomponent states whose parent Hamiltonians were previously unknown, and we verify the results by numerically computing their entanglement properties.

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Section: Introductionmentioning

confidence: 99%

Geometric Construction of Quantum Hall Clustering Hamiltonians

2015

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“…The most striking features of FQH state are the topological ground state degeneracy on torus and the emerging quasiparticles obeying fractional statistics 1,2 . Recently, it has been demonstrated that FQH states can also be realized in various topological flat-band (TFB) models without Landau levels [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] . In such an interacting system, an explicit demonstration of topological order and quasiparticle statistics is still highly desired, which has attracted lots of recent interests [22][23][24][25][26][27][28][29][30][31] .…”

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Minimal entangled states and modular matrix for fractional quantum Hall effect in topological flat bands

Zhu

Haldane

2013

Phys. Rev. B

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We perform an exact diagonalization study of the topological order in topological flat band models through calculating entanglement entropy and spectra of low energy states. We identify multiple independent minimal entangled states, which form a set of orthogonal basis states for the groundstate manifold. We extract the modular transformation matrices S (U) which contains the information of mutual (self) statistics, quantum dimensions and fusion rule of quasiparticles. Moreover, we demonstrate that these matrices are robust and universal in the whole topological phase against different perturbations until the quantum phase transition takes place.

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“…Since interactions on a lattice are intrinsically directional, FCIs can possess anisotropic PPs that are even larger than the isotropic ones. The PP decomposition of a FCI requires a basis mapping between the single-body orbitals of the FCI and FQH systems [39][40][41][42][43][44]. We adopt the mapping introduced in Ref.…”

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confidence: 99%

Generalized Pseudopotentials for the Anisotropic Fractional Quantum Hall Effect

Yang

Lee

et al. 2017

Phys. Rev. Lett.

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We generalize the notion of Haldane pseudopotentials to anisotropic fractional quantum Hall (FQH) systems that are physically realized, e.g., in tilted magnetic field experiments or anisotropic band structures. This formalism allows us to expand any translation-invariant interaction over a complete basis, and directly reveals the intrinsic metric of incompressible FQH fluids. We show that purely anisotropic pseudopotentials give rise to new types of bound states for small particle clusters in the infinite plane, and can be used as a diagnostic of FQH nematic order. We also demonstrate that generalized pseudopotentials quantify the anisotropic contribution to the effective interaction potential, which can be particularly large in models of fractional Chern insulators. DOI: 10.1103/PhysRevLett.118.146403 The fractional quantum Hall (FQH) system is host to a wide variety of topological phases of matter [1]. This complexity belies the deceivingly simple microscopic Hamiltonian containing only the effective Coulomb interaction projected to a single Landau level (LL) [2]. The understanding of different topological states was greatly facilitated by the concept of pseudopotentials (PPs) introduced by Haldane [3,4]. This formalism allows one to expand any rotation-invariant interaction over the complete basis of the PPs, which are projection operators onto two-particle states with a given value of relative angular momentum. Furthermore, a combination of a small number of PPs naturally defines parent Hamiltonians for some FQH model states, such as the Laughlin states [5,6]. The method has also been generalized to many-body PPs [7,8], which form the parent Hamiltonians of the non-Abelian FQH states [9,10]. In many cases, the ground state of these model Hamiltonians is believed to be adiabatically connected to the actual ground state of the experimental system. Thus, the relatively simple (and to some degree analytically tractable) model wave functions and Hamiltonians give much insight into the nature of the experimentally realized FQH states.Recently, interest in the FQH effect has been renewed due to emerging connections between topological order, geometry, and broken symmetry. An early precursor of these ideas was the realization that rotational invariance is not necessary for the FQH effect [4]. This lead to the conclusion that FQH states possess new "geometrical" degrees of freedom [11], uncovering a more complete description of their low-energy properties [12][13][14]. The notion of geometry has also inspired the construction of a more general class of Laughlin states with the nonEuclidean metric [15], which was shown to be physically relevant in situations where the band mass or dielectric tensor is anisotropic [16][17][18], or in the tilted magnetic field [19]. On the other hand, an intriguing coexistence of topological order with broken symmetry [20,21], leading to the nematic FQH effect, has also been proposed [22,23]. The nematic order is expected to arise due to spontaneous symmetry breaking, as suggested ...

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Bloch Model Wave Functions and Pseudopotentials for All Fractional Chern Insulators

Cited by 108 publications

References 48 publications

Geometric Construction of Quantum Hall Clustering Hamiltonians

Geometric Construction of Quantum Hall Clustering Hamiltonians

Minimal entangled states and modular matrix for fractional quantum Hall effect in topological flat bands

Generalized Pseudopotentials for the Anisotropic Fractional Quantum Hall Effect

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