Zero modes, Bosonization and Topological Quantum Order: The Laughlin State in Second Quantization

Abstract

We introduce a "second-quantized" representation of the ring of symmetric functions to further develop a purely second-quantized -- or "lattice" -- approach to the study of zero modes of frustration free Haldane-pseudo-potential-type Hamiltonians, which in particular stabilize Laughlin ground states. We present three applications of this formalism. We start demonstrating how to systematically construct all zero-modes of Laughlin-type parent Hamiltonians in a framework that is free of first-quantized polynomial wave functions, and show that they are in one-to-one correspondence with dominance patterns. The starting point here is the pseudo-potential Hamiltonian in "lattice form", stripped of all information about the analytic structure of Landau levels (dynamical momenta). Secondly, as a by-product, we make contact with the bosonization method, and obtain an alternative proof for the equivalence between bosonic and fermionic Fock spaces. Finally, we explicitly derive the second-quantized version of Read's non-local (string) order parameter for the Laughlin state, extending an earlier description by Stone. Commutation relations between the local quasi-hole operator and the local electron operator are generalized to various geometries.

version published in PRB, editor's suggestion

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Cited by in corpus (18)

• Geometric construction of Quantum Hall clustering Hamiltonians
• Entangled Pauli Principles: the DNA of Quantum Hall Fluids
• Local two-body parent Hamiltonians for the entire Jain sequence
• Algebraic approach to the study of zero modes of Haldane pseudopotentials
• The Jain-2/5 parent Hamiltonian: structure of zero modes, dominance patterns, and zero mode generators
• The Neutral Excitations in the Gaffnian State
• Composite fermions in Fock space: Operator algebra, recursion relations, and order parameters
• Statistical interactions and boson-anyon duality in fractional quantum Hall fluids
• Bosonization of Majorana modes and edge states
• Bounds for low-energy spectral properties of center-of-mass conserving positive two-body interactions
• Mechanism for particle fractionalization and universal edge physics in quantum Hall fluids
• Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders
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• From frustration-free parent Hamiltonians to off-diagonal long-range order: Moore-Read and related states in second quantization
• Fusion mechanism for quasiparticles and topological quantum order in the lowest Landau level
• Linking infinite bond-dimension matrix product states with frustration-free Hamiltonians
• Second-quantized approach to the study of Halperin state in fractional quantum Hall effect
• Excitation and tunneling spectra of a fractional quantum Hall system in the thin cylinder limit