The effects of interactions on the topological classification of free fermion systems

TL;DR

The paper presents a concrete counterexample to the free-fermion topological classification by showing that eight coupled Majorana chains with an unusual Time-Reversal symmetry (T^2=1) admit an interacting, gapped path linking two free phases with equal free invariants modulo 8, thereby reducing the Z classification to Z8. It provides an explicit Quartic, T-invariant interaction W that gaps boundary modes and, together with an adiabatic continuation analysis, demonstrates the connection through a strongly interacting region. The continuum analysis maps the problem to the SO(8)_1 WZW model and reveals a rich Ising-like phase structure, with RG flows showing a transition in the 2D Ising universality class and a phase diagram featuring first-order, second-order, and no-transition regions. These results illustrate how interactions can fundamentally alter topological classifications and provide a detailed framework for understanding interacting topological phases in 1D. The findings have implications for the classification of topological phases beyond free-fermion theories and highlight the role of symmetry and strong coupling in shaping boundary physics and phase diagrams.

Abstract

We describe in detail a counterexample to the topological classification of free fermion systems. We deal with a one dimensional chain of Majorana fermions with an unusual T symmetry. The topological invariant for the free fermion classification is an integer, but with the introduction of interactions it becomes well defined only modulo 8. We illustrate this in the microscopic model of the Majorana chain by constructing an explicit path between two distinct free phases whose topological invariants are equal modulo 8, along which the system remains gapped. The path goes through a strongly interacting region. We also find the field theory interpretation of this phenomenon. There is a second order phase transition between the two phases in the free theory which can be avoided by going through the strongly interacting region. We show that this transition is in the 2D Ising universality class, where a first order phase transition line, terminating at a second order transition, can be avoided by going through the analogue of a high temperature paramagnetic phase. In fact, we construct the full phase diagram of the system as a function of the thermal operator (i.e. the mass term that tunes between the two phases in the free theory) and two quartic operators, obtaining a first order Peierls transition region, a second order transition region, and a region with no transition.

Figures (5)

• Figure 1: Eigenvalues of $H = t \, T + (1-t) \, W_{\text{tot}}$ as a function of $t$. The system remains gapped throughout the path.
• Figure 2: Eigenvalues of $\tilde{H}=V + t\,T + w\,W_{\text{tot}}$ along the path $(t,w) = (0.1 \cos \theta, 0.1 \sin \theta), \, \theta \in [0, \pi]$. The degeneracy is broken and the system remains gapped.
• Figure 3: Smoothly connecting the $m>0$ and $m<0$ phases through an Ising transition region
• Figure 4: RG flow for $B$ ($x$-axis) and $A$ ($y$-axis).
• Figure 5: Phase diagram for the transition from negative to positive $m$, indicating dependence on $B$ ($x$-axis) and $A$ ($y$-axis)