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Register a new userAn Exactly Solvable Model for a $4+1D$ Beyond-Cohomology Symmetry Protected Topological Phase
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We construct an exactly solvable commuting projector model for a $4+1$ dimensional ${mathbb Z}_2$ symmetry-protected topological phase (SPT) which is outside the cohomology classification of SPTs. The model is described by a decorated domain wall construction, with three-fermion Walker-Wang phases on the domain walls. We describe the anomalous nature of the phase in several ways. One interesting feature is that, in contrast to in-cohomology phases, the effective ${mathbb Z}_2$ symmetry on a $3+1$ dimensional boundary cannot be described by a quantum circuit and instead is a nontrivial quantum cellular automaton (QCA). A related property is that a codimension-two defect (for example, the termination of a ${mathbb Z}_2$ domain wall at a trivial boundary) will carry nontrivial chiral central charge $4$ mod $8$. We also construct a gapped symmetric topologically-ordered boundary state for our model, which constitutes an anomalous symmetry enriched topological phase outside of the classification of arXiv:1602.00187, and define a corresponding anomaly indicator.
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We propose a general construction of commuting projector lattice models for 2D and 3D topological phases enriched by U(1) symmetry, with finite-dimensional Hilbert space per site. The construction starts from a commuting projector model of the topological phase and decorates U(1) charges to the state space in a consistent manner. We show that all 2D U(1) symmetry-enriched topological phases which allow gapped boundary without breaking symmetry, can be realized through our construction. We also construct a large class of 3D topological phases with U(1) symmetry fractionalized on particles or loop excitations.
We construct fixed-point wave functions and exactly solvable commuting-projector Hamiltonians for a large class of bosonic symmetry-enriched topological (SET) phases, based on the concept of equivalent classes of symmetric local unitary transformations. We argue that for onsite unitary symmetries, our construction realizes all SETs free of anomaly, as long as the underlying topological order itself can be realized with a commuting-projector Hamiltonian. We further extend the construction to anti-unitary symmetries (e.g. time-reversal symmetry), mirror-reflection symmetries, and to anomalous SETs on the surface of three-dimensional symmetry-protected topological phases. Mathematically, our construction naturally leads to a generalization of group extensions of unitary fusion categories to anti-unitary symmetries.
The computation of certain obstruction functions is a central task in classifying interacting fermionic symmetry-protected topological (SPT) phases. Using techniques in group-cohomology theory, we develop an algorithm to accelerate this computation. Mathematically, cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear basis, known as the resolutions of the group. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinite to finite. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups.
We study entanglement in the Hatsugai-Kohmoto model, which exhibits a continuous interaction-driven Mott transition. By virtue of the all-to-all nature of its center-of-mass conserving interactions, the model lacks dynamical spectral weight transfer, which is the key to intractability of the Hubbard model for $d>1$. In order to maintain a non-trivial Mott-like electron propagator, SU(2) symmetry is preserved in the Hamiltonian, leading to a ground state that is mixed on both sides of the phase transition. Because of this mixture, even the metal in this model is unentangled between any pair of sites, unlike free fermions whose ground state carries a filling-dependent site-site entanglement. We focus on the scaling behavior of the one- and two-site entropies $s_1$ and $s_2$, as well as the entropy density $s$, of the ground state near the Mott transition. At low temperatures in the two-dimensional Hubbard model, it was observed numerically (Walsh et al., 2018, arXiv:1807.10409) that $s_1$ and $s$ increase continuously into the metal, across a first-order Mott transition. In the Hatsugai-Kohmoto model, $s_1$ acquires the constant value $ln4$ even at the Mott transition. The ground states non-trivial entanglement structure is manifest in $s_2$ and $s$ which decrease into the metal, and thereby act as sharp signals of the Mott transition in any dimension. Specifically, we find that in one dimension, $s_2$ and $s$ exhibit kinks at the transition while in $d=2$, only $s$ exhibits a kink.
Symmetry protected topological (SPT) phases in free fermion and interacting bosonic systems have been classified, but the physical phenomena of interacting fermionic SPT phases have not been fully explored. Here, employing large-scale quantum Monte Carlo simulation, we investigate the edge physics of a bilayer Kane-Mele-Hubbard model with zigzag ribbon geometry. Our unbiased numerical results show that the fermion edge modes are gapped out by interaction, while the bosonic edge modes remain gapless at the $(1+1)d$ boundary, before the bulk quantum phase transition to a topologically trivial phase. Therefore, finite fermion gaps both in the bulk and on the edge, together with the robust gapless bosonic edge modes, prove that our system becomes an emergent bosonic SPT phase at low energy, which is, for the first time, directly observed in an interacting fermion lattice model.
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