Charge pumps, pivot Hamiltonians and symmetry-protected topological phases

TL;DR

This work investigates non-contractible loops in the space of symmetric gapped Hamiltonians, introducing pivot loops generated by a pivot Hamiltonian and showing these loops can pump charge and reveal symmetry-protected topological (SPT) content. It establishes deep links between charge pumps, SPTs, and anomalies, notably through strict circular loops tied to the Dolan–Grady relation and Onsager algebras, with concrete realizations in Onsager-integrable chiral clock models and Z_N×Z_N cluster/dipolar systems. The authors develop a general framework using MPS, group cohomology, and equivariant families to classify pumps, identify when pumps imply distinct SPTs at high-symmetry points, and explain boundary transitions and domain-wall decorations. The results illuminate when pumps constrain phase diagrams, relate to boundary phenomena, and suggest a unified perspective on how topological and anomalous structures arise from unitary loops in quantum many-body systems, with potential applications to higher-dimensional SPTs and Floquet-type dynamics.

Abstract

Generalised charge pumps are topological obstructions to trivialising loops in the space of symmetric gapped Hamiltonians. We show that given mild conditions on such pumps, the associated loop has high-symmetry points which must be in distinct symmetry-protected topological (SPT) phases. To further elucidate the connection between pumps and SPTs, we focus on closed paths, `pivot loops', defined by two Hamiltonians, where the first is unitarily evolved by the second `pivot' Hamiltonian. While such pivot loops have been studied as entanglers for SPTs, here we explore their connection to pumps. We construct families of pivot loops which pump charge for various symmetry groups, often leading to SPT phases -- including dipole SPTs. Intriguingly, we find examples where non-trivial pumps do not lead to genuine SPTs but still entangle representation-SPTs (RSPTs). We use the anomaly associated to the non-trivial pump to explain the a priori `unnecessary' criticality between these RSPTs. We also find that particularly nice pivot families form circles in Hamiltonian space, which we show is equivalent to the Hamiltonians satisfying the Dolan-Grady relation -- known from the study of integrable models. This additional structure allows us to derive more powerful constraints on the phase diagram. Natural examples of such circular loops arise from pivoting with the Onsager-integrable chiral clock models, containing the aforementioned RSPT example. In fact, we show that these Onsager pivots underlie general group cohomology-based pumps in one spatial dimension. Finally, we recast the above in the language of equivariant families of Hamiltonians and relate the invariants of the pump to the candidate SPTs. We also highlight how certain SPTs arise in cases where the equivariant family is labelled by spaces that are not manifolds.

Figures (10)

• Figure 1: Paths used in the argument for showing that a pump implies an SPT in the Ising pivot. We consider paths in the space of $\tilde{G}=\mathbb{Z}_2$-symmetric Hamiltonians between the $G=\mathbb{Z}_2\times\mathbb{Z}_2$-symmetric points $H_0$ and $H_\pi$. Assuming the existence of a $G$-symmetric path, then consistency implies that the loop constructed by following first $H_{\tilde{G}}(\theta)$ then the reverse of $P_{\mathrm{even}}H_{\tilde{G}}(\theta)P_{\mathrm{even}}$ must be a trivial $\tilde{G}$-charge pump. The pivot loop $H_\theta$ in \ref{['eq:circleintro']} is of this form, which means we can conclude from the non-trivial $\mathbb{Z}_2$ pump that $H_0$ and $H_\pi$ are in distinct $G$-SPT phases.
• Figure 2: Visualising a strict circular loop \ref{['eq:circle']} in the space of Hamiltonians. Here the path is generated by pivoting an initial Hamiltonian $H_0$ with a 'pivot' Hamiltonian $\tilde{H}$, giving rise to $H_\theta = \mathrm{e}^{-\mathrm{i} \theta \tilde{H}} H_0 \mathrm{e}^{\mathrm{i} \theta \tilde{H}}$. At the half-way point, we have $H_\pi = \mathrm{e}^{-\mathrm{i} \pi \tilde{H}} H_0 \mathrm{e}^{\mathrm{i} \pi \tilde{H}}$, which is sometimes in a distinct SPT phase from $H_0$. If this loop is a non-trivial pump, then the centre $H_\star = \frac{H_0 + H_\pi}{2}$ axis cannot be SRE and there will be a diabolical locus inside the loop. Hamiltonians equidistant from the $H_\star$ axis are isospectral (related by a unitary pivot). In this work we relate the structure of such pivot loops, pumps, and SPTs at high-symmetry points.
• Figure 3: Generalisation of \ref{['fig:paths0']}: paths in the space of $\tilde{G}$-symmetric Hamiltonians between $H_0$ and $H_\pi$. If $H_0$ and $H_\pi$ are $G$-symmetric, where $\tilde{G}$ is a proper normal subgroup of $G$, we can take a $\tilde{G}$-symmetric path $H_{\tilde{G}}(\theta)$ to $g_0^{}H_{\tilde{G}}(\theta)g_0^{\dagger}$. Assuming that $H_0$ and $H_\pi$ are in the same $G$-SPT phase implies the existence of a $G$-symmetric path between them. Using this path we can find constraints on the charge pumped by the loop constructed from $H_{\tilde{G}}(\theta)$ followed by the reverse of $g_0^{}H_{\tilde{G}}(\theta)g_0^{\dagger}$. If this charge does not satisfy the constraint, we can conclude that $H_0$ and $H_\pi$ are in different $G$-SPT phases.
• Figure 4: Relation between pumps and SPTs for loops generated by a pivot Hamiltonian $H_\theta = \mathrm{e}^{-\mathrm{i} \theta \tilde{H}} H_0 \mathrm{e}^{\mathrm{i} \theta \tilde{H}}$. $H_0$ and $H_\pi$ are $G$-symmetric, while $\tilde{H}$ (and hence the family $H_\theta$) is $\tilde{G}\subsetneq G$-symmetric. The Hamiltonian $H_\star = (H_0+H_\pi)/2$ is the midpoint of the interpolation between $H_0$ and $H_\pi$. Double arrows are immediate consequences of the appropriate definition.
• Figure 5: Visualising a generic loop in the space of Hamiltonians (to be contrasted with \ref{['fig:circularloop']}). The $h_n$ indicate additional orthogonal directions in the space.

Theorems & Definitions (2)

• Definition 1
• proof