Charge pumps, boundary modes, and the necessity of unnecessary criticality

Abstract

We link the presence of "unnecessary" quantum critical surfaces within a single gapped phase of matter to the non-trivial topology of families of gapped Hamiltonians that encircle the critical surface. We study a specific set of one-dimensional spin models where each such family forms a one-parameter loop in a two-dimensional phase diagram. Foliating the non-critical region by such loops identifies "radial" and "angular" coordinates in the phase diagram that respectively parametrize different families and different members of a single family. We show that each one-parameter family is a generalized Thouless charge pump, all with the same topological index, and hence the gapped phase undergoes one or more nontrivial boundary phase transitions as we vary the angular coordinate in a loop through members of one family. Tuning the radial coordinate generates loci of boundary critical points that terminate at endpoints of the bulk unnecessary critical line within the gapped phase. We discuss broader implications of our results and possible extensions to higher dimensions.

Figures (11)

• Figure 1: Unnecessary critical (a) and multicrital (b) phase diagrams in a spin ladder (c), for $J_M \approx \mp 5.2, \Delta \approx -0.05$. (a), (b) show evolution of ground states near one end of an open ladder using graphical notation described in the text. Over the pumping cycle the ladder remains in a single bulk gapped phase, but its boundary transforms under distinct irreducible representations of layer exchange ($\mathbb{Z}_2^L$) and spin-rotation ($O(2)$) symmetry for $\delta>0$ and $J_U <-|J_M|$ or $J_U > |J_M|$. These are separated by a single boundary transition (dashed line in (b)) or a non-trivial boundary phase involving a 2D boundary irrep of $O(2)$ (blue shaded region in (a)). Dotted lines in (a), (b) represent crossovers between 'RG basins' of symmetric field theory vacua T$_{1,\ldots,4}$ of \ref{['tab:vacuumsummary']}.
• Figure 2: Interpolations at the T$_3$-T$_1$ interface. For $J_M<0$ and $[{\cal V}_-]<[{\cal W}_-]$, $\phi_{1,2}$ wind in the same sense, so $\phi_1+\phi_2$ evolves smoothly from $0$ to $\pm2\pi$; this binds $U(1)$ charge $Q=\pm1$ to the interface, which is hence an $O(2)$ doublet. For $J_M<0$ and $[{\cal V}_-]>[{\cal W}_-]$ or $J_M>0$, $\phi_{1,2}$ wind oppositely; tunneling between the resulting $Q=0$ configurations via $J_U\cos(\theta_1-\theta_2)$ leads to a unique ground state whose $\mathbb{Z}_2^L$ charge is $-\text{sign}(J_U)$.
• Figure 3: Minimum eigenvalue of each symmetry sector of the four qubit Hamiltonian in \ref{['appeq:H_fourqubit']} for $\Delta > \Delta_c$ (top row) which shows no level crossing and $\Delta < \Delta_c$ (bottom row) which does. $\Delta_c \approx -0.35$ for $J_M = -5$, is the critical value of $\Delta$,defined in \ref{['appeq:Deltacrit']} above which the ground state is unique.
• Figure 4: Left, Middle: Plots of eigenvalues of the two-qubit boundary Hamiltonian in \ref{['appeq:Hboundary']}. Right: Graphical summary of the ground state irrep in various parameter regimes.
• Figure 5: A schematic representation of how the four pinned-field configurations leading to the trivial phase are generally placed in our phase diagrams. The dotted lines schematically indicate crossovers where the pinned values change without encountering singularities.