In addition to possessing fractional statistics, anyon excitations of a 2D topologically ordered state can realize symmetry in distinct ways , leading to a variety of symmetry enriched topological (SET) phases. While the symmetry fractionalization must be consistent with the fusion and braiding rules of the anyons, not all ostensibly consistent symmetry fractionalizations can be realized in 2D systems. Instead, certain `anomalous SETs can only occur on the surface of a 3D symmetry protected topological (SPT) phase. In this paper we describe a procedure for determining whether an SET of a discrete, onsite, unitary symmetry group $G$ is anomalous or not. The basic idea is to gauge the symmetry and expose the anomaly as an obstruction to a consistent topological theory combining both the original anyons and the gauge fluxes. Utilizing a result of Etingof, Nikshych, and Ostrik, we point out that a class of obstructions are captured by the fourth cohomology group $H^4( G, ,U(1))$, which also precisely labels the set of 3D SPT phases, with symmetry group $G$. We thus establish a general bulk-boundary correspondence between the anomalous SET and the 3d bulk SPT whose surface termination realizes it. We illustrate this idea using the chiral spin liquid ($U(1)_2$) topological order with a reduced symmetry $mathbb{Z}_2 times mathbb{Z}_2 subset SO(3)$, which can act on the semion quasiparticle in an anomalous way. We construct exactly solved 3d SPT models realizing the anomalous surface terminations, and demonstrate that they are non-trivial by computing three loop braiding statistics. Possible extensions to anti-unitary symmetries are also discussed.
It is well known that unitary symmetries can be `gauged, i.e. defined to act in a local way, which leads to a corresponding gauge field. Gauging, for example, the charge conservation symmetry leads to electromagnetic gauge fields. It is an open question whether an analogous process is possible for time reversal which is an anti-unitary symmetry. Here we discuss a route to gauging time reversal symmetry which applies to gapped quantum ground states that admit a tensor network representation. The tensor network representation of quantum states provides a notion of locality for the wave function coefficient and hence a notion of locality for the action of complex conjugation in anti-unitary symmetries. Based on that, we show how time reversal can be applied locally and also describe time reversal symmetry twists which act as gauge fluxes through nontrivial loops in the system. As with unitary symmetries, gauging time reversal provides useful access to the physical properties of the system. We show how topological invariants of certain time reversal symmetric topological phases in $D=1,2$ are readily extracted using these ideas.
The surfaces of three dimensional topological insulators (3D TIs) are generally described as Dirac metals, with a single Dirac cone. It was previously believed that a gapped surface implied breaking of either time reversal $mathcal T$ or U(1) charge conservation symmetry. Here we discuss a novel possibility in the presence of interactions, a surface phase that preserves all symmetries but is nevertheless gapped and insulating. Then the surface must develop topological order of a kind that cannot be realized in a 2D system with the same symmetries. We discuss candidate surface states - non-Abelian Quantum Hall states which, when realized in 2D, have $sigma_{xy}=1/2$ and hence break $mathcal T$ symmetry. However, by constructing an exactly soluble 3D lattice model, we show they can be realized as $mathcal T$ symmetric surface states. The corresponding 3D phases are confined, and have $theta=pi$ magnetoelectric response. Two candidate states have the same 12 particle topological order, the (Read-Moore) Pfaffian state with the neutral sector reversed, which we term T-Pfaffian topological order, but differ in their $mathcal T$ transformation. Although we are unable to connect either of these states directly to the superconducting TI surface, we argue that one of them describes the 3D TI surface, while the other differs from it by a bosonic topological phase. We also discuss the 24 particle Pfaffian-antisemion topological order (which can be connected to the superconducting TI surface) and demonstrate that it can be realized as a $mathcal T$ symmetric surface state.
One-way quantum computing achieves the full power of quantum computation by performing single particle measurements on some many-body entangled state, known as the resource state. As single particle measurements are relatively easy to implement, the preparation of the resource state becomes a crucial task. An appealing approach is simply to cool a strongly correlated quantum many-body system to its ground state. In addition to requiring the ground state of the system to be universal for one-way quantum computing, we also want the Hamiltonian to have non-degenerate ground state protected by a fixed energy gap, to involve only two-body interactions, and to be frustration-free so that measurements in the course of the computation leave the remaining particles in the ground space. Recently, significant efforts have been made to the search of resource states that appear naturally as ground states in spin lattice systems. The approach is proved to be successful in spin-5/2 and spin-3/2 systems. Yet, it remains an open question whether there could be such a natural resource state in a spin-1/2, i.e., qubit system. Here, we give a negative answer to this question by proving that it is impossible for a genuinely entangled qubit states to be a non-degenerate ground state of any two-body frustration-free Hamiltonian. What is more, we prove that every spin-1/2 frustration-free Hamiltonian with two-body interaction always has a ground state that is a product of single- or two-qubit states, a stronger result that is interesting independent of the context of one-way quantum computing.
Measurement based quantum computation (MBQC), which requires only single particle measurements on a universal resource state to achieve the full power of quantum computing, has been recognized as one of the most promising models for the physical realization of quantum computers. Despite considerable progress in the last decade, it remains a great challenge to search for new universal resource states with naturally occurring Hamiltonians, and to better understand the entanglement structure of these kinds of states. Here we show that most of the resource states currently known can be reduced to the cluster state, the first known universal resource state, via adaptive local measurements at a constant cost. This new quantum state reduction scheme provides simpler proofs of universality of resource states and opens up plenty of space to the search of new resource states, including an example based on the one-parameter deformation of the AKLT state studied in [Commun. Math. Phys. 144, 443 (1992)] by M. Fannes et al. about twenty years ago.
