Fermionic and bosonic Laughlin state on thick cylinders

Jansen, Sabine

doi:10.1063/1.4768250

Cited by 10 publications

(14 citation statements)

References 17 publications

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“…As such, it might be useful for generating approximate matrixproduct state (MPS) expressions for any type of quantum Hall state, given that alternative methods [46,47] depend sensitively on the type of CFT. Unfortunately, it is known that MPS based on the first-order thin torus expansion [27][28][29][30] is quite different from the alternate one based on CFT, and yields a poor description of the state in the isotropic (2D) limit [47]. An open question remains whether higher-order corrections near the thin-torus limit can generally be organized in a tractable way to achieve an accurate MPS for L y ≫ ℓ B , perhaps in a form of generalized Schrieffer-Wolff [61] or continuous unitary transformations [62] that have been applied successfully to the Hubbard model.…”

Section: Discussionmentioning

confidence: 99%

Solvable models for unitary and nonunitary topological phases

Papić

2014

Phys. Rev. B

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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 Ly of the Hall surface becomes comparable to the magnetic length ℓB. 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 neutral excitation spectrum. We study a large class of Abelian and non-Abelian states in this limit, including the Read-Rezayi Z k series, as well as states deriving from non-unitary or irrational conformal field theories -the "Gaffnian", "Haffnian", Haldane-Rezayi, and the "permanent" state. We find that the thin-cylinder limit of unitary (rational) states is "classical": their effective Hamiltonians reduce to only Hartree-type terms, the ground states are trivial insulators, and excitation gaps result from simple electrostatic repulsion. In contrast, for states deriving from non-unitary or irrational conformal field theories, the thin-cylinder limit is found to be intrinsically quantum -it contains hopping terms that play an important role in the structure of the ground states and in the energetics of the low-lying neutral excitations.

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Section: Discussionmentioning

confidence: 99%

Solvable models for unitary and nonunitary topological phases

Papić

2014

Phys. Rev. B

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“…Accessing the set of topologically degenerate ground states further allows us to extract the modular S; T matrices that encode all topological information about the quasiparticles [32,33]. Furthermore, parent Hamiltonians on a cylinder or torus can be used to derive solvable models for quantum Hall states when one of the spatial dimensions becomes comparable to the magnetic length [34][35][36][37][38][39]. It has been shown that such models can be used to construct "matrix-product state" representations for quantum Hall states, and in some cases, they can be used to study the physics of "nonunitary" states and classify their gapless excitations [39][40][41].…”

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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“…Although Eqs. (12) and (14) coincide at q = 3 (q = 2) for fermions (bosons), they are different in general. However, we may interpret the truncated Hamiltonian of (1) with (14) as an approximation of the full Hamiltonian with (12) for L 1 → 0 limit (see Appendix A).…”

Section: E Relationship With Laughlin Statesmentioning

confidence: 96%

“…Appendix A: Matrix elements of the pseudo potential [Eqs. (12) and (14)] We consider a model of N interacting electrons on a torus with lengths L l in x l directions (l = 1, 2). When the torus is pierced by N s magnetic flux quanta, boundary conditions require L 1 L 2 = 2πN s .…”

Section: Acknowledgmentsmentioning

confidence: 99%

One-dimensional lattice model with an exact matrix-product ground state describing the Laughlin wave function

Wang

Nakamura

2013

Phys. Rev. B

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We introduce one-dimensional lattice models with exact matrix-product ground states describing the fractional quantum Hall (FQH) states in Laughlin series (given by filling factors ν = 1/q) on torus geometry. Surprisingly, the exactly solvable Hamiltonian has the same mathematical structure as that of the pseudopotential for the Laughlin wave function, and naturally derives the general properties of the Laughlin wave function such as the Z2 properties of the FQH states and the fermion-boson relation. The obtained exact ground states have high overlaps with the Laughlin states and well describe their properties. Using the matrix product method, density functions and correlation functions are calculated analytically. Especially, obtained entanglement spectra reflects gapless edge states as was discussed by Li and Haldane.

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Fermionic and bosonic Laughlin state on thick cylinders

Cited by 10 publications

References 17 publications

Solvable models for unitary and nonunitary topological phases

Solvable models for unitary and nonunitary topological phases

Geometric Construction of Quantum Hall Clustering Hamiltonians

One-dimensional lattice model with an exact matrix-product ground state describing the Laughlin wave function

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