Hopf symmetry-protected topological phase at the intersection of magnetic and spin loop-current order

TL;DR

This work shows that Hopf theta terms, traditionally studied in relativistic contexts, generically appear in two-dimensional metals without spin-orbit coupling at the intersection of magnetic and spin loop-current order, yielding a symmetry-protected topological phase protected by SU(2) that is gapped in the bulk with chiral edge modes and a quantized spin-Hall response. The authors derive the Hopf term via a gradient expansion in a Hund-coupled spin-ordered insulator, establish the quantization of the Hopf angle in nonrelativistic systems, and demonstrate that a finite Hopf term emerges when TR-odd and TR-even orders coexist without reflection symmetry. They connect the theta angle to the Chern number as $\theta = -\pi C$ and provide explicit lattice models that realize the Hopf SPT, including altermagnetic and antiferromagnetic variants with tunable Chern numbers. The study also highlights limitations of gradient methods in capturing all Hopf terms and discusses boundary theories described by Wess-Zumino-Witten models, underlining potential pathways to experimental realization and further theoretical exploration.

Abstract

Hopf terms are topological theta terms that are associated with a host of interesting physics, including anyons, statistical transmutation, chiral edge states, and the quantum spin-Hall effect. Here, we show that Hopf terms generically appear in two-dimensional metals without spin-orbit coupling at the intersection of magnetic and spin loop-current order. In the locally ordered, but globally disordered, phase the system is governed by the Hopf term and realizes a Hopf symmetry-protected topological phase. This phase is protected by the $\mathrm{SU}(2)$ spin rotation symmetry, is gapped in the bulk, has chiral gapless edge states, and its spin-Hall conductance is quantized. Lattice models that realize this phase are introduced. In addition, we provide an elementary proof that the $θ$ angle of the Hopf term must be quantized to multiples of $π$ in non-relativistic systems, thereby precluding anyonic skyrmions in condensed matter systems.

Figures (4)

• Figure 1: Schematic temperature $T$ vs. tuning parameter $p$ phase diagram of a crossing between a magnetic and spin loop-current insulator in 2+1D. Due to Hohenberg-Mermin-Wagner's theorem Hohenberg1967Mermin1966, the magnetic (dark red) and spin loop-current (dark blue) phases, which break the $\mathop{\mathrm{SO}}\nolimits(3)$ spin rotation symmetry, only set in at $T = 0$. At finite $T$, the corresponding order parameters $\bm{\hat{n}}$ and $\bm{\hat{n}}'$, respectively, fluctuate, as indicated by the light red and blue shading. In between, the time-reversal-odd Ising variable $\bm{\hat{n}} \bm{\cdot} \bm{\hat{n}}'$ may condensed, resulting in a time-reversal symmetry-breaking (TRSB) phase (light purple). Below it, if the spin rotation symmetry stays unbroken at $T = 0$, a Hopf symmetry-protected topological (SPT) phase (dark purple) will generically appear.
• Figure 2: An example of a field configuration $\bm{\hat{n}}(x)$ that has a unit Hopf number $Q_{\text{Hopf}} = +1$. Everywhere outside of the lightly blue-shaded torus, $\bm{\hat{n}}(x)$ points along $+ \bm{\hat{e}}_3$ to within an angle of $\pi / 3$. Within the torus, the thick lines denote regions where $\bm{\hat{n}}(x)$ is oriented along the directions indicated by the legend. One way of interpreting this field configuration is as the creation of a skyrmion--anti-skyrmion pair that then winds around itself before getting annihilated.
• Figure 3: Partition of compactified spacetime $\mathcal{S}^1_{\tau} \times \mathcal{S}^2_{\bm{r}}$ into $V_1 = V_1' - \Delta$ (red region) and $V_2 = V_2' + \Delta$ (blue and purple regions). Since spacetime is periodic in imaginary time $\tau \equiv x^0$, which corresponds to height in the figure, the drawn cylinders represent toruses. Here $\Delta$ is the volume change under the deformation $V_n \to V_n'$. $A_{12} = \partial V_1 = - \partial V_2$, $A_{12}' = \partial V_1' = - \partial V_2'$, and $\Sigma$ are 2D surfaces, whereas $L_{12}$, $L_{12}'$, and $\tilde{L}_{12}$ are 1D lines. $\partial \Delta = A_{12}' - A_{12}$ and $\partial \Sigma = L_{12} - L_{12}'$.
• Figure 4: A square lattice with $\bm{Q} = (\pi, \pi)$ antiferromagnetic ordering. The $t$ arrows indicate the hoppings of the kinetic part of the Hamiltonian. The $g$ ($g'$) arrows indicate the momentum dependence of the coupling to $\bm{\hat{n}}$ ($\bm{\hat{n}}'$). After ordering, the induced hoppings related to $g$ and $g'$ acquire additional signs that depend on the colors of the sites.