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Register a new userCommuting-projector Hamiltonians for 2D topological insulators: edge physics and many-body invariants
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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.
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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.
We discuss how strongly interacting higher-order symmetry protected topological (HOSPT) phases can be characterized from the entanglement perspective: First, we introduce a topological many-body invariant which reveals the non-commutative algebra between flux operator and $C_n$ rotations. We argue that this invariant denotes the angular momentum carried by the instanton which is closely related to the discrete Wen-Zee response and fractional corner charge. Second, we define a new entanglement property, dubbed `higher-order entanglement, to scrutinize and differentiate various higher-order topological phases from a hierarchical sequence of the entanglement structure. We support our claims by numerically studying a super-lattice Bose-Hubbard model that exhibits different HOSPT phases.
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 prove that neither Integer nor Fractional Quantum Hall Effects with nonzero Hall conductivity are possible in gapped systems described by Local Commuting Projector Hamiltonians.
A gapped many-body system is described by path integral on a space-time lattice $C^{d+1}$, which gives rise to a partition function $Z(C^{d+1})$ if $partial C^{d+1} =emptyset$, and gives rise to a vector $|Psirangle$ on the boundary of space-time if $partial C^{d+1} eqemptyset$. We show that $V = text{log} sqrt{langlePsi|Psirangle}$ satisfies the inclusion-exclusion property $frac{V(Acup B)+V(Acap B)}{V(A)+V(B)}=1$ and behaves like a volume of the space-time lattice $C^{d+1}$ in large lattice limit (i.e. thermodynamics limit). This leads to a proposal that the vector $|Psirangle$ is the quantum-volume of the space-time lattice $C^{d+1}$. The inclusion-exclusion property does not apply to quantum-volume since it is a vector. But quantum-volume satisfies a quantum additive property. The violation of the inclusion-exclusion property by $V = text{log} sqrt{langlePsi|Psirangle}$ in the subleading term of thermodynamics limit gives rise to topological invariants that characterize the topological order in the system. This is a systematic way to construct and compute topological invariants from a generic path integral. For example, we show how to use non-universal partition functions $Z(C^{2+1})$ on several related space-time lattices $C^{2+1}$ to extract $(M_f)_{11}$ and $text{Tr}(M_f)$, where $M_f$ is a representation of the modular group $SL(2,mathbb{Z})$ -- a topological invariant that almost fully characterizes the 2+1D topological orders.
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