Symmetry protected topological orders and the group cohomology of their symmetry group (1106.4772v6)

Abstract: Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H, G_{\Psi}, H^{1+d}[G_{\Psi}, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_{\Psi} the symmetry group of the ground states.

• The paper introduces a group cohomology framework to classify interacting bosonic SPT phases via explicit cohomology constructions.
• It demonstrates that SPT orders exhibit full bulk energy gaps while maintaining gapless or degenerate edge states under symmetry preservation.
• The study links SPT phases to topological insulators, offering insights for fault-tolerant quantum protocols and future explorations in condensed matter physics.

Symmetry Protected Topological Orders and Group Cohomology

The paper of quantum phases of matter, particularly those that are not captured by the conventional Landau symmetry-breaking paradigm, continues to be a key focus in condensed matter physics. This paper explores Symmetry Protected Topological (SPT) orders, which emerge as unique phases of matter exhibiting non-trivial topological properties that are protected by some global symmetry. The authors leverage group cohomology to systematically characterize these phases for interacting bosonic systems in various dimensions.

SPT phases are characterized by the presence of a full energy gap in the bulk and potentially gapless edge states, the latter due to the preservation of certain symmetries. Unlike phases with intrinsic topological order, which display anyonic excitations and topological ground state degeneracy, SPT phases can transition into a trivial phase without closing the gap if the protecting symmetry is broken. The relevance of SPT phases is exemplified by the Haldane phase in one-dimensional spin chains and topological insulators, highlighted for their protected edge states due to time-reversal, charge-conservation, and spin-rotation symmetries.

Main Contributions and Results

1. Group Cohomology Classification: The primary contribution of the paper is the development of a framework to classify interacting bosonic SPT phases using group cohomology theory. Specifically, distinct SPT phases in $d$-dimensional systems with a symmetry group $G$ are classified by $\cH^{1+d}(G, U_T(1))$, the group cohomology classes of $G$.
2. Explicit Construction: The authors construct explicit representatives of these cohomology classes, leading to concrete models of the corresponding SPT phases. They demonstrate that these constructions result in states whose boundary excitations are either gapless or form degenerate states if symmetries are preserved.
3. Examples and Generality: The paper provides examples for various common symmetry groups, detailing the classification results in multiple dimensions. For instance, it is shown that systems with $SO(3)$ symmetry in three dimensions have seven distinct SPT phases, while those with $U(1)\rtimes Z_2^T$ exhibit three non-trivial phases.
4. Relation to Topological Insulators and Superconductors: The framework naturally extends to bosonic analogues of fermionic topological insulators and superconductors. These arise in contexts where additional symmetries, such as time-reversal or compositional symmetries (e.g., $U(1)$), interplay with the topological protection mechanisms.

Theoretical and Practical Implications

Such a classification not only advances the theoretical understanding of quantum phases and topological order but also sets the stage for future explorations of how symmetries protect non-local quantum entanglements. The implications are broad, affecting quantum computation and information, where edge states and SPT phases may play critical roles in robust quantum protocols.

From a practical standpoint, identifying and classifying SPT phases allow experimental physicists to design systems that exploit these properties, such as constructing systems where quantum bits remain entangled and robust due to symmetry protection against local perturbations.

Future Directions

The research invites several avenues for future exploration, including the extension of these methods to fermionic systems using group super-cohomology, exploring interactions between SPT phases and intrinsic topological orders, and developing a comprehensive theory that unifies these concepts within a single framework of symmetry-enriched topological phases. Furthermore, the role of symmetry and topology in quantum systems continues to drive innovation in topological quantum computing and fault-tolerant quantum state engineering.

In conclusion, this paper provides a sophisticated approach to classify and construct SPT phases, offering deep insights into the richness of quantum topological phenomena protected by symmetry, and opening pathways towards leveraging these phases for advanced technological applications.