[Paper Review] The Free Fermion Anomaly and Representations of the Pin Groups

Yakov Landau|arXiv (Cornell University)|Sep 13, 2018

Topological Materials and Phenomena17 references3 citations

TL;DR

This paper establishes a duality between a free complex fermion and a boson on a ring with half-integer flux, showing both exhibit a discrete anomaly tied to the Pin±(2) groups. The key contribution is a novel method to distinguish Pin₊(2n) and Pin₋(2n) irreducible representations using the commutation relation between time reversal and charge conjugation: TCT = ±C, which correlates with C² = ±1 and determines the anomaly type.

ABSTRACT

We consider a duality between a boson on a ring and a free fermion and show that they have an anomaly which corresponds to the states transforming under double covers of O(2). There are two (in general not isometric) double covers of O(2), known as Pin$_+(2)$ and Pin$_-(2)$. These can in general be distinguished at the group level by checking whether reflections square to $\pm 1$. We show that in irreducible representations in complex Hilbert spaces the commutation of time reversal and charge conjugation gives another method for distinguishing Pin$_+(2)$ and Pin$_-(2)$. While we only demonstrate the duality for a single fermion, the anomaly is also present in any number of free fermions. For an even number of fermions we show that the two double covers of O(2n) are isomorphic. For an odd number of fermions we show that the distinction between irreducible representations of Pin$_+$ and Pin$_-$ can still be detected by the commutation of time reversal and charge conjugation namely Pin$_\pm(2n)$ will have $TCT = \pm C$.

Motivation & Objective

• To resolve the ambiguity in distinguishing Pin₊(2) and Pin₋(2) structures in the context of a free fermion anomaly.
• To extend the anomaly classification to systems of multiple free fermions, particularly for odd and even numbers of fermions.
• To establish a method for distinguishing irreducible representations of Pin₊(2n) and Pin₋(2n) using time reversal and charge conjugation symmetry.
• To confirm that the same anomaly appears in both the fermionic and bosonic realizations of the duality.

Proposed method

• Construct a duality between a free complex fermion and a boson on a ring with half-integer magnetic flux, showing identical ground states and symmetries.
• Define two double covers of O(2), Pin₊(2) and Pin₋(2), distinguished by whether reflections square to +1 or -1.
• Use the commutation relation TCT = ±C to distinguish irreducible representations of Pin₊(2n) and Pin₋(2n), where T is time reversal and C is charge conjugation.
• Construct explicit Clifford algebra representations in ℂ(2ⁿ) using γ matrices, and define time reversal as T = θσ_y⊗⋯⊗σ_y with T² = (-1)ⁿ.
• Introduce anti-linear operators J₊ = TM and J₋ = TP, where M and P are products of γ matrices over specific indices, to determine whether a representation is real or quaternionic.
• Show that J₊ is equivariant only for Pin₊(2n) and J₋ only for Pin₋(2n), and compute J² to determine the real/quaternionic structure based on n mod 4.

Experimental results

Research questions

• RQ1 How can the Pin₊(2) and Pin₋(2) structures be distinguished beyond group-level relations?
• RQ2 What is the role of time reversal and charge conjugation in distinguishing irreducible representations of Pin₊(2n) and Pin₋(2n)?
• RQ3 Does the anomaly in a system of multiple free fermions remain detectable via TCT = ±C, and how does it depend on the fermion number?
• RQ4 Can the same discrete anomaly be consistently realized in both the fermionic and bosonic dual theories?
• RQ5 How does the real or quaternionic structure of the representation relate to the anomaly type in Pin±(2n)?

Key findings

• The anomaly in the free fermion and boson-on-a-ring system is classified by the Pin₊(2) or Pin₋(2) structure, distinguished by whether C² = 1 or C² = -1.
• For n odd, the irreducible representations of Pin₊(2n) and Pin₋(2n) are distinguished by the commutation relation TCT = C or TCT = -C, respectively.
• When n ≡ 1 mod 4, Pin₊(2n) is quaternionic and Pin₋(2n) is real; when n ≡ 3 mod 4, the roles reverse, as determined by J₊² and J₋².
• The duality between the free fermion and the boson on a ring preserves the anomaly, confirming that both theories share the same discrete symmetry anomaly.
• For an even number of fermions, the double covers Pin₊(2n) and Pin₋(2n) are isomorphic, but the distinction between them persists in the representation theory via TCT = ±C.
• The method of using TCT = ±C to distinguish Pin±(2n) representations provides a robust alternative to group-level definitions, especially in irreducible complex representations.