1d lattice models for the boundary of 2d "Majorana" fermion SPTs: Kramers-Wannier duality as an exact $Z_2$ symmetry

TL;DR

This work probes the boundaries of beyond-(super)cohomology 2D fermion SPTs with symmetry Z2 × Zf2 by constructing an exact 1D boundary model that lives in a constrained Hilbert space rather than a local tensor product. The boundary dynamics are encoded by a simple 3-site Hamiltonian and a Z2 self-duality symmetry implementing Kramers-Wannier duality, with evidence from ED and DMRG that the edge flows to an Ising CFT at criticality. The study connects the edge theory to a bulk Tarantino-Fidkowski model and derives the boundary action rigorously, including NS/R spin-structure sectors and their parity properties, demonstrating how non-onsite symmetries can capture boundary anomalies not realizable with onsite symmetries. The results generalize to other ν in Z8 and to related symmetry classes, offering a concrete framework to understand boundaries of beyond-cohomology SPTs and raising questions about the role of constrained Hilbert spaces in classifying such phases.

Abstract

It is well known that symmetry protected topological (SPT) phases host non-trivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite symmetry action. This raises the question: can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of 2+1D fermion SPTs. We focus on the root SPT phase for the symmetry group $G =Z_2 \times Z^f_2$. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective 1d lattice model for the boundary. Crucially, the Hilbert space of this 1d model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the $Z_2$ symmetry on this Hilbert space and find a simple 3-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising CFT where the $Z_2$ symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the 2+1D topological superconductor protected by time-reversal symmetry ${\cal T}$ with ${\cal T}^2 = (-1)^{\cal F}$.

Figures (15)

• Figure 1: Boundary of the $n =1$$Z_2\times Z^f_2$ SPT: a domain wall between two opposite domains of the $Z_2$ symmetry carries a Majorana mode.
• Figure 2: An example of an admissible state on a segment (segment boundaries are shown with double lines). Each link is labelled as $1$, $\sigma$ or $f$. The leftmost link corresponds to $i = 0$ and the rightmost link to $i = N_{seg}+1$. Domain walls are marked in solid blue: each domain wall supports a Majorana fermion (purple circles). The fusion product of boundary Majoranas on each $"-"$ domain (top line) is determined by the product of the $1, f$ labels on the $"+"$ domains to the immediate left and right.
• Figure 3: Sectors of the model on the circle. The fermion parity symmetry of the microscopic fermion model is related to the $S$ symmetry of our 1d model. NS and R stand for Neveu-Schwarz and Ramond spin-structures of the fermion model.
• Figure 4: The action of the $Z_2$ symmetry $U_{11}$ for NS spin structure and $(-1)^{\cal F} = 1$. We illustrate the case with $N_d = 3$$+/-$ domains. The line is periodic. The consecutive $"-"$ domains in the initial state stretch from $i_k$ to $j_k$, $k = 1 \ldots N_d$. $\mu_k/\nu_k \in \{0, 1\}$ label the $"+"$ domains in the initial/final states as either $1$ ($\mu = 0$) or $f$ ($\mu = 1$). The final state is a sum over all $\{\nu\}$ with coefficients given by Eq. (\ref{['U11']}).
• Figure 5: The action of the $Z_2$ symmetry $U_{22}$ for NS spin structure and $(-1)^{\cal F} = -1$. We illustrate the case with $N_d = 3$$+/-$ domains. We work on the double-cover of the circle. The consecutive $"-"$ domains in the initial state stretch from $i_k$ to $j_k$, $k = 1 \ldots N_d$. On the double-cover, we, thus, also have $"-"$ domains stretching from $\tilde{i}_k = i_k + N$ to $\tilde{j}_k = j_k + N$, $k = 1 \ldots N_d$. The consecutive $"+"$ domains in the initial/final states on the double cover are labelled by $\mu_k$/$\nu_k \in \{0,1\}$, $k =1 \ldots 2N_d$, with $\mu_{k+N_d} = 1-\mu_{k+N_d}$, $\nu_{k+N_d} = 1-\nu_{k+N_d}$. The final state is a sum over all $\{\nu\}$ with coefficients given by Eq. (\ref{['U22']}).

Theorems & Definitions (4)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4