# Investigations of the size distribution and magnetic properties of
nanoparticles of Cu2OSeO3

S. J. R. Holt<EMAIL_ADDRESS>University of Warwick, Department of
Physics, Coventry, CV4 7AL, United Kingdom  A. Štefančič  University of
Warwick, Department of Physics, Coventry, CV4 7AL, United Kingdom  J. C.
Loudon  University of Cambridge, Department of Materials Science and
Metallurgy, Cambridge, CB3 0FS, United Kingdom  M. R. Lees  University of
Warwick, Department of Physics, Coventry, CV4 7AL, United Kingdom  G.
Balakrishnan<EMAIL_ADDRESS>University of Warwick, Department
of Physics, Coventry, CV4 7AL, United Kingdom

###### Abstract

Skyrmions in confined geometries have been a subject of increasing interest
due to the different properties that they exhibit compared to their bulk
counterparts. In this study, nanoparticles of skyrmion-hosting
$\text{Cu}_{2}\text{OSeO}_{3}$ have been synthesised using a precipitation
method followed by thermal treatment. This enables us to produce nanoparticles
whose mean size varies from tens of nanometers to a few micrometers by varying
the temperature and duration of the thermal decomposition of the precursor.
These sizes span the $\sim 63$ nm diameter of skyrmions in
$\text{Cu}_{2}\text{OSeO}_{3}$, allowing investigations into how the magnetic
state changes when the size of the geometrical confinement is similar to and
smaller than the size of an isolated magnetic skyrmion. AC susceptibility
measurements performed on nanoparticles with a size distribution from 15 to
250 nm show a change in the magnetic phase diagram compared to bulk
$\text{Cu}_{2}\text{OSeO}_{3}$.

##  1 Introduction

Magnetic skyrmions are topologically stable magnetic spin textures which have
properties of quasi-particles [1]. These magnetic vortex-like structures are
of great interest to the research community, offering many potential uses in
spintronic devices and for applications such as data storage [2, 3]. Magnetic
skyrmions have been observed in a variety of forms such as Bloch and Néel type
skyrmions, both with a topological charge of $+1$ and a vorticity of $\pi/2$
and $0$ respectively [4, 5]. The counterpart to these are antiskyrmions with a
topological charge of $-1$ which have been observed in magnets with D2d
symmetry [6, 1]. Alongside these skyrmions, there are a plethora of other
predicted topologically non-trivial magnetic quasi-particles [7] such as
higher-order skyrmions [8], skyrmionium [9, 10], bimerons [11, 12, 13], chiral
bobbers [14, 15], and hopfions [16, 17]. Some of this rich selection of
magnetic phases have the potential to be stabilised in different magnetic
materials.

Bloch-skyrmions in B20 chiral magnets manifest as a two-dimensional skyrmions
that extend in a third dimension creating a tube like objects. Rather than
being completely transitionally invariant along this third dimension, these
skyrmion tubes are modulated near the surface of the sample in a way known as
a chiral twist [18, 19, 20]. This chiral twist occurs in the vicinity of the
surface of the sample on a length scale comparable to that of the magnetic
modulation length of the skyrmions, reducing the total energy of the system
[20, 21, 22]. At the surface of the skyrmion tube there is also a change in
the angle of magnetisation known as surface twists which have an additional
Néel contribution [18, 20]. These edge effects allow for the stabilisation of
particle-like states such as chiral bobbers which manifest in a similar way to
a skyrmion tube, but rather than the tube propagating all the way through the
material, they end in a Bloch point [14, 15]. These chiral bobbers have a
characteristic depth which is less than half of the magnetic modulation length
and can be stabilised in a wide range of magnetic phase diagrams, highlighting
the importance of surface effects along the axis of skyrmion tubes. There are
also additional confinement effects originating from the interaction of the
samples edge with the skyrmion tube. Often, skyrmions form a hexagonal
lattice, in a region just below the Curie temperature $T_{\textrm{C}}$, but
due to their topological nature, skyrmions can also be stabilised as isolated
quasi-particles [4]. Skyrmions have a repulsive interaction with the edge of
the sample [23] leading to the confinement of skyrmions within a sample. In
order to study these confinement effects the size of the particles in which
the skyrmions are stabilised have to be comparable to the average size or
modulation length of the skyrmions in these materials, which has been well
established in most bulk materials. One such route to study the confinement
effects is in the investigation of the behaviour of nanoparticles of skyrmion
hosts, making nanoparticles a prime contender for the confinement of magnetic
skyrmions.

Skyrmions in B20 materials have a diameter determined by the material
parameters of the Heisenberg exchange and Dzyaloshinskii–Moriya interactions,
these sizes are typically on the scale of tens of nanometers. For example,
Bloch-type skyrmions have been experimentally identified in B20 materials such
as MnSi with a size of 17 nm, FeGe with a size of 70 nm, and Cu2OSeO3 with a
size of 63 nm [4, 24, 25]. It has been observed that physically changing the
size and shape of a material can dramatically change the region of the phase
diagram over which of skyrmions are stabilised. For example, in Lorentz
transmission electron microscopy (LTEM) studies, thinning the sample
dramatically increases the range of temperatures and magnetic fields over
which skyrmions are stable [25]. This highlights the considerable effects that
confinement along the axis of a skyrmion tube have on the stabilisation of
skyrmions. In MnSi, investigations have demonstrated the effect of confinement
on skyrmions by comparing thin films [26], nanowires [27], nanoparticles [28],
and bulk [4]. Nanowires exhibit the largest enhancement to the extent of the
skyrmion phase in the magnetic phase diagram. The skyrmion phase in thin films
is found at a reduced temperature and higher magnetic fields compared to the
bulk. In nanoparticles, previous observations indicate an increase in the
temperature at which skyrmions are observed, making them stable only under
very small fields [28].

In this study, we focus on the skyrmion-hosting material Cu2OSeO3, and the
effects of three-dimensional geometric confinement of the skyrmions in
nanoparticles of this material. Cu2OSeO3 is a multiferroic insulator that has
been investigated extensively for multiple reasons including the use of an
external electric field to manipulate skyrmions [29, 25], exhibiting two
distinct skyrmion phases [30], and the ability to study the effects of
chemically substitution of Cu atoms with Zn and Ag [31, 32]. Cu2OSeO3
crystallises in the chiral $P2_{1}3$ space group ($T$ point group), a symmetry
group in which the majority of known skyrmion materials have been found [33].
There are two crystallographically inequivalent Cu sites present in a $1:3$
ratio with a spin $1/2$ pointing in opposite directions on each site, creating
a ferrimagnetic lattice [34, 35, 36] where superexchange via the O atoms
mediates the magnetic interactions [37, 38].

Recently, there have been micromagnetic investigations into the confinement of
skyrmions in nanoparticles of FeGe [39]. Due to the similar magnetic
modulation period of both FeGe and Cu2OSeO3, micromagnetic simulations of FeGe
also describe the ferrimagnetic Cu2OSeO3 system well [40]. This means that
results from micromagnetic studies on nanoparticles in FeGe systems are
relevant to Cu2OSeO3 nanoparticles. Pathak and Hertel [39] found that the
magnetic phases such as skyrmions, helices, merons, chiral bobbers, and
saturated states can be stabilised across a variety of nanoparticle sizes
under different applied magnetic fields compared to the bulk system. For
example, the skyrmion phase in a small applied magnetic field has been shown
to give way to a meron phase and finally a saturated magnetic state upon
reduction of the nanoparticle size. Our study sets out to investigate if any
changes in the magnetic state can be experimentally observed in nanoparticles
of Cu2OSeO3 as the size of the nanoparticle approaches, and is reduced smaller
than the size of an isolated magnetic skyrmion.

##  2 Experimental Details

Powder X-ray diffraction (PXRD) was performed on a Panalytical X-Pert Pro
diffractometer operating in Bragg-Brentano geometry equipped with a
monochromatic Cu Kα1 source and a solid-state PIXcel one-dimensional detector.
A Zeiss SUPRA 55-VP scanning electron microscope with an accelerating voltage
of 10 kV was used for scanning electron microscopy (SEM) experiments.
Transmission electron microscopy (TEM) was performed using a FEI Titan
electron microscope. A Mastersizer 2000 particle size analyser was used to
measure the size of the Cu2OSeO3 nanoparticles by laser diffraction. AC
susceptibility measurements were performed using a Quantum Design Magnetic
Property Measurement System (MPMS5) superconducting quantum interference
device (SQuID) magnetometer as a function of DC applied field. These AC
susceptibility measurements were performed following a zero-field-cooling
procedure with an AC field of 10 Hz and 0.3 mT.

##  3 Sample Preparation

Cu2OSeO3 nanoparticles were prepared using a precipitation method followed by
thermal treatment [41]. Copper(II) chloride dihydrate (CuCl2.2H2O) and sodium
selenite decahydrate (Na2SeO3.10 H2O) were dissolved separately in distilled
water with a small amount of HCl added to the CuCl2 solution to reduce the pH.
The CuCl2 solution was slowly added to the Na2SeO3 solution producing a
precipitate which gradually changed from milky green to blue upon adding
additional CuCl2. This precipitate, which forms via the reaction

$\displaystyle\begin{split}\ce{Na2SeO3_{(aq)}+&CuCl2_{(aq)}->}\\\
\ce{&2NaCl_{(aq)}+CuSeO3.2H2O_{(s)}}\end{split}$ (1)

was confirmed to be CuSeO3.2H2O using powder X-ray diffraction. The solid
CuSeO3.2H2O precipitate was filtered from the solution and washed with
distilled water, isopropanol, and acetone before drying at 50 ∘C on air for 18
hours.

Cu2OSeO3 nanoparticles were synthesised by heating CuSeO3.2H2O in an alumina
crucible under a flow of O2 in a tube furnace. At the initial heating stage, a
dehydration process takes place in multiple steps leading to the overall
reaction [42]

$\ce{CuSeO3.2H2O->[\Delta T]CuSeO3+2H2O.}$ (2)

A thermal decomposition then takes place to convert the CuSeO3 into Cu2OSeO3
nanoparticles via the following reactions:

$\displaystyle\ce{4CuSeO3->[\Delta T]Cu4O(SeO3)3+SeO2,}$ (3)
$\displaystyle\ce{Cu4O(SeO3)3->[\Delta T]2Cu2OSeO3+SeO2.}$ (4)

One of the key stages in the process is the thermal decomposition of
Cu4O(SeO3)3 to Cu2OSeO3 as during this process the material goes through an
amorphous phase [41]. The temperatures used here for the decomposition varied
between 400 ∘C and 470 ∘C depending on the particle size desired. Fours sets
of nanoparticles, hereafter referred to as Samples A, B, C, and D, were
produced. The procedures used and the size of the nanoparticles produced
(determined in the following sections) are detailed in Table 1. It is worth
noting that at higher temperatures or long time treatments, special care has
to be taken not to thermally decompose the Cu2OSeO3 further via [42]

$\ce{2Cu2OSeO3->[\Delta T]4CuO+2SeO2.}$ (5)

For comparison, a polycrystalline sample of $\text{Cu}_{2}\text{OSeO}_{3}$
(Sample SS) was synthesised by thoroughly grinding together stoichiometric
amounts of CuO ($99.99\%$, metals basis, Alfa Aesar), $\text{SeO}_{2}$
($99.999\%$, trace metal basis, Acros Organics) inside an argon-filled glove
box. The mixtures of powders were transferred into silica tubes, evacuated and
sealed. Samples were then heated at a rate of 3.5 ∘C/h to 650
${}^{\circ}\text{C}$, kept at this temperature for 96 h, followed by water
quench cooling.

Table 1: Synthesis conditions of samples and sizes of Cu2OSeO3 particles. Samples A - D were prepared by precipitation and thermal treatment. Sample SS was prepared via a solid state reaction. Sample | Temperature | Dwell time | Particle diameter
---|---|---|---
| $(^{\circ}\text{C})$ | (h) | PXRD | SEM/TEM
A | 400 then 420 | 288 then 48 |  $\sim$ $3.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | 1-8 $\text{\,}\mathrm{\SIUnitSymbolMicro m}$
B | 420 | 72 |  $\sim$ $3.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | 0.8-2 $\text{\,}\mathrm{\SIUnitSymbolMicro m}$
C | 450 | 24 |  $\sim 112$ nm | 15-250 nm
D | 470 | 24 |  $\sim 126$ nm | -
SS | 650 | 96 | - | 10-100 $\text{\,}\mathrm{\SIUnitSymbolMicro m}$

##  4 Characterisation

###  4.1 Powder X-ray Diffraction

The phase purity of the Cu2OSeO3 nanoparticles was examined by powder X-ray
diffraction. All peaks could be indexed in the $P2_{1}3$ space group, in
excellent agreement with the literature [31]. Figure 1 shows an X-ray
diffraction pattern of Cu2OSeO3 nanoparticles from Sample D. By measuring a Si
standard, the instrumental broadening of the diffraction peaks can be
measured, and hence the sample broadening can be isolated so that the
crystallite domain size and the strain can be quantified using the Debye-
Scherrer equation [43].

Figure 2 shows a comparison of Sample SS, Cu2OSeO3 powder made using solid
state synthesis, and nanoparticles from Samples B, C, and D synthesised via
the precipitation method followed by thermal treatment. A clear broadening can
be seen in the Bragg peaks of the nanoparticles of Cu2OSeO3 compared to the
powder prepared by solid state synthesis. This indicates that the
nanoparticles have a smaller crystallite size. If it is assumed that the
crystallites are spherical, with a size distribution that follows a log-normal
distribution [44], the volume-weighted domain size can be related to the radii
of the crystallites [45]. Quantification of the crystallite sizes using
Rietveld analysis can be seen in Table 1.

Figure 1: Powder X-ray diffraction profile of Cu2OSeO3 nanoparticles (Sample
D). The experimentally-obtained diffraction profile at ambient temperature
(blue open circles), refinement based on the model obtained from single
crystal X-ray diffraction at room temperature (red solid line), difference
(olive green solid line) and predicted peak positions (black tick marks).
Figure 2: Powder X-ray diffraction of the (311) peak comparing Cu2OSeO3
produced by solid synthesis (Sample SS), and Samples B, C, and D obtained by
the precipitation method followed by thermal treatment.

###  4.2 Scanning Electron Microscopy

Scanning electron microscopy was used to image the particles in order to
investigate their size and shape. Table 1 details the sizes of nanoparticles
imaged by SEM. Figure 3 shows the SEM images of nanoparticles from Samples B
and C. It can be seen that there is a size distribution of particles which
have an average diameter of $\sim 1$ $\text{\,}\mathrm{\SIUnitSymbolMicro m}$
in Sample B, in agreement with the powder X-ray diffraction measurements. The
particles in Sample C can be seen to be significantly smaller with most
particles ranging in diameter from 60 to 250 nm. In both samples the particles
are quite rough and tend to be seen in larger aggregates which can be up to
$\sim 20$ $\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in diameter.

Figure 3: SEM images of Cu2OSeO3 nanoparticles from (a) Sample B and (b)
Sample C. The sizes of the individual particles and aggregates can be clearly
seen.

###  4.3 Laser diffraction

Laser diffraction was used to measure the particle size in the powders. This
technique is based on the optical size of the object, hence aggregates appear
as a larger particle rather than collections of smaller particles. Figure 4
shows the measured size distributions for Cu2OSeO3 nanoparticles from Sample C
suspended in distilled water. With the application of ultrasound, the larger
aggregates with a peak at $\sim 1500$ $\text{\,}\mathrm{\SIUnitSymbolMicro m}$
break up into particles with sizes below $\sim 100$
$\text{\,}\mathrm{\SIUnitSymbolMicro m}$. These smaller particle sizes are
similar to what is observed using TEM and SEM. However, it can be seen that
even with ultrasonication the measured size does not approach the crystallite
size measured with X-ray diffraction or SEM, indicating that the majority of
the nanoparticles remain in the form of aggregates. Similar behaviour is seen
for all of the Cu2OSeO3 nanoparticles synthesised.

Figure 4: Particle size distribution of Cu2OSeO3 nanoparticles in Sample C
determined from laser diffraction. The ultrasonication process has broken up
the larger aggregates.

###  4.4 Transmission electron microscopy

Transmission electron microscopy was used to image the nanoparticles to gain
more information on their size and shape. Nanoparticles were suspended in
methanol and then subject to ultrasonication to disperse the nanoparticles and
break up some of the larger aggregates. A small part of the solution was then
placed on a lacey carbon film. Table 1 details the sizes of nanoparticles
imaged by TEM. Figure 5 shows a TEM image of the nanoparticles (Sample C).
There are a variety of different diameters of nanoparticles ranging from $\sim
15$ to $\sim 250$ nm. Most of the particles are relatively rough as opposed to
having well-defined crystallographic faces. The sizes of these particles are
in the range of interest, as they span the size of a skyrmion ($\sim 63$ nm)
in Cu2OSeO3.

Figure 5: TEM image of the Cu2OSeO3 nanoparticles in Sample C highlighting
their irregular shapes and sizes.

##  5 Magnetism

###  5.1 AC susceptibility

AC magnetometry was performed on nanoparticles of Samples C and D as these had
the smallest average size, and the results were compared with measurements on
Sample SS, in order to observe the differences in the magnetic phases
exhibited by the bulk sample and the nanoparticles. AC magnetometry is
sensitive to the dynamics of the magnetism in the sample. If a different
magnetic state is realised this may result in a change in the AC
susceptibility response.

Figure 6: (a-c) In-phase and (d-f) out-of-phase AC susceptibility as a
function of temperature and applied DC field for polycrystalline (Sample SS)
and nanopaticles (Samples C and D) of Cu2OSeO3. The magnetic phases are given
for polycrystalline Cu2OSeO3 where H is the helical phase, C is the conical
phase, and S is the skyrmion lattice. FP and P are the field polarised and
paramagnetic states, respectively. The lines are given as a guide to the eye.
The crystallite sizes indicated are obtained from PXRD (see Table 1).

Figure 6 shows the in-phase, $\chi^{\prime}$, and out-of-phase component,
$\chi^{\prime\prime}$, of the AC susceptibility as a function of temperature
and magnetic feld for the nanoparticles (Samples C and D), compared to
polycrystalline Cu2OSeO3 produced by solid state synthesis (Sample SS). For
$\chi^{\prime}$, a clear difference can be seen in the phase diagrams, as the
pocket associated with skyrmions in the polycrystalline Cu2OSeO3 is no longer
visible for the nanoparticle samples. The signals associated with the conical
and helical phase are also dramatically changed as there are no longer clear
plateaus that are indicative of both of these phases.

A comparison of the phase diagrams obtained from $\chi^{\prime\prime}$ for the
different particles shows that there are some similar features present in all
of them. For the polycrystalline Cu2OSeO3 there is a peak just below 58 K
associated with the transition to a skyrmion phase with a long ridge
stretching from 58 K downwards linked to the helical to conical transition.
For the nanoparticles, the peak in $\chi^{\prime\prime}$ has shifted to $\sim
54$ K and a slightly higher DC magnetic field. The signal is also larger in
magnitude, indicating higher dissipation. A ridge in $\chi^{\prime\prime}$ can
also be seen in Samples C and D that extends from the peak at $\sim 54$ K
downward in temperature.

Figure 7: In-phase AC susceptibility $\chi^{\prime}$ versus applied DC
magnetic field at 53 K and 57.5 K for polycrystalline Cu2OSeO3 and
nanoparticles of Cu2OSeO3 taken from Sample C and D. The crystallite sizes
indicated are obtained from PXRD.

Figure 7 shows an AC susceptibility field sweep that compares the
nanoparticles with polycrystalline Cu2OSeO3. It is clear that the plateau of
the helical phase in the polycrystalline material is no longer present in the
nanoparticles and the magnitude of the susceptibility is higher compared to
the magnitude of the conical phase in the polycrystalline Cu2OSeO3. However,
there is a maximum at $\sim 12$ mT in $\chi^{\prime}$ for the nanoparticles
that occurs at the same point as the helical to conical transition. There is
also a shoulder at $\sim 45$ mT that is close to the conical to field
polarised transition seen in polycrystalline Cu2OSeO3. At 57.5 K the features
seen in the nanoparticles occur at the same fields as the peaks surrounding
the skyrmion pocket at $\sim 8.5$ mT and $\sim 26$ mT. These features suggest
that these or similar transitions occur in at least some of the nanoparticles.

##  6 Discussion and Summary

Nanoparticles of Cu2OSeO3 have been synthesised via a precipitation method
followed by thermal treatment with a degree of control over the size of
nanoparticles produced. X-ray diffraction was used to confirm the phase purity
of the synthesised nanoparticles and the broadening of the diffraction peaks
was used to calculate the crystallite sizes. A clear correlation between
particle size and synthesis conditions have been observed with thermal
decomposition taking place at a higher temperature for a shorter amount of
time producing smaller nanoparticles. The ability to tune the nanoparticle
crystallite size is likely due to the amortisation of the intermediate phase
that occurs during the thermal decomposition process.

The particle shape and size distribution was also investigated using SEM and
TEM, which corroborated the link between nanoparticle size and the synthesis
conditions. However, these techniques also revealed that the nanoparticles
tend have rough surfaces and form in larger aggregates. In addition, laser
diffraction was used to determine the particle size distribution, but due to
this technique being based on the optical size of particles rather than the
crystallite size, it yielded results pointing to sizes that were larger than
the actual nanoparticle size. This means that this technique is limited to
measuring the size of the aggregates rather than the size of the
nanoparticles. In addition, the size distribution produced with laser
diffraction was approximately consistent for all of the samples produced by
precipitation method followed by thermal treatment.

AC susceptibility was performed to examine the magnetic phases present. It is
not possible to unambiguously identify a magnetic phase based on the response
from AC susceptibility alone so either further measurements using different
techniques such as electron holography are needed to identify magnetic phases
or a comparison has to be made to. Comparing the AC susceptibility phase
diagrams of the nanoparticles with that of polycrystalline Cu2OSeO3, where the
magnetic phases are known it appears that there are significant changes in the
magnetic states stabilised. There are, howvever, some important some
similarities between in both the polycrystalline and nanoparticles AC
susceptibility phase diagrams. For example, $\chi^{\prime\prime}$ for the
nanoparticles has a set of peaks at 54 K. This could be an indication that
skyrmions still form in some of the nanoparticles, but are suppressed to a
lower temperature. A prominent ridge is also observed in $\chi^{\prime\prime}$
for the nanoparticles. There are several possible causes of the ridge. It
could be due to a helical to conical phase transition, as in the bulk
material. However, the ridge could also be due to a correlation between the
temperature at which a peak in $\chi^{\prime\prime}$ appears and the
crystallite size. This would mean a distribution of particles sizes will cause
a smearing of a peak into a ridge.

The investigations presented here clearly show the ability to tune the size of
the nanoparticles formed depending on the thermal decomposition temperatures
and dwell times. It has also been suggested that a ferrimagnetic ground state
is stabilised in nanoparticles of size smaller than the size of a skyrmion
[46]. While it is not possible from the investigation presented here to
unequivocally confirm this, the AC susceptibility does suggest a distinct
change in the magnetic phases present depending on the size of nanoparticles.
This is corroborated by micromagnetic simulations on a similar material, FeGe
[39].

While definite changes to the magnetism within nanoparticles have been
observed, further experimental investigations of the effects on the magnetic
structure of the nanoparticle size are required. If Cu2OSeO3 nanoparticles can
be produced that have a very narrow spread of sizes, or nanoparticles can be
easily separated by size, there is potential for further experiments to
investigate the magnetic structure - nanoparticle size relationship.
Investigation should also be conducted into nanoparticles of other skyrmion
hosting materials, such as FeGe [24] and Co-Zn-Mn [47], which are simpler
systems to study the effects of confinement in nanoparticles. In addition,
other materials which host different types of topological magnetism such as
Néel [5, 48, 49] and anti-skyrmions [6] provide a fruitful route for further
investigations.

## Acknowledgements

This work is financially supported by the EPSRC UK Skyrmion Project Grant
EP/N032128/1. We would like to thank Steve York (University of Warwick) for
assistance with the SEM measurements and David W. Lester (University of
Warwick) for help with the laser diffraction measurements.

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[a,b]Tamas G. Kovacs

# Topology and the Dirac Spectrum in Hot QCD

###### Abstract

It is known that contrary to expectations, the order parameter of chiral
symmetry breaking, the Dirac spectral density at zero virtuality does not
vanish above the critical temperature of QCD. Instead, the spectral density
develops a pronounced peak at zero. We show that the spectral density in the
peak has large violations of the expected volume scaling. This anomalous
scaling and the statistics of these eigenmodes is consistent with them being
produced by mixing instanton and antiinstanton zero modes. Consequently, we
show that a nonvanishing topological susceptibility implies a finite density
of eigenvalues around zero, which can have implications on the restoration of
chiral symmetry above the critical temperature.

##  1 Introduction

The spectrum of the quark Dirac operator has long been known to provide
information about the physics of QCD. In particular, the long-distance physics
is encoded in the low-lying, infrared part of the Dirac spectrum. The spectral
quantity that has probably the simplest direct connection to the physics is
the spectral density at zero virtuality, $\rho(0)$, which – by the Banks-
Casher relation – is the order parameter of chiral symmetry breaking [1].
Indeed, according to the standard picture of the finite temperature transition
in QCD, in the low temperature, hadronic phase $\rho(0)$ is nonzero, signaling
the spontaneous breaking of chiral symmetry. As the system is heated and
enters the high temperature, quark-gluon plasma phase and chiral symmetry gets
restored, $\rho(0)$ becomes zero.

Strictly speaking, the above picture applies only to an imaginary world, where
quarks are massless and the chiral symmetry of the Lagrangian is exact. In the
real world the quark masses are very small compared to the QCD scale, and
chiral symmetry is only approximate. As a result, the finite temperature
transition is only a crossover, and instead of jumping to zero at a well
defined critical temperature, $\rho(0)$ only rapidly decreases through the
crossover region, as the system enters the quark-gluon plasma state.

This simple picture, however, was later challenged. The first sign of a more
complicated behavior started to emerge, when in the very early days of the
overlap Dirac operator, it was noticed that in the high temperature phase of
the quenched theory the spectral density of the overlap Dirac operator
develops an unexpected sharp peak at zero virtuality [2]. Even though, this
was in the quenched theory, where chiral symmetry is not straightforward to
interpret, this behavior of the spectral density was rather counterintuitive.
These simulations were performed on rather coarse lattices ($N_{t}=4$, just
above $T_{c}$), nevertheless the tentative interpretation of the unexpected
spectral peak was that it was due to topology-related would-be zero modes of
the Dirac operator, and resolving them was made possible only by the overlap
operator, thanks to its exact chiral symmetry. We emphasize that these are not
the chiral zero modes, corresponding to the net topological charge – those can
be easily distinguished in the overlap spectrum. Rather, the peak in the
spectral density was suggested to be due to mixing zero modes of instantons
and antiinstantons111In the present work, for simplicity we will refer to
lumps of unit topological charge as (anti)instantons, even though close to
$T_{c}$ in the high temperature phase their charge profile might not even be
close to that of calorons. For our discussion, the only important property of
these objects is that they are isolated lumps of topological charge of unit
magnitude., eigenmodes that come in complex conjugate pairs and can be close
to zero if their splitting due to mixing is small [3]. One could also argue
that these unusual small modes could be just cutoff effects caused by the
coarse lattice, or a quenched artifact, since fluctuations of the topological
charge are expected to be suppressed by dynamical quarks. This is, however,
not the case, as was shown in several further works using much finer lattices,
as well as light dynamical quarks [4, 5, 6]. It was even suggested that the
approximate $1/\lambda$ shape of the spectral peak signals a previously
unnoticed “phase” of QCD, intermediate between the hadronic and the quark-
gluon plasma phase. In the present work we will show further evidence that the
initial interpretation of the spectral peak as mixing topological near zero
modes was correct.

##  2 The spectral peak

We work in the quenched approximation, but eventually will argue that our
results also apply to the case with dynamical fermions, at least in a
qualitative fashion. Let us first look at how the spectral density of the
overlap Dirac operator changes across the finite temperature phase transition.
In the quenched case with $SU(3)$ gauge group, this is a genuine first order
phase transition, and tuning the coupling to the critical point allows us to
sample both phases at the same value of the coupling. In Fig. 1 we show the
spectral density for an ensemble of $N_{t}=8$ lattices at Wilson gauge
coupling $\beta=6.063$, which is the critical value for the given temporal
lattice extension. Based on the magnitude of the Polyakov loop, we separated
the configurations into two sets, belonging to the two phases, and computed
the spectral density separately in the two phases222In the deconfined phase,
only configurations in the “physical”(real) Polyakov loop sector were used. In
the complex sectors the spectral density and the properties of the
corresponding eigenmodes are very different [7].. The difference is quite
clear. For most of the spectrum the spectral density is much lower in the
deconfined phase, a behavior, consistent with the naive expectation of the
spectral density jumping to a lower value when the system enters the quark-
gluon plasma phase. However, very close to zero virtuality, the deconfined
phase shows a marked peak that goes much higher than the spectral density in
the confined phase.

Figure 1:  The spectral density of the overlap Dirac operator on an ensemble
of $N_{t}=8$ quenched lattices at the critical point. Based on the magnitude
of the Polyakov loop, the confined and deconfined phases were separated, and
their spectral densities are plotted separately.

Let us further examine the peak by looking at its volume dependence. To be
able to do so in a cleaner situation where we do not have to separate the two
phases, we chose to do it at a slightly higher temperature, $T=1.045T_{c}$. In
Fig. 2 we show the spectral density at $N_{t}=8$, Wilson $\beta=6.09$ for
ensembles of four different linear sizes, $L=32,40,48,56$. As expected, the
(volume normalized) spectral density is volume independent, except at the
spectral peak where it shows a rather strong volume dependence.

Figure 2:  The spectral density of the overlap Dirac operator on four
ensembles of $N_{t}=8$ lattices at $T=1.045T_{c}$. The spatial linear lattice
sizes range from $L=32$ to 56.

Figure 3:  The integrated spectral density of the four ensembles of Fig. 2. In
the left panel the exact zero modes were not included in the integral, while
in the right panel they were.

The lack of volume scaling at the very low-end of the spectral density is also
manifest in the left panel of Fig. 3, where we show the integrated spectral
density. It is important to keep in mind that both in Fig. 2, showing the
spectral density and in the left panel of Fig. 3, showing its integral, we did
not include the exact zero modes. In the density they would show up as a Dirac
delta at the origin, which would shift the integrated density curves upward.
To check how this affects the integrated spectral densities, in the right
panel of Fig. 3 we also included these shifts due to the zero mode Dirac
deltas, and – as can be seen in the figure – the resulting curves for the
different volumes are thus shifted on top of one another. In other words, this
means that the total number of low modes scales properly with the volume, and
the “deficit” we observe in the near zero modes in the smaller volumes is made
up by the relatively larger density of exact zero modes there.

This indicates that the strong volume dependence of the spectral density is
connected to the zero modes and topological charge. However, these large
finite-size corrections to the density still seem quite counter-intuitive, and
it is not even clear from Fig. 2 whether the spectral density at zero
converges to a finite value. Assuming that the modes in the spectral peak are
mixing topological zero modes, we can have a more quantitative assessment of
the expected finite-size effects in the spectral density. Indeed, the density
of topological objects is proportional to the volume, and – on average – half
of them are instantons, half antiinstantons. If in a configuration there are
$n_{\scriptscriptstyle+}$ instantons and $n_{\scriptscriptstyle-}$
antiinstantons, their would be zero modes combine into
$|n_{\scriptscriptstyle+}-n_{\scriptscriptstyle-}|$ exact chiral zero modes,
and the remaining would be zero modes will mix into non-chiral eigenmodes
close to zero, making up the spectral peak. $\langle
n_{\scriptscriptstyle+}+n_{\scriptscriptstyle-}\rangle$ scales with the
volume, but

$\langle|n_{\scriptscriptstyle+}-n_{\scriptscriptstyle-}|\rangle=\langle|Q|\rangle\propto
V^{1/2},$ (1)

where $Q$ is the topological charge. So the volume scaling of the number of
non-chiral close to zero modes is expected to have a large finite-size
correction, vanishing in the thermodynamic limit only as $V^{-1/2}$. This is
because some of the zero modes carried by the (anti)instantons combine into
exact zero modes and do not show up in the spectral density arbitrarily close
to zero. This deficit of close to zero modes is seen in the spectral density
and causes the large finite-size correction. The missing modes, however, are
not lost, they make up the exact zero modes, and – as we already saw – by
their inclusion in the integrated spectral density, we can remove the large
finite-size correction.

##  3 Ideal instanton gas

Above the critical temperature, the topological susceptibility rapidly
decreases [8, 9, 10], the instanton gas becomes dilute and to a good
approximation the (anti)instantons form an ideal (non-interacting) gas [11,
12]. In such a dilute gas, topological objects are typically far away from one
another and the splitting from the origin of the non-chiral near zero modes of
topological origin is expected to be small. Therefore, it is a reasonable
assumption that these eigenvalues of topological origin are the ones closest
to zero. They occupy a spectral region $[-\lambda_{zmz},\lambda_{zmz}]$ that
we call the zero mode zone.

Figure 4:  The spectral density of the overlap Dirac operator on four
ensembles of $N_{t}=8$ lattices at $T=1.045T_{c}$. The blue vertical bar
indicates $\lambda_{zmz}$, the boundary of the zero mode zone, its width is
the uncertainty.

Using the ideal instanton gas picture, we can estimate $\lambda_{zmz}$ by
noting that in an ideal gas the number of instantons and antiinstantons follow
independent and identical Poisson distributions. The parameter of this
distribution, giving the density of topological objects is $\chi V$, where
$\chi$ is the topological susceptibility and $V$ is the space-time volume of
the system. It follows from the properties of Poisson distributions that the
density of topological objects is

$\langle n_{\scriptscriptstyle+}+n_{\scriptscriptstyle-}\rangle=\chi V.$ (2)

From the number of exact zero modes of the overlap Dirac operator, the
topological susceptibility can be easily computed, and it is also related to
the average number of topological objects as

$\chi V=\langle
Q^{2}\rangle=\langle(n_{\scriptscriptstyle+}-n_{\scriptscriptstyle-})^{2}\rangle.$
(3)

Using the susceptibility, obtained from the exact zero modes, we can estimate
$\lambda_{zmz}$ by assuming that to each instanton and antiinstanton, there is
a corresponding eigenvalue in the zero mode zone. This is done by setting
$\lambda_{zmz}$ such that on average there be $\chi V$ eigenvalues in the
$[-\lambda_{zmz},\lambda_{zmz}]$ interval, including the exact zero modes. In
Fig. 4 we show $\lambda_{zmz}$, the boundary of the zero mode zone, computed
in the above described way. As can be seen in the figure, the zero mode zone
includes the whole of the spectral peak and the lower part of the wide valley
in the spectral density, located between the peak at zero and the bulk of the
spectrum, where the density starts to rise rapidly.

If all these eigenmodes in the zero mode zone are associated to topological
lumps of unit charge that form an ideal gas, then the distribution of the
number of these eigenmodes should follow a Poisson distribution with mean
$\chi V$. We already fixed the mean to $\chi V$, but it is a nontrivial test
whether the distribution is actually Poissonian. This can be easily checked in
the lattice overlap spectra by counting the number of eigenvalues in the zero
mode zone, configuration by configuration. We show the result in Fig. 5. Also
shown in the figure is the expected Poisson distribution, that is the
distribution of the number of topological lumps in an ideal instanton gas with
the given susceptibility. Notice that there is no further fitting involved
here, as the only parameter of the distribution, $\chi V$ has already been
determined from the exact zero modes of the overlap. The agreement with the
lattice data is fairly good, indicating that the topological objects form a
non-interacting gas.

Figure 5:  The distribution of the number of eigenmodes in the zero mode zone.
The red crosses are the lattice data on an ensemble of $N_{t}=8,V=32^{3}$
lattices at a temperature of $T=1.045T_{c}$. The blue circles represent the
expected ideal gas distribution of the number of topological objects with
their density equal to the density of eigenvalues in the zero mode zone.
Figure 6:  The chi squared per degree of freedom of a fit to a Poisson
distribution of the number of eigenvalues in an interval
$[-\lambda_{cut},\lambda_{cut}]$ as a function of $\lambda_{cut}$. The blue
bar represents the previously determined value of $\lambda_{zmz}$ with its
width being the uncertainty.

At this point one could also ask the question whether $\lambda_{zmz}$ is
really a special point of the spectrum in the sense that the number of
eigenvalues in the $[-\lambda_{zmz},\lambda_{zmz}]$ interval is Poisson
distributed, but for other intervals it cannot be described with a Poisson
distribution. To see this, we tried to fit the distribution of the number of
eigenvalues in intervals $[-\lambda_{cut},\lambda_{cut}]$ for different values
of $\lambda_{cut}$. The chi squared per degree of freedom as a function of
$\lambda_{cut}$ is shown in Fig. 6. Indeed, $\lambda_{zmz}$ is in the middle
of the spectral region where the chi squared of the fit is acceptable, and the
distribution is close to a Poisson distribution. Notice that this region is
rather wide because of the simple reason that the spectral density is very
small there, and by increasing $\lambda_{cut}$, so few eigenvalues are added
to the data set that they do not influence the quality of the fit
significantly.

##  4 Conclusions

In the present paper we studied the peak at zero virtuality in the spectral
density of the QCD Dirac operator. We saw that the spectral density at the
peak has unexpectedly large finite size corrections. However, this volume
dependence of the spectral density can be understood if we assume that the
eigenmodes in the spectral peak are mixing instanton-antiinstanton zero modes,
originating from a non-interacting gas of topological objects. The
distribution of the number of eigenmodes in the peak also supports this
picture.

The most important consequence of our findings is that there is a strong
connection between the topological susceptibility and the total number of
eigenvalues in the zero mode zone, most of which are in the spectral peak. In
particular, a nonzero topological susceptibility implies the presence of a
corresponding spectral peak, which means that chiral symmetry is not
immediately restored in the high temperature phase.

We have to add some qualifications to the above statement. The numerical
results we presented, were obtained in the quenched approximation. Dynamical
fermions are expected to suppress fluctuations of the topological charge, and
will definitely affect both the topological susceptibility and the spectral
peak. However, as long as the fermions do not induce strong interactions among
topological objects, the connection between the susceptibility and the
spectral peak remains valid. Moreover, even if light dynamical quarks induce
strong interactions among instantons, we expect this effect to suppress the
net topological charge, i.e. the susceptibility, more than the near zero modes
in the peak. This means that even in the presence of fermion-induced
interactions, we expect a nonzero topological susceptibility to imply the
presence of a spectral peak. It is a dynamical question how at a given value
of the quark mass the breaking of chiral symmetry due to the spectral peak
compares to the explicit breaking by the quark mass. Presently there is
ongoing work to explore how dynamical quarks influence the spectral peak.

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