Topological Phases of One-Dimensional Fermions: An Entanglement Point of View

Abstract: The effect of interactions on topological insulators and superconductors remains, to a large extent, an open problem. Here, we describe a framework for classifying phases of one-dimensional interacting fermions, focusing on spinless fermions with time-reversal symmetry and particle number parity conservation, using concepts of entanglement. In agreement with an example presented by Fidkowski \emph{et. al.} (Phys. Rev. B 81, 134509 (2010)), we find that in the presence of interactions there are only eight distinct phases, which obey a $\mathbb{Z}_8$ group structure. This is in contrast to the $\mathbb{Z}$ classification in the non-interacting case. Each of these eight phases is characterized by a unique set of bulk invariants, related to the transformation laws of its entanglement (Schmidt) eigenstates under symmetry operations, and has a characteristic degeneracy of its entanglement levels. If translational symmetry is present, the number of distinct phases increases to 16.

• The paper presents a novel classification method using entanglement analysis to reduce the topological phases of interacting 1D fermions from ℤ to ℤ8.
• It employs entanglement spectra to identify eight unique phases based on symmetry invariants, with potential expansion to 16 when translational symmetry is included.
• The work establishes a ℤ8 group structure that informs phase combinations and guides the design of quantum systems with robust topological properties.

Analyzing Topological Phases of One-Dimensional Fermions Through Entanglement

This paper, titled "Topological Phases of One-Dimensional Fermions: An Entanglement Point of View," explores the complex classifications of one-dimensional interacting fermions, with specific focus on topological phases, using entanglement as an analytical framework. The authors, Ari M. Turner, Frank Pollmann, and Erez Berg, confront the challenges posed by electron-electron interactions and propose a robust method to categorize distinct phases in systems characterized by time-reversal symmetry and particle number parity conservation.

Background and Objective

The landscape of topological insulators and superconductors, particularly in non-interacting systems, has been substantially charted through past research, highlighting systems characterized by the topology of Bloch bands or fermionic quasi-particle spectra. However, the introduction of interactions obscures this classification, as the Hamiltonian cannot be reduced to a single particle matrix. This paper aims to fill this gap by developing a classification strategy for one-dimensional fermionic systems, including interactions, utilizing the entanglement properties of ground states.

Main Findings

1. Reduction of Classification from $\mathbb{Z}$ to $\mathbb{Z}_8$: The paper corroborates the findings of Fidkowski and Kitaev, demonstrating that interactions reduce the classification of one-dimensional topological phases from infinite $\mathbb{Z}$ to a finite $\mathbb{Z}_8$. The authors articulate that only eight distinct phases exist when interactions are considered.
2. Characterization Through Entanglement Spectra: The work extends the concept of entanglement spectra to classify phases. Through the symmetry properties of entanglement (Schmidt) eigenstates, the paper identifies eight topologically distinct phases, each defined by a unique set of invariants related to symmetry operations. The paper demonstrates that these phases are not smoothly connectable unless a bulk phase transition occurs.
3. Implications for Translational Symmetry: The authors note that if translational symmetry is present, the number of distinct phases increases from eight to sixteen. This highlights the complexity introduced by additional symmetries and reflects the deep interrelation between physical and entanglement-induced topological properties.
4. Group Structure of Phases: The phases obey a $\mathbb{Z}_8$ group structure, which helps in constructing and understanding the relationships between these phases. This structure emerges from rules for combining phases with different invariants.

Implications and Future Directions

The findings have important implications for the understanding of topological phases in interactive systems. Practically, these results could affect the design of quantum systems and materials, where the robustness and stability of topological phases are essential. Theoretically, this framework opens up avenues to explore more complex topological structures in higher dimensions and across other Altland-Zirnbauer symmetry classes. Future research may focus on expanding these topological insights to higher-dimensional systems or those with different symmetry constraints, considering the revealed impact of interaction-induced modifications on topological invariants.

The paper provides a compelling framework to understand how interactions reshape the landscape of topological phases, moving from an infinite to a finite classification, and establishing significant links between localized symmetries in entanglement spectra and global topological features. This approach enriches our comprehension of one-dimensional fermionic systems, offering detailed insights that may pave the way for further studies in related areas.