Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $σ$ models and a special group supercohomology theory (1201.2648v4)

Abstract: Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of super cocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (ie, non-on-site) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric} boundary must be gapless for 1+1D boundary, and must be gapless or topologically ordered beyond 1+1D. As an application of this general result, we construct a new SPT phase in 3D, for interacting fermionic superconductors with coplanar spin order (which have $T^2=1$ time-reversal $Z_2^T$ and fermion-number parity $Z_2^f$ symmetries described by a full symmetry group $Z_2^T\times Z_2^f$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion-pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in 2D with a full symmetry group $Z_2\times Z_2^f$. Those 2D fermionic SPT phases all have central-charge $c=1$ gapless edge excitations, if the symmetry is not broken.

• The paper introduces group supercohomology theory to systematically classify interacting fermionic SPT phases in 1D, 2D, and 3D systems.
• The paper employs a discrete space-time path integral approach to construct ideal ground state wave functions and bulk Hamiltonians.
• The paper identifies new fermionic SPT phases with gapless edge states under time-reversal and cyclic symmetries, offering clear experimental predictions.

Overview of the Paper

The paper "Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory" by Gu and Wen explores the theoretical framework of symmetry-protected topological (SPT) phases applicable to fermions. This paper extends the utilization of group cohomology theory from bosons to more complex interacting fermionic systems. By introducing a specialized group supercohomology theory, the authors aim to construct a systematic approach to describe and classify fermionic SPT phases across various dimensions.

Primary Contributions

1. Group Supercohomology Theory: The paper proposes a generalization of the standard group cohomology theory, known as group supercohomology theory, specifically designed to account for interacting fermionic systems. This theory incorporates data from super cocycles, enabling the construction of ideal ground state wave functions and bulk Hamiltonians for fermionic SPT phases.
2. Dimensional Analysis and Constraints: Through thorough exploration in 1D, 2D, and 3D systems, the authors identify conditions under which certain boundary symmetries necessitate gapless edge states or lead to topological orders. The paper highlights examples where these fermionic SPT states exhibit features unattainable by free fermions or interacting bosons alone.
3. New Fermionic SPT Phases: A notable achievement is the construction of previously unidentified SPT phases, specifically in 3D systems with time-reversal symmetry and 2D systems with cyclic symmetries. The paper provides examples and Hamiltonians to realize these phases in theoretical models.
4. Rigorous Computational Methods: The paper employs a discrete space-time path integral approach, translating physical theories into computational models that withstand renormalization group transformations, maintaining topological properties across scales.
5. Interplay with Symmetry and Stability: An essential aspect discussed is the relationship between symmetry operations on fermions and the mathematical formalism involving function transformations via group supercohomology, ensuring a consistent theoretical model aligned with physical symmetries.

Numerical Results and Theoretical Implications

• Classification of Phases: The paper outlines the fermionic phases manifested in the presence of certain symmetries and computes their exact classifications. For instance, $\mathcal{H}^d[G_f, U_T(1)]$ is calculated for various symmetry groups indicating the possibility of multiple, distinct fermionic phases not previously categorized.
• Gapless Edge States: The theoretical framework predicts conditions where edge states remain gapless, protected by non-on-site symmetries, leading to experimental implications in surface conductivity and quantum information robustness.

Future Directions and Speculations

While the scope of this paper is extensive in classifying and understanding fermionic SPT orders, it opens numerous avenues for future research:

• Extensions Beyond Special Supercohomology: The authors note the limitation of their formulation to scenarios where fermions form one-dimensional representations of symmetry groups. Extending this theory to encompass cases involving higher-dimensional representations remains a crucial future goal.
• Experimental Realizations: Although primarily theoretical, the findings have potential experimental applications in quantum materials and topological superconductors, especially in systems where conventional symmetry-based classifications fail.
• Integration with Quantum Information: The intricate relationship between topological phases and quantum computation hints at possible advancements in error-resistant qubit systems and quantum entanglement paradigms.

In summary, Gu and Wen's paper presents a groundbreaking method to explore fermionic SPT phases through an innovative theoretical lens, employing computational and algebraic formalism with profound implications for both theoretical physics and potential experimental applications.