Optimal anti-ferromagnets for light dark matter detection Angelo Esposito1 2 3 and Shashin Pavaskar4 1Dipartimento di Fisica Sapienza Universit a di Roma Piazzale Aldo Moro 2 I-00185 Rome Italy

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Optimal anti-ferromagnets for light dark matter detection

Angelo Esposito

1, 2, 3, ∗

and Shashin Pavaskar

4, †

Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 2, I-00185 Rome, Italy

INFN Sezione di Roma, Piazzale Aldo Moro 2, I-00185 Rome, Italy

School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA

Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA

(Dated: August 3, 2023)

We propose anti-ferromagnets as optimal targets to hunt for sub-MeV dark matter with spin-

dependent interactions. These materials allow for multi-magnon emission even for very small

momentum transfers, and are therefore sensitive to dark matter particles as light as the keV. We use

an eﬀective theory to compute the event rates in a simple way. Among the materials studied here,

we identify nickel oxide (a well-assessed anti-ferromagnet) as an ideal candidate target. Indeed, the

propagation speed of its gapless magnons is very close to the typical dark matter velocity, allowing

the absorption of all its kinetic energy, even through the emission of just a single magnon.

I. INTRODUCTION

There is today overwhelming evidence that most of the

matter in the Universe is dark. Despite that, the question

about its nature arguably remains among the biggest ones

in fundamental physics. In particular, the possible dark

matter mass spans a range of several orders of magnitude.

In light of stringent constraints on heavy WIMPs [e.g.,

–

], recent years have witnessed an increasing interest

in models for sub-GeV dark matter [e.g.,

–

], also mo-

tivating new detection ideas. In particular, dark matter

candidates in the keV to GeV range, while still heavy

enough to be considered as particles, cannot release appre-

ciable energy via standard nuclear recoil. They therefore

require detectors with low energy thresholds, such as

semiconductors [

–

], superconductors [

–

], Dirac

materials [

–

], lower dimensional materials [

–

and so on (see also [39–41]).

Among these, the proposals based on superﬂuid

4He

[

–

] and solid crystals [

–

] aim at detecting the

collective excitations (phonons) produced by the spin-

independent interaction of dark matter with the nuclei in

the material—for an overview see [

–

]. These collective

modes have typical energies below

(100 meV), and are

therefore sensitive to particles as light as

mχ∼ O

(

keV

Diﬀerent proposals for the detection of single phonons

have been recently put forth [63–66].

The targets above are, however, not the most suitable

ones to probe possible scenarios where spin-dependent

interactions of dark matter with the Standard Model are

dominant over the spin-independent ones. In this regard,

it has been proposed to use ferromagnets [

–

], i.e. ma-

terials that exhibit a non-zero macroscopic magnetization

in their ground state.

The dark matter can interact

∗angelo.esposito@uniroma1.it

†spavaska@andrew.cmu.edu

The materials presented in [

] are actually insulating ferri-

magnets. This makes no diﬀerence in our discussion [

]. We

refer to ferromagnets, which are conceptually simpler.

with the individual spins in the target, exciting their local

precession: a propagating collective mode called magnon.

The proposals to detect single magnons involve either

calorimetric readout [

], using TES or MKID, or quan-

tum sensors, which instead couple the magnon mode to

a superconducting qubit [

–

]. A generic ferromagnet

features several magnon types (branches). However, for

suﬃciently light dark matter (

mχ≲

10 MeV, for the

typical material [

]), the momentum transfer becomes

smaller than the inverse separation between the spins. In

this regime the event rate is dominated by the emission of

gapless magnons which, for ferromagnets, are character-

ized by a quadratic dispersion relation,

(

) =

q2/

mθ

with

mθ

a mass scale set by the properties of the mate-

rial under consideration. Moreover, as we argue below,

conservation of total magnetization implies that, when

only gapless magnons are allowed, no more than one can

be produced in each event. Thus, for

mχ≲

10 MeV, the

maximum energy that can be released to a ferromagnet is

ωmax

= 4

Tχx/

(1 +

)

, with

Tχ

the dark matter kinetic

energy and

x≡mθ/mχ

. Typically,

mθ∼ O

(

MeV

) (e.g.,

mθ≃

5 MeV for

Y3Fe5O12

[

], see also [

]), and

a sub-MeV dark matter will not deposit all its energy to

the target.

In this work, we show that, instead, anti -ferromagnets

are optimal materials to probe the spin-dependent inter-

actions of light dark matter. Similarly to ferromagnets,

they also exhibit magnetic order in the ground state,

but the spins are anti-aligned, leading to a vanishing

macroscopic magnetization. This leads to two crucial

diﬀerences: (1) gapless magnons have a linear dispersion

relation,

(

) =

vθq

, and (2) the interaction with the dark

matter can excite any number of them. If only one magnon

is emitted, the maximum energy that can be transferred

to the anti-ferromagnet is

ω1,max

= 4

Tχy

−y

), with

y≡vθ/vχ

. One of the anti-ferromagnets we consider

here, nickel oxide, features magnons with a propagation

speed surprisingly close to the typical dark matter veloc-

ity, which allows it to absorb most of the kinetic energy

even through a single magnon mode. This is a well-known

and well-studied material, which makes it a particularly

ideal target. Moreover, the possibility of exciting several

arXiv:2210.13516v2 [hep-ph] 1 Aug 2023

FIG. 1. Schematic representation of the spins in the ground

state of an anti-ferromagnet.

magnons in a single event relaxes the kinematic con-

straints above, allowing any anti-ferromagnet to absorb

the totality of the dark matter kinetic energy, hence being

sensitive to masses down to mχ∼ O(keV).

In what follows, we describe anti-ferromagnets and their

interaction with dark matter via an eﬀective ﬁeld theory

(EFT) [

]. This elucidates the role played by

conservation laws in allowing multi-magnon emission and

allows the computation of the corresponding event rates

in a simple way, bypassing the diﬃculties encountered

with more traditional methods [e.g., 78–80].

Conventions: We work in natural units,

ℏ

= 1, and

employ the indices

i, j, k

= 1

3 for spatial coordinates

and a, b = 1,2 for the broken SO(3) generators.

II. THE EFT

A. Magnons alone

One can often picture an atom in a magnetic material as

having a net spin coming from the angular momentum of

the electrons localized around it. The Coulomb interaction

between electrons pertaining to diﬀerent atoms induces a

coupling between diﬀerent spins which, in turns, causes

magnetic order in the ground state [e.g.,

]. In an anti-

ferromagnet these interactions are such that the spins are

anti-aligned along one direction (Figure 1), which from

now on we take as the

-axis. One can then deﬁne an order

parameter, the so-called N´eel vector, as

N≡Pi

(

−

iSi

where

is the

-th spin and (

−

is positive for those

sites pointing ‘up’ in the ordered phase, and negative for

those pointing ‘down’. In the ground state the N´eel vector

acquires a non-zero expectation value,

⟨N⟩ ̸

= 0 [e.g.,

In the non-relativistic limit, a system of 3-dimensional

spins enjoys an internal

(3) symmetry. The ground

state described above breaks it spontaneously down to

only the rotations around the

-axis,

(3)

→SO

(2),

and the gapless magnons are nothing but the associated

Goldstone bosons. As such, at suﬃciently low energies,

they are described by a universal EFT, very much analo-

gous to the chiral Lagrangian in QCD. A convenient way

of parametrizing the magnons is as ﬂuctuations of the or-

der parameter around its equilibrium value,

n≡eiθaSa·ˆ

with

= 1

2. Here

θa

(

) is the magnon ﬁeld and

are

the broken SO(3) generators.

The EFT Lagrangian is derived purely from symmetry

considerations. First of all, one notes that under time

reversal each spin changes sign,

Si→ −Si

. If combined

with a translation by one lattice site, which swaps spin

‘up’ with spin ‘down’, this leaves the ground state un-

changed. The eﬀective Lagrangian for anti-ferromagnets

must then be invariant under the joint action of these

two symmetries. At large distances, translations by one

lattice site do not aﬀect the system, and the only re-

quirement is time reversal: the Lagrangian must feature

an even number of time derivatives [

]. Moreover, the

underlying crystal lattice spontaneously breaks boosts.

Assuming, for simplicity, that the material is homoge-

neous and isotropic at long distances, this implies that

there must be explicit invariance under spatial transla-

tions and rotations, but that space and time derivatives

can be treated separately.

Since

|ˆ

= 1, the most gen-

eral low-energy Lagrangian for the gapless magnons

an anti-ferromagnet is then [70,76],

Lθ=c1

2∂tˆ

n2−c2

2∂iˆ

n2

=c1

2˙

θa2−c2

2∂iθa2+. . . ,

(1)

where in the second equality we expanded in small ﬂuctu-

ations around equilibrium. The coeﬃcients

c1,2

depend

on the details of the anti-ferromagnet under consideration,

and cannot be determined purely from symmetry.

One recognizes Eq.

(1)

as the real representation of

the Lagrangian of a complex scalar, corresponding to two

magnons with linear dispersion relation,

(

) =

vθq

, and

propagation speed

c2/c1

. The two magnons are com-

pletely analogous to relativistic particle and anti-particle,

and they carry opposite charge under the unbroken

(2).

As shown in [

], the action for a ferromagnet, instead,

contains only one time derivative and it is analogous to

that of a non-relativistic particle, which does not feature

excitations with opposite charge. This is the reason why,

when coupled to light dark matter, anti-ferromagnets al-

low for the emission of more than one magnon in each

event, while ferromagnets do not. We discuss this more

in Section II B.

As far as our application is concerned, a central role is

played by the spin density, which is the time-component

of the Noether current associated to the original

(3)

symmetry [

]. This rotates the

vector (i.e.,

ˆni→

Rij ˆnj

), and the current can be computed with standard

We treat the underlying solid as a background which sponta-

neously breaks some spacetime symmetries. The corresponding

Goldstone bosons, the phonons, realize these symmetries nonlin-

early and can be included in the description if necessary [70].

Crystalline anisotropy, which arises from spin-orbit coupling, can

result in a small gap for magnons. This is analogous to the pion

obtaining a mass due to the explicit breaking of chiral symmetry.

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Optimalanti-ferromagnetsforlightdarkmatterdetectionAngeloEsposito1,2,3,∗andShashinPavaskar4,†1DipartimentodiFisica,SapienzaUniversit`adiRoma,PiazzaleAldoMoro2,I-00185Rome,Italy2INFNSezionediRoma,PiazzaleAldoMoro2,I-00185Rome,Italy3SchoolofNaturalSciences,InstituteforAdvancedStudy,Princeton,NJ08540,U...

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Optimal anti-ferromagnets for light dark matter detection Angelo Esposito1 2 3 and Shashin Pavaskar4 1Dipartimento di Fisica Sapienza Universit a di Roma Piazzale Aldo Moro 2 I-00185 Rome Italy

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