Zero modes, bosonization, and topological quantum order: The Laughlin state in second quantization

Mazaheri, Tahereh; Ortíz, Gerardo; Nussinov, Zohar; Seidel, Alexander

doi:10.1103/physrevb.91.085115

Cited by 26 publications

(59 citation statements)

References 81 publications

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“…From this very fact the entire zero mode structure can be derived, also by using only the second quantized algebraic setting used here. 19,76 We believe that these results deepen our insights into the structure of fractional quantum Hall and Chern insulator type of parent Hamiltonians, as well as frustration free lattice Hamiltonians in general, and in particular, long ranged ones with matrix product ground states of unbounded bond dimension. Their application to other known as well as possibly novel parent Hamiltonians is left as an interesting direction for the future.…”

Section: Discussionmentioning

confidence: 75%

“…The relation between the latter and the e n is not of any importance in the following, and will be clarified by work in parallel. 76 In terms of the e n , we now define new operators…”

Section: The Generators Of Edge Excitations and Relevant Commutatimentioning

confidence: 99%

“…In this special case, the relation (18) we seek turns out to be a second quantized rendering of Read's recursive formula for the Laughlin state using the "string order parameter". 75 The relation of our formalism to the string order parameter will be explored by work in parallel, 76 and will not be elaborated further in the following.…”

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confidence: 99%

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Algebraic approach to the study of zero modes of Haldane pseudopotentials

Chen

Seidel

2015

Phys. Rev. B

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We consider lattice Hamiltonians that arise from putting Haldane pseudo-potentials into a second quantized or "guiding-center-only" form. These are fascinating examples for frustration free lattice Hamiltonians. This is so since even though their highest density zero energy ground states, the Laughlin states, are known to have matrix-product structure (with unbounded bond dimension), the frustration free character of these lattice Hamiltonians seems obscure, unless one goes back to the original first quantized picture of analytic lowest Landau level wave functions. This step involves putting back additional degrees of freedom associated with dynamical momenta, and one wonders whether the addition of these degrees of freedom is truly necessary to recognize the frustration free character of the underlying lattice Hamiltonian. Fundamentally, these degrees of freedom have nothing to do with spectrum of a "guiding-center-only" Hamiltonian. Moreover, such constructions are unfamiliar and not available in the study of simpler (finite range) frustration free lattice Hamiltonians with matrix product ground states (of finite bond dimension). That the zero mode properties of "lattice versions" of pseudo-potentials can be understood from a polynomial-free, intrinsically lattice point of view is also suggested by the fact that these pseudo-potentials are constructed from an algebra of reasonably simply looking operators. Here we show that zero mode properties, and hence the frustration free character, of these lattice Hamitlonians can be understood as a consequence of algebraic structures that these operators are part of. We believe that our results will deepen insights into parent Hamiltonians of matrix product states with infinite bond dimensions, as could be of use, especially, in the study of fractional Chern insulators.

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Section: Discussionmentioning

confidence: 75%

“…The relation between the latter and the e n is not of any importance in the following, and will be clarified by work in parallel. 76 In terms of the e n , we now define new operators…”

Section: The Generators Of Edge Excitations and Relevant Commutatimentioning

confidence: 99%

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Algebraic approach to the study of zero modes of Haldane pseudopotentials

Chen

Seidel

2015

Phys. Rev. B

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“…(In all aspects, such an operator-based approach has been constructed by some of us previously for the n = 1 case related to the 1/3-Laughlin state, and in fact for all the Laughlin states. 27,36,37 We will comment more on the situation below. )…”

Section: Quantized Operators Q (I)mentioning

confidence: 99%

“…Then the one-toone correspondence between CF-states and edge states at fixed N i applies to all CF-states whose angular momentum relative to |N 0 , N 1 CF is smaller than a cutoff given by particle number: ∆L N i (c.f., e.g., Ref. 37). That is, the number of such CF zero modes of given N i and ∆L relative to |N 0 , N 1 CF is equal to the number of edge states described by Eq.…”

Section: B Edge Mode Countingmentioning

confidence: 99%

Jain-2/5 parent Hamiltonian: Structure of zero modes, dominance patterns, and zero mode generators

2017

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We analyze general zero mode properties of the parent Hamiltonian of the unprojected Jain-2/5 state. We characterize the zero mode condition associated to this Hamiltonian via projection onto a four-dimensional two-particle subspace for given pair angular momentum, for the disk and similarly for the spherical geometry. Earlier numerical claims in the literature about ground state uniqueness on the sphere are substantiated on analytic grounds, and related results are derived. Preference is given to second quantized methods, where zero mode properties are derived not from given analytic wave functions, but from a "lattice" Hamiltonian and associated zero mode conditions. This method reveals new insights into the guiding-center structure of the unprojected Jain-2/5 state, in particular a system of dominance patterns following a "generalized Pauli principle", which establishes a complete one-to-one correspondence with the edge mode counting. We also identify one-body operators that function as generators of zero modes.

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Mechanism for particle fractionalization and universal edge physics in quantum Hall fluids

et al. 2022

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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. We present a second-quantization framework that reveals an exact fusion mechanism for particle fractionalization in FQH fluids, and uncovers the fundamental structure behind the condensation of non-local operators characterizing topological order in the lowest-Landau-level. We show the first exact analytic computation of the quasielectron Berry connections leading to its fractional charge and exchange statistics, and perform Monte Carlo simulations that numerically confirm the fusion mechanism for quasiparticles. We express the sequence of (bosonic and fermionic) Laughlin second-quantized states, highlighting the lack of local condensation, and present a rigorous constructive subspace bosonization dictionary for the bulk fluid. Finally, we establish universal long-distance behavior of edge excitations by formulating a conjecture based on the DNA, or root state, of the FQH fluid.

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Zero modes, bosonization, and topological quantum order: The Laughlin state in second quantization

Cited by 26 publications

References 81 publications

Algebraic approach to the study of zero modes of Haldane pseudopotentials

Algebraic approach to the study of zero modes of Haldane pseudopotentials

Jain-2/5 parent Hamiltonian: Structure of zero modes, dominance patterns, and zero mode generators

Mechanism for particle fractionalization and universal edge physics in quantum Hall fluids

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