# Quantum skyrmion Hall effect

Ashley M. Cook∗ Max Planck Institute for Chemical Physics of Solids,
Nöthnitzer Strasse 40, 01187 Dresden, Germany  Max Planck Institute for the
Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany

###### Abstract

We consider the problem of magnetic charges in $(2+1)$ dimensions for a torus
geometry in real-space, subjected to an inverted Lorentz force due to an
external electric field applied normal to the surface of the torus. We compute
the Hall conductivity associated with transport of these charges for the case
of negligible gapless excitations and global $\mathrm{U}(1)$ charge
conservation symmetry, and find it is proportional to an integer-valued
topological invariant $\mathcal{Q}$, corresponding to a magnetic quantum Hall
effect (MQHE). We identify a lattice model realizing this physics in the
absence of an external electric field. Based on this, we identify a
generalization of the MQHE to be quantized transport of magnetic skyrmions,
the quantum skyrmion Hall effect (QSkHE), with a $\mathrm{U}(1)$ easy-plane
anisotropy of magnetic skyrmions and effective conservation of charge
associated with magnetic skyrmions yielding incompressibility, provided a
hierarchy of energy scales is respected. As the lattice model may be
characterized both by a total Chern number and the topological invariant
$\mathcal{Q}$, we furthermore outline a possible field theory for electric
charges, magnetic charges, and correlations between magnetic and electric
charges approximated as composite particles, on a two-torus, to handle the
scenario of intermediate-strength correlations between electric and magnetic
charges modeled as composite particles. We map this problem to a generalized
$(4+1)$D theory of the quantum Hall effect for the composite particles.

## I Introduction

The quantum Hall effect (QHE), in which Hall conductivity quantizes to
rational numbers in units of fundamental constants for a two-dimensional
electron gas subjected to an external magnetic field applied normal to the
plane of the gas, is foundational to understanding of the role of topology in
condensed matter. Remarkably, its experimental signatures Klitzing  _et al._
(1980); Tsui  _et al._ (1982) have been characterized effectively by
theoretical developments Kallin and Halperin (1984); Halperin  _et al._
(1993); Halperin (1982, 1984); Wen (1991). In particular, Laughlin’s discovery
of wavefunctions capturing this physics Laughlin (1983) ultimately led to a
microscopically-derived field theory for the QHE Zhang  _et al._ (1989) and a
theory of composite fermions for understanding of many fractional quantum Hall
states Jain (1989). The theory has also very successfully been applied to
understanding of multi-component quantum Hall systems Lee and Kane (1990);
Sondhi  _et al._ (1993) and, more recently, to understanding of the
topological insulators and semimetals Qi  _et al._ (2008a); Ryu  _et al._
(2010); Schnyder  _et al._ (2008).

Many of these concepts of non-trivial topology of the QHE have also been
applied to understanding of topological magnetization textures, known as
magnetic skyrmions Bogdanov and Yablonskii (1989); Yu  _et al._ (2010);
Mühlbauer  _et al._ (2009); Schulz  _et al._ (2012). While experimental
realization and manipulation of magnetic skyrmions has advanced considerably,
most study is strongly guided by experiments, which remain focused on
overcoming challenges in realization, detection, and manipulation of large,
classical, topological magnetization textures Parkin  _et al._ (2008); Sampaio
_et al._ (2013); Donnelly  _et al._ (2020); Zhang  _et al._ (2020);
Maccariello  _et al._ (2018). Topological transport signatures have therefore
focused primarily on exploiting the classical magnetization textures to
generate the quantum anomalous Hall effect via strong Hund’s coupling between
local moments and itinerant electrons via the topological Hall effect Neubauer
_et al._ (2009); Bruno  _et al._ (2004); Lee  _et al._ (2009), or transverse
motion of classical magnetic skyrmions in response to external fields in the
skyrmion Hall effect Jiang  _et al._ (2017); Zang  _et al._ (2011), with
unquantized transport given the classical nature of the magnetic skyrmions
considered.

In this work, we generalize theory of the QHE to identify topological
transport signatures of insulating and also itinerant magnets, in which
magnetic—rather than electric—charges exhibit quantized transport in response
to an external electric—rather than magnetic—field. Such magnetic charges are
not relevant to Maxwell’s equations in the vacuum, but magnetic monopoles are
realized in condensed matter settings as magnetic skyrmions Castelnovo  _et
al._ (2008); Yu  _et al._ (2010); Mühlbauer  _et al._ (2009); Schulz  _et al._
(2012). Rather than classical magnetic skyrmions, we consider possible
transport of quantum magnetic skyrmions Sotnikov  _et al._ (2018). We first
approximate them as point magnetic charges, with a global $\mathrm{U}(1)$
magnetic charge conservation symmetry. We consider the problem of transport of
such magnetic charges on a two-torus in real-space—rather than the real plane
$\mathbb{R}^{2}$ as is typically considered in study of magnetic skyrmions—in
the presence of an external electric field. This scenario is dual to the QHE
Laughlin (1981); Niu  _et al._ (1985), and we find quantized transport and
incompressibility in the MQHE as a result.

We then identify topological phases of spin, realized in lattice models with
spin and orbital degrees of freedom (dof), with the MQHE Cook (2023). This
suggests a possible generalization of the MQHE to quantized transport of
magnetic skyrmions, and we identify generalizations of the ingredients in the
MQHE to a quantum skyrmion Hall effect (QSkHE), or quantized transport of
magnetic skyrmions in response to an external electric field. This requires
generalization of the $\mathrm{U}(1)$ charge conservation symmetry important
to the incompressibility of the QHE, in particular, to a $\mathrm{U}(1)$ easy-
plane anisotropy, corresponding to anisotropic Dzyaloshinskii-Moriya (DM)
interactions Rowland  _et al._ (2016); Hanada and Sawado (2023). Based on
this, we also generalize the theory of multi-component quantum Hall systems to
an effective action containing terms for electric charges and magnetic charges
individually coupling to gauge fields, as well as coupling of their
correlations, modeled as composite particles, to gauge fields. This theory
more robustly handles the scenario of intermediate-strength correlations
between electric and magnetic charges, or where electric/magnetic charge
conservation individually break down while composite charge conservation is
retained.

## II Results

### II.1 Magnetic quantum Hall effect

We consider the problem of magnetic point charges on a two-torus, with
$\mathrm{U}(1)$ global charge conservation symmetry, subjected to an external
electric field applied normal to the surface of the torus. The external
electric field couples to the magnetic charges via an inverted Lorentz force,
as Faraday’s law is generalized to include a magnetic charge current. Such
magnetic charges are realized in condensed matter settings as emergent
quasiparticles and topological magnetization textures Castelnovo  _et al._
(2008); Yu  _et al._ (2010); Mühlbauer  _et al._ (2009). Assuming the $j$th
magnetic charge to have mass $\mu_{j}$ and topological charge $m_{j}$, the
effective Hamiltonian may therefore be written as

$\displaystyle H$ $\displaystyle=\sum_{j=1}^{N}\left[{1\over
2\mu_{j}}\left(-i\hbar{\partial\over\partial x_{j}}\right)^{2}+{1\over
2\mu_{j}}\left(-i\hbar{\partial\over\partial
y_{j}}-m_{j}Ex_{j}\right)^{2}\right]$ (1)
$\displaystyle+\sum_{j=1}^{N}U(x_{j},y_{j})+\sum_{\kappa}^{N}\sum_{\delta}^{N}V\left(|r_{\kappa}-r_{\delta}|\right),$

where $E$ is the strength of the external electric field applied normal to the
surface of the torus, $U$ is an effective background potential, and $V$ is a
two-body interaction term. In the following discussion, we neglect
interactions governed by $V$ and assume $\mu_{j}=\mu$ and $m_{j}=m$ for each
$j$. We take boundary conditions to be

$\displaystyle\Gamma_{j}(L_{1},\hat{x})\psi(x_{j})$ $\displaystyle=e^{i\alpha
L_{1}}\psi(x_{j}),$ (2) $\displaystyle\Gamma_{j}(L_{2},\hat{y})\psi(y_{j})$
$\displaystyle=e^{i\beta L_{2}}\psi(y_{j}),$ (3)

where $j=1,2,...,N$ and $\Gamma_{j}(L_{1},\hat{x})$ and
$\Gamma_{j}(L_{2},\hat{y})$ are the single-particle electric translation
operators in the $x$ and $y$ directions.

### II.2 Quantized transport of magnetic charges

We start from the Kubo formula expression for Hall conductivity considered by
Niu  _et al._ Niu  _et al._ (1985), but for particles of magnetic charge $m$
in the presence of electric field $\boldsymbol{E}$ and background
magnetization $n(x,y)$ effectively encoded as a state of $H$,

$\sigma_{m}={im^{2}\over
A\hbar}\sum_{n(>0)}{(\nu_{1})_{0n}(\nu_{2})_{n0}-(\nu_{2})_{0n}(\nu_{1})_{n0}\over\left(E_{0}-E_{n}\right)^{2}}.$
(4)

where $A=L_{1}L_{2}$ is the system size of the torus (of length $L_{1}$ in the
$x$-direction, and $L_{2}$ in the $y$-direction), $n$ labels excited states,
$(\nu_{i})_{0n}$ is the velocity matrix element for the $i$th direction
between the ground state $|\phi_{0}\rangle$ and excited state
$|\phi_{n}\rangle$, $E_{0}$ is the energy of the ground state, and $E_{n}$ is
the energy of the $n$th excited state.

We perform a unitary transformation as
$\phi_{n}=exp\left[-i\alpha(x_{1}+\ldots+x_{N})\right]exp\left[-i\beta(y_{1}+\ldots+y_{N})\right]\psi_{n}$,
rewriting $\sigma_{m}$ as

$\displaystyle\sigma_{m}$ $\displaystyle={im^{2}\over
A\hbar}\sum_{n,(>0)}\left({\langle\phi_{0}|{\partial\tilde{H}\over\partial\alpha}|\phi_{n}\rangle\langle\phi_{n}|{\partial\tilde{H}\over\partial\beta}|\phi_{0}\rangle\over(E_{0}-E_{n})^{2}}\right.$
(5)
$\displaystyle-\left.{\langle\phi_{0}|{\partial\tilde{H}\over\partial\beta}|\phi_{n}\rangle\langle\phi_{n}|{\partial\tilde{H}\over\partial\alpha}|\phi_{0}\rangle\over(E_{0}-E_{n})^{2}}\right)$

where $\tilde{H}$ is the transformed Hamiltonian.

Using the transformed velocity operators in terms of coordinates $\alpha$ and
$\beta$, and then taking $\theta=\alpha L_{1}$ and $\varphi=\beta L_{2}$, we
simplify the expression to the following,

$\sigma_{m}={im^{2}\over\hbar}\left[\langle{\partial\phi_{0}\over\partial\theta}|{\partial\phi_{0}\over\partial\varphi}\rangle-\langle{\partial\phi_{0}\over\partial\varphi}|{\partial\phi_{0}\over\partial\theta}\rangle\right]$
(6)

As in Niu  _et al._ Niu  _et al._ (1985), we assume a finite energy gap
separating the ground state from excited states. $\sigma_{m}$ is computed by
averaging over $\theta$ and $\varphi$ as

$\displaystyle\sigma_{m}$ $\displaystyle=\bar{\sigma}_{m}$ (7)
$\displaystyle={m^{2}\over h}\int_{0}^{2\pi}\int_{0}^{2\pi}d\theta
d\varphi{1\over 2\pi
i}\left[\langle{\partial\phi_{0}\over\partial\theta}|{\partial\phi_{0}\over\partial\varphi}\rangle-\langle{\partial\phi_{0}\over\partial\varphi}|{\partial\phi_{0}\over\partial\theta}\rangle\right]$

As $\theta$ or $\varphi$ changes by $2\pi$, the ground state must return to
itself (up to overall phase factor), unless the ground state is not uniquely
determined by the periodic boundary conditions of the torus.
$\bar{\sigma}_{m}$ is then quantized to integer $\mathcal{Q}$ times $m^{2}/h$,
for non-degenerate ground state and finite magnetization gap.

### II.3 Incompressibility

These arguments may also be applied in the case of open boundary conditions in
one direction, and periodic boundary conditions in the second, to revisit
Laughlin’s gedanken experiment Laughlin (1981) and investigate whether the
magnetic charges form an incompressible liquid. We first look for the solution
to the following problem of the magnetic point particles for the simplest case
of magnetic point particles with fermionic exchange statistics (setting
$c=1$):

$\mathcal{H}=\sum_{j}^{N}\left[{1\over 2\mu}\left({\hbar\over
i}\vec{\nabla}_{j}-{m}\vec{A}_{m}(\vec{r}_{j})\right)^{2}+Bmy_{j}\right],$ (8)

where here $\vec{A}_{m}(\vec{r})=Ey\hat{x}$ is the gauge field for the case of
magnetic charges, yielding out-of-plane electric field of strength $E$.

Here, we measure lengths in multiples of
$\ell=\sqrt{{\hbar\over\mu\omega_{c}}}=\sqrt{{\hbar\over mE}}$, where
$\omega_{c}={mE\over\mu}$. The solution is a Slater determinant of orbitals,

$\displaystyle\psi_{k,\gamma}={1\over\sqrt{2^{\gamma}\gamma!\sqrt{\pi}L_{x}}}e^{ikx}e^{(y+y_{0}-k)^{2}/2}\left({\partial\over\partial
y}\right)^{\gamma}e^{-(y+y_{0}-k)^{2}},$ (9)

where $y_{0}=mBl/\hbar\omega_{c}$. The corresponding energies are

$E_{k,\gamma}=(\gamma+1/2)\hbar\omega_{c}+\hbar k\left({B\over
E}\right)-{\mu\over 2}\left({B\over E}\right)^{2}.$ (10)

We consider an in-plane magnetic field $B$ is small relative to the gap
between these effective Landau levels, and assume a finite energy barrier to
gapless excitations (the magnetic charge equivalent of a chemical potential
being adjusted to lie within the gap between Landau levels). The number of
magnetic charges in the sample is $N=\gamma L_{x}L_{y}/2\pi l^{2}$, the
magnetic charge density is $\rho_{m}={\gamma m\over 2\pi l^{2}}$.

The Hall conductivity for transport of the magnetic charges, in terms of
magnetic charge current $J_{m}$, is then

$\displaystyle\sigma_{m,xy}={J_{m}\over B}={\rho_{m}\over E}={\gamma
m^{2}\over h}$ (11)

We therefore find that

$J_{m}={\gamma m^{2}\over\hbar}B.$ (12)

Taking the gauge field $\vec{A}_{m}\rightarrow\vec{A}_{m}+A_{m,o}\hat{x}$,
with $A_{m,o}$ constant, we may further write the current operator as the
derivative of the Hamiltonian $\mathcal{H}$ with respect to the vector
potential,

$\displaystyle{m\over\mu}\sum_{j}\left[{\hbar\over i}{\partial\over\partial
x}-{m}A_{x}(\vec{r}_{j})\right]=-{\partial\mathcal{H}\over\partial A_{m,o}}$
(13)

If the sample is periodic in one direction as we consider here, $A_{m,o}$ is
an electric flux $\phi_{E}=A_{m,o}L_{x}$ threaded through the loop.

We may solve the eigen problem of the Hamiltonian for a given $\phi_{E}$ as
$\mathcal{H}_{\phi_{E}}=|\Psi_{\phi_{E}}\rangle=\lambda_{\phi_{E}}|\Psi_{\phi_{E}}\rangle$.
By Hellman’s theorem, we have
$\langle\Psi_{\phi_{E}}|{\partial\mathcal{H}_{\phi_{E}}\over\partial\phi_{E}}|\Psi_{\phi_{E}}\rangle={\partial\lambda_{\phi_{E}}\over\partial\phi_{E}}$.

Assuming current $J_{m}$ changes negligibly during the flux insertion, we may
write $J_{m}=\Delta E/\Delta\phi_{E}$, with $\Delta\phi_{E}$ the electric flux
quantum $h/m$. Under these conditions, $\mathcal{H}_{\Delta\phi}$ equals
$\mathcal{H}_{0}$ up to a gauge transformation. As $\phi_{E}$ is tuned from
$0$ to the electric flux quantum $\Delta\phi_{E}$, one state (one magnetic
charge) per effective Landau level is transferred from the left side of the
sample to the right side. Turning on a sufficiently small impurity potential
such that gaps remain between the effective Landau levels, adiabatic evolution
of $\phi_{E}$ still transfers exactly one magnetic charge.

### II.4 Counterpart of MQHE on a lattice

Given that it is possible to realize topological phases characterized by non-
trivial Chern numbers without an external magnetic field in lattice-based
systems Haldane (1988), it is expected that a lattice-based counterpart of the
magnetic quantum Hall effect can be realized without external electric field.
We provide a minimal toy model for the topological skyrmion phases of matter
Cook (2023) as an example, as the topological skyrmion phases of matter are
the lattice-based counterparts of the MQHE and a corresponding generalization
discussed in the Results, section E. For a system with multiple dofs, these
are topological phases associated with subset(s) of these dofs: they have so
far specifically been realized in lattice models as topological phases
associated with the spin dof in systems also possessing an orbital dof and/or
a generalized particle-hole dof. They are characterized by the formation of a
magnetic skyrmion in the ground state spin expectation value texture over the
Brillouin zone, even when spin is not conserved, due to an additional orbital
degree of freedom and non-negligible atomic spin-orbit coupling. We discuss
them here as candidates for lattice counterparts of the MQHE without an
external electric field, in itinerant magnets.

We first introduce some helpful expressions for three different embeddings of
vectors of Pauli matrices into $3\times 3$ matrix representations,
$\boldsymbol{\sigma}_{\alpha}=\langle\sigma_{\alpha,x},\sigma_{\alpha,y},\sigma_{\alpha,z}\rangle$,
with components defined as

$\displaystyle\sigma_{\alpha,x}$
$\displaystyle=\begin{pmatrix}0&\delta_{\alpha 1}&\delta_{\alpha 2}\\\
\delta_{\alpha 1}&0&\delta_{\alpha 3}\\\ \delta_{\alpha 2}&\delta_{\alpha
3}&0\\\ \end{pmatrix},\sigma_{\alpha,y}=i\begin{pmatrix}0&-\delta_{\alpha
1}&-\delta_{\alpha 2}\\\ \delta_{\alpha 1}&0&-\delta_{\alpha 3}\\\
\delta_{\alpha 2}&\delta_{\alpha 3}&0\\\ \end{pmatrix}$ (14)
$\displaystyle\sigma_{\alpha,z}$ $\displaystyle=\mathrm{diag}(\delta_{\alpha
1}+\delta_{\alpha 2},-\delta_{\alpha 1},-\delta_{\alpha 2}),$

where $\delta_{\alpha\beta}=1$ if $\alpha=\beta$ and zero otherwise.

We may then write toy models as Hamiltonians of the form

$\mathcal{H}=\sum_{\boldsymbol{k}}\psi^{\dagger}_{\boldsymbol{k}}\mathcal{H}(\boldsymbol{k})\psi_{\boldsymbol{k}},$
(15)

where
$\psi_{\boldsymbol{k}}=\left(c_{\boldsymbol{k},\uparrow,\alpha},c_{\boldsymbol{k},\downarrow,\beta},c_{\boldsymbol{k},\downarrow,\kappa}\right)^{\top}$,
where $c_{\boldsymbol{k},\sigma,\ell}$ annihilates a fermion with spin
$\sigma$ in orbital $\ell$. The inspiration for these minimal toy models are
tight-binding Hamiltonians describing transition metal oxides such as Sr2RuO4
as studied in past work Cook (2023), and the t2g orbitals of transition metal
ions subjected to crystal field splitting could serve as the three-fold
orbital degree of freedom considered here. We take the spin to be $1/2$ and
effective orbital angular momentum to be $1$. The Bloch Hamiltonian
$\mathcal{H}(\boldsymbol{k})$ takes the following form:

$\displaystyle\mathcal{H}(\boldsymbol{k})=\boldsymbol{d}_{1}(\boldsymbol{k})\cdot\boldsymbol{\sigma}_{1}+\boldsymbol{d}_{2}(\boldsymbol{k})\cdot\boldsymbol{\sigma}_{2}.$
(16)

Even richer topology is realized by adding terms
$\mathcal{H}_{3}(\boldsymbol{k})=\boldsymbol{d}_{3}(\boldsymbol{k})\cdot\boldsymbol{\sigma}_{3}$.

We compute the spin expectation value using the following spin operators
introduced in past work Cook (2023),

$\displaystyle S_{x}={1\over 2}\begin{pmatrix}0&1&1\\\ 1&0&1\\\ 1&1&0\\\
\end{pmatrix}\hskip 2.84526ptS_{y}={1\over 2}\begin{pmatrix}0&-i&-i\\\
i&0&-i\\\ i&i&0\\\ \end{pmatrix}\hskip 2.84526ptS_{z}={1\over
2}\begin{pmatrix}2&0&0\\\ 0&-1&0\\\ 0&0&-1\\\ \end{pmatrix}$ (17)

Rich phase diagrams are realized by choosing each $\boldsymbol{d}$-vector to
individually characterize a two-band Chern insulator. The system can then at
least be characterized in terms of a total Chern number $\mathcal{C}$ and
skyrmion number $\mathcal{Q}$ as in past work Cook (2023); Liu  _et al._ .
Here, we consider the example of $\boldsymbol{d}_{1}$ to be a modified QWZ
$\boldsymbol{d}$-vector Qi  _et al._ (2006)

$\displaystyle d_{1,x}(\boldsymbol{k})$ $\displaystyle=2\sin(k_{y})$ (18)
$\displaystyle d_{1,y}(\boldsymbol{k})$ $\displaystyle=2\sin(k_{x})$ (19)
$\displaystyle d_{1,z}(\boldsymbol{k})$
$\displaystyle=m_{1}-2\cos(k_{x})-2\cos(k_{y}),$ (20)

and $\boldsymbol{d}_{2}$ to be the same as $\boldsymbol{d}_{1}$ except for
exchange of sine functions and cosine functions (e.g.,
$\sin{k_{x}}\rightarrow\cos(k_{x})$) and a second, independent mass $m_{2}$ in
place of $m_{1}$. We also include a term $\lambda\sigma_{3,x}$ in this
example.

We first compute the topological charge $\mathcal{Q}$ of the ground state spin
expectation value texture over the Brillouin zone, for the lowest band
occupied, as

$\displaystyle\mathcal{Q}={1\over 4\pi}\int
d\boldsymbol{k}\langle\hat{S}(\boldsymbol{k})\rangle\cdot\left(\partial_{k_{x}}\langle\hat{S}(\boldsymbol{k})\rangle\times\partial_{k_{y}}\langle\hat{S}(\boldsymbol{k})\rangle\right),$
(21)

where $\langle\hat{S}(\boldsymbol{k})\rangle$ is the normalized ground state
spin expectation value at momentum $\boldsymbol{k}$. We first compute $\langle
1,\boldsymbol{k}|S_{i}|1,\boldsymbol{k}\rangle$ for $i\in\\{x,y,z\\}$ and
$S_{i}$ the spin operators given in Eqn. 17, with $|1,\boldsymbol{k}\rangle$
the lowest-energy eigenstate at momentum $\boldsymbol{k}$. We show an example
of the spin texture winding in Fig. 1, corresponding to $\mathcal{Q}=2$.

Figure 1: Characterization of spin topology in the bulk: winding of $x$, $y$
and $z$ components of normalized ground state spin expectation value texture
$\langle\hat{S}\rangle$ over the Brillouin zone for lowest band occupied are
shown in a), b), and c), respectively, for $m_{1}=m_{2}=-1.8$, $\lambda=0.6$,
with topological charge of $\mathcal{Q}=2$. d) Mapping of normalized ground
state spin expectation value texture over the Brillouin zone to the surface of
a two-sphere for same coordinates.

For this parameter set, we then open boundary conditions in the
$\hat{x}$-direction, and compute the slab energy spectrum, shown in Fig. 2 a).
Gapless boundary modes in the slab spectrum are high-lighted in Fig. 2 b).
They are not chiral, corresponding to total Chern number of zero, and do not
fully traverse the gap, but possess gaplessness as two topologically-protected
crossings. The probability density and the natural logarithm of the
probability density for each in-gap state is plotted as a function of
eigenvector index in Fig. 2 c) and d), respectively, showing exponential
localization of the edge states, despite proximity of one of the in-gap states
to bulk states for the chosen $k_{y}=-2.5$.

Figure 2: a) Full slab energy spectrum for open boundary conditions in
$\hat{x}$-direction (system size $L_{x}=80$) and periodic boundary conditions
in the $\hat{y}$ direction. b) slab energy spectrum for same boundary
conditions as in a), over smaller interval in slab BZ, highlighting gapless
boundary modes. c) Probability density for gapless boundary modes at
$k_{y}=-2.5$ highlighted in b) versus eigenvector index $i$, d) Natural
logarithm of probability densities for gapless boundary modes shown in b) and
c), illustrating exponential decay.

We compute the unnormalized spin expectation value components for these two
edge states in Fig. 3 a) and b), respectively. We observe a net pumping of the
$\hat{y}$-component of the spin angular momentum over the interval between
crossings, corresponding to pumping of spin angular momentum from the bulk
valence to bulk conduction bands. At the same time, the structure of the edge
states yields no net pumping of charge across the bulk gap, corresponding to
total Chern number of zero.

Figure 3: Spin expectation value components vs. $k_{y}$ for gapless boundary
modes a) $|\psi_{80}\rangle$ and b) $|\psi_{81}\rangle$, respectively, shown
in bulk gap in Fig. 2 b).

We compute entanglement spectra of two-point, equal-time spin correlators as a
generalization of Peschel’s method Peschel (2003) for this cylinder geometry
and band occupancy, tracing out half of the cylinder, shown in Fig. 4. Over
the interval in $k_{y}$ between the points at which the edge states cross, we
observe a state strongly-localized on one edge of the cylinder for the
$\hat{x}$\- and $\hat{y}$\- correlators, but not for the $\hat{z}$-correlator.
However, the edge state in the $\hat{y}$-correlator merges with the bulk,
corresponding to pumping of the $\hat{y}$-component of spin angular momentum
observed in Fig. 3, while the edge state in the $\hat{x}$\- correlator does
not merge with the bulk, corresponding to no pumping of the
$\hat{x}$-component of spin angular momentum across the cylinder. Neither of
these edge states is present for periodic boundary conditions in both the
$\hat{x}$\- and $\hat{y}$-directions, indicating they are due to bulk-boundary
correspondence of non-trivial $\mathcal{Q}$. We also note that open boundary
conditions in the $\hat{y}$-direction and periodic boundary conditions in the
$\hat{x}$-direction would instead yield pumping of the $\hat{x}$-component of
the spin angular momentum.

We finally briefly examine a type-II topological phase transition across which
$\mathcal{Q}$ changes from $2$ to $1$ for the chosen parameter set. That is,
$\mathcal{Q}$ changes from one integer value to another as a result of the
minimum magnitude of the ground state spin expectation value going to zero,
without the closing of the minimum direct bulk energy gap: non-negligible
atomic spin-orbit coupling and the orbital dof instead yield total angular
momentum that is entirely orbital in character for some
$\boldsymbol{k}$-value(s) in the Brillouin zone. The transition is shown in
Fig. 5. We observe, for this particular parameter set, that the crossings
between the edge states occur over an extended region of phase space, rather
than being fine-tuned. The in-gap states eventually separate, however, at
which point the edge states in the two-point, equal-time spin correlators are
lost.

Figure 4: Two-point, equal-time spin correlator entanglement spectra for a)
$\langle S_{x,i}S_{x,j}\rangle$, b) $\langle S_{y,i}S_{y,j}\rangle$, c)
$\langle S_{z,i}S_{z,j}\rangle$, d) Natural logarithm of probability density
versus eigenvector index $i$ for highest energy state $|\phi_{x}\rangle$,
$|\phi_{y}\rangle$, $|\phi_{z}\rangle$ in each of a), b), c), respectively.
Parameter values are $\lambda=0.6$, $m_{1}=m_{2}=-1.8$, and $k_{y}=-2.8$.

We note that the spin skyrmion characterized by $\mathcal{Q}$ is retained upon
tracing out the orbital degree of freedom. For the case of non-negligible
atomic spin-orbit coupling, the spin degree of freedom is coupled to the
orbital degree of freedom, and can therefore be interpreted as an open system
upon tracing out the orbital degree of freedom. After performing this partial
trace over the orbital degree of freedom, the spin degree of freedom is the
only degree of freedom. We may then write an effective Bloch Hamiltonian of
the spin subsystem in terms of the ground state spin texture,
$\langle\boldsymbol{S}(\boldsymbol{k})\rangle$, as
$\mathcal{H}(\boldsymbol{k})=-\langle\boldsymbol{S}(\boldsymbol{k})\rangle\cdot\boldsymbol{\sigma}$.
As is done in the case of a two-band Chern insulator, we may compute a Hall
conductivity $\sigma_{xy}$ in terms of this ground state spin texture
$\langle\boldsymbol{S}(\boldsymbol{k})\rangle$. As in the case of the two-band
Chern insulator, this Hall conductivity is associated with quantized transport
of charge, but is proportional to $\mathcal{Q}$. $\mathcal{Q}$ may then be
interpreted as an effective Chern number for the open spin subsystem. This
Hall conductivity may be distinct from that associated with a total Chern
number computed for the full three-band model, and the charge transport
associated with Hall conductivity of this effective two-band model is
physically distinct from the charge transport of the full three-band Bloch
Hamiltonian. That is, non-trivial $\mathcal{Q}$ is associated with quantized
transport and bulk-boundary correspondence of this spin charge of the open
spin subsystem.

Figure 5: Slab energy spectra through type-II topological phase transition
(without closing of minimum direct bulk energy gap) from $\mathcal{Q}=2$ to
$\mathcal{Q}=1$ for $\lambda=0.6$ and a) $m_{1}=m_{2}=-1.8$, b)
$m_{1}=m_{2}=-1.6$, c) $m_{1}=m_{2}=-1.4$, d) $m_{1}=m_{2}=-1.2$.

As well, we briefly consider an interesting case of partial filling of the
second-lowest energy bulk band. For finite temperature, $\mathcal{Q}$ can
actually be integer-quantized, even for a partially-filled band. We show a
representative winding of the ground state spin expectation value components
in Fig. 6 a), b), and c), respectively. The normalized ground state spin
expectation value texture is also shown mapped to the surface of a two-sphere
in Fig. 6 d), demonstrating non-trivial wrapping of the two-sphere. Although
the ground state spin expectation value texture for partially-filled bands
exhibits singularities in the partial derivatives with respect to $k_{x}$ or
$k_{y}$, these singularities are removed at finite temperature $k_{B}T\neq 0$.
For spin expectation value finite in magnitude for each $\boldsymbol{k}$ in
the Brillouin zone, $\mathcal{Q}$ stabilizes to an integer. That is,
$\mathcal{Q}$ can characterize a topologically stable and non-trivial metal.
We will explore the consequences of this in greater detail in future work.
This is an important demonstration that $\mathcal{Q}$ is a global topological
invariant Avron  _et al._ (1983).

Figure 6: Characterization of spin topology for metallic bulk (chemical
potential partially intersects second-lowest energy band): winding of $x$, $y$
and $z$ components of normalized ground state spin expectation value texture
$\langle\hat{S}\rangle$ over the Brillouin zone for lowest-energy band
occupied and second lowest-energy band partially occupied are shown in a), b),
and c), respectively, for $m_{1}=m_{2}=-1.2$, $\lambda=0.6$, chemical
potential $\mu=-1.6$, $k_{B}T=0.05$, with topological charge of
$\mathcal{Q}=1$. d) Mapping of normalized ground state spin expectation value
texture over the Brillouin zone to the surface of a two-sphere for same
coordinates.

### II.5 Relevance of MQHE to magnetic skyrmions

To identify a possible generalization of the MQHE, we define a magnetization
texture realizing magnetic skyrmions over a two-torus $T^{2}$ in real-space of
length $L_{1}$ in the $x$ direction and $L_{2}$ in the $y$ direction as
$M(x,y)$. We first consider $M$ corresponding to $N$ strongly-localized
magnetic skyrmions (of radius $r$ much smaller than the minimum of $L_{1}$ and
$L_{2}$, as well as the spacing between these magnetic skyrmions) against a
ferromagnetic background magnetization texture. We assume $N$ is fixed,
corresponding to the energy barrier for formation or deletion of these small
magnetic skyrmions being much higher than other energy scales of the problem.
That is, charge associated with the $N$ small magnetic skyrmions is conserved,
provided the energy barrier that must be overcome to create or delete these
small magnetic skyrmions—which also includes the energy cost to increase their
radius—is much greater than other energy scales in the problem. This scenario
might correspond, for instance, to short-range skyrmion density-density
interactions dominating over other terms relevant to the problem as minimal
ingredients for skyrmion formation, such as ferromagnetic exchange coupling,
DM interactions (here we consider anisotropic DM interactions as the out-of-
plane external electric field corresponds to a Rashba-type spin-orbit coupling
(SOC) Rowland  _et al._ (2016); Hanada and Sawado (2023)), and effective
Zeeman field term. We note that such anisotropic DM terms yield a
$\mathrm{U}(1)$ easy-plane anisotropy of the magnetic skyrmions, which
generalizes the $\mathrm{U}(1)$ charge conservation symmetry. That is, the
charge conservation symmetry generalizes to conservation of angular momentum.
The lattice model provides some evidence of this, exhibiting pumping of spin
angular momentum across the bulk gap, without net pumping of charge.

Although gapless modes, such as phonon modes, may couple to the magnetization
texture, we assume the system is at sufficiently low temperature such that
these modes are so low in energy that they do not couple to the high-energy
internal degrees of freedom of these small magnetic skyrmions. For
sufficiently small magnetic skyrmions, we may furthermore treat them as
quantum mechanical, with quantized effective angular momentum associated with
their curvature. Such quantum magnetic skyrmions may have already been
realized in past work Sotnikov  _et al._ (2018).

For this magnetization texture possessing $N$ small magnetic skyrmions, we may
integrate the curvature associated with the $j$th magnetic skyrmion to yield
its topological charge $m_{j}$. Given a surrounding ferromagnetic
magnetization texture and non-negligible exchange coupling, we also assign it
an effective mass $\mu_{j}$: propagation of a small magnetic skyrmion involves
deformation of the magnetization texture, and this deformation costs energy
due to terms in the Hamiltonian for the magnetization texture, such as the
exchange coupling, spin-orbit coupling, and Zeeman field terms. The effective
mass in the language of the magnetization texture is the factor by which the
velocity of a small magnetic skyrmion is converted to an energy cost.

We will now create an additional magnetic skyrmion, with radius on the scale
of the full system, and therefore comparable in scale to $L_{1}$ and $L_{2}$.
The energy scale of creating this large skyrmion is set by the couplings in
the Hamiltonian for the magnetization texture: the large magnetic skyrmion
forming at lower energies than the small magnetic skyrmions requires that
high-frequency change in orientation of the magnetization be energetically
costly in comparison to low-frequency changes in orientation of the
magnetization, in combination with skyrmion density-density interactions
favoring large inter-skyrmion spacing for the small magnetic skyrmions. Under
these conditions, we form an additional large magnetic skyrmion by applying an
external electric field normal to the surface of the torus, of sufficient
strength to overcome the energy barrier to formation of the large magnetic
skyrmion, which is then also of strength weak compared to the energy scale of
creating/deleting the small magnetic skyrmions and the energy scale of
changing the radius of a skyrmion from $r$ to the scale of $L_{1}$ or $L_{2}$
or vice versa: in simple cases, this external electric field couples to the
magnetization texture as an additional anisotropic DM term corresponding to
Rashba spin-orbit coupling Rowland  _et al._ (2016), and can generate an
additional large magnetic skyrmion in combination with other terms such as
exchange coupling and effective Zeeman field.

The topological charge of the large magnetic skyrmion, or
$\boldsymbol{M}_{BG}(x,y)$, must quantize when the magnetization is everywhere
finite in magnitude on the two-torus. Defining a normalized magnetization
vector
$\hat{\boldsymbol{M}}_{BG}(x,y)=\boldsymbol{M}_{BG}(x,y)/|\boldsymbol{M}_{BG}(x,y)|$,
we define mappings from the two-torus $T^{2}$ to the space of normalized
magnetization vectors $S^{2}$. Such mappings are contractible to mappings
$S^{2}\rightarrow S^{2}$, with non-trivial homotopy group
$\pi_{2}(S^{2})=\mathbb{Z}$. We may define a curvature of the magnetization as
$\Omega_{M_{BG}}(x,y)=\hat{\boldsymbol{M}}_{BG}(x,y)\cdot\left(\partial_{x}\hat{\boldsymbol{M}}_{BG}(x,y)\times\partial_{y}\hat{\boldsymbol{M}}_{BG}(x,y)\right)/4\pi$,
and integrate this curvature over the two-torus to find
$\int_{T^{2}}\Omega_{M_{BG}}(x,y)dA=\mathcal{Q}_{BG}$ an integer.

In the language of the MQHE, magnetic charges experience a local electric flux
resulting from coupling to the external electric field. In the language of the
magnetization texture, the local electric flux after integrating out the
internal degrees of freedom of the small magnetic skyrmions is
$\Omega_{M_{BG}}(x,y)$. These two curvatures are therefore equal at a given
point on the two-torus, despite being computed from different physical
quantities.

Although we can neglect coupling of gapless excitations to the small magnetic
skyrmions by considering the case of high-energy internal dofs of the small
magnetic skyrmions, and the system at sufficiently low temperatures such that
gapless excitations do not couple to these high-energy internal dofs, there
remains the possibility that gapless excitations could couple to the large
magnetic skyrmion. Considering the system at sufficiently low temperature such
that these gapless excitations cannot change the topological charge of this
large magnetic skyrmion, $\mathcal{Q}_{BG}$, the remaining potential concern
is the role played by local perturbations of the system due to gapless
excitations. This question has already been explored in the context of quasi-
topological phases by Bonderson and Nayak Bonderson and Nayak (2013): the
large magnetic skyrmion is in a ground state protected by finite energy
barrier, $\Delta$, to change in its topological charge. Gapless excitations
could yield a set of pseudo-groundstates $|a\rangle$, $a=1,2,...,N_{\Sigma}$,
in the language of Bonderson and Nayak, with energy $E_{a}$ satisfying the
following relation:

$\displaystyle\delta
E_{0}\equiv\mathrm{max}|(E_{a}-E_{a^{\prime}})|=\mathcal{O}(e^{-L/\varepsilon}),$
(22)

where $L$ is the system size and $\varepsilon$ is the correlation length of
the system. Local perturbations at energies below $\Delta$ act within this
manifold of pseudo-groundstates: the system can therefore acquire an overall
phase at most and quantum information encoded in this pseudo-ground state
subspace is “topologically protected” at zero temperature in the thermodynamic
limit. At non-zero temperature $T$, the error rate $\Gamma$ is exponentially
suppressed by the gap, $\Gamma\sim e^{-\Delta/T}$.

For our purposes, then, gapless excitations corresponding to changes in local
curvature of the magnetization texture, which preserve the topological charge
of the large magnetic skyrmion and do not couple to the high-energy internal
degrees of freedom of the small magnetic skyrmions, are permitted. In the
language of the magnetic charges, this corresponds to local deformation of the
electric flux while remaining within a given Landau level.

Under these conditions, we identify quantized transport of small magnetic
skyrmions with quantized transport of magnetic charges. This quantum skyrmion
Hall effect (QSkHE) is therefore a generalization of the MQHE, in that the
magnetic charges of the MQHE are generalized to small magnetic skyrmions, the
$\mathrm{U}(1)$ charge conservation symmetry of the MQHE is generalized to a
$\mathrm{U}(1)$ easy-plane anisotropy and charge conservation symmetry reliant
on maintenance of a hierarchy of energy scales, and the background electric
flux is generalized to a large background magnetic skyrmion. A lattice
counterpart of the QSkHE is the topological phase realized by the lattice
model Eqn. 16 for $\mathcal{Q}\neq 0$, which corresponds to the lattice
counterpart of the MQHE after partial trace over orbital dof. We also note
that the flat-band limit assumption, first stated in its modern form in Avron
_et al._ Avron  _et al._ (1983), therefore derives from an earlier assumption
of point charges in the theory of the QHE Laughlin (1981), which can be
relaxed to generalize the QHE as the QSkHE.

### II.6 Role of external field in the quantum skyrmion Hall effect

For applied field strength comparable to the energy scale of the large
background magnetic skyrmion, changing the external field strength corresponds
to flux insertion and pumping of charges to the boundary due to
incompressibility. For applied field strength comparable to the energy scale
of creating/deleting small magnetic skyrmions, changing the external field
strength corresponds to tuning the chemical potential of the magnetic charges,
corresponding to proliferation of small magnetic skyrmions due to
incompressibility.

### II.7 Two-dimensional gas of both electric and magnetic charges

We now explore the role of the QSkHE in the context of the quantum Hall
problem in $(2+1)$ dimensions, generalized to the case of a 2D gas of
electric—but also magnetic—charges. That is, what is the potential role of the
QSkHE in the response and topology of an itinerant magnetic system confined to
a 2D plane? This investigation is also motivated by past study of topological
skyrmion phases in lattice systems, where the Chern insulator phase coexists
with topological phases characterized by spin skyrmion number $\mathcal{Q}$
Cook (2023); Liu  _et al._ , as well as past work on multiplicative
topological phases, which exhibit non-trivial topology of single particle
systems and also of the correlations between single particle systems Cook and
Moore (2022).

First, we note that the MQHE is dual to the QHE: exchanging magnetic charges
for electric charges, and the applied electric field for a magnetic field in
the MQHE, we map to the QHE. We therefore expect the field theory of the
itinerant magnet, for the case of both electric field and magnetic field
applied normal to the plane of the system, to be self-dual: exchanging
magnetic charge for electric charge and external electric field for external
magnetic field should yield the same field theory.

Such self-duality is a property of the $(4+1)$ dimensional quantum Hall
effect, in that there is a symmetry in the field theory between how particles
respond to external magnetic versus electric fields Qi  _et al._ (2008b). We
show that the $(4+1)$D QHE can be mapped to the problem of a 2D system of
electric and magnetic charges. To do so, we reinterpret the particles coupled
to external fields in the $(4+1)$D field theory for the QHE, $q_{c}$, as
composite particles, composed of a particle with electric charge, $q_{e}$, and
a particle with magnetic charge, $q_{m}$, as $q_{c}=q_{e}+q_{m}$.

By reinterpreting the axes $x$, $y$, $z$, $\omega$ of four-dimensional space
as $x$, $y$, $x^{\prime}$, $y^{\prime}$, where $x$ and $y$ are the coordinates
of the electric charges, and $x^{\prime}$ and $y^{\prime}$ are the coordinates
of the magnetic charges, we map the $(4+1)$D theory of composite particles to
a $(2+2+1)$D theory of electric and magnetic charges of a two-dimensional gas.
For consistency in realizing an $\boldsymbol{E}\cdot\boldsymbol{B}$ coupling
for this interpretation of particles in the $(4+1)$D theory as composite, we
introduce an operator
$f^{\dagger}_{c,\boldsymbol{r},\boldsymbol{r^{\prime}}}=f^{\dagger}_{e,\boldsymbol{r}}f^{\dagger}_{m,\boldsymbol{r}^{\prime}}$
to create composite particles at position $\boldsymbol{r}=(x,y)$,
$\boldsymbol{r}^{\prime}=(x^{\prime},y^{\prime})$, where
$f^{\dagger}_{e,\boldsymbol{r}}$ creates a particle with electric charge at
position $\boldsymbol{r}$, and $f^{\dagger}_{m,\boldsymbol{r}^{\prime}}$
creates a particle with magnetic charge at position $\boldsymbol{r}^{\prime}$.
For electric and magnetic charges each fermionic, the composite particle is
then a composite boson Combescot and Shiau (2015).

We consider external magnetic and electric field each applied normal to the
plane of the gas, rather than just a magnetic field or electric field.
$\boldsymbol{E}$ and $\boldsymbol{B}$ are therefore parallel. In this case,
the $(4+1)$D theory includes the terms of the $(2+1)$D theory of the QHE and
MQHE: $\boldsymbol{E}$ exerts a force on electric charge, but only in the
$\hat{z}$ direction, so this coupling is neglected and only $\boldsymbol{B}$
coupling to electric charge is non-trivial. Similarly, $\boldsymbol{B}$ can
exert a force on the magnetic charges, but only in the $\hat{z}$-direction,
such that it is trivial and only the $\boldsymbol{E}$ coupling to the magnetic
charges is non-trivial. However, with $\boldsymbol{E}$ and $\boldsymbol{B}$
parallel, the magnetoelectric term of the $(4+1)$D theory of composite
particles, corresponding to a topologically non-trivial magnetoelectric
response of the system expressed in terms of the second Chern form Qi  _et
al._ (2008b), is relevant but unnecessary to fully characterize the topology
of the system in simple cases.

We previously considered a $\mathrm{U}(1)$ globally-symmetric MQHE, and its
generalization to the QSkHE with $\mathrm{U}(1)$ easy-plane anisotropy due to
anisotropic DM interaction: a charge conservation symmetry effectively holds
in this case, provided a certain hierarchy of energy scales is maintained, in
which the energetic cost of creation/deletion or increasing the radius of a
small magnetic skyrmion is significantly greater than other operations. We now
consider the scenario where correlations between electric and magnetic charges
are intermediate in strength, such that electric and magnetic charges are not
necessarily at the same positions, and these correlated states are not
necessarily tensor products of states of the electric charge subsystem and
states of the magnetic charge subsystem. Additionally, we consider the
scenarios in which electric charge conservation and magnetic charge
conservation each hold in some parameter regimes, but are potentially lost
while composite charge conservation is potentially retained in some parameter
regimes.

In these regimes, a third contribution to the action, associated with the
correlations between electric and magnetic charges modeled as composite
particles, more fully captures the quantum Hall physics of this multi-
component system. Such higher-dimensional topology of correlations has been
realized previously as multiplicative topological phases Cook and Moore
(2022), in particular as the perpendicular multiplicative Kitaev chain Pal
_et al._ (2023), although these are simple cases where the topology of the
correlations derives from the single particle topology of each parent phase.

We therefore consider an effective action $S_{\mathrm{eff}}$ for the
$(2+2+1)$D theory in terms of three gauge fields, one for the coupling of
external fields to electric charges (capturing the fact that only the
$\boldsymbol{B}$ field couples in this case), $A_{e,i}$, one for coupling of
the external fields to the magnetic charges (capturing the fact that only the
$\boldsymbol{E}$ field couples to these particles, but as $\boldsymbol{B}$
couples to the electric charges), $A_{m,i}$, and a gauge field $A_{c,i}$,
capturing the coupling of fields to the composite particles,
$S_{\mathrm{eff}}=S_{e,\mathrm{eff}}+S_{m,\mathrm{eff}}+S_{c,\mathrm{eff}}$,
where

$\displaystyle S_{e,\mathrm{eff}}$ $\displaystyle={C\over 4\pi}\int
d^{2}x_{e}dt\varepsilon^{\mu\nu\rho}A_{e,\mu}\partial_{\nu}A_{e,\rho}$ (23)
$\displaystyle S_{m,\mathrm{eff}}$ $\displaystyle={\mathcal{Q}\over 4\pi}\int
d^{2}x_{m}dt\varepsilon^{\mu\nu\rho}A_{m,\mu}\partial_{\nu}A_{m,\rho}$ (24)
$\displaystyle S_{c,\mathrm{eff}}$ $\displaystyle={C_{2}\over 24\pi^{2}}\int
d^{4}xdt\varepsilon^{\mu\nu\rho\sigma\tau}A_{c,\mu}\partial_{\nu}A_{c,\rho}\partial_{\sigma}A_{c,\tau}$
(25)

In the $(4+1)$D QHE, one can pump $C_{2}$ charges by applying magnetic fields
corresponding to gauge field $\boldsymbol{A}=\left(By,0,B\omega,0\right)$ Qi
_et al._ (2008b). In the $(2+2+1)$D theory, this corresponds to
$\boldsymbol{A}_{c}=\left(By,0,Ey^{\prime},0\right)$, which will correspond to
pumping of $C_{2}$ electric and magnetic charges, corresponding to pumping of
$C_{2}$ composite particles as required.

## III Discussion and Conclusion

We study the problem of magnetic charges in $(2+1)$ dimensions subjected to an
external electric field, which couples to the magnetic charges as an inverted
Lorentz force. Considering this problem on a two-torus with $\mathrm{U}(1)$
global symmetry corresponding to conservation of this magnetic charge, we find
the Hall conductivity governing transport of the magnetic charges quantizes in
the case of negligible gapless excitations, being proportional to an integer-
valued topological invariant $\mathcal{Q}$, to realize a magnetic quantum Hall
effect (MQHE).

We present results on a counterpart lattice problem, realizing physics of the
MQHE in the absence of an external electric field, characterized by a three-
band Bloch Hamiltonian. We characterize its topology in terms of a total Chern
number $\mathcal{C}$ and a winding number $\mathcal{Q}$ for the ground state
spin expectation value as in past work on topological skyrmion phases Cook
(2023), which we identify as the lattice counterparts of the MQHE. We mean
this in the same sense that the Chern insulator is the lattice counterpart of
the QHE realized in the absence of an external magnetic field Haldane (1988),
and note that the topological skyrmion phases discussed here can be understood
as generalizations of Chern insulators. We consider a topological phase
characterized by total Chern number $\mathcal{C}=0$ and skyrmion number
$\mathcal{Q}=2$, demonstrating the quantized topological charge of the ground
state spin expectation value texture over the bulk Brillouin zone, and a bulk-
boundary correspondence in which non-trivial $\mathcal{Q}$ yields gapless
modes localized on the system boundary. Here, gaplessness is ensured by the
crossing(s) of edge states in the slab spectrum that permit pumping of spin
angular momentum without pumping of net charge.

We develop a generalization of Peschel’s method Peschel (2003) for computing
entanglement spectra, to compute entanglement spectra for two-point, equal-
time spin correlators of the lattice model, in which we observe a bulk-
boundary correspondence when the gapless edge states are present in the slab
energy spectrum: edge states in the entanglement spectra correspond to pumping
of the $\hat{y}$-component of spin angular momentum for sample edges oriented
in the $\hat{y}$-direction.

We also show an example of a type-II topological phase transition Cook (2023)
over which the gapless boundary modes are lost without the closing of the
minimum direct bulk energy gap, which may be helpful in efforts to
experimentally confirm these topological phases of matter. As well, we
demonstrate quantized winding of the ground state spin expectation value in
the bulk Brillouin zone for the case of the lattice model describing a bulk
metal at small but finite temperature. This illustrates the fact that the
skyrmion number is a global topological invariant Avron  _et al._ (1983), as
well as potentially a topological invariant for characterizing topological
metals in addition to semimetals and insulators, which we will explore in
greater detail in future work.

As the lattice model realizes a counterpart of the MQHE after tracing out an
orbital degree of freedom, as well as a generalization of the MQHE prior to
performing a partial trace, we also identify a generalization of the MQHE
without a lattice, the quantum skyrmion Hall effect (QSkHE). The magnetic
point charge generalizes to a small magnetic skyrmion, and the $\mathrm{U}(1)$
charge conservation generalizes to a $\mathrm{U}(1)$ easy-plane anisotropy
corresponding to anisotropic DM interactions and charge conservation symmetry
maintained by a hierarchy of energy scales, in which the energetic costs of
creating/deleting small magnetic skyrmions or increasing their radius are far
greater than energetic costs of other operations. At sufficiently low
temperatures, gapless excitations do not couple to the high-energy internal
dofs of the small magnetic skyrmions, but can yield local perturbations of the
large magnetic skyrmion which do not interfere with incompressibility. The
lattice model counterpart of the QSkHE and MQHE for a spin $1/2$ dof in a
system also possessing an orbital dof and non-negligible atomic SOC
furthermore suggests this physics persists in non-equilibrium settings, as the
spin dof may be interpreted as an open system, coupling to an orbital dof
environment via non-negligible atomic SOC. We also note that the
generalization from a $\mathrm{U}(1)$ charge conservation symmetry to the
$U(1)$ easy-plane anisotropy corresponds to generalization of pumping of
charge to pumping of angular momentum, which is observed in the full lattice
model.

We then consider a field theory for an itinerant magnet in $(2+1)$ dimensions
inspired by both the topological skyrmion phases Cook (2023) and the
multiplicative topological phases Cook and Moore (2022), which are topological
phases of correlations between single-particle systems. This field theory
includes terms in the effective action for the QHE of electric charges and the
MQHE of magnetic charges relevant in cases where $\mathrm{U}(1)$ charge
conservation holds for each type of charge individually, as well as terms of a
$(4+1)$D QHE of composite particles consisting of electric and magnetic charge
pairs, effectively capturing topology of their correlations relevant in cases
of intermediate-strength correlations between electric and magnetic charges,
and where the system is not well-described as a direct product of a system of
electric charges and a system of magnetic charges.

In the case of the itinerant magnet, it is not clear whether a more
comprehensive characterization of topology beyond the three topological
invariants considered here is required, and this will be explored in future
work. Importantly, the assumption of point-like charges could, more generally,
be relaxed, with a more general field theory expressed in terms of underlying
polarization and magnetization fields. Related to this, we note that the QSkHE
can potentially be realized in vector fields of myriad observables, including
the polarization, if these vector fields can satisfy the requirements outlined
here.

Past work on the quantum Hall ferromagnet has already shown magnetic skyrmions
form in the context of the quantum Hall effect Sondhi  _et al._ (1993); Girvin
and MacDonald (1996). Our results are likely relevant to this problem. We may
also extend these results to a fractional quantum skyrmion Hall effect
(FQSkHE), similarly to the treatment of the fractional quantum Hall effect in
Niu  _et al._ Niu  _et al._ (1985), by adjusting system size $A$ relative to
magnetic particle number $N$ to realize overall partial filling.

Acknowledgements—We would like to thank C. Nayak, J. E. Moore, X. L. Qi, S.
Raghu, E. Altman and S. A. Kivelson for helpful discussions. We would also
like to thank R. Moessner for many helpful discussions and careful reading of
the manuscript.

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