Topological phases of fermions in one dimension (1008.4138v2)

Abstract: In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of 1D systems. We focus on the TR-invariant Majorana chain (BDI symmetry class). While the band classification yields an integer topological index $k$, it is known that phases characterized by values of $k$ in the same equivalence class modulo 8 can be adiabatically transformed one to another by adding suitable interaction terms. Here we show that the eight equivalence classes are distinct and exhaustive, and provide a physical interpretation for the interacting invariant modulo 8. The different phases realize different Altland-Zirnbauer classes of the reduced density matrix for an entanglement bipartition into two half-chains. We generalize these results to the classification of all one dimensional gapped phases of fermionic systems with possible anti-unitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.

• The paper demonstrates that interactions reduce the non-interacting Z invariant to a Z8 classification in 1D fermionic systems.
• Authors employ the entanglement spectrum via matrix product states to analyze time-reversal and fermion parity symmetries.
• The study links interacting invariants to Wall’s theorem, revealing a deep connection between group cohomology and topological phases.

Overview of "Topological Phases of Fermions in One Dimension"

The paper "Topological Phases of Fermions in One Dimension" by Lukasz Fidkowski and Alexei Kitaev contributes to the theoretical understanding of topological phases in one-dimensional (1D) fermionic systems. The authors investigate how the classification of topological phases in 1D systems, particularly those described by the symmetry class BDI, is affected by interactions. Specifically, they address the TR-invariant Majorana chain and propose that the apparent topological distinctions captured by band theory are modified when interactions are considered, leading to a reduction from an integer classification to $\mathbb{Z}_8$.

Key Contributions

1. Identification of $\mathbb{Z}_8$ Classification: The paper expands on known topological classifications of non-interacting fermionic systems by accounting for interacting effects. The authors demonstrate that the $k$-index used to characterize phases in band theory only corresponds to distinct phases modulo 8 when interactions are allowed. This notion is a significant adjustment from the purely band-theory-oriented viewpoint, which considers $k$ as $\mathbb{Z}$.
2. Entanglement Spectrum Approach: The authors employ the concept of the entanglement spectrum as a diagnostic tool for analyzing topological phases. By examining the reduced density matrix from a spatial bipartition, they assess the symmetry properties related to time-reversal and fermionic parity. This approach mirrors techniques used for topological insulators and the fractional quantum Hall effect, reinforcing the entanglement spectrum's role in identifying topological order beyond edge modes.
3. Projective Representations and Symmetries: The use of matrix product states (MPS) allows a systematic paper of symmetries on the entanglement Hilbert space. The MPS formalism provides a framework in which the interplay of fermionic parity and time-reversal symmetry can be naturally explored, yielding projective representations that extend to anti-unitary symmetries. The paper uses this method to furnish rigorous arguments for their reduction in classification.
4. Interacting Invariants and Wall’s Theorem: By connecting the entanglement-based classification to Wall's theorem, the authors frame interaction effects in topological phases within the mathematical structure of central extensions and relevant group cohomology. Their work builds a bridge between these theoretical constructs and practical case studies of fermionic chains, extending their results to potentially describe all 1D gapped phases with symmetry considerations.

Implications and Speculations on Future Developments

The reconception of topological classification in interacting systems has profound implications for condensed matter physics and quantum computing. The findings of this paper might spur further exploration of interaction-driven phase transitions in topological insulators and superconductors. Future developments could involve extending these results to higher-dimensional systems, considering more complex symmetry classes, or exploring similar classifications in systems with additional degrees of symmetry.

From a practical viewpoint, interfacing this theoretical advancement with experimental observations in cold atom setups or solid-state contexts could offer new insights into realizing robust quantum states for qubits, thereby impacting quantum technology developments.

In speculative terms, such insights into 1D systems could hint at complex phenomena in multi-dimensional and multi-domain quantum systems. The adaptability of entanglement-based techniques to broader and more complicated systems might mark a significant advancement in understanding quantum matter's interacting and hierarchical structures.

Overall, the paper significantly impacts the theoretical groundwork for interacting topological phases in one-dimensional systems, offering a context that dovetails mathematical rigor with physical intuition and applicability.