Gappability Index for Quantum Many-Body Systems

We propose an index ${\mathcal{I}}_G$ which characterizes the degree of gappability, namely the difficulty to induce a unique ground state with a nonvanishing excitation gap, in the presence of a symmetry $G$. ${\mathcal{I}}_G$ represents the dimension of the subspace of ambient uniquely gapped theories in the entire $G$-invariant “theory space.” The celebrated Lieb-Schultz-Mattis theorem corresponds, in our formulation, to the case ${\mathcal{I}}_G=0$ (completely ingappable) for the symmetry $G$ including the lattice translation symmetry. We illustrate the usefulness of the index by discussing the phase diagram of spin-$1/2$ antiferromagnets in various dimensions, which do not necessarily have the translation symmetry.

Figure 1

(a) $\mathcal{I}$’s of NUG Hamiltonians (denoted by dots) in a three-dimensional parameter space. Here the curve, surface, and solid cube represent the NUG phase to which a NUG Hamiltonian belongs while blank regions are uniquely gapped phases. (b) A path of uniquely gapped Hamiltonians exists once the dimension extension of the NUG phase ceases. (c) Two $S^{m-1}$’s with distinct topological assignments are not deformable to each other and an extending NUG phase with ${\mathcal{I}}_G\le m$ obstructs the contractibility of $S^{m-1}$ if $Q_{1,2}\ne Q_0$.

Figure 2

(a) The definition of $\theta (r_H)$ and the featureless phases along the loop within which gapless points take place. (b) An odd number of undimerized spins on the boundary $r_V=1$ where $\theta (r_H)$ changes from $\pi /2-\epsilon$ to $\pi /2+\epsilon$ around $r_H=x_0$. (c) Boundary spins are dimerized into bulk where $\theta (r_H)\approx -\pi /2$.

Figure 3

$\overline{\mathcal{H}}(r_{H;1},r_{H;2})$ wraps the torus $T^2$ around $\mathcal{C}\cong S^2$. The position of undimerized boundary spins is indicated by the red open circle.