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arXiv:1505.02576 · Physical Review B

A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations

Sagar Vijay, Jeongwan Haah, Liang Fu

Published Dec 21, 2015• 401 citations
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Abstract

Exactly solvable models often provide valuable insights in theoretical studies of topological phases. Here the authors introduce exactly solvable models of interacting Majorana fermions each with extensive topological ground-state degeneracy and a hierarchy of pointlike, topological excitations that are only free to move within submanifolds of the full lattice. These very different models make up a new kind of topological quantum order.

DOI
10.1103/physrevb.92.235136

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Exactly solvable models often provide valuable insights in theoretical studies of topological phases. Here the authors introduce exactly solvable models of interacting Majorana fermions each with extensive topological ground-state degeneracy and a hierarchy of pointlike, topological excitations that are only free to move within submanifolds of the full lattice. These very different models make up a new kind of topological quantum order.
We introduce exactly solvable models of interacting (Majorana) fermions in d≥3 spatial dimensions that realize a new kind of fermion topological quantum order, building on a model presented by S. Vijay, T. H. Hsieh, and L. Fu [Phys. Rev. X 5, 041038 (2015)10.1103/PhysRevX.5.041038]. These models have extensive topological ground-state degeneracy and a hierarchy of pointlike, topological excitations that are only free to move within submanifolds of the lattice. In particular, one of our models has fundamental excitations that are completely stationary. To demonstrate these results, we introduce a powerful polynomial representation of commuting Majorana Hamiltonians. Remarkably, the physical properties of the topologically ordered state are encoded in an algebraic variety, defined by the common zeros of a set of polynomials over a finite field. This provides a “geometric” framework for the emergence of topological order.
Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.
Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this Letter, we present a nontrivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wave functions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the nonapproximate study of topological one-dimensional parafermionic chains with spatial inversion and time-reversal symmetry in the absence of strong edge modes.
We construct an exactly solvable Hamiltonian acting on a three-dimensional lattice of spin-12\frac{1}{2}21​ systems that exhibits topological quantum order. The ground state is a string net and a membrane-net condensate. Excitations appear in the form of quasiparticles and fluxes, as the boundaries of strings and membranes, respectively. The degeneracy of the ground state depends on the homology of the 3-manifold. We generalize the system to $D\ensuremath{\geqslant}4$, where different topological phases may occur. The whole construction is based on certain special complexes that we call colexes.
Recently, there has been much effort in understanding topological phases of matter with gapless bulk excitations, which are characterized by topological invariants and protected intrinsic boundary states. Here we show that topological semimetals of Majorana fermions arise in exactly solvable Kitaev spin models on a series of three-dimensional lattices. The ground states of these models are quantum spin liquids with gapless nodal spectra of bulk Majorana fermion excitations. It is shown that these phases are topologically stable as long as certain discrete symmetries are protected. The corresponding topological indices and the gapless boundary states are explicitly computed to support these results. In contrast to previous studies of noninteracting systems, the phases discussed in this work are novel examples of gapless topological phases in interacting spin systems.
We discuss an extension of higher-order topological phases to include bosonic systems. We present two spin models for a second-order topological phase protected by a global ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. One model is built from layers of an exactly solvable cluster model for a one-dimensional ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological phase, while the other is built from more conventional spin couplings (XY or Heisenberg). These models host gapped, symmetry-protected topological phases on their edges, and corner modes that fall into a projective representation of the symmetry. Using Jordan-Wigner transformations we show that both our models are related to a bilayer of free Majorana fermions that form a fermionic second-order topological phase. We also discuss how our models can be extended to three dimensions to form a third-order topological phase.