# Many-body non-Hermitian Skin Effect for Multipoles

Jacopo Gliozzi  Giuseppe De Tomasi  Taylor L. Hughes  Department of Physics
and Institute for Condensed Matter Theory, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801-3080, USA

###### Abstract

In this work, we investigate the fate of the non-Hermitian skin effect in one-
dimensional systems that conserve the dipole moment and higher moments of an
associated global $\text{U}(1)$ charge. Motivated by field theoretical
arguments and lattice model calculations, we demonstrate that the key feature
of the non-Hermitian skin effect for $m$-pole conserving systems is the
generation of an $(m+1)$th multipole moment. For example, in contrast to the
conventional skin effect where charges are anomalously localized at one
boundary, the dipole-conserving skin effect results in charges localized at
both boundaries, in a configuration that generates an extremal _quadrupole_
moment. In addition, we explore the dynamical consequences of the $m$-pole
skin effect, focusing on charge and entanglement propagation. Both numerically
and analytically, we provide evidence that long-time steady-states have Fock-
space localization and an area-law scaling of entanglement entropy, which
serve as quantum indicators of the skin effect.

Introduction.— A central postulate of quantum mechanics is the unitarity of
time evolution, which implies the conservation of probability during dynamics.
However, interactions between a quantum system and its environment can induce
dissipation, decoherence, or wavefunction collapse, destroying unitarity.
Recent advances in controlled experimental techniques that allow for the
manipulation of single qubits, along with the need to overcome decoherence in
quantum devices, have led to increasing interest in such non-unitary dynamics
as a paradigm to describe open and monitored quantum systems [1, 2, 3, 4, 5].
Relaxing unitarity allows for a rich phenomenology that ranges from novel
topological phases [6, 7, 8, 9, 10, 11, 12, 13] to phase transitions in the
entanglement structure of non-equilibrium steady-states [14, 15, 16, 17, 18,
19, 20].

One approach to non-unitary dynamics is through non-Hermitian Hamiltonians,
which model quantum systems subject to dissipation [21, 22] or continuous
measurements and postselection [23, 24, 25, 26, 27]. Remarkably, non-Hermitian
couplings can lead to unique phenomena with no Hermitian counterpart, such as
the non-Hermitian skin effect, an anomalous localization of extensively many
eigenstates at the boundary of a system [28, 29, 30, 31], which has been
observed experimentally [32, 33, 34, 35]. While the boundary localization of
particles is an essentially classical effect due to non-reciprocal hopping, it
also has implications for the spread of quantum correlations during time
evolution. For example, the skin effect has been shown to hinder entanglement
growth and can trigger a volume-to-area law transition in the entanglement
entropy of long-time steady-states [36]. In addition, the skin effect
challenges the traditional understanding of a bulk-boundary correspondence in
topological systems [37, 38].

Previous studies of the non-Hermitian skin effect have been mostly restricted
to non-interacting Hamiltonians with $\text{U}(1)$ particle-number symmetry.
Moreover, the few extensions to interacting systems have relied on the
existence of a non-interacting limit [39, 40, 41, 42]. In this work, we
investigate the fate of the non-Hermitian skin effect in intrinsically
interacting systems that conserve multipole moments of a $\text{U}(1)$ charge.
Such conservation laws constrain particle motion and can dramatically affect
the out-of-equilibrium dynamics [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53].
For instance, in systems where the total dipole moment is conserved, isolated
charges cannot move, leading to anomalously slow charge transport and even
ergodicity breaking.

Figure 1: (a-b) Schematic of many-body non-Hermitian charge and dipole skin
effect, which generate a dipole and quadrupole moment, respectively. (c-d)
Time evolution of the local charge density $\rho_{\text{charge}}$ from a
random uniform state for (c) charge-conserving and (d) dipole-conserving
Hamiltonians with a non-Hermitian imbalanced kinetic term.

The interplay between multipole conservation laws, which restrict particle
motion, and non-reciprocal hoppings, which pump particles to a boundary
through the non-Hermitian skin effect, creates a complex scenario for charge
transport. For example, charge flows to a single boundary with only
$\text{U}(1)$ symmetry, but in dipole-conserving systems charges must flow
symmetrically, e.g., outwards to both boundaries (see Fig. 1). Motivated by
the hierarchical structure of multipoles, we are able to make significant
progress by defining the $m$-pole skin effect in terms of an $(m+1)$-pole
moment in the charge distribution of eigenstates. We also investigate the
dynamical consequences of this generalized skin effect by examining charge and
entanglement propagation following a global quench. In particular, we provide
analytical and numerical evidence that the $m$-pole skin effect leads to an
area-law scaling of entanglement entropy in the long-time steady-state.
Importantly, our definition also encompasses the standard “charge” skin
effect, and helps identify experimentally accessible probes in systems with
strong interactions.

Charge skin effect.— Before discussing higher multipole moments, let us first
revisit the non-Hermitian skin effect in the context of $\text{U}(1)$ charge
conservation. We focus on one-dimensional fermionic chains of length $L$. The
paradigmatic non-Hermitian model having the skin effect is the Hatano-Nelson
Hamiltonian [54, 55]:

$H_{\text{HN}}(g)=-\sum_{j}(e^{g}c^{\dagger}_{j}c_{j+1}+e^{-g}c^{\dagger}_{j+1}c_{j}),$
(1)

where $c^{\dagger}_{j}$ is the fermionic creation operator at site $j,$ and
$g$ parameterizes the left-right imbalance in charge hopping. With periodic
boundary conditions (PBCs), the single-particle spectrum of (1) winds around
$E=0$ in the complex plane, and the eigenstates are extended plane waves. In
contrast, for open boundary conditions (OBCs) the spectrum is real-valued, and
all single-particle eigenstates are exponentially localized at one boundary.
This stark difference between PBC and OBC spectra is the hallmark of the non-
Hermitian skin effect for non-interacting systems [28, 29].

To extend this definition to interacting systems, we first need a many-body
diagnostic of the skin effect. Unlike the single-particle case, an extensive
number of particles cannot be exponentially localized at a boundary in a many-
body state because of the Pauli exclusion principle, which forbids fermions
from clustering at the same site [56, 57]. Nevertheless, features of the
single-particle phenomenology persist. For instance, the $N$-particle spectrum
of (1) winds in the complex plane when the PBCs are twisted by a phase. In
OBCs, the spectrum still becomes real-valued, and the eigenstates exhibit a
left-right imbalance of charge. This asymmetry in the charge density of open
eigenstates has been interpreted as a many-body extension of the skin effect
[58].

While such diagnostics have been used for the Hatano-Nelson model with Hubbard
interactions [41, 42], we aim for a definition that applies even in the
absence of a non-interacting limit. Independently of particle number or
interactions, the skin effect arises from the imbalanced kinetic term, which
preferentially moves electrons to one edge and holes to the other. The
resulting charge distribution, which is depicted in Fig. 1(a), exhibits an
electric _dipole_ moment, which is defined by

$P^{(1)}=\sum_{j=1}^{L}j\rho_{j},$ (2)

where $\rho_{j}$ is the deviation of the local density at site $j$,
$n_{j}=c^{\dagger}_{j}c_{j}$, from the average density per site,
$\bar{n}=N/L$. We then define the non-Hermitian skin effect as the presence of
an extensive dipole moment in the $N$-particle eigenstates with OBCs,
formalizing the previous notion of a left-right asymmetry in density.
Importantly, such a dipole moment arises both in the distribution of
eigenstates and dynamically from time evolution, as shown in Fig. 1(c). We
will see below that this definition allows for a direct generalization to
systems that conserve arbitrary $\text{U}(1)$ multipole moments.

For the Hatano-Nelson model, the relation between imbalanced charge-hopping
and dipole moment can be understood by applying an “imaginary gauge
transformation” [54, 55]:

$c_{j}^{\dagger}\mapsto R^{-1}c_{j}^{\dagger}R=e^{gj}c_{j}^{\dagger},\quad
R=e^{-gP^{(1)}},$ (3)

where $P^{(1)}$ is the dipole moment in Eq. (2). In a chain with OBCs, this is
a similarity transformation that maps $H_{\text{HN}}(g)$ to
$H_{\text{HN}}(0)$, the latter being a _Hermitian_ Hamiltonian without
imbalance. An immediate consequence of this map is that the OBC spectrum of
$H_{\text{HN}}(g)$ must be real. Furthermore, single-particle eigenstates of
the Hermitian $H_{\text{HN}}(0)$ are standing wave-like, so applying the map
in reverse yields eigenstates of the original Hamiltonian that are
exponentially localized at an edge. Indeed, a single-particle state
$\ket{\psi_{0\alpha}}=\sum_{j}\alpha_{j}c^{\dagger}_{j}\ket{0}$ is mapped back
to $\ket{\psi_{g\alpha}}=\sum_{j}e^{gj}\,\alpha_{j}c^{\dagger}_{j}\ket{0}$,
which is localized within $\sim\absolutevalue{g}^{-1}$ of the boundary as long
as $\alpha_{j}$ is not fine-tuned to counteract the exponential dependence.
Regardless of particle number, the operator $R$ tends to project onto states
that maximize or minimize the dipole operator (depending on the sign of $g$).
Hence, while a generic many-body eigenstate $\ket{\psi_{0}}$ of
$H_{\text{HN}}(0)$ does not exhibit charge imbalance, the corresponding
eigenstate of $H_{\text{HN}}(g)$, $\ket{\psi_{g}}=R^{-1}\ket{\psi_{0}}$,
necessarily has an extremal dipole moment 111The many-body eigenstates of
$H_{\text{HN}}(g)$ are Slater determinants of the single-particle states
$\ket{E(g)}=R^{-1}\ket{E(0)}$..

Although the map in (3) is applicable only to Hamiltonians having a real-
valued spectrum, we anticipate that imbalanced charge hopping also induces a
dipole moment in more general models. We can argue this heuristically using
the effective field theory approach for non-Hermitian systems developed in
Ref. 60. There, the authors couple the kinetic term of a non-Hermitian
Hamiltonian to a background $\text{U}(1)$ gauge field through a Peierls factor
$c^{\dagger}_{j}e^{iA_{j,j+1}}c_{j+1}$, and derive an effective response
action for the vector potential $A_{x}$ in the continuum limit,

$S_{E}=W(E)\int dx\,A_{x},$ (4)

where $E$ is a reference energy, and $W(E)$ is the winding of the PBC spectrum
around $E$. This effective action encodes the response of quasiparticle modes
of energy $E$, and though $W(E)$ depends on the underlying Hamiltonian, the
form of the action is fixed by $\text{U}(1)$ symmetry. The main observable
associated with Eq. (4) is a persistent unidirectional current:
$j^{x}(E)\equiv\delta S_{E}[A_{x}]/\delta A_{x}=W(E)$ 222With open boundaries,
the effective action must include boundary degrees of freedom to preserve
gauge invariance. Physically, these are external sources and sinks of charge
that allow a steady-state current to exist.. Expressing the current in terms
of the total dipole moment, $j^{x}=\partial_{t}P^{(1)}$, it follows that the
physical response of non-Hermitian $\text{U}(1)$-conserving system is the
generation of a dipole moment when $W(E)$ is non-vanishing.

Figure 2: (a) Many body energy spectrum of $H_{\text{dip}}$ at half-filling
and with $P^{(1)}=0$, for system size $L=12$ and $g_{1}=2g_{2}/3\equiv
g=-1,t=0.6$. For OBCs (red dots), the spectrum is real, while for PBCs (empty
dots) the spectrum is complex and winds as the boundary conditions are twisted
(blue lines). (b) Quadrupole moment, $P^{(2)}$ in Eq. (6), of the right
eigenstates of the open chain with (red) and without (grey) imbalance. All
eigenstates with $g\neq 0$ exhibit close to maximal quadrupole moment (dashed
line), a signature of the dipole skin effect. (c) Local charge density and (d)
local dipole density of 30 mid-spectrum states for $L=20$ and OBCs. The charge
distribution exhibits a quadrupole moment as opposite dipoles are piled up at
opposite edges. The distribution of some eigenstates of the analogous
Hermitian model is shown in black dashed lines for comparison.

Multipole skin effect.— Having defined the charge skin effect in terms of the
dipole operator, we now turn to the higher multipole case. In particular, we
will show that the dipole-conserving skin effect is associated with a
_quadrupole moment_ in the charge distribution. This generalizes to a
hierarchical structure in which the $m$-pole skin effect is defined through
the $(m+1)$-pole moment. To demonstrate, we begin by considering an
analytically tractable model that conserves both $\text{U}(1)$ charge
$N=\sum_{j}n_{j}$ and dipole moment $P^{(1)}$ 333The dipole moment is only
conserved modulo the system size $L$ in periodic boundary conditions.,

$H_{4}(g)=-\sum_{j}\left(e^{g}\,c^{\dagger}_{j}c_{j+1}c_{j+2}c^{\dagger}_{j+3}+e^{-g}\,c_{j}c^{\dagger}_{j+1}c^{\dagger}_{j+2}c_{j+3}\right).$
(5)

This Hamiltonian generalizes the Hatano-Nelson model to imbalanced dipole-
hopping terms, where $d^{\dagger}_{j}\equiv c^{\dagger}_{j}c_{j+1}$ creates a
local dipole at site $j$ (see Fig. 1(b)). Like the Hatano-Nelson model, the
many-body spectrum of $H_{4}(g)$ is complex for PBCs and real for OBCs.

Taking $L$ to be divisible by four for simplicity, we consider the model at
half-filling $N=L/2$ and fixed dipole moment $P^{(1)}=0$, i.e., the largest
symmetry sector. Because $H_{4}$ exhibits strong Hilbert space fragmentation,
this symmetry sector itself splits into exponentially many disconnected Krylov
sub-sectors [47, 63]. The largest of these sub-sectors, which is still
exponentially small in $L$, is integrable and can be mapped to a free-fermion
chain of length $L/2$ [63]. The mapping proceeds by grouping sites $2j-1$ and
$2j$ of the original chain into a new composite site, which hosts four degrees
of freedom associated with the occupations of the original sites:
$\ket{00},\ket{01},\ket{10},\ket{11}$. Composite sites are exactly singly-
occupied in the largest sub-sector, and the allowed states $\ket{10}$ and
$\ket{01}$ can be interpreted as a local dipole and anti-dipole, respectively.
As a result, the dipole creation and annihilation operators on odd sites,
$d^{\dagger}_{2j-1}$ and $d_{2j-1}$, act as genuine fermionic operators on
this subspace.

Only terms in Eq. (5) having odd $j$ act nontrivially in this sector, and the
resulting effective Hamiltonian can be written as
$H_{4}^{\text{sector}}=\sum_{k}(e^{g}d^{\dagger}_{k}d_{k+1}+e^{-g}d_{k}d^{\dagger}_{k+1})$,
where $k$ labels the composite sites [63]. This is just the Hatano-Nelson
model, though with the fermions interpreted as local dipoles instead of
charges. As a result, at least in this sector, the non-Hermitian skin effect
is manifest as a dipole moment of _dipoles_ , and therefore a quadrupole
moment of charges. The buildup of dipoles (anti-dipoles) at the right (left)
end of an open chain is depicted in Fig. 1(b), and generates an extensive
quadrupole moment:

$P^{(2)}=\sum_{j=1}^{L}j^{2}\rho_{j}.$ (6)

In contrast to the ordinary skin effect, there is no asymmetry in the charge
distribution for eigenstates with OBCs, and particles are localized at both
boundaries.

A few comments about the generality of this model are in order. Although this
quadrupolar response is particularly clear in the example above, the mapping
between $H_{4}(g)$ and a free-fermion chain holds only in a few integrable
subsectors. Furthermore, the strong Hilbert space fragmentation precludes
ergodicity, and thus makes it difficult to observe the skin effect
dynamically. Finally, because the dipole operator $P^{(1)}$ does not commute
with translations [64], local dipoles themselves are not always well-defined.
For example, a sequence of alternating positive and negative charges can be
viewed as a chain of either local dipoles _or_ anti-dipoles.

To circumvent these restrictions, we consider a more general dipole-conserving
Hamiltonian:

$\displaystyle H_{\text{dip}}(g_{1},g_{2})$
$\displaystyle=-\sum_{j}\left(h^{(2)}_{j}(g_{1})+t\,h^{(3)}_{j}(g_{2})\right),$
(7) $\displaystyle h^{(k)}(g)$
$\displaystyle=e^{g}c^{\dagger}_{j}c_{j+1}c_{j+k}c^{\dagger}_{j+k+1}+e^{-g}\times\text{h.c.},$

which reduces to Eq. (5) for $t=0$, but otherwise contains longer range
dipole-hopping terms that prevent strong Hilbert space fragmentation [47].
Once again, we fix $N=L/2$ and $P^{(1)}=0$. The resulting PBC spectrum winds
around the complex plane when the boundary conditions are twisted, as shown
via exact diagonalization in Fig. 2(a).

Many-body spectral windings generically diverge in the thermodynamic limit
[41], so we focus on OBCs. The model in Eq. (7) is not integrable and cannot
be mapped to a free-fermion Hamiltonian. Nevertheless, for the particular
choice of parameters $g_{2}=3g_{1}/2$, there exists a similarity
transformation analogous to Eq. (3) that maps $H_{\text{dip}}(g,3g/2)$ to the
Hermitian Hamiltonian $H_{\text{dip}}(0,0)$:

$c_{j}^{\dagger}\mapsto
R^{-1}c_{j}^{\dagger}R=e^{gj^{2}/4}c_{j}^{\dagger},\quad R=e^{-gP^{(2)}/4}.$
(8)

The spectrum of $H_{\text{dip}}(g,3g/2)$ is therefore real, and the
eigenstates of the non-Hermitian model are related to the Hermitian
eigenstates by $R^{-1}$, which manifestly projects onto states with large
positive or negative quadrupole moment $P^{(2)}$. We claim that this is a
fundamental signature of the “dipole skin effect”, which, as we have seen,
arises in generic dipole-conserving Hamiltonians with non-reciprocal hoppings.
Motivated by the charge- and dipole-conserving cases, we can generalize the
definition to systems that conserve up to $m$th multipole moments: the non-
Hermitian $m$-pole skin effect is marked by the generation of an extensive
$(m+1)$-pole moment in the open-boundary eigenstates 444Multipole moments are
origin-dependent when lower, conserved moments do not vanish, but for any
choice of origin, the $m$-pole skin effect always tends to extremize the
$(m+1)$-pole moment..

We confirm the dipole skin effect through exact diagonalization of
$H_{\text{dip}}(g_{1},g_{2})$: the real OBC spectrum is shown in Fig. 2(a) for
$L=12$, while the quadrupole moments of the eigenstates plotted in Fig. 2(b)
are near-maximal. The charge density profiles of a few mid-spectrum
eigenstates for $L=20$ are shown in Fig. 2(c): here the quadrupole moment is
manifest as a buildup of positive charge at the system boundaries and negative
charge in the middle. Following Refs. [66, 67], we can also define a bounded
local dipole density $\rho^{\text{dip}}_{j}=-\sum_{k\leq j}\rho_{k}$ 555This
comes from expressing the dipole operator as
$P^{(1)}=-\sum_{j=1}^{L}(L-j)\rho_{j}$ (since $\sum_{j}\rho_{j}=0$ by
definition) and then rewriting the sum as
$-\sum_{j=1}^{L-1}\sum_{k=1}^{j}\rho_{j}$, which we plot for the same
eigenstates in Fig. 2(d). The imbalance in dipole hopping yields a dipole
moment of dipoles, i.e., a quadrupole moment of the charge, which should be
contrasted with the Hermitian case, where $\rho_{j}$ and
$\rho^{\text{dip}}_{j}$ are uniformly distributed across the sample (dashed
lines).

The similarity transformation in (8) helps prove the existence of the skin
effect, but generic non-reciprocal Hamiltonians with complex OBC spectra
exhibit the same phenomenology. Indeed, we have checked numerically that our
results are qualitatively unchanged for arbitrary values of $g_{1}$ and
$g_{2}$. This stability can be justified by an effective field theory that we
propose for dipolar response. The background gauge field for dipole symmetry,
$A_{xx}$, is a (one-component) rank-2 tensor and gives the Peierls phase
associated with a dipole-hopping process [69]. By analogy with (4) we expect a
response action of $S_{E}=\int dxj^{xx}A_{xx}$ for generic dipole-conserving
non-reciprocal systems, whose primary observable is a dipole current
$j^{xx}=\partial_{t}P^{(2)}$ that generates a quadrupole moment in open
boundaries [70].

Figure 3: Dynamical probes of the late-time steady-state after evolution with
a charge-conserving (inset: dipole-conserving) non-Hermitian Hamiltonian with
non-reciprocal hopping imbalance $g$. (a) The half-chain von Neumann
entanglement entropy exhibits area-law scaling. (b) The inverse participation
ratio converges with system size, indicating Fock space localization. Both
quantities are averaged over random initial product states.

Dynamical signatures.—Having defined and examined the skin effect in systems
that conserve multipole moments, we now focus on its implications for
dynamics. Dynamical probes are especially important because they can be
implemented in experimental platforms to characterize the skin effect.
Concretely, we will study the state of a system at time $t$ undergoing
dynamics modeled by a non-Hermitian Hamiltonian $H$:

$|\psi(t)\rangle=\frac{e^{-itH}|\psi(0)\rangle}{\|e^{-itH}|\psi(0)\rangle\|}.$
(9)

We have already discussed the dynamics of charge: the $m$-pole skin effect
causes charged particles to flow and generate an $(m+1)$-pole moment. Now, we
turn to the evolution of quantum correlations, focusing on the entanglement
scaling. Entanglement is quantified by the von Neumann entropy
$S(t)=-\text{Tr}[\rho_{L/2}\log{\rho_{L/2}}]$, where $\rho_{L/2}$ is the
bipartite reduced density matrix of the evolved state $|\psi(t)\rangle$ for a
spatial cut the divides the system in half. Generic closed quantum systems,
such as $H_{\text{dip}}(0,0)$, thermalize under their own time evolution. It
follows that generic states evolve to have volume-law entanglement,
$S\sim{O}(L)$. This behavior should be distinguished from area-law
entanglement, $S\sim O(1)$, which is typically found only for ground states.

In addition to entanglement, we also investigate the spread of
$|\psi(t)\rangle$ in Fock space through the inverse participation ratio,
$IPR_{2}(t)=\sum_{\overline{n}}|\langle\overline{n}|\psi(t)\rangle|^{4}$,
where $\\{|\overline{n}\rangle\\}$ is the many-body computational basis. The
evolved state spans the entire Hilbert space if $IPR_{2}\sim 2^{-D_{2}L}$ with
fractal dimension $D_{2}=1$, and is localized if $D_{2}=0$. Let us consider a
generic $m$-pole conserving Hamiltonian having asymmetric hopping such that a
similarity transformation $R^{-1}H^{m}(g)R=H^{m}(0)$ exists, with
$R=e^{-\beta(g)P^{(m+1)}}=e^{-\beta(g)\sum_{j}j^{m+1}\rho_{j}}$. Starting from
a random product state in the computational basis,
$\ket{\psi(0)}=\prod_{\\{s_{j}\\}}c_{s_{j}}^{\dagger}\ket{0}$, we can use the
transformation to write

$IPR_{2}(t)=\frac{\sum_{\underline{n}}e^{-4\beta\sum_{j}j^{m+1}n_{j}}|\langle\underline{n}|e^{-itH^{(m)}(0)}|\psi(0)\rangle|^{4}}{(\sum_{\underline{n}}e^{-2\beta\sum_{j}j^{m+1}n_{j}}|\langle\underline{n}|e^{-itH^{(m)}(0)}|\psi(0)\rangle|^{2})^{2}}.$
(10)

Assuming that the corresponding Hermitian evolution reaches a generic state,
e.g., via thermalization, such that
$|\langle\underline{n}|e^{-itH^{(m)}(0)}|\psi(0)\rangle|^{2}\sim 2^{-L}$ for
large $t$, we obtain

$|{IPR_{2}(t\rightarrow\infty)}|\approx\prod_{j=1}^{L}\frac{(1+e^{-4\beta
j^{m+1}})}{(1+e^{-2\beta j^{m+1}})^{2}}.$ (11)

This expression is bounded by
$|\log{IPR_{2}}|\leq\sum_{j=1}^{L}\left|\log{\left(1-\frac{2e^{-2\beta
j^{m+1}}}{(1+e^{-2\beta j^{m+1}})^{2}}\right)}\right|$, which converges as
$L\rightarrow\infty$, implying localization in Fock space ($D_{2}\rightarrow
0$). This should be viewed as a generalization of the edge localization that
results from the single-particle skin effect. Although fermion repulsion
prevents many-body states from localizing at the physical boundary, the
$m$-pole skin effect extremizes the $(m+1)$-pole moment of any initial state,
driving time evolution to a highly restricted region of the Fock space.
Finally, localization in Fock space implies that the entanglement entropy of
the steady-state has area-law scaling [71].

We test our conjecture by performing exact numerical simulations of charge-
and dipole-conserving time evolution. Fig. 3 (a-b) shows
$S(t\rightarrow\infty)$ and $IPR_{2}(t\rightarrow\infty)$ for the $U(1)$
charge skin effect using $H_{\text{HN}}(g)$ in Eq. 1 (for various system sizes
as a function of the non-reciprocity $g$). In the insets of Fig. 3 (a-b), we
consider the case of the dipole skin effect with $H_{4}(g)$ in Eq. 5. In
agreement with our analytical analysis, we find that the final states exhibit
area-law scaling for the entanglement entropy and are localized in Fock space.

Conclusions— Nonreciprocal hopping leads to the localization of charge at one
boundary of a 1D chain. On the other hand, $m$-pole conservation imposes
strong constraints on charge transport. The competition between these two
effects gives rise to a new class of non-Hermitian skin effects in
intrinsically interacting systems. In this work, we studied these generalized
skin effects using lattice models, field theory arguments, and numerics. Our
central result, that eigenstates exhibiting the $m$-pole skin effect have an
$(m+1)$-pole moment, naturally implies a definition of the ordinary skin
effect beyond the non-interacting regime.

Furthermore, this definition provides an experimentally accessible signature
of the $m$-pole skin effect, which may then be observed in ultra-cold atom
platforms. For example, non-reciprocal charge hopping can be engineered by
coupling an optical lattice to a collective one-body loss term induced by
dissipation [72]. Placing this lattice in a strongly tilted external field
then effectively imposes dipole-conservation [73, 74, 75], and the ensuing
dynamics is governed by the dipole skin effect. Looking beyond local densities
of conserved charges, we have also provided a truly quantum characterization
of the skin effect, showing that it results in area-law scaling of the
entanglement entropy in long-time steady-states.

We have seen an intimate connection between the symmetries of a system and its
non-Hermitian dynamics. To this end, it would be interesting to consider
models with other types of symmetries, like higher-form or subsystem
symmetries, as well as patterns of symmetry-breaking. It may also be possible
to extend the $m$-pole skin effect to higher dimensions and connect it to
many-body topological invariants, both fruitful directions for future
research.

G. D. T. acknowledges the support from the EPiQS Program of the Gordon and
Betty Moore Foundation. T.L.H. and J. G. acknowledge support from the U.S.
Office of Naval Research (ONR) Multidisciplinary University Research
Initiative (MURI) under Grant No. N00014-20-1-2325 on Robust Photonic
Materials with Higher-Order Topological Protection.

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###### Abstract

Magnetic helicity plays a tremendously important role when it is different
from zero on average. Most notably, it leads to the phenomenon of an inverse
cascade. Here, we consider decaying magnetohydrodynamic (MHD) turbulence as
well as some less common examples of magnetic evolution under the Hall effect
and ambipolar diffusion, as well as cases in which the magnetic field
evolution is constrained by the presence of an asymmetry in the number density
of chiral fermions, whose spin is systematically either aligned or anti-
aligned with its momentum. In all those cases, there is a new conserved
quantity: the Hosking integral. We present quantitative scaling results for
the magnetic integral scale as well as the magnetic energy density and its
spectrum. We also compare with cases were a magnetic version of the Saffman
integral is initially finite. Rotation in MHD turbulence tends to suppress
nonlinearity and thereby also inverse cascading. Finally, the role of the
Hosking and magnetic Saffman integrals in shell models of turbulence is
examined.

###### keywords:

decaying turbulence; MHD turbulence; chiral magnetic effect; Hall effect;
shell models

1 1 0 Academic Editor: Boris Galperin, Annick Pouquet, and Peter Sullivan
https://doi.org/ Turbulence with Magnetic Helicity that is Absent on Average
Turbulence with Magnetic Helicity that is Absent on Average Axel Brandenburg
1,2,3,4,†,‡, and Gustav Larsson 1,2 Axel Brandenburg and Gustav Larsson
Brandenburg, A.; Larsson, G. Correspondence<EMAIL_ADDRESS>
##  1 Introduction

This paper is part of a special issue commemorating the work of Jack Herring.
His scientific career started off with papers on the effect of the solar wind
on the lunar atmosphere in 1959 Herring and Licht (1959). In 1961, he extended
this work to exoplanet atmospheres Herring and Kyle (1961). He also worked on
stellar opacities Arking and Herring (1963). In all those cases, he was very
much ahead of its time. At the time of Parker’s prize-winning paper on the
discovery of non-static solutions (Parker, 1958), the physical reality and
properties of the solar wind were still rather unclear and under-appreciated.
Likewise, Herring’s work on stellar opacities was well before proper numerical
stellar structure and evolution models became available; the Henyey method
Henyey et al. (1964) (solving a matrix equation instead of using an iterative
shooting method from both ends) became known only in 1964. Subsequently,
Herring turned to hydrodynamic convection and turbulence – topics that then
determined much of his future work. During his career, he never really worked
on magnetic fields or helicity, but he did interact with people on a daily
basis, who were very much involved in these subjects, both early on Pouquet et
al. (1976) and also later during his career Pouquet et al. (2019). It is
therefore not surprising that this special issue also extends to topics
involving magnetic fields and helicity.

Having helicity in a system usually requires external factors such as
stratification and rotation (Moffatt, 1978; Parker, 1979; Krause and Rädler,
1980). In this sense, the absence of helicity may be regarded as the more
generic situation. It may therefore also seem natural that helicity does not
play an important role when it is absent on average. This is believed to be
the case in hydrodynamic turbulence, but it changes when magnetic fields are
involved. Although both kinetic and magnetic helicities are ideal invariants,
only the magnetic helicity has a non-ideal dissipation that is slower than
that of the magnetic energy. By contrast, the dissipation of kinetic helicity
is faster than that of kinetic energy Matthaeus and Goldstein (1982);
Brandenburg and Subramanian (2005). Therefore, in the magnetic case, helicity
plays a very important role in a way that is unknown in the hydrodynamic
context. But is this still true when the net magnetic helicity is actually
zero?

The physical situations of interest here include the decay of primordial
magnetic fields in the early Universe during the radiation-dominated era, when
the electric conductivity is high and the initially generated magnetic field
can only decay. When the plasma is hot enough, the chirality of fermions also
plays an important role, leading to an interplay with magnetic helicity.
Another situation of interest is when only the Hall effect plays a role, so
there are then no fluid motions, but just the flow of electrons. This is
relevant in neutron star crusts, which are solid, so the ions are immobile.
Another application is to the solar wind on scales below the proton gyroscale,
where the ions create a smooth motionless background. The induction equation
with just the Hall effect included leads to interesting decay
dynamics—remarkably similar to ordinary magnetohydrodynamics (MHD). In all
those cases, magnetic helicity can play a role even when it vanishes on
average. In those cases, magnetic helicity fluctuations may be responsible for
driving an inverse cascade similar to the case of nonvanishing mean magnetic
helicity.

Less obvious examples of the dynamics discussed above include the Sun, because
here the magnetic helicity is usually nonvanishing on average Rust and Kumar
(1996). Even in the solar wind, where the conditions resemble those of
decaying turbulence, the magnetic helicity is observed to be nonvanishing on
average and systematically of opposite signs in the northern and southern
hemispheres Brandenburg et al. (2011). Near the ecliptic, however, the
magnetic helicity fluctuates around zero Matthaeus and Goldstein (1982) and
may also be in a state of decay, so this may be another example where magnetic
helicity fluctuations play an important role.

##  2 Nonhelical Turbulence and the Hosking Integral

In this section, we discuss the Hosking integral and why it is crucial to
understanding nonhelical MHD turbulence with strong magnetic fields. Unlike
the case of weak magnetic fields, when the dynamics is still controlled by the
presence of hydrodynamic effects, we are dealing here with effects that are
specific to the presence of magnetic fields, albeit with zero average. We
focus on decaying turbulence.

###  2.1 Nonhelical Inverse Cascading and Scaling Relations

Already in 2001, it was noted that, even in the nonhelical case of a
turbulently decaying magnetic field, there is a small amount of inverse
cascading in the sense that for wavenumbers below the peak, the magnetic
energy spectrum rises with time uniformly for all lower $k$ Christensson et
al. (2001). The actual amount of this rise was small and one could have argued
that it was just because of numerical inaccuracies. Subsequent simulations
Kahniashvili et al. (2013), however, confirmed such inverse cascading and
those authors discussed the potential interplay between the shallower kinetic
energy spectrum proportional to $k^{2}$ and the steeper magnetic energy
spectrum proportional to $k^{4}$. The qualitative idea was that the shallower
velocity spectrum pushes the magnetic spectrum upward, which then would drive
more kinetic energy at small $k$, and so forth.

The choice of the initial magnetic energy being proportional to $k^{4}$ is
important here. When such a spectrum was used in the first numerical
simulations Christensson et al. (2001), the authors made reference to the
early work in Ref. Durrer et al. (1998), where causality arguments were put
forward. Nowadays, however, Ref. Durrer and Caprini (2003) has become the
standard reference for the choice of an initial $k^{4}$ spectrum. Later, it
turned out that with a shallower initial $k^{2}$ spectrum, no inverse
cascading can be found Reppin and Banerjee (2017); Brandenburg et al. (2017).
The reason for this particular aspect will be discussed in more detail in this
paper.

In 2014, the idea of an inverse cascade in the nonhelical case with a $k^{4}$
spectrum became really very clear Brandenburg et al. (2015). This paper was on
the arXiv since April 2014, but the paper was published only in February 2015.
The results were reproduced in the relativistic context in Ref. Zrake (2014).
Their work was on the arXiv since July 2014 and made reference to the 2015
paper. The significance of both findings is that it presents early support for
the subsequent discovery of the Hosking integral as a new invariant in MHD
turbulence at large magnetic Reynolds numbers.

When the Hosking integral was discovered in Ref. Hosking and Schekochihin
(2021), it was originally called the “Saffman helicity invariant”. As already
pointed out in Ref. Zhou et al. (2022), H. K. Moffatt informed the community
of the fact that this term may be misleading, because the term ‘helicity
invariant’ is reserved for integrals that are chiral in character. He also
recalled that Saffman never considered helicity in his papers. The term
“magnetic helicity density correlation integral” may be more appropriate, but
it is rather clumsy. Following Schekochihin (2022), where this quantity was
called the Hosking integral, this term continued being used by others Uchida
et al. (2022); Sharma and Brandenburg (2022). It should also be noted that
‘integral’ instead of ‘invariant’ is appropriate since applications to
turbulence apply always to finite Reynolds and Lundquist numbers. In this
connection, it should be emphasized that the Hosking integral tends to decay
with time in a power-law fashion and that the exponent decreases with
increasing Lundquist number Lu approximately as $\mbox{\rm Lu}^{-1/4}$ Zhou et
al. (2022).

The energy decay in turbulence is usually characterized by the energy spectrum
$E(k,t)$. In the following, we sometimes add the subscripts K and M for
kinetic and magnetic energy spectra and other quantities. We focus here on
magnetic energy spectra, $E_{\rm M}(k,t)$, which are defined such that $\int
E_{\rm M}(k,t)\,{\rm d}{}k=\langle\bm{B}^{2}\rangle/2\mu_{0}\equiv{\cal
E}_{\rm M}(t)$ is the magnetic energy, and $\mu_{0}$ is the vacuum
permeability. The decay can then be parameterized by ${\cal E}_{\rm M}(t)$ and
the magnetic integral scale, which is defined in terms of the magnetic energy
spectrum as

$\xi_{\rm M}(t)=\left.\int_{0}^{\infty}k^{-1}E_{\rm
M}(k,t)\,dk\right/\int_{0}^{\infty}E_{\rm M}(k,t)\,dk.$ (1)

One can always attempt to describe the relations for $\xi_{\rm M}(t)$ and
${\cal E}_{\rm M}(t)$ through power laws. In addition, the spectrum can evolve
underneath an envelope,

$E_{\rm M}(k,t)\leq{\rm const}{}\times k^{\beta},$ (2)

which is in general different from the initial subinertial range spectrum,
$E_{\rm M}(k,t_{0})={\rm const}{}\times k^{\alpha}$, where $\alpha$ is the
subinertial range slope. The three relations for $\xi_{\rm M}(t)$, ${\cal
E}_{\rm M}(t)$, and $E_{\rm M}(k,t)$ can then be constrained through
dimensional arguments once we have a good idea about the relevant dimensional
quantity that governs the decay.

In 2017, the decay of a nonhelical turbulent magnetic field is found to be
described by an exponent $\beta$ that was determined to be between $\beta=1$
Brandenburg and Kahniashvili (2017) and $\beta=2$ Brandenburg et al. (2017),
but it was unclear why any of those two possibilities, or any other one, would
have to be expected. This is what the Hosking integral now explains, namely
that $\beta=3/2$.

Figure 1a shows magnetic energy spectra at four different times for a
nonhelical magnetically dominated run corresponding to Run K60D1bc in Ref.
Zhou et al. (2022). Here, $k$ is normalized by the initial peak wavenumber
$k_{0}$. We clearly see that the spectrum exhibits inverse cascading in that
the spectral magnetic energy increases with time at small $k$, as indicated by
the upward arrow on the left. The overall energy does of course decay, as
indicated by the decline of the spectral peak and the decrease of spectral
energy at large $k$, as indicated by the downward arrow on the right.

To quantify the temporal changes of $\xi_{\rm M}(t)$ and ${\cal E}_{\rm
M}(t)$, it is convenient to compute the instantaneous scaling exponents
Brandenburg and Kahniashvili (2017)

$q(t)={\rm d}{}\ln\xi_{\rm M}/{\rm d}{}\ln t\quad\mbox{and}\quad p(t)=-{\rm
d}{}\ln{\cal E}_{\rm M}/{\rm d}{}\ln t;$ (3)

see Figure 1b. We see that with time (larger red symbols), the solution
evolves toward the point $(q,p)=(4/9,\,10/9)$, as is also theoretically
expected Zhou et al. (2022). Although we mainly focus on the case of
nonhelical magnetic fields, we also compare in Figure 1b with the expected
solution for the fully helical case (orange), and include solutions for
hydrodynamic turbulence that are governed either by the Loitsyansky or the
Saffman integrals.

Figure 1:  (a) magnetic energy spectra, normalized by $I_{\rm
H}^{-1/2}k_{0}^{-3/2}$; the dashed dotted line shows the envelope
$0.028\,(k/k_{0})^{3/2}$ under which the spectrum evolves. The times are
$ck_{1}t=3$, 7, 17, and 58. (b) $qp$ diagram showing as red dots the
convergence of $p(t)$ versus $q(t)$ toward the Hosking attractor
$(q,p)=(4/9,\,10/9)$. The blue symbols denote the Loitsyansky and Saffman
attractors, respectively, and the orange symbol denotes the magnetic helicity
attractor.

Before we continue, it is useful to clarify the concept of what we often refer
to as a ‘governing quantity’. Take, for example, standard hydrodynamic
Kolmogorov turbulence. Here, the rate of energy transfer per unit mass
$\epsilon$ (which is the rate of energy input and also the rate of energy
dissipation) is such a quantity and the relevant physical scaling laws can be
expressed in terms of powers of $\epsilon$ and other relevant variables such
as the wavenumber $k$ itself. This then yields for the energy spectrum per
unit mass the expression $E(k)=C_{\rm K}\epsilon^{2/3}k^{-5/3}$, where $C_{\rm
K}$ is a dimensionless coefficient of order unity (the Kolmogorov constant;
typically $C_{\rm K}\approx 1.6$). Other such governing quantities include the
mean magnetic helicity density $\langle h\rangle$ and some other quantities
that are crucial to the physics. They are usually constant or well conserved.

###  2.2 The Loitsyansky and Saffman Integrals in Hydrodynamics

In the hydrodynamic case, the decay of turbulence can follow different
behaviors depending on the relevant conservation law. (In practice, conserved
quantities are usually not perfectly conserved under turbulent conditions, and
some are better conserved than others. Which one is the most relevant quantity
depends on the relative conservation properties under different
circumstances.) One such conserved quantity is the Loitsyansky integral
Davidson (2000, 2009),

$I_{\rm
L}=-\int\langle\bm{u}(\bm{x})\cdot\bm{u}(\bm{x}+\bm{r})\rangle\,r^{2}\,{\rm
d}{}^{3}\bm{r},$ (4)

which is believed to play an important role. This integral reflects the local
conservation of angular momentum and has dimensions $[I_{\rm L}]=\,{\rm
m}^{7}\,{\rm s}^{-2}$. If this quantity governs the decay of turbulence, the
time dependence of the growth of the integral scale can be motivated by
dimensional arguments as $\xi(t)\propto I_{\rm L}^{a}t^{b}$, where the
exponents $a$ and $b$ must be, on dimensional grounds, $a=1/7$ and $b\equiv
q=2/7$. The kinetic energy then obeys ${\cal E}_{\rm K}\propto I_{\rm
L}^{2/7}t^{-10/7}$, i.e., $p=10/7$. The envelope under which the peak of the
spectrum evolves obeys $E_{\rm K}(k,t)\leq C_{\rm L}I_{\rm L}k^{4}$.

Another conserved quantity is the Saffman integral,

$I_{\rm S}=\int\langle\bm{u}(\bm{x})\cdot\bm{u}(\bm{x}+\bm{r})\rangle\,{\rm
d}{}^{3}\bm{r},$ (5)

which has dimensions $[I_{\rm S}]=\,{\rm m}^{5}\,{\rm s}^{-2}$. Similarly, if
this quantity governs the decay of turbulence, the time dependence of $\xi$
must be $\xi(t)\propto I_{\rm S}^{a}t^{b}$, where $a=1/5$ and $b\equiv q=2/5$
on dimensional grounds. The kinetic energy then obeys ${\cal E}_{\rm K}\propto
I_{\rm L}^{2/5}t^{-6/5}$, i.e., $p=6/5$. The envelope under which the peak of
the spectrum evolves obeys in this case $E_{\rm K}(k,t)\leq C_{\rm S}I_{\rm
S}k^{2}$.

Whether $I_{\rm L}$ or $I_{\rm S}$ determine the decay depends on the
existence of long-range correlations, as can be seen from the Taylor expansion
of the kinetic energy spectrum as Hosking and Schekochihin (2021); Davidson
(2000)

$2E_{\rm K}(k\to 0)\equiv\mbox{\rm Sp}(\bm{u})(k\to 0)=\frac{I_{\rm
S}}{2\pi^{2}}k^{2}+\frac{I_{\rm L}}{12\pi^{2}}k^{4}+...,$ (6)

where an initially non-vanishing Saffman integral automatically implies a
$k^{2}$ scaling in the subinertial range. Thus, the decay does depend on the
infrared part of the initial kinetic energy spectrum. In that case, the slope
is the same as that required for the initial spectrum so that the Saffman
integral is indeed nonvanishing. Furthermore, as pointed out in Ref. Hosking
and Schekochihin (2021), owing to the invariance of $I_{\rm S}$ and $I_{\rm
L}$, both an initial $k^{2}$ and a $k^{4}$ spectrum will remain unchanged.
This implies that there can be no inverse cascading in hydrodynamics.

###  2.3 The Magnetic Saffman Integral: Comparison with the Hosking Integral

As already pointed out in Ref. Hosking and Schekochihin (2021), the
formulation of Section 2.2 can also be applied to the magnetic field, except
that there is no reason for the magnetic version of the Loitsyansky integral
to be conserved. The magnetic Saffman integral (hereafter $I_{\rm SM}$), on
the other hand, might indeed be conserved. Physically, it would reflect the
local conservation of magnetic flux. Again, when $I_{\rm SM}$ is non-vanishing
initially, we expect a quadratic magnetic energy spectrum, which would also
persist at later times. For a steeper $k^{4}$ subinertial range magnetic
energy spectrum, however, the magnetic Saffman integral must vanish and the
Hosking integral is then expected to play a dominant role. It is defined as

$I_{\rm H}=\int\langle h(\bm{x})h(\bm{x}+\bm{r})\rangle\,{\rm d}{}^{3}\bm{r},$
(7)

where $h=\bm{A}\cdot\bm{B}$ is the magnetic helicity density with dimensions
$[h]=[B]^{2}[x]$. In ordinary MHD, we can express the magnetic field as an
Alfvén velocity, i.e., we write the magnetic field in Alfvén units, so
$[B]=\,{\rm m}\,{\rm s}^{-1}$. Therefore, $[h]=[x]^{3}[t]^{-2}$, and thus
$[I_{\rm H}]=[B]^{4}[x]^{5}=[x]^{9}[t]^{-4}$. If $I_{\rm H}$ plays a governing
role in the decay, we expect therefore $\xi_{\rm M}(t)\propto I_{\rm
H}^{1/9}t^{4/9}$, ${\cal E}_{\rm M}\propto I_{\rm H}^{2/9}t^{-10/9}$, and
$E_{\rm M}(k,t)\leq C_{\rm H}I_{\rm H}k^{3/2}$.

The Hosking integral is in general expected to be different from zero Hosking
and Schekochihin (2021). This automatically implies a quadratic scaling of the
helicity variance spectrum, $\mbox{\rm Sp}(h)$. Here, $\mbox{\rm
Sp}(h)=\oint_{4\pi}|\tilde{h}|^{2}\,k^{2}{\rm d}{}\Omega_{k}/(2\pi L)^{3}$
denotes the shell-integrated spectrum, a tilde marks a quantity in Fourier
space, and $\Omega_{k}$ is the solid angle in Fourier space, so that
$\int\mbox{\rm Sp}(h)\,{\rm d}{}k=\langle h^{2}\rangle$. The quadratic scaling
for a finite Hosking integral follows from the expansion

$\mbox{\rm Sp}(h)(k\to 0)=\frac{I_{\rm H}}{2\pi^{2}}k^{2}+...$ (8)

In three dimension, a quadratic spectrum corresponds to white noise. We also
know that the spectrum of a quadratic quantity cannot be more blue than that
of white noise Brandenburg and Boldyrev (2020), so it seems impossible to have
a helicity variance spectrum whose subinertial range is steeper than $k^{2}$.

In Figure 2, we show magnetic energy and magnetic helicity variance spectra
for initial spectra of the form

$E_{\rm M}(k,t_{0})={\rm
const}{}\times\frac{k^{\alpha}}{1+(k/k_{0})^{\alpha+5/3}}\propto\begin{cases}k^{\alpha}&\mbox{for}\;k\ll
k_{0},\\\ k^{-5/3}&\mbox{for}\;k\gg k_{0},\end{cases}$ (9)

for $\alpha=2$ and $\alpha=4$. We solve the isothermal compressible MHD
equations using the Pencil Code Pencil Code Collaboration et al. (2021) with
$1024^{3}$ mesh points. As expected, and as pointed out previously Reppin and
Banerjee (2017); Brandenburg et al. (2017), there is inverse cascading only
for $\alpha=4$, but not for $\alpha=2$. Nevertheless, we see that $\mbox{\rm
Sp}(h)$ retains a $k^{2}$ spectrum at low wavenumbers in both cases. This
suggests that the Hosking integral is indeed always conserved; see Figure
2b,d. It may, however, be less dominant than the magnetic Saffman integral.

Figure 2:  Comparison of $\mbox{\rm Sp}(\bm{B})$ (a,c) and $\mbox{\rm Sp}(h)$
(b,d) for $\alpha=2$ (a,b) and $\alpha=4$ (c,d). The dashed-dotted lines
indicate $k^{3/2}$ scaling in (c) and $k^{2}$ scaling otherwise.

To determine the relevant integrals, $I_{\rm H}$ and $I_{\rm SM}$, it is
convenient to plot compensated spectra. Specifically, to determine $I_{\rm
SM}$ and $I_{\rm H}$, we scale both $\mbox{\rm Sp}(\bm{B})$ and $\mbox{\rm
Sp}(h)$ by $2\pi^{2}/k^{2}$. The result is shown in Figure 3. Thus, in
summary, we have

$\xi_{\rm M}(t)\approx 0.16\,I_{\rm SM}^{1/5}t^{2/5},\quad{\cal E}_{\rm
M}(t)\approx 4.2\,I_{\rm SM}^{2/5}t^{-6/5},\quad E_{\rm M}(k)\approx
0.037\,I_{\rm SM}(k/k_{0})^{2}.$ (10)

If the initial spectrum is not $\propto k^{2}$, but $\propto k^{4}$, we have

$\xi_{\rm M}(t)\approx 0.15\,I_{\rm H}^{1/9}t^{4/9},\quad{\cal E}_{\rm
M}(t)\approx 3.8\,I_{\rm H}^{2/9}t^{-10/9},\quad E_{\rm M}(k)\approx
0.025\,I_{\rm H}^{1/2}(k/k_{0})^{3/2}.$ (11)

It is remarkable that the prefactors for the Saffman and Hosking scalings are
very close to each other; see Table 2.3 for a summary of the nondimensional
prefactors in the relations

$\xi_{\rm M}(t)=C_{i}^{(\xi)}\,I_{i}^{\sigma}t^{q},\quad{\cal E}_{\rm
M}(t)=C_{i}^{({\cal E})}\,I_{i}^{2\sigma}t^{-p},\quad E_{\rm
M}(k)=C_{i}^{(E)}\,I_{i}^{(3+\beta)/\sigma}(k/k_{0})^{\beta},$ (12)

where the index $i$ on the integrals $I_{i}$ and the coefficients
$C_{i}^{(\xi)}$, $C_{i}^{({\cal E})}$, and $C_{i}^{({E})}$ stands for SM or H
for magnetic Saffman and Hosking scalings, respectively, and $\sigma$ is the
exponent with which length enters in $I_{i}$: $\sigma=5$ for the magnetic
Saffman integral ($i={\rm SM}$) and $\sigma=9$ for the Hosking integral
($i={\rm H}$). Interestingly, while $\beta$ and $p$ can uniquely be related to
$q$ via $\beta=2/q-3$ and $p=2(1-q)$ Brandenburg and Kahniashvili (2017), the
exponent $\sigma$ is not uniquely linked to $q$ and we have $\sigma q=2$ for
Saffman scaling and $\sigma q=4$ for Hosking scaling.

The value of $C_{\rm SM}^{(E)}$ only makes sense when $\alpha=\beta=2$, while
that of $C_{\rm H}^{(E)}$ only makes sense when $\alpha=4$ and $\beta=3/2$.
For the other cases, the subinertial range spectrum is not parallel to
$k^{\beta}$, so $\alpha$ and $\beta$ are said to be incompatible with each
other (see Table 2.3) and the given values of $C_{\rm SM}^{(E)}$ and $C_{\rm
H}^{(E)}$ only yield crossings in the middle of the subinertial range.

Figure 3:  Compensated spectra for $\alpha=2$ (a,c,e) and $\alpha=4$ (b,d,f).
From (a,b), the horizontal dashed-dotted lines indicate that $I_{\rm
SM}\approx 0.23$ and 0.09, respectively, and from (c,d) they indicate that
$I_{\rm H}\approx 2\times 10^{-3}$ and $5\times 10^{-4}$, respectively. The
asymptotic values estimated from (e,f) are discussed in the text and in Table
2.3.

[t] Summary of nondimensional prefactors in the relations for $\xi_{\rm
M}(t)$, ${\cal E}_{\rm M}(t)$, and $E_{\rm M}(k,t)$. The numbers in
parentheses indicate that the slope $\beta$ is incompatible with the value of
$\alpha$.  $\alpha$ $\beta$ $C_{\rm SM}^{(\xi)}$ $C_{\rm H}^{(\xi)}$ $C_{\rm
SM}^{({\cal E})}$ $C_{\rm H}^{({\cal E})}$ $C_{\rm SM}^{(E)}$ $C_{\rm
H}^{(E)}$ 2 2 0.16 0.15 4.2 3.8 0.025 (0.05) 4 3/2 0.15 0.13 4.0 3.5 (0.02)
0.037

We see from Figure 3a that for $\alpha=2$, the compensated value
$(2\pi/k^{2})\,\mbox{\rm Sp}(\bm{B})\to I_{\rm SM}\approx 0.2$. For
$\alpha=4$, on the other hand, we only see a flat envelope, i.e.,
$(2\pi/k^{2})\,\mbox{\rm Sp}(\bm{B})\leq 0.1$, i.e., $2E_{\rm M}(k,t)\leq
0.1/(2\pi^{2}/k^{2})\,(k/k_{0})^{2}$. From Figure 3c,d, we see that
$(2\pi/k^{2})\,\mbox{\rm Sp}(h)\to I_{\rm H}\approx 0.001$ in both cases,
i.e., for $\alpha=2$ and $\alpha=4$, respectively.

Given that we now know the values of $I_{\rm SM}$ and $I_{\rm H}$, we can
compensate the time evolutions of $\xi_{\rm M}(t)\propto t^{q}$ with
$q=2/5=0.4$ and $q=4/9\approx 0.44$, and those of ${\cal E}_{\rm M}(t)\propto
t^{-p}$ with $p=6/5=1.2$ and $p=10/9\approx 1.1$. The results for the
corresponding coefficients in Equation (12) are summarized in Table 2.3.

###  2.4 The Effect of Rotation

Rotation suppresses hydrodynamics turbulence. This is modelled by including
the Coriolis force, $-2\bm{\Omega}\times\bm{u}$, on the right-hand side of the
momentum equation, which then reads

$\frac{{\rm D}{}\bm{u}}{{\rm D}{}t}=-c_{\rm
s}^{2}{\bm{\nabla}}\ln\rho-2\bm{\Omega}\times\bm{u}+\frac{1}{\rho}\left[\bm{J}\times\bm{B}+{\bm{\nabla}}\cdot(2\rho\nu\mbox{\boldmath${\sf
S}$}{})\right],$ (13)

where ${\rm D}{}/{\rm D}{}t=\partial/\partial t+\bm{u}\cdot{\bm{\nabla}}$ is
the advective derivative, $c_{\rm s}$ is the isothermal sound speed,
$\bm{\Omega}$ is the angular velocity,
$\bm{J}={\bm{\nabla}}\times\bm{B}/\mu_{0}$ is the current density, $\mu_{0}$
is the permeability, $\rho$ is the density, $\nu$ is the viscosity, and ${\sf
S}_{ij}=(\partial_{i}u_{j}+\partial_{j}u_{i})/2-\delta_{ij}{\bm{\nabla}}\cdot\bm{u}/3$
are the components of the rate-of-strain tensor.

In Figure 4, we show $\mbox{\rm Sp}(\bm{B})$ and $\mbox{\rm Sp}(h)$ for
$\Omega/c_{\rm s}k_{0}=10^{-3}$, 0.01, 0.1, and $1$ for runs with $\alpha=4$,
which are otherwise the same as that of Figure 2c,d. We see a clear
suppression of inverse cascading already for $\Omega/c_{\rm s}k_{0}=10^{-3}$
and a very strong suppression when this parameter is unity. This is caused by
the suppression of the turbulent velocity and thereby of the
$\bm{u}\times\bm{B}$ nonlinearity in the induction equation.

To express the angular velocity in a physically more meaningful way, we note
that in the run with $\Omega/c_{\rm s}k_{0}=1$, the rms Mach number, $u_{\rm
rms}/c_{\rm s}$, drops below $10^{-3}$ by the end of the run, while for
$\Omega/c_{\rm s}k_{0}=10^{-3}$, it still stays well above $10^{-3}$. This
means that in the latter, the Coriolis number, $\mbox{\rm Co}\equiv
2\Omega/u_{\rm rms}k_{0}$, is around unity when rotational suppression becomes
appreciable. At $t=10^{4}$, the values of Co are 1.2, 22, 500, and $10^{4}$
for our four runs in Figure 4.

Figure 4:  $\mbox{\rm Sp}(\bm{B})$ (a,c,e,g) and $\mbox{\rm Sp}(h)$ (b,d,f,h)
for $\Omega/c_{\rm s}k_{0}$ from $10^{-3}$ (a,b: $\mbox{\rm Co}=1.2$) to
$\Omega/c_{\rm s}k_{0}=1$ (g,h: $\mbox{\rm Co}=10^{4}$). The times are 220
(black), 1000 (blue), 4600 (orange), and 22,000 (red).

##  3 Extensions of the Hosking Idea

Equation (7) is the Hosking integral in its original form. In the meantime,
two further variants of $I_{\rm H}$ have been considered. One is where $h$ has
been replaced by $h_{\rm tot}=\bm{A}\cdot\bm{B}+2\mu_{5}/\lambda$, where
$\mu_{5}$ is the chiral chemical potential (here in units of an inverse
length) and $\lambda$ is a coefficient that quantifies the coupling between
fermions and electromagnetic fields. The case $\langle h_{\rm tot}\rangle=0$
has been studied recently in Ref. Brandenburg et al. (2023). Another variant
of the Hosking integral is that in the case where the magnetic field is
controlled by the electromagnetic induction from the Hall effect, which we
discuss next.

###  3.1 Hall Effect

In neutron star crusts, the ions are immobile and the current is only carried
by electrons with the velocity $\bm{u}_{\rm e}=-\bm{J}/en_{\rm e}$, where $e$
is the electric charge and $n_{\rm e}$ is the electron density. As alluded to
in the introduction, a similar situation occurs in the solar wind on scales
below the proton gyroscale, where the ions constitute a smooth background Cho
(2011). The induction equation with the induction from $\bm{u}_{\rm
e}\times\bm{B}$ therefore takes the form Goldreich and Reisenegger (1992)

$\frac{\partial\bm{B}}{\partial t}={\bm{\nabla}}\times\left(-\frac{1}{en_{\rm
e}}\bm{J}\times\bm{B}-\eta\mu_{0}\bm{J}\right),$ (14)

where $\eta$ is the magnetic field diffusivity. In the presence of magnetic
helicity, one finds an inverse cascade with an overall decay of the magnetic
field and a growth of spectral magnetic energy at small wavenumbers below that
of the peak of the spectrum Cho (2011). In the present paper, however, the
focus is on the nonhelical case, which was already considered in B20, but
understood mathematically only later Brandenburg (2023).

In this context, it is important to note that the natural dimensions of the
magnetic field here are no longer $\,{\rm m}\,{\rm s}^{-1}$, but $\,{\rm
m}^{2}\,{\rm s}^{-1}$. This was already emphasized in Ref. Brandenburg (2020),
who used $e=1.6\times 10^{-19}\,{\rm A}\,{\rm s}$, $\mu_{0}=4\pi\times
10^{-7}\,{\rm T}\,{\rm m}\,{\rm A}^{-1}$, and $n_{\rm e}\approx 2.5\times
10^{40}\,{\rm m}^{-3}$ for neutron star crusts, so we have $en_{\rm
e}\mu_{0}\approx 5\times 10^{15}\,{\rm T}\,{\rm s}\,{\rm m}^{-2}$, and
therefore

$\frac{B}{en_{\rm e}\mu_{0}}=\frac{B}{5\times 10^{15}\,{\rm T}}\,\frac{\,{\rm
m}^{2}}{\,{\rm s}},$ (15)

which is why we say $B$ has dimensions of $\,{\rm m}^{2}\,{\rm s}^{-1}$ in the
Hall cascade. This modifies all the dimensional arguments related to $\bm{B}$
correspondingly. In particular, the units of the magnetic helicity are
$[h]=\,{\rm m}^{5}\,{\rm s}^{-2}$ and those of energy spectra are also $\,{\rm
m}^{5}\,{\rm s}^{-2}$. Therefore, one has $q=p=2/5$. This scaling was
confirmed in Ref. Brandenburg (2020).

In the nonhelical case, the modified Hosking integral has dimensions
$m^{13}\,{\rm s}^{-4}$, and therefore $q=4/13$ (instead of $4/9$ in MHD).
Furthermore, $p=10/13$ (instead of $10/9$ in MHD), but still $\beta=3/2$ (just
like in MHD). While such a scaling was already seen in the original
simulations of Ref. Brandenburg (2020), the work in Ref. Brandenburg (2023)
showed that the modified Hosking integral is indeed conserved. In Figure
5a,b,d,e, we show that, also for Hall dynamics, the Saffman scaling is obeyed
for $\alpha=2$, while Hosking scaling is obeyed for $\alpha=4$.

It should be noted that earlier work on the Hall cascade focussed on the
concept of Whistler turbulence Gary et al. (2008), where the Whistler time
$t_{\rm w}$ was identified as the governing timescale Cho (2011). The
definition of $t_{\rm w}$ in Ref. Cho (2011) involved the electron plasma
frequency and the electron gyrofrequency such that the electron mass drops
out. Therefore, $t_{\rm w}$ can more easily be written as

$t_{\rm w}=\frac{L^{2}}{B_{\rm rms}/en_{\rm e}\mu_{0}},$ (16)

which is just the characteristic time based on the magnetic field expressed in
units of a diffusivity; see Equation (15). Earlier interpretations in terms of
Whistler waves Cho (2011) seem therefore artificial and obscured the relevant
interpretation of the magnetic field as a quantity with dimensions of $\,{\rm
m}^{2}\,{\rm s}^{-1}$.

Figure 5:  $\mbox{\rm Sp}(\bm{B})$ (a–c) and $\mbox{\rm Sp}(h)$ (d–f) for Hall
dynamics with $\alpha=2$ (a,d) and $\alpha=4$ (b,e), and for ambipolar
diffusion with $\alpha=4$ (c,f). Note the presence of inverse cascading for
$\alpha=4$ in panels (b,c), although $\mbox{\rm Sp}(h)$ changes at $k/k_{0}\ll
1$ in all cases. The straight lines indicate $k^{3/2}$ scaling in (b) and
$k^{2}$ scaling otherwise.

Observationally, the inverse cascade with helicity may be more easily
accessible, and this has already been tried in the context of neutron stars
Sarin et al. (2023). Measuring inverse cascading in the solar wind is
conceptually harder, because the system is statistically steady and more
advanced stages of decay would only correspond to larger distances from the
Sun. On the other hand, most of the spacecrafts are located near the ecliptic
and might therefore allow better insight into nonhelical inverse cascading.
Unfortunately, most of the observations focus on the high frequency range of
the spectrum Smith et al. (2006), while the low frequency range is dominated
by noise, making it nearly impossible to say anything about inverse cascading
to even larger scales.

###  3.2 Ambipolar Diffusion

The Hall effect is a two-fluid effect where the two components are the
positive and negative charge carriers. Another two-fluid effect is ambipolar
diffusion where the charged fluid with positive and negative charge carriers
is taken as one component and neutrals are taken as the other component. The
governing equation is

$\frac{\partial\bm{B}}{\partial
t}={\bm{\nabla}}\times\left(-\frac{\bm{J}\times\bm{B}}{\rho_{\rm i}\nu_{\rm
in}}\times\bm{B}-\eta\mu_{0}\bm{J}\right),$ (17)

where $\rho_{\rm i}$ is the ion density and $\nu_{\rm in}$ is the ion–neutral
collision frequency.

Unlike the Hall effect in neutron star crusts, where the magnetic field is
said to have dimensions of $\,{\rm m}^{2}\,{\rm s}^{-1}$, we can here write

$\frac{B}{\sqrt{\rho_{\rm i}\mu_{0}}}=\frac{B}{1.5\times 10^{-16}\,{\rm
T}}\,\frac{\,{\rm m}}{\,{\rm s}},$ (18)

where we used $\rho_{\rm i}=1.7\times 10^{-26}\,{\rm kg}\,{\rm m}^{-3}$ for
the interstellar medium with an ionization fraction of $10^{-5}$ and a neutral
density of one proton per cubic centimeter. This is why we say that with
ambipolar diffusion, just like in MHD, $B$ has dimensions of $\,{\rm m}\,{\rm
s}^{-1}$. For this reason, we also see in Figure 5c,f qualitatively the same
decay behavior as in ordinary MHD.

###  3.3 Chiral MHD

For chiral MHD, the induction equation attains an extra term under the curl
that leads to a contribution to the electric field proportional to the product
of the magnetic field and a pseudoscalar given by the chiral chemical
potential, expressed here as a wavenumber Rogachevskii et al. (2017).

$\mu_{5}=24\,\alpha_{\rm em}\,(n_{\rm L}-n_{\rm R})\,(\hbar c/k_{\rm
B}T)^{2},$ (19)

where $\alpha_{\rm em}\approx 1/137$ is the fine structure constant, and
$n_{\rm L}$ and $n_{\rm R}$ are the number densities of left- and right-handed
fermions, respectively. The uncurled induction equation takes then the form

$\frac{\partial\bm{A}}{\partial
t}=\eta(\mu_{5}\bm{B}-\mu_{0}\bm{J})+\bm{u}\times\bm{B},\quad\bm{J}={\bm{\nabla}}\times\bm{B}/\mu_{0}.$
(20)

The term $\eta\mu_{5}\bm{B}$ leads to a growth of the magnetic field for
wavenumbers $k<\mu_{5}$, just in the same way as in mean-field dynamo theory
(Moffatt, 1978; Parker, 1979; Krause and Rädler, 1980), but here no mean-field
theory is invoked. The generated magnetic field is fully helical, but the
relevant quantity is now the total chirality density

$h_{\rm tot}=\bm{A}\cdot\bm{B}+2\mu_{5}/\lambda,$ (21)

and it is its volume average that is conserved, i.e., $\langle h_{\rm
tot}\rangle={\rm const}{}$, provided the boundary conditions are periodic
and/or closed, i.e., perfectly conducting. As the magnetic field grows,
$\mu_{5}$ decreases. The rate of this change is proportional to the parameter
$\lambda$, which we take here as an adjustable parameter, but in reality is it
given by an expression involving the temperature; see Equation (49) of Ref.
Rogachevskii et al. (2017).

It is important to point out that the physical chiral chemical potential
(which has the units of an energy) is sometimes defined differently. First,
the authors of Ref. Brandenburg et al. (2023) used Lorentz-Heaviside units,
which implies another factor of $4\pi$ in the numerator of the conversion
factor (or rather the lack of a $4\pi$ factor in the denominator), and,
second, there is also a factor of $2$ in the denominator, so $\hbar
c/8\alpha_{\rm em}$ instead of $\hbar c/4\alpha_{\rm em}$ for the conversion
factor of Brandenburg et al. (2023), because they defined their physical
chiral chemical potential as half the difference between the physical right-
and left-handed chiral chemical potentials. In addition, there is a sign
difference between Refs. Rogachevskii et al. (2017) and Brandenburg et al.
(2023), but this affects only the physical chiral chemical potential and not
our equations, where $\mu_{5}$ has the units of a wavenumber.

It turns out, perhaps not surprisingly, that in this case, when $\langle
h_{\rm tot}\rangle=0$, the turbulence decays again in such a way that $q=4/9$
and $p=10/9$ and, again, $\beta=3/2$. This is just like in ordinary (but
nonhelical) MHD. In this case, however, $\langle h\rangle\neq{\rm const}{}$,
but its modulus decays $\propto t^{-r}$ in a way that is compatible with the
real-space realizability condition, $|\langle h\rangle|\leq 2{\cal E}_{\rm
M}\xi_{\rm M}$, i.e., $r=p-q=(10-4)/9=2/3$. This was also confirmed in Ref.
Brandenburg et al. (2023). This study was then applied to the problem of
baryogenesis Brandenburg et al. (2023), where one tries to explain the small
excess of matter over antimatter in the Universe, which is referred to as
baryon asymmetry.

The Hosking scaling was confirmed for $\langle\mu_{5}\rangle\xi_{\rm M}\gg 1$
Brandenburg et al. (2023), but in the opposite limit of
$\langle\mu_{5}\rangle\xi_{\rm M}\ll 1$ the Hosking scaling was no longer
obeyed and then both $\langle\mu_{5}\rangle$ and $\xi_{\rm M}$ are believed to
be approximately independently conserved Brandenburg et al. (2023). Trying to
understand this more thoroughly must be a goal for future studies, where one
may hope to reach much larger scale separation between the different relevant
wavenumbers in the system, such as the wavenumber $k_{0}$ of the peak of the
magnetic energy spectrum and the value of $\mu_{5}$.

In Figure 6, we plot magnetic energy and magnetic helicity spectra, as well as
magnetic helicity variance spectra for a chiral MHD run with balanced
chirality and an initial $k^{4}$ spectrum for the magnetic field. We see
standard inverse cascading with $\beta=3/2$. Next we compare with the case of
an initial $k^{2}$ spectrum; see Figure 7. In this case, there is still weak
inverse cascading, which is probably a consequence of the strong contribution
from mean magnetic helicity conservation over extended spatial patches.

Figure 6:  (a) Magnetic energy (solid lines) and magnetic helicity spectra
(dotted lines), and (b) magnetic helicity variance spectra for a chiral MHD
run with balanced chirality and an initial $k^{4}$ spectrum for the magnetic
field. In (a), positive (negative) magnetic helicities are indicated by small
red (blue) dots. The four large dots denote the positions of $\xi_{\rm
M}^{-1}$. Their colors are the same as those of the solid lines in (b) and
correspond to the times 1,500, 5,000, 15,000, and 50,000.  Figure 7:  Similar
to Figure 6, but with an initial $k^{2}$ spectrum. Note the presence of slight
inverse cascading in (a), although $\mbox{\rm Sp}(h_{\rm tot})={\rm const}{}$
at $k/k_{0}\ll 1$ in (b). The different colors refer to the same times as in
Figure 6.

Departures from the conservation of the Hosking integral based on $h_{\rm
tot}$ have been seen when $\mu_{5}<k_{0}$ Brandenburg et al. (2023), but here
we have $\mu_{5}>k_{0}$. To understand more thoroughly the regime where
$\mu_{5}\gg k_{0}$, we would need to have much larger numerical resolution. A
possible alternative is to use shell models Plunian et al. (2013), as will be
discussed next. However, it is unclear whether such models can capture the
relevant effects related to the Hosking integral or the chiral magnetic
effect.

##  4 Hosking Integral in Shell Models of Chiral MHD

Shell models describe turbulence through real or complex scalar variables on
concentric shells in wavenumber space such that certain conservation laws are
obeyed. In MHD, the relevant conservation laws are those of total chirality,
total (magnetic plus kinetic) energy, and cross helicity. The Hosking integral
describes helicity fluctuations over different scales, and does not have a
direct counterpart at the level of shell models. However, the scaling
properties resulting from its conservation, could still be manifest in shell
models describing the decay of MHD turbulence.

The Hosking integral is particularly important in cases where the mean
chirality vanishes. It is also conserved otherwise when the mean total
chirality is non-vanishing, but then the conservation of the mean chirality is
usually more important. It is also important that the magnetic field is
strong, because otherwise the decay properties are dominated by the
hydrodynamic turbulent decay. Our goal here is to investigate the decay of
magnetic fields with vanishing net chirality in chiral MHD using shell models.

In a shell model, we describe the state of the system in shells of
logarithmically spaced wavenumbers $k_{n}=2^{n}$, where $n=0$, 1, 2, …, $N$
denotes the shell and $N$ is the truncation level. For $N=30$, for example, we
can span ten orders of magnitude in wavenumber. In MHD, one usually considers
complex variables $B_{n}$ and $u_{n}$ for the magnetic and velocity fields.
The mean magnetic and kinetic energy densities are given by

${\cal E}_{\rm M}={\textstyle{1\over
2}}\sum_{n=0}^{N}|B_{n}|^{2}\quad\mbox{and}\quad{\cal E}_{\rm
K}={\textstyle{1\over 2}}\sum_{n=0}^{N}|u_{n}|^{2}.$ (22)

In shell models, the fluid density is constant and therefore not indicated in
the definition of the kinetic energy. Also the permeability factor in the
magnetic energy has been omitted.

Magnetic helicity is a signed quantity, i.e., it can be positive or negative.
How to describe this in a standard shell model is a matter of convention. One
approach is to associate even and odd shells with the decomposition into
positively and negatively polarized modes of the field. This idea was first
developed for the kinetic helicity Kadanoff et al. (1995). This then leads to
the definition of the magnetic helicity as Brandenburg et al. (1996); Frick
and Sokoloff (1998); Basu et al. (1998)

${\cal H}_{\rm M}=\sum_{n=0}^{N}(-1)^{n}|B_{n}|^{2}/k_{n},$ (23)

which satisfies the realizability condition

$k_{n}|{\cal H}_{\rm M}(k_{n})|\leq 2E_{\rm M}(k_{n}).$ (24)

To preserve the preferential growth of positively (negatively), polarized
modes on even (odd) shells, we write

$\left[\eta k\left(k_{n}-(-1)^{n}\mu_{5}\right)+\frac{{\rm d}{}}{{\rm
d}{}t}\right]B_{n}={\textstyle{1\over 6}}{\rm
i}k_{n}\,\left[M(u,B)-M(B,u)\right],$ (25)

where the $\mu_{5}$ term leads to a growth of $|b_{n}|^{2}$ for even (odd)
values of $n$ when $\mu_{5}$ is positive (negative), and $M(x,y)$ is a
nonlinear functional, where $x$ and $y$ stand for the full $n$-dependent
arrays. The essence of shell models is to couple only nearest and next-nearest
neighbors. We refer to this model as type I. This prescription then leads to

$M(x,y)=x_{n+1}y_{n+2}+x_{n-1}y_{n+1}+x_{n-2}y_{n-1}\quad\mbox{(type~{}I)}.$
(26)

As already emphasized in Section 2.4, the velocity plays a crucial role in
producing an inverse cascade. It is governed by the Navier-Stokes equation
with the Lorentz force included. There are then two further quadratic
nonlinearities for $u$ and $B$; see Refs. Brandenburg et al. (1996); Frick and
Sokoloff (1998); Basu et al. (1998) for details.

Another approach to treat helicity is to write the equations separately for
the positively and negatively polarized modes and thus have evolution
equations for $u_{n}^{\pm}$ and $B_{n}^{\pm}$. We refer to this model as type
II. The helicity density can then be written as Brandenburg et al. (1997)

${\cal H}_{\rm
M}=\sum_{n=0}^{N}\left.\left(|B^{+}_{n}|^{2}-|B^{-}_{n}|^{2}\right)\right/k_{n},$
(27)

and the magnetic energy is ${\cal E}_{\rm
M}=\sum_{n=0}^{N}(|B^{+}_{n}|^{2}+|B^{-}_{n}|^{2})$. The evolution equations
for $B_{n}^{\pm}$ take then the form

$\left[\eta k\left(k_{n}\mp\mu_{5}\right)+\frac{{\rm d}{}}{{\rm
d}{}t}\right]B^{\pm}_{n}={\textstyle{1\over 6}}{\rm
i}k_{n}\,\left[M_{\pm}(u,B)-M_{\pm}(B,u)\right]\quad\mbox{(type~{}II)},$ (28)

where Brandenburg et al. (1997)

$M_{\pm}(x,y)=x^{\mp}_{n+1}y^{\pm}_{n+2}+x^{\mp}_{n-1}y^{\mp}_{n+1}+x^{\pm}_{n-2}y^{\mp}_{n-1}.$
(29)

Note that for the intermediate terms, the signs in the superscripts are the
same, i.e., $u^{-}_{n-1}B^{-}_{n+1}$ appear in the evolution of $B^{+}_{n}$
and $u^{+}_{n-1}B^{+}_{n+1}$ in the evolution of $B^{-}_{n}$; see also Ref.
Lessinnes et al. (2009), where such models were proposed independently.

In Figure 8, we present models of types I and II with $N=30$ shells using
$\lambda=10^{10}$, $k_{0}=2^{14}=16384\approx 1.6\times 10^{4}$,
$\nu=\eta=5\times 10^{-11}$, and $\mu_{5}$ is computed as $\mu_{5}=-\mu_{\rm
M}$, where $\mu_{\rm M}\equiv{\cal H}_{\rm M}\lambda/2\approx 1.8\times
10^{5}$ is the chiral chemical potential equivalent of the magnetic helicity.

Figure 8:  Evolution of $E_{\rm M}(k,t)$ from shell models of (a) type I and
(b) type II. The times are 10 (red), 1 (orange), 0.1 (green), 0.01 (blue), and
earlier times are denoted by black lines of different line types. Note the
presence of inverse cascading in both cases.

In all cases, we start with a $k^{2}$ spectrum, so we expect to see no inverse
cascading. Looking at the results of Figure 8, however, this does not seem to
be the case. Our results are still preliminary, but our conclusion so far is
that shell models may not capture the same inverse cascade behavior that we
have found in the direct numerical simulations. On the other hand, more
parameter studies are warranted before one can draw more firm conclusions. One
must also remember that departures from the conservation of the Hosking
integral have been seen in certain direct numerical simulations Brandenburg et
al. (2023).

##  5 Conclusions

In this paper, we have presented a discussion of the Hosking integral in
various contexts in which it has been considered so far: ordinary MHD, MHD
with chiral fermions, as well as just the induction equation – either with
Hall nonlinearity or with ambipolar diffusion nonlinearity. When the total
chirality vanishes (non-chiral case with zero magnetic helicity or chiral case
with finite magnetic helicity balancing the fermion chirality) it is the
correspondingly adapted Hosking integral that governs the decay of ${\cal
E}_{\rm M}(t)\propto t^{-p}$ and the increase of $\xi_{\rm M}(t)\propto t^{q}$
with $p=10/9$ and $q=4/9$ for both ordinary MHD and also just the induction
equation with ambipolar diffusion. When the nonlinearity is given by the Hall
effect, on the other hand, we have $p=10/13$ and $q=4/13$. The case with
chiral fermions is somewhat special, because now the magnetic field is
actually fully helical, but this helicity is balanced by fermion chirality.
Again, in that case the Hosking integral determines the decay behavior.
However, there is also another decaying quantity: the mean magnetic helicity
density, which is now actually finite and balanced by fermion chirality. It is
found to decay like $t^{-2/3}$.

In previous work on decaying turbulence, the decay properties of hydrodynamic
and MHD turbulence were motivated by the use of self-similarity and invariance
of the governing equations under rescaling Olesen (1997); Brandenburg and
Kahniashvili (2017). This is different in the present work where we have just
made use of dimensional arguments. Still, the use of invariance under
rescaling is necessary to motivate the equilibrium line $p=2(1-q)$ in the $qp$
diagram in Figure 1b. It will therefore be interesting to find out whether the
existence of this line could also be motivated by other means. It probably
can, as implied by the derivation of Equation (12), and this might reveal a
more basic relation to the parameter that there was called $\sigma$.

An open question is whether the Hosking integral can also play a role in
driven MHD turbulence, for example. One possibility could be the production of
inverse cascade behavior where magnetic energy grows on wavenumbers below the
energy injection wavenumber. This could then leads to a turbulent subinertial
range scaling of the form

$E_{\rm M}(k)\propto I_{\rm H}^{a}k^{b}.$ (30)

Using dimensional arguments, we would find $3=9a-b$ and $2=4a$ for balancing
the dimensions of length and time, respectively. Therefore, $a=1/2$ and
$b=3/2$. Thus, $b$ is positive and equal to the Kazantsev slope known in
kinematic nonhelical small-scale dynamos Kazantsev (1968). Whether or not
there is actually a connection with Kazantsev’s small-scale dynamo theory
remains another open question.

In our work we have also examined whether some aspects of the Hosking integral
might be reproducible with shell models. At the moment, this does not seem to
be the case, but this could well be a consequence of not having performed
sufficiently extensive parameter studies. Thus, more work might be warranted.

Conceptualization, A.B. and G.L.; methodology, A.B.; software, A.B. and G.L.
All authors have read and agreed to the published version of the manuscript.

This research was funded by Vetenskapsrådet grant number 2019-04234 and NASA
ATP award number 80NSSC22K0825.

The source code used for the simulations of this study, the Pencil Code
(Pencil Code Collaboration et al., 2021), is freely available on
https://github.com/pencil-code/ (accessed on 24 May 2023). The DOI of the code
is https://doi.org/10.5281/zenodo.2315093 (accessed on 24 May 2023). The
simulation setups and the corresponding secondary data are available on
http://norlx65.nordita.org/~brandenb/projects/Hosking-Shell (accessed on 24
May 2023).

The authors declare no conflict of interest.

References

yes

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