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Register a new userLocal Commuting Projector Hamiltonians and the Quantum Hall Effect
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We prove that neither Integer nor Fractional Quantum Hall Effects with nonzero Hall conductivity are possible in gapped systems described by Local Commuting Projector Hamiltonians.
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We consider the process of flux insertion for ground states of almost local commuting projector Hamiltonians in two spatial dimensions. In the case of finite dimensional local Hilbert spaces, we prove that this process cannot pump any charge and we conclude that the Hall conductance must vanish.
We introduce a family of commuting-projector Hamiltonians whose degrees of freedom involve $mathbb{Z}_{3}$ parafermion zero modes residing in a parent fractional-quantum-Hall fluid. The two simplest models in this family emerge from dressing Ising-paramagnet and toric-code spin models with parafermions; we study their edge properties, anyonic excitations, and ground-state degeneracy. We show that the first model realizes a symmetry-enriched topological phase (SET) for which $mathbb{Z}_2$ spin-flip symmetry from the Ising paramagnet permutes the anyons. Interestingly, the interface between this SET and the parent quantum-Hall phase realizes symmetry-enforced $mathbb{Z}_3$ parafermion criticality with no fine-tuning required. The second model exhibits a non-Abelian phase that is consistent with $text{SU}(2)_{4}$ topological order, and can be accessed by gauging the $mathbb{Z}_{2}$ symmetry in the SET. Employing Levin-Wen string-net models with $mathbb{Z}_{2}$-graded structure, we generalize this picture to construct a large class of commuting-projector models for $mathbb{Z}_{2}$ SETs and non-Abelian topological orders exhibiting the same relation. Our construction provides the first commuting-projector-Hamiltonian realization of chiral bosonic non-Abelian topological order.
Inspired by a recently constructed commuting-projector Hamiltonian for a two-dimensional (2D) time-reversal-invariant topological superconductor [Wang et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector model that describes an interacting yet exactly solvable 2D topological insulator. We explicitly show that both the gapped and gapless boundaries of our model are consistent with those of band-theoretic, weakly interacting topological insulators. Interestingly, on certain lattices our time-reversal-symmetric models also enjoy $mathcal{CP}$ symmetry, leading to intuitive interpretations of the bulk invariant for a $mathcal{CP}$-symmetric topological insulator upon putting the system on a Klein bottle. We also briefly discuss how these many-body invariants may be able to characterize models with only time-reversal symmetry.
Using the decorated domain wall procedure, we construct Finite Depth Local Unitaries (FDLUs) that realize Fermionic Symmetry-Protected Topological (SPT) phases. This results in explicit full commuting projector Hamiltonians, where full implies the fact that the ground state, as well as all excited states of these Hamiltonians, realizes the nontrivial SPT phase. We begin by constructing explicit examples of 1+1D phases protected by symmetry groups $G=mathbb Z_2^T times mathbb Z_2^F$ , which also has a free fermion realization in class BDI, and $G=mathbb Z_4 times mathbb Z_4^F$, which does not. We then turn to 2+1D, and construct the square roots of the Levin-Gu bosonic SPT phase, protected by $mathbb Z_2 times mathbb Z_2^F$ symmetry, in a concrete model of fermions and spins on the triangular lattice. Edge states and the anomalous symmetry action on them are explicitly derived. Although this phase has a free fermion representation as two copies of $p+ip$ superconductors combined with their $p-ip$ counterparts with a different symmetry charge, the full set of commuting projectors is only realized in the strongly interacting version, which also implies that it admits a many-body localized realization.
We report on our systematic attempts at finding local interactions for which the lowest-Landau-level projected composite-fermion wave functions are the unique zero energy ground states. For this purpose, we study in detail the simplest non-trivial system beyond the Laughlin states, namely bosons at filling $ u=frac{2}{3}$ and identify local constraints among clusters of particles in the ground state. By explicit calculation, we show that no Hamiltonian up to (and including) four particle interactions produces this state as the exact ground state, and speculate that this remains true even when interaction terms involving greater number of particles are included. Surprisingly, we can identify an interaction, which imposes an energetic penalty for a specific entangled configuration of four particles with relative angular momentum of $6hbar$, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state, referred to as the $lambda$-state, is not identical to the projected composite-fermion state, but the following facts suggest that the two might be topologically equivalent: the two sates have a high overlap; they have the same root partition; the quantum numbers for their neutral excitations are identical; and the quantum numbers for the quasiparticle excitations also match. On the quasihole side, we find that even though the quantum numbers of the lowest energy states agree with the prediction from the composite-fermion theory, these states are not separated from the others by a clearly identifiable gap. This prevents us from making a conclusive claim regarding the topological equivalence of the $lambda$ state and the composite-fermion state. Our study illustrates how new candidate states can be identified from constraining selected many particle configurations and it would be interesting to pursue their topological classification.
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