Composite fermions in Fock space: Operator algebra, recursion relations, and order parameters

Abstract

We develop recursion relations, in particle number, for all (unprojected) Jain composite fermion (CF) wave functions. These recursions generalize a similar recursion originally written down by Read for Laughlin states, in mixed first/second-quantized notation. In contrast, our approach is purely second-quantized, giving rise to an algebraic, "pure guiding center" definition of CF states that de-emphasizes first-quantized many-body wave functions. Key to the construction is a secondquantized representation of the flux attachment operator that maps any given fermion state to its CF counterpart. An algebra of generators of edge excitations is identified. In particular, in those cases where a well-studied parent Hamiltonian exists, its properties can be entirely understood in the present framework, and the identification of edge state generators can be understood as an instance of "microscopic bosonization". The intimate connection of Read's original recursion with "non-local order parameters" generalizes to the present situation, and we are able to give explicit second-quantized formulas for non-local order parameters associated with CF states. arXiv:1812.08353v3 [cond-mat.str-el]

Inner workings of fractional quantum Hall parent Hamiltonians: A matrix product state point of view

“…Eq. ( 18) can be iterated arbitrarily many times to give the general edge excitation (17). This now establishes these states as a complete set of zero modes if we already knew that the 𝑝 𝑛 generate such when acting on |𝜓 𝑁 .…”

“…𝑎 𝑛 𝑀 . While the above motivates the expression (17), it does not establish this expression from a Hamiltonian principle, which we will achieve below. As a stepping stone, it is worthwhile to motivate Eq.…”

“…A radical departure from this is the study of FQH Hamiltonians in the LL occupation number basis, i.e., in second quantization. 12,[14][15][16][17][18] In this approach, FQH parent Hamiltonians are indeed formally cast as one dimensional lattice models, albeit, crucially, with interactions that are not strictly finite ranged. At the pure wave function level, the last decade has seen similar radically new directions, driven by the insight that FQH model wave functions are matrix product states (MPS).…”

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

“…Studying the long-range order of Read's operator [17] amounts to establishing that K M (z) † K M (0) approaches a non-zero constant at large |z| [18], or alternatively, the condensation of…”

“…e n M for l > 0, and 0 otherwise, as defined in [18,44]. Before we proceed with the proof we remark that our M corresponds to M + 1 in [18,44], and we are working with the disk geometry as opposed to the infinitely thick cylinder geometry used therein. Moreover, we use the phase (−1) M N which is appropriate both for fermions and bosons.…”