[Paper Review] Homotopy Theoretic Classification of Symmetry Protected Phases

Jonathan A. Campbell|arXiv (Cornell University)|Aug 14, 2017

Homotopy and Cohomology in Algebraic Topology15 references11 citations

TL;DR

This paper provides a homotopy-theoretic classification of symmetry-protected topological phases (SPTs) using spectra derived from cobordism theories, particularly computing low-dimensional homotopy groups of novel cobordism spectra such as $MT ext{Pin}^-$, $MT ext{Pin}^+$, and $M( ext{Spin} \times_{\mathbb{Z}/2} \mathbb{Z}/2^n)$. It confirms physical classification results via algebraic topology, offering a rigorous mathematical framework for SPT classification that aligns with physical computations, thereby validating the cobordism hypothesis in condensed matter physics.

ABSTRACT

We classify a number of symmetry protected phases using Freed-Hopkins' homotopy theoretic classification. Along the way we compute the low-dimensional homotopy groups of a number of novel cobordism spectra.

Motivation & Objective

• To provide a mathematically rigorous classification of symmetry-protected topological phases (SPTs) using homotopy theory.
• To compute the low-dimensional homotopy groups of novel cobordism spectra relevant to physical systems with various symmetry groups.
• To verify that the homotopy-theoretic classification via the cobordism hypothesis agrees with known physical classifications of SPTs.
• To extend the framework of Freed-Hopkins to include fermionic and time-reversal symmetric systems, particularly $\text{Spin} \times_{\mathbb{Z}/2} \mathbb{Z}/2^n$-structures.
• To explore the physical relevance of manifold representatives for cobordism classes, which remain unknown in most cases.

Proposed method

• Uses the cobordism hypothesis to model invertible topological quantum field theories (TQFTs) as maps of spectra from $MTG(n)$ to $\Sigma^{n+1}I_{\mathbb{Z}}$, where $I_{\mathbb{Z}}$ is the Anderson dual of the sphere spectrum.
• Applies the Adams spectral sequence to compute homotopy groups of cobordism spectra such as $MT\text{Pin}^\pm$, $MT\text{Pin}^{\tilde{c}\pm}$, and $M(\text{Spin} \times_{\mathbb{Z}/2} \mathbb{Z}/2^n)$.
• Employs Thom isomorphism and $\mathcal{A}(1)$-module structures on mod 2 cohomology to analyze the spectral sequence's $E_2$-page.
• Utilizes the May-Milgram theorem to identify differentials in the Adams spectral sequence, particularly $d_n$ as the $2^n$-Bockstein.
• Constructs Thom spectra via pullbacks from classifying spaces, e.g., $M(\text{Spin} \times_{\mathbb{Z}/2} \mathbb{Z}/2^n) \simeq M\text{Spin} \wedge (B\mathbb{Z}/2^{n-1})^{2\xi}$.
• Validates results by comparing with known physical classifications, showing exact agreement without relying on physical models.

Experimental results

Research questions

• RQ1 How can symmetry-protected topological phases be classified using stable homotopy theory and cobordism spectra?
• RQ2 What are the low-dimensional homotopy groups of $MT\text{Pin}^-$, $MT\text{Pin}^+$, and related spectra for physical symmetry groups?
• RQ3 To what extent does the homotopy-theoretic classification of invertible TQFTs reproduce known physical classifications of SPTs?
• RQ4 Can the Adams spectral sequence be effectively applied to compute homotopy groups of exotic cobordism spectra arising from group extensions like $\text{Spin} \times_{\mathbb{Z}/2} \mathbb{Z}/2^n$?
• RQ5 Why do the results from homotopy-theoretic methods agree precisely with those from physical lattice model computations, despite different methodologies?

Key findings

• The homotopy groups of $MT\text{Pin}^-$ are $\pi_0 = \mathbb{Z}/2$, $\pi_1 = 0$, $\pi_2 = \mathbb{Z}/2$, $\pi_3 = \mathbb{Z}/16$, $\pi_4 = 0$, $\pi_5 = 0$, matching known physical results.
• For $MT\text{Pin}^+$, the groups are $\pi_0 = \mathbb{Z}/2$, $\pi_1 = \mathbb{Z}/2$, $\pi_2 = \mathbb{Z}/8$, $\pi_3 = 0$, $\pi_4 = 0$, $\pi_5 = 0$, confirming physical classifications.
• The spectrum $M(\text{Spin} \times_{\mathbb{Z}/2} \mathbb{Z}/2^n)$ has $\pi_0 = \mathbb{Z}$, $\pi_1 = \mathbb{Z}/2^n$, $\pi_2 = 0$, $\pi_3 = \mathbb{Z}/2^{n-2}$, $\pi_4 = \mathbb{Z}$, $\pi_5 = \mathbb{Z}/16$ for $n \geq 3$, with explicit computation via Adams spectral sequence.
• The classification of bosonic SPTs with $U(1)$ symmetry in $d+1$ dimensions is given by $[MSO \wedge \mathbb{C}P^\infty_+, \Sigma^{n+1}I_{\mathbb{Z}}]$, yielding $\pi_0 = \mathbb{Z}$, $\pi_1 = 0$, $\pi_2 = \mathbb{Z} \oplus \mathbb{Z}$, $\pi_3 = 0$, $\pi_4 = \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}/2$.
• For systems with time-reversal symmetry, the classification via $[MO, \Sigma^{n+1}I_{\mathbb{Z}}]$ gives $\pi_0 = 0$, $\pi_1 = \mathbb{Z}/2$, $\pi_2 = 0$, $\pi_3 = (\mathbb{Z}/2)^2$, $\pi_4 = \mathbb{Z}/2$, matching physical expectations.
• The fermionic system with $\mathbb{Z}/2$ symmetry in $3+1$ dimensions has trivial classification, as $[M\text{Spin} \wedge B\mathbb{Z}/2_+, \Sigma^5 I_{\mathbb{Z}}] = 0$, implying no non-trivial SPT phases.