Spin Response of a Magnetic Monopole and Quantum Hall Response in Topological Lattice Models through Local Invariants and Light

TL;DR

This work develops a geometrical framework that connects magnetic monopole physics on the Bloch sphere to topological invariants in lattice models, enabling local reading of the quantum Hall response via spin markers and circularly polarized light. By mapping Brillouin-zone Dirac points to poles on a sphere and introducing the dressed angle and Berry gauge potentials, it defines local invariants that quantify topological phases and transitions, including a half-invariant at criticality. The approach is demonstrated for the Haldane honeycomb model and extended to coupled-planes systems, where even-odd layer sequences yield distinct $\mathbb{Z}$ and $\mathbb{Z}_2$ topologies and a Ramanujan-series interpretation of a $\tfrac{1}{2}$ invariant. The framework provides practical probes (spin polarization, light responses) and a bridge to real-space and momentum-space observables, with potential applications in quantum circuits and layered topological materials.

Abstract

Here, we elaborate on and develop the geometrical approach introduced in K. Le Hur, Physics Reports 1104 1-42 (2025) between the magnetic monopole created from a radial field, quantum physics and topological lattice models through quantum phase transitions. We introduce an effective magnetic moment for a monopole when applying an additional source field along z-direction which also mediates the quantum phase transition. We present its relation with the transverse pumped quantum Hall current. The magnetic susceptibility can be introduced as a measure of the topological invariant i.e. remains quantized within the topological phase until the transition. We show the relation with two-dimensional topological lattice models such as a honeycomb Haldane model in real space. We develop the theory and present a numerical analysis between local invariants in momentum space introduced from Dirac points, correlation functions and the responses to circularly polarized light. We develop the formalism for coupled-planes materials including the possibility of quantum spin Hall effect and address a relation between the Ramanujan infinite alternating series and an interface in real space with a topological number one-half.

Figures (8)

• Figure 1: Spin Response as a function of the external applied magnetic field $M$ on the Bloch sphere. When $M<B$, the sphere reveals the hedgehog structure related to a Skyrmion. At $M=B$ at the transition, properties of the spin response are mostly modified close to south pole where the polarization along z direction is effectively zero. When $M>B$ the spin polarization at south pole becomes identical to the spin polarization at north pole.
• Figure 2: (Left) Classical magnetic monopole. We generalize the classical analogue of $A_{\varphi}$ introduced in Eq. (10) of Ref. KLHReview when including the magnetic field $M$ along $z$ direction as $A_{\varphi}=-\frac{\cos\theta}{2}-\frac{M}{4}\cos(2\theta)$ (for a unit sphere). We verify in this way that the topological charge remains protected for the classical monopole due to the presence of $M$ as it produces a component to the Berry gauge potential such that $A_{\varphi}^M(0)=A_{\varphi}^M(\pi)$. (Right) Representation of the effective topological properties for the quantum situation when including the magnetic field $M$ through the (dressed) angle $\tilde{\theta}(\theta=\pi)$ as a function of $M$. For all the phases the north pole is identified with $\theta=0$ i.e. $\tilde{\theta}=0$. Within the topological phase $B<M$, the south pole corresponds to $\tilde{\theta}(\theta=\pi)$ and the magnetic field is applied on the whole (dressed) sphere. For $M>B$, the north and south poles are identical and the equivalent geometry does not encircle the topological charge at the origin. For $M=B$, geometrical properties are equivalent to a half sphere corresponding to a half monopole or a half Skyrmion.
• Figure 3: Numerical evaluation of $\frac{dI}{d\xi}=2\chi$ as a function of $\xi$, that reproduces the analytical result $\frac{4}{3}C$ with $C=1$ and $\frac{8}{3}C_{1/2}$ with $C_{1/2}=\frac{1}{2}$ at the transition; it remains perfectly quantized within the topological phase.
• Figure 4: (Top) Honeycomb lattice and Brillouin zone. The rhomboid Brillouin zone is equivalent to the hexagonal one. (Bottom) Within the correspondence for $t_2=0.1$ in units of $t$, we introduce the unit vector ${\bf n}_d=\frac{{\bf d}}{|{\bf d}|}$ in Eq. (\ref{['dvector']}). When the ${\bf d}$ vector covers the entire sphere then this corresponds to a topological number equal to one e.g. for $M=0$. In this case, Dirac points are located at different poles on the sphere. Within the trivial phase $M=2M_c$, favoring the occupancy on one sublattice, both Dirac points then map onto the same pole. In that case, the magnetic field (Berry curvature) does not wrap around the origin completely. This leads to a trivial winding number. (Bottom) Path with fixed azimuthal angle $\varphi$ with the parameter $M=0$ and $t_2=0.1$.
• Figure 5: Berry curvatures for different values of $M$ and $t_2$. (Top Left) This corresponds to the physical situation of graphene with a charge density wave substrate leading to Eq. (\ref{['Berryfunction']}) Eva. (Top Right) We are within the topological phase with $C=1$ and the numbers agree with Eq. (\ref{['Berryfunction']}) and also with numerical results in the literature Anton. (Bottom Left) Results within the non-topological phase with $C=0$. (Bottom Right) When increasing $t_2$ within the topological phase, the Berry curvature become small at the Dirac points traducing e.g. that the energy dispersion of the bands become important in other regions. However, interestingly the formula $C = A_{\varphi}({\bf K}') - A_{\varphi}({\bf K})=+1$ from the Dirac points yet works well.