Gapless Symmetry Protected Topological Order

TL;DR

This work introduces gapless symmetry-protected topological (gSPT) order achieved by decorating domain-wall condensates in gapless phases. Through exact constructions in 1D and 2D, it demonstrates that bulk criticality can coexist with symmetry-protected edge modes that are robust to symmetry-preserving perturbations. Using decorated-domain-wall techniques, strange correlator bulk-boundary analysis, and entanglement-spectrum calculations, the authors show edge phenomena including a c=1 edge in gapless trivial cases and a c=2 edge when decorated (gSPT), highlighting a novel interplay between topology and criticality. The results open pathways to broad generalizations and a framework for gapless topological matter with potential experimental relevance.

Abstract

We introduce exactly solvable gapless quantum systems in $d$ dimensions that support symmetry protected topological (SPT) edge modes. Our construction leads to long-range entangled, critical points or phases that can be interpreted as critical condensates of domain walls "decorated" with dimension $(d-1)$ SPT systems. Using a combination of field theory and exact lattice results, we argue that such gapless SPT systems have symmetry-protected topological edge modes that can be either gapless or symmetry-broken, leading to unusual surface critical properties. Despite the absence of a bulk gap, these edge modes are robust against arbitrary symmetry-preserving local perturbations near the edges. In two dimensions, we construct wavefunctions that can also be interpreted as unusual quantum critical points with diffusive scaling in the bulk but ballistic edge dynamics.

Figures (8)

• Figure 1: Representative states of each order in the 2D example. (a) Trivial: Paramagnetic spins on a triangular lattice with fluctuating domain walls. (b) SPT: Decorating the domain walls gives an SPT with a $c=1$ edge mode. (c) Gapless Trivial: Tuning the domain walls to criticality by restricting them to fully-packed loop configurations (defined below) closes the gap and gives a $c=1$ edge mode. (d) Gapless SPT: Doing both yields a gapless SPT with $c=1+1=2$ edge modes.
• Figure 2: Lattices used throughout this paper to construct SPT and gSPT wavefunctions in one (Top) and two dimensions (Bottom: Triangular and Union Jack lattices). The control-$Z$ twist operator used to obtain non-trivial SPT order gives a factor of $(-1)$ to links with two down spins in the 1D case and triangles with three down spins in the 2D case, as exemplified by the green shading.
• Figure 3: Edge magnetization of the critical $\sigma$ spins for typical gSPT and gTrivial groundstates as a function of a small magnetic field. The groundstates were computed on $L=200$$\sigma$ spins (and 200 $\tau$ spins) using DMRG, including small but arbitrary symmetry-preserving boundary perturbations. Inset: spatial magnetization profiles for a field $h=10^{-10}$.
• Figure 4: (a) Phase diagram showing a gSPT line separating the Haldane and ferromagnetic phases obtained from ED on 12 and 16 sites with $\Delta_\tau = 10$, $g_\tau = u_\sigma = 0.1$, $u_\sigma = \delta = 0.2$. (b) Magnetization profiles of the lowest four eigenstates via ED on 20 sites with the same parameters as (a), but fixing $\gamma = 0.1$, which implies $g_\sigma = g_\sigma^c \approx 1.421$ at the gSPT point. To break the symmetry, a small magnetic field $\sim{\rm e}^{-L}$ in the $z$ direction is applied. (c) Cartoon spectrum of $H_\text{gSPT}'$. Colors of states correspond to magnetization profiles.
• Figure 5: (a) Spectrum of $H_{\text{gTrivial}}^{\text{LL}}$. (b) Spectrum of the gSPT $H_{\text{gSPT}}^{\text{LL*}}$. For both cases, and spectra are normalized to be able to read off CFT operator dimensions. The conformal blocks are labelled by the magnetic charge sector $m$ and spaced horizontally, and small horizontal spacings show degenerate eigenvalues (up to exponential splitting). One can see that the states in the gSPT case are all doubly degenerate, due to the edge modes, but also that operator dimensions have changed relative to the gTrivial case. The numerical spectra were computed via DMRG ITensor on up to 32 sites with finite-size scaling, and the solid lines correspond to the exponents expected from boundary CFT using $\Delta_{\rm eff} = - \cos \pi g$toAppear. To improve convergence, the gap on the paramagnetic sector was increased from one to ten. (c) The phase diagram of $H_{\text{gSPT}}^{\text{LL*}}$, as computed via DMRG ITensor. Each line denotes a different eigenvalue crossing which accompanies a phase transition, and black crosses denote multicritical points. The Hamiltonian parameters used are $\Delta = -0.5, \alpha = 0.1$, $g_\tau = 0.3$, $u_\tau = 0.1$ for (a), (b) and (c).