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.
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.
Using an algebra of second-quantized operators, we develop local two-body parent Hamiltonians for all unprojected Jain states at filling factor n/(2np+1), with integer n and (half-)integer p. We rigorously establish that these states are uniquely stabilized and that zero mode counting reproduces mode counting in the associated edge conformal field theory. We further establish the organizing "entangled Pauli principle" behind the resulting zero mode paradigm and unveil an emergent SU(n) symmetry characteristic of the fixed point physics of the Jain quantum Hall fluid.
We present microscopic, multiple Landau level, (frustration-free and positive semi-definite) parent Hamiltonians whose ground states, realizing different quantum Hall fluids, are parton-like and whose excitations display either Abelian or non-Abelian braiding statistics. We prove ground state energy monotonicity theorems for systems with different particle numbers in multiple Landau levels, demonstrate S-duality in the case of toroidal geometry, and establish complete sets of zero modes of special Hamiltonians stabilizing parton-like states, specifically at filling factor \nu=2/3 ν = 2 / 3 . The emergent Entangled Pauli Principle (EPP), introduced in [Phys. Rev. B 98, 161118(R) (2018)] and which defines the “DNA” of the quantum Hall fluid, is behind the exact determination of the topological characteristics of the fluid, including charge and braiding statistics of excitations, and effective edge theory descriptions. When the closed-shell condition is satisfied, the densest (i.e., the highest density and lowest total angular momentum) zero-energy mode is a unique parton state. We conjecture that parton-like states generally span the subspace of many-body wave functions with the two-body M M -clustering property within any given number of Landau levels, that is, wave functions with M M th-order coincidence plane zeroes and both holomorphic and anti-holomorphic dependence on variables. General arguments are supplemented by rigorous considerations for the M=3 M = 3 case of fermions in four Landau levels. For this case, we establish that the zero mode counting can be done by enumerating certain patterns consistent with an underlying EPP. We apply the coherent state approach of [Phys. Rev. X 1, 021015 (2011)] to show that the elementary (localized) bulk excitations are Fibonacci anyons. This demonstrates that the DNA associated with fractional quantum Hall states encodes all universal properties. Specifically, for parton-like states, we establish a link with tensor network structures of finite bond dimension that emerge via root level entanglement.
The study of model Hamiltonians, whose exact ground states are well known quantum Hall states, have been investigated solely in the literature from the point of view of first quantized approach in the last few decades. However, a second quantized approach to such Hamiltonians has not been taken in the same measure. In this dissertation, we study Haldane pseudopotential, which is the parent Hamiltonian of 1/M Laughlin state, and Trugman-Kivelson pseudopotential, which is the parent Hamiltonian of 2/5 unprojected Jain state. We find that the study of these Hamiltonians in their second quantized forms not only reproduces all properties of their zero modes already known in first quantized approach, but also sheds lights on general studies of such structures of these Hamiltonians as frustration free lattice Hamiltonians.
We consider lattice Hamiltonians that arise from putting Haldane pseudopotentials into a second quantized or ``guiding-center-only'' form. These are fascinating examples of 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 pseudopotentials can be understood from a polynomial-free, intrinsically lattice point of view is also suggested by the fact that these pseudopotentials 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.
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 second-quantized 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 ``nonlocal order parameters'' generalizes to the present situation, and we are able to give explicit second-quantized formulas for nonlocal order parameters associated with CF states.
Number of zeros seen by a particle around small clusters of other particles is encoded in the root partition, and partly characterizes the correlations in fractional quantum Hall trial wavefunctions. We explore a generalization wherein we consider the counting of zeros seen by a cluster of particles on another cluster. Numbers of such zeros between clusters in the Laughlin wavefunctions are fully determined by the root partition. However, such a counting is unclear for general Jain states where a polynomial expansion is difficult. Here we consider the simplest state beyond the Laughlin wavefunction, namely a state containing a single quasiparticle of the Laughlin state. We show numerically and analytically that in the trial wavefunction for the quasiparticle of the Laughlin state, counting of zeros seen by a cluster on another cluster depends on the relative dimensions of the two clusters. We further ask if the patterns in the counting of zeros extend, in at least an approximate sense, to wavefunctions beyond the trial states. Using numerical computations in systems up to N=9, we present results for the statistical distribution of zeros around particle clusters at the center of an FQH droplet in the ground state of a Hamiltonian that is perturbed away from the V1 interaction (short-range repulsion). Evolution of this distribution with the strength of the perturbation shows that the counting of zeros is altered by even a weak perturbation away from the parent Hamiltonian, though the perturbations do not change the phase of the system.