Emanant and emergent symmetry-topological-order from low-energy spectrum

Abstract

Low-energy emanant and emergent symmetries can be anomalous, higher-group, or non-invertible. A way to systematically capture the properties of such symmetries is through the topological orders in one-higher dimension, known as symmetry topological orders (symTOs). Consequently, identifying the emergent or emanant symmetry of a system is not simply a matter of determining its group structure, but rather of computing the corresponding symTO. In this work, we develop a method to compute the symTO of 1+1D systems by analyzing their low-energy spectra under closed boundary conditions with all possible symmetry twists. Following this approach, we show that the gapless antiferromagnetic (AF) spin-$1/2$ Heisenberg model possesses an exact emanant symTO corresponding to the $D_8$ quantum double, when the global symmetry is restricted to the $\mathbb{Z}_2^x \times \mathbb{Z}_2^z$ subgroup of the $SO(3)$ spin-rotation symmetry and lattice translations. Moreover, this model exhibits an emergent $SO(4)$ symmetry, whose exact components are described jointly by automorphisms of the $D_8$ quantum double and the $SO(3)$ spin-rotations. Using the condensable algebras of the emanant symTO, we further identify several other phases that may be accessible by modifying interactions among low-energy excitations: (1) a gapped dimer phase, connected to the AF phase via an $SO(4)$ rotation, (2) a commensurate collinear ferromagnetic phase that breaks translation by one site with a $ω\sim k^2$ mode, (3) an incommensurate, translation-symmetric ferromagnetic phase featuring both $ω\sim k^2$ and $ω\sim k$ modes, (4) and an incommensurate ferromagnetic phase that breaks translation by one site with both $ω\sim k^2$ and $ω\sim k$ modes.

Figures (8)

• Figure 1: Two-dimensional slices of the phase diagram of antiferromagnetic Heisenberg chain \ref{['eq:Ham Heisenberg']} perturbed by the deformations \ref{['eq:NNN deformation']} and \ref{['eq:pert-neel-1']}. (a) Three Néel phases at the slice $J=0$ and $\sum_a \Delta_a \leq 1$, with $0\leq \Delta_a$ and the center of the triangle being the point $\Delta_a=0$. (b) Dimer and $\text{N\'eel}_a$ phases at the two-dimensional slice with $J, \Delta_a \neq 0$.
• Figure 2: ED spectra for the Heisenberg model with PBC for $L=20,21,22,23$. The momentum is normalized by $2 \pi$. We label the first few low-energy eigenstates with their corresponding anyons and degeneracies (denoted as $D$). The full lattice symmetry is $SO(3) \times \mathbb{Z}_L$. But as we noted the lattice model at $J = J_c$ has good emergent SO(4) symmetry even at relatively short distance, and this SO(4) symmetry leads to the nearly fourfold degeneracy for some of the low-energy states.
• Figure 3: ED spectra for the Heisenberg model under PBC with a chiral perturbation for $L=20,21,22,23$. The chiral perturbation term is $0.1\sum_j \mathbf S_j \cdot (\mathbf S_{j+1} \times \mathbf S_{j+2})$. The degeneracy of states labeled by $(k, S_z^\text{tot})$ is preserved. However, as the improper $\mathbb{Z}_2$ reflection symmetry is broken, 6-fold degeneracy of states with momentum $k$ and $-k$ is lifted.
• Figure 4: ED spectra for the Heisenberg model with $\mathbb{Z}^z_2$-twisted boundary conditions for $L=20,21,22,23$. The first few low-energy eigenstates are labeled with their corresponding anyons and degeneracies. The momentum is normalized by $\pi$. The full lattice symmetry is $\frac{\mathrm{U}(1)_z \times \mathbb{Z}^z_{2L}}{\mathbb{Z}^z_2} \rtimes \mathrm{Z}^x_2$.
• Figure 5: Charge sector anyons, evaluated at lattice size $L=20$ with PBCs. Each spectrum is labeled with the anyon type and quantum numbers of the projected subspace. For the corresponding irreps, see Table \ref{['tab:charge-anyon']}.