LSM and CPT

TL;DR

The paper analyzes 1+1d lattice models with anti-unitary CPT-like symmetries, focusing on how lattice CRT-like operators generate mod-8 and mod-2 anomalies that constrain low-energy behavior through Lieb–Schultz–Mattis-type constraints. It develops a unified framework linking lattice anomalies to continuum counterparts, including emergent or emanant internal Z_2 symmetries and Smith isomorphisms realized via interface constructions. Through detailed studies of the Majorana chain, Heisenberg spin chain, Levin-Gu edge model, and Ising-type noninvertible RT structures, it demonstrates how lattice symmetries and their anomalies map to continuum CPT/Theta and to emergent anomalies, guiding the possible IR phases and gaplessness. The results illuminate how crystalline symmetries on the lattice can encode nontrivial anomaly structure that persists or transforms in the continuum, with implications for anomaly matching, emergent symmetries, and higher-dimensional generalizations.

Abstract

We study a number of 1+1d lattice models with anti-unitary symmetries that simultaneously reflect space and reverse time. Some of these symmetries are anomalous, leading to Lieb-Schultz-Mattis-type constraints, thus excluding a trivially gapped phase. Examples include a mod 8 anomaly in the Majorana chain and various mod 2 anomalies in the spin chain. In some cases, there is an exact, non-anomalous lattice symmetry that flows in the continuum to CPT. In some other cases, the CPT symmetry of the continuum theory is emergent or absent. Depending on the model, the anomaly of the lattice model is matched in the continuum in different ways. In particular, it can be mapped to an emergent anomaly of an emanant symmetry.

Figures (3)

• Figure 1: A pictorial way to compute $\Theta_\mathsf{U}^2$. In panel (a), we show a $\mathbb{Z}_2^\mathsf{U}$ defect. Applying $\Theta_\mathsf{U}$, which reverses space and time, leads to the defect in panel (b). Applying $\Theta_\mathsf{U}$ again leads to the defect in panel (c). Alternatively, $\Theta_\mathsf{U}^2$, which is a $2\pi$-rotation in Euclidean spacetime, maps the configuration in panel (a) directly to the configuration in panel (c). Certain topological manipulations of the defect, such as those described in Bhardwaj:2017xupChang:2018iayLin:2019kpn, bring the defect back to a straight line, with a factor of $\epsilon = \pm1$, reflecting the anomaly. A precise discussion of this manipulation can depend on a possible counterterm at the intersection point.
• Figure 2: The lattice Hamiltonians in \ref{['HDW']} preserving the anti-unitary symmetry ${\tilde{\Pi}}=\mathcal{R} \mathcal{T}$. Each blue link represents a pairing of two Majorana fermions of the form $i \chi_\text{odd}^A\chi_\text{even}^A$, while each red link represents a pairing term $i \chi_\text{even}^A\chi_\text{odd}^A$. The gray link exists when $L=4n$ or $L=4n-1$.
• Figure 3: A physics realization of the Smith isomorphism. In the continuum Majorana fermion field theory, we turn on opposite mass terms in two halves of space with localized zero modes at the interfaces. This setup relates a unitary $\mathbb{Z}_2$ symmetry $\mathsf{C}$ in 1+1d to an anti-unitary time-reversal symmetry $\mathsf{R}\mathsf{T}$ in quantum mechanics Hason:2020yqfCordova:2019wpi. The short arrows near the localized zero modes indicate how the time coordinate of the effective quantum mechanical model reflects under these operators.