Gapless Symmetry-Protected Topological Order

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 wave functions that can also be interpreted as unusual quantum critical points with diffusive scaling in the bulk but ballistic edge dynamics.

Popular Summary

An overarching goal of condensed-matter physics is to identify and classify new phases of matter. A natural dichotomy is provided by the presence (or absence) of an energy gap. Materials with a gap (known as gapped systems) can only absorb packets of energy that exceed the size of the gap, whereas gapless systems can absorb arbitrarily small amounts of energy. The recent revolution in topological materials has provided a new metric based on topology. A material might act differently if formed into a doughnut, for example, versus a sphere.

While researchers have produced an incredible host of gapped topological materials, their gapless counterparts are rather unexplored, but they can be expected to give rise to intriguing new phenomena. The most prominent examples of gapless topological systems to date, Weyl semimetals, are described in terms of noninteracting particles and thus miss the full richness of strongly interacting quantum systems. We present a general theoretical framework for describing a class of strongly interacting gapless systems with topological properties.

Our construction generalizes the notion of a symmetry-protected topological (SPT) phase, a wide class of gapped systems with topological properties, to the gapless case. Encouragingly, many of the results and tools developed for gapped SPTs carry over very well. Leveraging these techniques, together with ideas from quantum criticality and spin liquids, we explore exactly solvable examples of gapless SPTs and demonstrate their topological nature.

These constructions should pave the way for the systematic study of gapless topological matter.

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, giving a 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), closing the gap and giving a $c=1$ edge mode. (d) 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 wave functions in one (top) and two dimensions (bottom: triangular and Union Jack lattices). The control-$Z$ twist operator used to obtain nontrivial 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 ground states as a function of a small magnetic field. The ground states were computed on $L=200$ $\sigma$ spins (and $200\hspace{0.17em}\tau$ 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 ${\mathrm{\Delta }}_{\tau }=10$, $g_{\tau }=u_{\sigma }=0.1$, $u_{\sigma }=\delta =0.2$. (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 $\sim {\mathrm{e}}^{-L}$ in the $z$ direction is applied. (c) Cartoon spectrum of $H_{\mathrm{gSPT}}^{\prime }$. Colors of states correspond to magnetization profiles.

Figure 5

(a) Spectrum of $H_{\text{gTrivial}}^{\mathrm{LL}}$. (b) Spectrum of the gSPT $H_{\text{gSPT}}^{\text{LL*}}$. For both cases, spectra are normalized to be able to read off CFT operator dimensions. The conformal blocks are labeled 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, because of the edge modes, but also that operator dimensions have changed relative to the gTrivial case. The numerical spectra were computed via DMRG [92] on up to 32 sites with finite-size scaling, and the solid lines correspond to the exponents expected from boundary CFT using ${\mathrm{\Delta }}_{\text{eff}}=-\cos \pi g$ [93]. 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 [92]. Each line denotes a different eigenvalue crossing, which accompanies a phase transition, and black crosses denote multicritical points. The Hamiltonian parameters used are $\mathrm{\Delta }=-0.5$, $\alpha =0.1$, $g_{\tau }=0.3$, and $u_{\tau }=0.1$ for (a), (b), and (c).

Figure 6

Entanglement entropy of the $1+1d$ ground state of the entanglement Hamiltonian $H_{\text{E}}=-\log \rho$, with $\rho$ the reduced density matrix for a bipartite cut on an infinite cylinder of circumference $\ell$. Blue points are for $|{\mathrm{\Psi }}_{\text{gTrivial}}⟩$, and red points are for $|{\mathrm{\Psi }}_{\text{gSPT}}⟩$. Data are offset so that $S(1/2)=1$.

Figure 7

The Union Jack lattice showing the $A$ sublattice (blue) and BC sublattices (red) on a cylinder. The empty sites on the bottom are identified with the full ones on top, giving a circumference $\ell =4$. The entanglement cut is denoted with a double line, and the triangles ABC that it breaks are highlighted in green. The top and bottom show the extent of the left, right, and boundary regions for each species of spins.

Figure 8

Entanglement spectral gaps for the gTrivial and gSPT for entanglement cuts of the $2+1d$ example on the cylinder of circumference $\ell$. The vertical axis is rescaled to be able to directly read off the operator dimensions of the excitations (up to a nonuniversal sound velocity). Only the first few excitations are well converged for this range of $\ell$.