# Expanding Microscopic Black Holes

Samuel Kováčik  Faculty of Mathematics, Physics and Informatics, Comenius
University in Bratislava,
Mlynská dolina F1, Bratislava, 842 48, Slovakia
<EMAIL_ADDRESS>Department of Theoretical Physics and
Astrophysics, Faculty of Science, Masaryk University
Kotlářská 267/2, Brno, 611 37, Czech Republic

###### Abstract

Two interesting hypotheses about black holes have been proposed. The older one
states that microscopic black holes can be accountable for the observed dark
matter density. The newer one states that black holes are coupled to the
expansion of the universe. Here, we combine those ideas and investigate the
behaviour of expanding microscopic black holes. We observe two temperatures at
which the radiation balances the expansion. While one of the balance points
might be important in the analysis of primordial scenarios, the other would
lead to a strong diffuse gamma radiation background, which is contradicted by
the lack of observations. This establishes another indirect evidence
disfavouring the hypothesis of cosmological coupling of black holes.

###### keywords:

Regular black holes; expanding black holes; Hawking radiation .

##  1 Introduction

Two of the most astonishing findings of Einstein’s theory of general
relativity are the expansion of the universe and the existence of black holes.
Each of the concepts is still being actively researched. In the case of black
holes, a series of papers investigates models with a regular inner structure
[1, 2, 3, 4, 5, 6, 7, 8], possibly effectively described as having a de Sitter
core.

Perhaps the most daunting open question related to the expansion of the
universe is the nature of the force driving it today — currently being
described as dark energy, which is being studied thoroughly but is far from
being properly understood [9, 10, 11].

Recently, several papers analyzed black holes and other relativistic objects
coupled to the expansions of the universe [12, 13, 14]; in Croker:2021duf, the
authors considered cosmological coupled black holes in the context of
detection of gravitational waves. In Croker:2019mup, the Friedman equation was
derived from the Einstein-Hilbert action with an explicit formulation of the
averaging procedure. A startling consequence of the analysis was that
relativistic compact objects, such as de Sitter-core black holes, should be
expanding with the universe. An exciting scenario becomes possible if the size
— and therefore the mass — of black holes scales with the cosmological scale
factor. The energy of a black hole may grow as $\sim a^{3}$, which would
cancel the $\sim a^{-3}$ decrease in the concentration of black holes in a
proper volume. As a result, the energy contribution of black holes to the
universe’s energy budget would be constant and could be the driving force of
its expansion. In Croker:2019kje, it was suggested that compact remnants of
population III stars would correspond to the currently observed contribution
of dark energy. Also, a recent observational study suggests the assumed
scaling factor [16] — an intriguing picture began to emerge.

So far, every black hole observed, directly or indirectly, had a mass above
the limit set by the stellar collapse scenario. In the current stage of the
universe, this seems to be the only way of creating a macroscopic black hole —
first being formed by a gravitational collapse of a stellar object and then
growing either by accretion or merger with other black holes [17, 18, 19].
However, it has been suggested that black holes of mass below the stellar
limit could have been formed during the universe’s earliest stages from local
overdensities [20, 21, 22]. While for stellar or supermassive black holes, the
Hawking radiation is negligible, it would be crucial for sufficiently small
black holes — the absence of a signal of this type applies substantial
restrictions on theories of small primordial black holes [23]. A hint of a
subsolar black hole was reported recently in Morras:2023jvb.

The temperature of the Hawking radiation from a Schwarzschild black hole is
inversely proportional to its mass. That means that as the black hole
evaporates, it heats up, and the evaporation process accelerates. This would
suggest that the presence of microscopic black holes can be disregarded, as
suggested by Hawking [25] for black holes with mass under $10^{15}g$. However,
it is now understood that quantum effects can change this picture profoundly.
For theories with a fundamental length scale $\lambda$, which is usually
assumed to be the Planck length, the behaviour of microscopic black holes
shifts drastically once their Schwarzschild radius reaches
$r_{H}\approx\lambda$. In Kovacik:2021qms, it was shown that a large class of
black holes that have a smeared singularity ––– so-called regular black holes
— have a vanishing Hawking temperature at this scale. This feature has been
observed in particular examples before [27, 28, 29]. Instead of evaporating
completely, they form a Planckian remnant, which is frozen and small enough
not to interact with other fields; its cross-section is on the order of
$\lambda^{2}$. Such objects were hypothesized to be plausible dark matter
constituents [29, 30, 31, 32, 33, 34]; recently, new details were discussed,
either on the exact moment of their formation [26, 35], the scale of the
underlying structure $\lambda$ [36] or their M-theory or stringy aspects [37,
38].

This paper investigates how microscopic black holes would behave had they been
expanding with the universe. We show that if small black holes exist and make
a significant proportion of the dark matter, they would, due to the expansion
of the space, produce a strong gamma signal, which is not being observed. We
also discuss how the cosmological coupling of the black holes would affect
primordial black hole formation.

##  2 Small black holes

Hawking deduced [39, 25] that black holes have a temperature inversely
proportional to their mass:

$T=\frac{\hbar c^{3}}{8\pi Gk_{\scriptsize{B}}M}.$ (1)

Hawking radiation has a negligible effect on stellar black holes but may be
substantial in late-age scenarios of the universe [40]. However, small black
holes have a temperature that tends to infinity in the final moments of the
evaporation process.

Primordial black holes (PBH) formed from matter overdensities in the
universe’s earliest stages and can have masses below the range of stellar
black holes [41]; observing a gravitational wave signal from them might be
within reach shortly either with space or ground-based detectors [42, 43, 44].
For such black holes, the radiation can be arbitrarily energetic — lack of
such signal places severe restrictions on the parameter space [23]. Recently,
pulsar time array data were used to put constraints on the primordial power
spectrum, which also limits the overproduction of PBH; however, additional
data are required to strengthen these claims. Another possible probe of PBH is
the analysis of dark-halo microlensing [45].

So far, our discussion has only concerned singular black holes. Various
theories considering the quantum effects predict that the singularity is
blurred — this is, for example, a rather general prediction of theories of
quantum space. In Kovacik:2021qms, three different classes of spherically
symmetrical regular matter densities were analysed:

$\displaystyle\rho_{1}(r)$ $\displaystyle\sim$ $\displaystyle
e^{-\left(r/\lambda\right)^{q}},$ $\displaystyle\rho_{2}(r)$
$\displaystyle\sim$ $\displaystyle\left(1+\left(r/\lambda\right)\right)^{-q},$
(2) $\displaystyle\rho_{3}(r)$ $\displaystyle\sim$
$\displaystyle\left(1+\left(r/\lambda\right)^{q}\right)^{-1}.$

Here, $r$ is the radial coordinate, $\lambda$ is the fundamental lengh-scale
and $q$ is a model-dependent parameter, and $\lambda$ is a constant with the
dimension of length that describes the scale of the quantumness of space. The
previously considered cases of microscopic black holes [27, 28, 29] fall
within $\rho_{1}$ with $q=1,2,3$. The similarity in their behaviour was
extended in Kovacik:2021qms to all three classes with $q$ up to $15$.
Therefore, the behaviour discussed in what follows can be considered robust.

It has been shown in this general setting that the temperature is regulated
when the size of the horizon approaches the fundamental length scale,
$\lambda$:

$T_{\mbox{\scriptsize{reg}}}(m)=Q_{\scriptsize{\lambda}}(m)\times
T_{\mbox{\scriptsize{sing}}}(m),$ (3)

where $T_{\mbox{\scriptsize{sing}}}$ is the temperature of a black hole with a
singular mass density and $T_{\mbox{\scriptsize{reg}}}$ is the temperature of
a black hole with regular matter distribution. For the considered cases,
$Q_{\scriptsize{\lambda}}(m)$ is a monotonic function that grows from
$Q_{\scriptsize{\lambda}}(m_{0})=0$ to $Q_{\scriptsize{\lambda}}(\infty)=1$,
see figure 1. Here, $m_{0}$ is the minimal mass of the black hole. For
$m<m_{0}$, the object has no event horizons; for $m>m_{0}$, there are two, and
the radiation temperature from the outer horizon is positive. For $m_{0}$, the
two horizons overlap, and consequently, the temperature vanishes. Instead of
evaporating with unbounded temperatures, black holes with blurred singularity
have a maximal temperature below which they quickly freeze, and the radiation
ceases. This happens as they reach the fundamental length scale size, which is
assumed to be the Planck length — so these objects are sometimes referred to
as Planck size remnants or Planckian black holes [46, 47, 48]. They have a
negligible cross-section and, therefore, do not interact with matter or
radiation, so they do not accrete. Being massive and not interacting with
ordinary matter or radiation makes them a viable dark-matter candidate. To
make up for the observed dark matter density, their concentration needs to be
of the order of $n\sim 10^{-20}\ m^{-3}$, see Kovacik:2017tlg, and unless the
fundamental scale is far away from the Planck length [36]; their production
has to be completed at the age of the universe around $t\approx 10^{-28}\ s$,
see Kovacik:2021qms, Lehmann:2021ijf.

Figure 1: Comparison of the rescaled temperature profiles $T/T_{M}$ of
Schwarzschild (dotted) and regular (solid) black holes. Values were obtained
for the model where the mass density inside the black hole behaves as
$\rho(r)\sim\exp\left(-r/\lambda\right)$ but the shape of the curve is
general. In this case, the minimal horizon value is $r_{\scriptsize{0}}\approx
3.38\lambda$ and $T_{M}\approx 0.01\frac{\hbar c}{k_{\scriptsize{B}}\lambda}$.
Figure 2:  Comparison of the event horizon radius of the Schwarzschild black
hole, $r_{h}=2m$, (dotted) and a regular black hole (solid). Values were
obtained for the model where the mass density inside the black hole is
$\rho(r)\sim\exp\left(-r/\lambda\right)$. In this case, the minimal horizon
value is $r_{\scriptsize{0}}\approx 3.38\ \lambda$. The plot shows that taking
$r_{h}=r_{\scriptsize{0}}$ for $m\approx m_{\scriptsize{0}}$ and $r_{h}=2m$
elsewhere is justified — the difference is a factor of the order of $1$.

There are two essential properties of regular black holes. The first is the
form of the solution that resembles, for a nonrotating and electrically
neutral source, the Schwarzschild solution for $r\gg\lambda$ while for
$r\approx\lambda$, it tends to the de Sitter solution. This is the effect of a
quantum space that prevents a further collapse and acts as an effective
outward pressure. The second important property is the general shape of the
temperature profile, as discussed in the previous paragraph.

Therefore, we will assume this general behaviour and investigate the
consequences of coupling a microscopic black hole to the universe’s expansion.
A similar case ignoring the radiation, therefore valid for large regular black
holes, was investigated recently [49]. The focus of our analysis will be the
interaction of two competing effects — the expansion due to the expansion of
space and their shrinking due to Hawking radiation. There is an implicitly
time-dependent temperature where these two effects are balanced:

$\left.\frac{dm}{dt}\right|_{\mbox{\scriptsize{exp}}}=3mH=4\pi
r_{\scriptsize{h}}^{2}\sigma_{B}c^{-2}T^{4}=-\left.\frac{dm}{dt}\right|_{\mbox{\scriptsize{rad}}},$
(4)

where we took the mass of the black hole to be coupled to the expansion of the
universe as $m(t)=m(t_{i})\left(a/a_{i}\right)^{3}$ as suggested in
Farrah:2023opk and $r_{\scriptsize{h}}$ denotes the outer black hole horizon.
Also, $H=\dot{a}/a$ is the time-dependent Hubble parameter.

Let us denote $T=T_{\scriptsize{B}}$ the temperature for which equation 4
holds. A balance can be reached as long as $T_{B}<T_{M}$, where $T_{M}$ is the
maximal temperature of a black hole. For the classes of solutions considered
here, the maximal temperature is typically two or three orders below the
Planck temperature; see Kovacik:2021qms. Because of the general form of the
temperature profile $T(m)$ shown in figure 1, there are two possible masses at
which the balance is reached; we denote them $m_{\scriptsize{1}}$ and
$m_{\scriptsize{2}}$. Because the balance in equation 4 is of the form $\alpha
r_{\scriptsize{h}}^{-1}=T^{4}$, it holds that
$T(m_{\scriptsize{1}})>T(m_{\scriptsize{2}})$. Typically $m_{\scriptsize{1}}$
is close to the minimal mass, $m_{\scriptsize{0}}$, where the temperature is
zero but grows rapidly above it, and $m_{\scriptsize{2}}$ is considerably
larger:

$m_{\scriptsize{0}}\lesssim m_{\scriptsize{1}}\ll m_{\scriptsize{2}}.$ (5)

We will now consider those two cases of balancing masses separately.

##  3 Balancing at $m_{\scriptsize{1}}$ and background radiation from
expanding microscopic black holes

As expressed in equation 3, the Hawking temperature vanishes as the minimal
mass $m_{\scriptsize{0}}$ is reached. In cosmological models, the Planckian
remnants with $m=m_{\scriptsize{0}}$ are assumed to be frozen and not
radiating. However, if the black holes are coupled to the universe’s
expansion, a different balance point is reached; this is expressed in equation
4. Close to the minimal mass, $m\gtrsim m_{\scriptsize{0}}$, the temperature
rapidly grows from zero. In Kovacik:2017tlg, it was calculated that the
temperature close to this point behaves as

$T(m)\sim
T_{M}\left(\frac{m-m_{\scriptsize{0}}}{m_{\scriptsize{0}}}\right)^{1/2},$ (6)

where $T_{M}$ is the maximal temperature reachable by a black hole — typically
a few orders below the Planck temperature [26]. The numerical factor in this
relation is model-dependent but typically is of the order of $1$; for example,
for the regular matter density, $\rho_{1}(r)$ with $q=1$ as given in 2 is the
numerical factor $4.09$. Also, there is a non-trivial factor in the relation
between the mass and the horizon radius, $r_{h}$, which is of the same order
and which we will ignore here, see LABEL:figure2. The balance is now reached
at

$3\ m_{\scriptsize{1}}c^{2}H=4\pi
r_{h}^{2}\sigma_{B}T_{M}^{4}\left(\frac{m_{\scriptsize{1}}-m_{\scriptsize{0}}}{m_{\scriptsize{0}}}\right)^{2}.$
(7)

Since the mass dependence of Hawking radiation is different around the small
mass $m_{1}$, see 6, and the large mass $m_{2}$, where we can take $T(m)\sim
m^{-1}$, so is different the nature of the balance. For the larger mass, the
balance in equation 4 is expressed in the form
${\beta_{\scriptsize{1}}(m)=m-\alpha_{1}m^{-2}=0}$ which is unstable since
${\left.\frac{d\beta_{\scriptsize{1}}}{dm}\right|_{m_{\scriptsize{2}}}>0}$. An
increase in mass leads to an increase in the growth rate and vice versa.
However, the second balancing equation 7 is of the form
$\beta_{\scriptsize{2}}(m)=m-\alpha_{2}m^{2}=0$ and since
$\left.\frac{d\beta_{\scriptsize{2}}}{dm}\right|_{m_{\scriptsize{1}}}<0$ the
opposite happens — increasing mass leads to increased radiation, and the
system is returned to its equilibrium in a Le Chatelier way. Equation 7 can be
put in the form:

$\gamma\
m_{\scriptsize{1}}=\left(m_{\scriptsize{1}}-m_{\scriptsize{0}}\right)^{2},$
(8)

where $\gamma=\frac{3Hm_{\scriptsize{0}}^{2}c^{2}}{4\pi
r_{\scriptsize{0}}^{2}\sigma_{B}T_{M}^{4}}$. For the current value of the
Hubble parameter, we have $\gamma_{\scriptsize{0}}=2.85\times 10^{-61}kg$. The
solution to this is, within the leading order approximation,
$m_{\scriptsize{1}}=m_{\scriptsize{0}}+\sqrt{m_{\scriptsize{0}}\gamma_{\scriptsize{0}}}$,
where the correction is indeed minimal $\frac{\delta
m_{1}}{m_{\scriptsize{0}}}\approx 2.25\times 10^{-27}$. If we evaluate the
temperature as expressed in equation 6, we obtain $T_{\scriptsize{B}}\approx
6.32\times 10^{16}K$ — which is considerably large.

That means that if the dark matter is made of Planckian remnants and those
remnants are coupled to the universe’s expansion, they should be radiating
restlessly with the temperature of $6.32\times 10^{16}K$. This translates into
the energy of radiated particles $16$ orders below the Planck energy. If we
compare the power at this temperature with the peak particle energy, we obtain
that an emission happens with the frequency of $f\approx 0.013\ s^{-1}$.

In standard physical models, ultra-energetic are attenuated photons travelling
through space due to interaction with low-energy background photons [50, 51].
A distant, $z=0.151$, the ultra-energetic signal has been observed recently,
[52, 53, 54, 55, 56, 57, 58]. Possible explanations of this phenomenon are
particle physics beyond the standard model [59, 60] or non-trivial vacuum
dispersion [61, 62, 63]. However, will focus only on local sources here; it
was estimated in Posti that the mass of the dark matter in the inner part of
the Milky Way is of the order of $\approx 10^{11}{M_{\odot}}$ which is
$\approx 10^{49}m_{\scriptsize{\lambda}}$, recall that
$m_{\scriptsize{\lambda}}$ is the Planck mass. If $\approx 10^{49}$
microscopic black holes were radiating thermal radiation with temperature
$6.32\times 10^{16}K$ with the peak energy $~{}15.3TeV$, each having surface
area $4\pi\lambda^{2}$, they would be radiating $\approx 0.01$ high-energy
particles isotropically every second, the total flux on Earth would be
$1.82\times 10^{5}m^{-2}s^{-1}$. Such an ultra-energetic gama background is,
however, not being observed.

##  4 Balancing at $m_{\scriptsize{2}}$ and primordial formation of black
holes

For sufficiently large masses, the regular and singular solutions become
nearly indistinguishable. In the considered models is the minimal mass of the
order of the Planck mass, $m_{\scriptsize{0}}\approx
m_{\scriptsize{\lambda}}=\frac{\hbar}{\lambda c}$. At the larger balancing
mass, we have $m_{\scriptsize{2}}\gg m_{\scriptsize{0}}$, and so we can take
the Hawking temperature as expressed in equation 1. The value of
$m_{\scriptsize{2}}$ for which the balancing temperature is reached is given
by the equation

$3m_{\scriptsize{2}}c^{2}H=\frac{\hbar c^{6}}{15360\pi
G^{2}m_{\scriptsize{2}}^{2}}.$ (9)

This can be expressed as

$m_{2}=\eta\ H^{-1/3},$ (10)

where $\eta=\left(\frac{c^{4/3}\hbar^{1/3}}{46080^{1/3}g^{2/3}}\right)\approx
109\ 807\ \mbox{kg}\ \mbox{s}^{-1/3}$. Today, with $H_{0}\approx 2.2\times
10^{-18}\ s^{-1}$, is the balancing mass $m_{\scriptsize{2,0}}\approx
8.41\times 10^{10}\ kg$.

This balancing point is unstable. If the mass increases,
$m>m_{\scriptsize{2}}$, the expansion overtakes the radiation, the black hole
keeps expanding, and the Hawking radiation becomes negligible. The black hole
mass keeps scaling as $\sim a^{3}$, therefore mimicking the behaviour of the
dark energy. In addition to that, the growth can be accelerated by the Bondi
accretion [65, 66, 67]. On the other hand, if the mass decreases,
$m<m_{\scriptsize{2}}$, the evaporation takes over. It dominates until the
point where quantum effects occur, and the temperature reduces according to
equation 3. That means the black holes are driven from the balancing mass
$m_{\scriptsize{2}}$.

At the current stage of the universe, the observed black holes are well above
the balancing mass 10. This discussion can be, however, relevant when
considering the earliest stages of the universe, where some of the radiation
and expansion scales were of the same order. The growth equation takes the
form:

$\dot{m}=3Hm-\Gamma m^{-2},$ (11)

and whether the radiation or expansion dominates depends on the Hubble
parameter, $H$, see figure 4. As a result, the primordial black hole spectrum
would be split into two parts, small black holes would be driven towards the
mass $m_{1}$ while the large ones would grow indefinitely. Had we not
concluded in the previous section that the existence of small microscopic
black holes coupled to the expansion of the universe is contradicted by the
lack of a strong gamma background, this would be a daunting model of
primordial formation of both the dark matter (small black holes with mass
$m_{1}$) and dark energy (large mass being driven away from $m_{2}$).

Figure 3: Mass evolution during a scenario when the expansion and radiation
scales are comparable. There are three options: either the expansion dominates
at every moment (dot-dashed line), it is surpassed at a certain point by the
radiation (solid line), or the radiation always dominates (dashed line).

##  5 Conclusion

Microscopic black holes have been of some theoretical interest in recent years
due to their peculiar properties — the most important of which is their
vanishing Hawking temperature at a sufficiently small scale. The black hole
needs to have a regular, usually de Sitter, core for this modification to be
present. This behaviour is a general feature of quantum space models;
therefore, it is reasonable to assume that the relevant scale is that of
Planck units.

In this paper, we have discussed the interplay of two hypothetical concepts:
black holes coupled to the expansion of the universe and microscopic black
holes with regular matter distribution. On certain scales, the effects of
expansion due to cosmological coupling and shrinking due to Hawking radiation
are of the same magnitude. There are two different masses for which an exact
balance is reached. They are not alike — as balancing around the smaller mass
is stable while around the larger one is unstable.

Expansion of the universe would be moving Planck-size black holes from the
frozen state, making them radiate restlessly. If the dark matter at our
galactic core was composed of radiating black holes, they would produce a
gamma-background flux that has not been observed. We have considered only dark
matter in our galaxy, but perhaps in this model, all dark matter in the
universe would create a cosmic gamma background. Also, if the dark energy was
sourced from PBH it would affect the delicate process of early nucleosynthesis
and source primordial perturbations — the exact observational signatures of
this scenario require more research. Still, apriori, it doesn’t seem easy to
incorporate this into current models of the early-stage universe.

There are a couple of possible explanations why this has not been observed:
small black holes have never been formed, or black holes are not coupled to
the universe’s expansion and are therefore not forced to expand. However,
there are other options; for example, Hawking radiation can be in the form of
unobservable particles, or the fundamental scale that needs to be considered
is not the Planck one. Also, details of the radiation process for microscopic
black holes are not fully understood, and the scenario needs to be
investigated using a proper model of quantum gravity.

The other balancing point around the mass $m_{2}>m_{1}$ turned out to be
unstable. The primordial mass spectrum would be split into two classes of
black holes: one coupled to the expansion of the universe and the other of
black holes that evaporated and their mass tended to $m_{1}$.

In this paper, we have discussed the case of
$m(t)=m(t_{i})\left(a/a_{i}\right)^{3}$ as this is the option most frequently
present in the literature. Other choices of the scaling coefficient would not
change our results considerably.

The bottom line is that abundant expanding microscopic black holes seem
incompatible with current observations not detecting TeV photon background.
This means that either Planck scale black holes do not exist or the black
holes are not coupled to the universe’s expansion, as other authors suggested
recently for various reasons [68, 69, 70, 71, 72, 73].

## Acknowledgement

This research was supported by VEGA 1/0025/23 and the MUNI Award for Science
and Humanities, funded by the Grant Agency of Masaryk University. The author
would like to thank J. Tekel and P. Mészáros for their valuable comments.

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††thanks: These authors contributed equally††thanks: These authors contributed
equally, Corresponding author††thanks: Corresponding author

# Imaging the coherent propagation of collective modes in the excitonic
insulator Ta2NiSe5 at room temperature

Hope M. Bretscher  Paolo Andrich  Cavendish Laboratory, University of
Cambridge, Cambridge CB3 0HE, United Kingdom  Yuta Murakami  Department of
Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan  Denis
Golež  Center for Computational Quantum Physics, Flatiron Institute, New York,
New York 10010, USA  Faculty of Mathematics and Physics, University of
Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia  Jožef Stefan Institute,
Jamova 39, SI-1000, Ljubljana, Slovenia  Benjamin Remez  Cavendish Laboratory,
University of Cambridge, Cambridge CB3 0HE, United Kingdom  Prachi Telang
Anupam Singh  Luminita Harnagea  Department of Physics, Indian Institute of
Science Education and Research, Pune, Maharashtra 411008, India  Nigel R.
Cooper  Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE,
United Kingdom  Andrew J. Millis  Center for Computational Quantum Physics,
Flatiron Institute, New York, New York 10010, USA  Department of Physics,
Columbia University, New York, New York 10027, USA  Philipp Werner  Department
of Physics, University of Fribourg, Fribourg 1700, Switzerland  A. K. Sood
Department of Physics, Indian Institute of Science, Bangalore, Karnataka
560012, India  Akshay Rao  Cavendish Laboratory, University of Cambridge,
Cambridge CB3 0HE, United Kingdom

(August 28, 2024)

Excitonic insulators host a condensate of electron-hole pairs at equilibrium,
giving rise to collective many-body effects. Although several materials have
emerged as excitonic insulator candidates, evidence of long-range coherence is
lacking and the origin of the ordered phase in these systems remains
controversial. Here, using ultrafast pump-probe microscopy, we investigate the
possible excitonic insulator Ta2NiSe5. Below 328 K, we observe the anomalous
micrometer-scale propagation of coherent modes at velocities of $\sim$105 m/s,
which we attribute to the hybridization between phonon modes and the phase
mode of the condensate. We develop a theoretical framework to support this
explanation and propose that electronic interactions provide a significant
contribution to the ordered phase in Ta2NiSe5. These results allow us to
understand how the condensate’s collective modes transport energy and interact
with other degrees of freedom. Our study provides a unique paradigm for the
investigation and manipulation of these properties in strongly correlated
materials.

##  I Introduction

The excitonic insulator (EI) phase is a state of matter that was first
proposed more than 50 years ago [(1; 2; 3)]. It was theorized that below a
critical temperature (Tc), weakly screened Coulomb interactions in small
bandgap semiconductors and semimetals could lead to the formation of an
exciton condensate in the ground state (Fig. 1A). In the proposed theoretical
framework, condensation would result from the spontaneous breaking of a U(1)
continuous symmetry, which describes the separate conservation of charges in
the valence and conduction bands. Therefore, this macroscopic state would be
accompanied by characteristic collective excitations, including an amplitude
(Higgs) mode and a gapless phase (Goldstone) mode, as is the case for other
correlated systems, like superconductors. The emergence of the latter
collective mode would lead to intriguing quantum phenomena such as
superfluidic transport [(4; 5)].

A number of small bandgap or semimetallic transition metal chalcogenide (TMC)
compounds have recently attracted interest as promising excitonic insulator
candidates [(6; 7; 8; 9; 10; 11; 12; 13; 14)]. The reduced dimensionality of
TMCs results in weak screening of the Coulomb interaction and consequently
large exciton binding energies, which could lead to condensation at non-
cryogenic temperatures. Among this family of materials, particular attention
has been devoted to Ta2NiSe5 following hints of the potential existence of an
EI phase below 328 K. [(8; 9; 10; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21;
22)]. These works have provided insights into the properties in Ta2NiSe5, for
instance, by using equilibrium [(8; 12)] and time-resolved [(9; 23; 18; 17;
21)] angle-resolved photoemission spectroscopy (ARPES), by analyzing the
effect of physical and chemical pressure [(14; 22)], doping [(16)], and by
detecting anomalies in charge transport [(14)], optical signatures [(24; 25)]
and phonon properties [(10; 15; 19; 20; 22; 26)] as a function of temperature.

However, despite numerous experimental and theoretical studies, the
predominant driving mechanism of symmetry breaking in Ta2NiSe5, and thus the
existence and character of the EI phase itself, remains controversial. In real
condensed matter systems, contrary to the idealized picture presented above,
the Hamiltonian’s continuous symmetry may be further reduced to a discrete
symmetry due to electron-phonon interactions or a hybridization between the
conduction and valence bands [(27; 28)]. Additionally, the observed structural
phase transition can be driven both by electronic interactions and by
electron-phonon coupling [(19; 26; 25; 21; 24; 20)]. All these interactions
are in principle present in a real system, and quantitative differences in
their relative magnitudes have a significant, qualitative impact on the
properties of the collective modes in the ordered phase. For instance, the
presence of additional symmetry breaking terms in the Hamiltonian forces a gap
at $k\simeq$ 0 in the dispersion of the phase mode, suppressing the proposed
supertransport properties [(29)]. Theory suggests that if the transition is
largely due to electron-phonon interactions, one would even expect the low-
energy, acoustic-like phase mode to be largely suppressed, and an excitonic
insulator phase would not be supported by the material [(30; 28)]. These
different pictures could be untangled through the detection of the
condensate’s collective modes and the analysis of their transport properties
and real-space dynamics. If the phase mode remains gapless or minimally
gapped, then the continuous symmetry must remain largely conserved and the
transition is predominantly driven by electronic interactions, meaning the
system is close to the idealised EI phase.

In this work, we use ultrafast, spatially-resolved pump-probe microscopy to
study the real space dynamics of Ta2NiSe5. We show that photoinduced
excitations in Ta2NiSe5 propagate coherently over distances up to 1 $\mu$m and
with electron-like velocities, which we attribute to collective modes of the
excitonic condensate and their hybridization with the lattice degrees of
freedom. These results provide novel insights towards understanding the nature
of the ordered phase in Ta2NiSe5, potentially establishing a pathway for its
manipulation.

##  II Results

We probe the spatio-temporal dynamics of Ta2NiSe5 following photoexcitation
using a recently developed femtosecond optical pump-probe microscopy technique
[(31; 32; 33)], which provides sub 10 fs time-resolution and 10 nm spatial
precision (for a characterization of the sample with standard, non-spatially
resolved pump-probe spectroscopy, see Supplementary Materials Sec. S1). In
these measurements (see Fig. 1B) we perturb a small area of the sample using a
diffraction limited ($\simeq$400 nm full-width half-maximum) optical pulse of
$\simeq$12 fs duration. We measure the resulting change in transmission using
a wide-field (15 $\mu$m full-width half-maximum), $\simeq$10 fs probe pulse,
which is projected onto an EMCCD camera. The large probe allows us to study
both the area directly excited, and the surrounding material, imaging how
excitations propagate in space and time as the presence of quasiparticles or
oscillatory modes in the system modulate the probe transmission. Fig. 1B
displays a schematic of the microscopy setup at the position of the sample and
an example of a two-dimensional plot of the photoinduced transmissivity change
($\Delta$T/T) as measured 300 fs after the arrival of the pump pulse.
$\Delta$T/T is obtained from (T${}_{\text{pump on}}$-T${}_{\text{pump
off}}$)/(T${}_{\text{pump off}}$), where T${}_{\text{pump on}}$ and
T${}_{\text{pump off}}$ indicate the transmission measured in the presence and
in the absence of the pump pulse respectively. Crucially, as we describe
below, we primarily focus our analysis on the regions of the material where we
do not directly inject hot quasiparticles.

The kinetic recorded at the center of the excitation spot as a function of the
pump-probe delay time is shown in the inset of Fig. 1D and is analogous to
those obtained in previous works using non-spatially resolved pump-probe
spectroscopy [(10; 15)]. The signal is characterized by an exponentially
decaying electronic component superimposed with a strong oscillatory signal.
In Fig. 1D we additionally report the Fourier Transform (FT) of this
oscillatory component. The main recognizable features are the peaks at 1, 2.9
and 3.6 THz, which have been commonly associated with phonon modes in Ta2NiSe5
[(10; 15)].

The same FT analysis is now applied to the signal in areas away from the pump
position, where no excitations are generated directly by the pump pulse. In
this case, to focus on coherent propagation out of the photoexcited region and
to maximize the signal-to-noise ratio, we average the $\Delta$T/T signal over
rings of pixels equidistant from the center (see Supplementary Materials Sec.
S2 for details on the nearly isotropic nature of the signal). Through this
procedure, we determine the spatial decay of the oscillatory modes by plotting
the FT power density as a function of the ring distance, as shown in Fig. 2A
for the 1, 2.9, and 3.6 THz modes. The resulting Gaussian shape is a
convolution of the profile of the pump pulse (shown as a reference in Fig.
2B-D) and of the spatial propagation of the excited modes.

In Fig. 2B, we show the dependence of the FT power density spatial profile for
the 2.9 THz mode on the pump fluence. From a Gaussian fit of these curves we
extract the half width at half maximum (HWHM), which is plotted in Fig. 2C. We
observe a highly nonlinear dependence of HWHM against the field strength, as
the spatial extent of the FT grows after $\simeq$ 0.55 mJ/cm2 and reaches
twice its original size, before finally saturating when we approach the
material breaking point. This result suggests that at low fluences the signal
is dominated by the profile of the pump pulse and the propagation of the mode
outside the pumped region becomes visible only at high fluences.
Interestingly, while an analogous behavior is observed for the 3.6 THz mode
(see Supplementary Materials Sec. S3), a rather different fluence dependence
emerges for the 1 THz mode (Fig. 2C). This result hints at different
microscopic mechanisms underlying the excitation of the oscillations,
indicating nonlinear effects may be required to excite the higher frequency
modes.

We next compare (Fig. 2D) the spatial distribution of the FT power density of
the 2.9 THz mode above and below Tc (see the Supplementary Materials Sec. S3
for analogous behavior in the 1 and 3.6 THz modes). While the data is
normalized for ease of comparison, the fluence, the overall $\Delta$T/T
signal, and the absolute FT power density at the center of the pumped region
are comparable for all these measurements. As Tc is approached from below, the
FT profile narrows. Once the temperature surpasses Tc, the FT profile has
shrunk to the size of the pump spot size, suggesting that above Tc the
coherent propagation of the mode breaks down.

Returning to room temperature data, we further analyze the time traces for the
different rings by performing continuous wavelet transforms (CWT) (see
Supplementary Materials Sec. S4 for more details). This procedure allows us to
determine the amplitude of the oscillatory components as a function of time
and distance from the photoexcited area. In Fig. 3A we show the results of
this analysis for a few rings. It can be seen that the strongest amplitude for
the 2.9 THz mode (the only one that is clearly traceable within the signal to
noise limit) progressively shifts toward later times as we move farther away
from the pump region. The position of this region in time is represented in
Fig. 3B for all the rings. By performing a linear fit of this curve, focusing
on ring radii residing outside of the directly pumped region, we extract a
propagation velocity of 1.51 $\pm$ 0.11 x 105 m/s. This means that below Tc
the 2.9 THz mode propagates coherently, at a velocity characteristic of
electronic excitations, for distances of up to 1 $\mu$m.

##  III Discussion

We now discuss possible origins of the anomalous spatial propagation at room
temperature. First, we note that the measured propagation velocity is orders
of magnitude larger than typical values both for optical phonons, which are
commonly characterized by rather flat energy dispersions, and acoustic
phonons, whose velocity is typically below 103 m/s [(34)]. This suggests that
the propagating signal does not have a purely phononic origin. Another
potential explanation of the observed behavior is the excitation of rapidly
propagating bulk or surface phonon-polaritons [(35; 36)]. However, as a result
of the crystal inversion symmetry, the bulk phonon modes are Raman and not IR
active, and therefore cannot participate in the formation of phonon-polariton
modes. In addition, for this type of excitation we would expect to measure
higher propagation velocities ($>$5$\times 10^{6}$ m/s) due to the light-like
nature of these modes (see Supplementary Materials Section S8). Finally, the
disappearance of the propagation above TC is unlikely to be explained by this
scenario, as the observed phonon modes characterize both the low and the high
temperature phase of this material, which several recent works suggest being
an insulator even above TC [(37; 25)]. We next consider the possibility of the
propagation of quasiparticles (QPs) emanating from the excitation spot with
velocity $v_{k}=(\partial\epsilon_{k}/\partial k)$ of the order of 105 m/s,
which can excite phonons in their wake. However, at room temperature, the
carrier relaxation time of QPs is usually only of the order of a few tens of
femtoseconds, corresponding to a mean free path of a few nanometers [(38;
39)]. In addition, the emission of phonons by the QPs would generate
oscillations that are not in phase as we average rings around the photoexcited
region. In our experiments coherent behavior is observed on a much wider
spatial range and on timescales extending to a few picoseconds. We contend
that more coherent electronic oscillations (such as the case of plasmons) are
unlikely to be excited in this experiment as the material is an insulator at
equilibrium. Even after photoexcitation we therefore do not expect the probed
region to host the required carrier population to support these modes. An
exciton-polariton, formed by uncondensed excitons at the conduction-band edge,
is a final potential candidate that could explain the observed behavior
without requiring the existence of any exotic ground state in the system. Yet,
long-range and long-lived (several picosecond) excitations such as the ones
that we detect require significant coupling between the light and matter modes
and significant confinement. Considering that for comparable van der Waals
materials the estimated Q factors for a 60 nm flake are $\leq$5 [(36)], we
would not expect to observe modes with lifetimes longer than $\sim$10 fs in
our system (see Supplementary Materials Section S8). Very strong light-matter
coupling (resulting in a Rabi splitting of the excitonic line larger than 100
meV) also would be required to achieve experimentally relevant velocities, yet
no hints of this are observed in equilibrium measurements [(40)]. Finally, we
are unaware of a mechanism by which these modes could be coherently excited
with our pumping scheme, where the injection of quasiparticles with energies
much larger than the bandgap will relax to the bandedge through successive
stochastic relaxation processes. We therefore believe that these
considerations make the exciton-polariton scenario rather remote, but
analogous measurements in the mid-IR would be useful to more conclusively
address this scenario.

One remaining possibility for the anomalous propagation is the coupling
between the phonon modes and the EI’s collective phase mode. A phase mode is
characterized by a linear dispersion and group velocities on the electronic
scale. If electron-phonon coupling is present, the phase and the phonon modes
can hybridize resulting in excitations of mixed electronic and phononic nature
which can propagate at velocities close to that of the pure phase mode (see
Fig. 1D). Indeed, the presence of long-range exciton-phonon complexes
supporting acoustic-like low energy excitations is consistent with recent
measurements of the optical response functions of Ta2NiSe5[(40; 41)]. We
expect that the lifetime of the phase mode can be significantly longer than
that of QPs, as is supported by a theoretical calculation of the effects of
the possible disorder on the QPs and the phase mode (see the Supplementary
Materials Sec. S9 and [(42)]). While the overall relaxation rate of the
phonon-phase mode might not differ drastically from that of the phonons, the
significantly higher speed can result in long propagation lengths
($\simeq\mu$m) for these excitations. The hybrid modes can then carry a
phononic signature ballistically over long distances. This coupled phonon-
phase mode scenario is also consistent with the suppression of propagation
above Tc, where the collective phase mode should disappear. We note that a
hybrid phonon-phase mode with an electronic-like group velocity is realized as
long as the gap of the phase mode is comparable or smaller than the phonon
energy. This occurs regardless of the origin of the gap, even where additional
electronic terms directly breaking the symmetry [(27; 28)] also contribute to
the formation of the gap.

Returning to Fig. 2C, we can tentatively interpret the different behavior of
the two oscillatory modes as the result of a different coupling process
between these phonons and the phase mode (see Supplementary Materials Sec.
S6). As the hybridization between the two modes occurs at larger wavevector
with increasing phonon frequency, we expect that a higher order process, like
Raman scattering involving multiple phonon modes, could be required to excite
the more energetic modes at wavevectors that are not provided by the focused
laser beam (see Supplementary Materials Sec. S5). Intriguingly, the saturation
and drop in the 1 THz extension at high fluences could be the signature of the
enhancement of the gap in the phase mode dispersion. If raised above the
phonon energy, the phase mode would indeed not contribute to the formation of
the hybrid modes. Further studies that go beyond the scope of this work are
required to address these hypotheses.

To support our interpretations, we develop a theoretical model within a two
band approximation. We start from a typical Hubbard-type Hamiltonian with
additional terms accounting for the electron–phonon coupling and the phonon
energy,

$H=H_{\text{kin}}+H_{\text{int}}+H_{\text{el-ph}}+H_{\text{ph}}$ (1)

where $H_{\text{kin}}$ represents the electronic bands, $H_{\text{int}}$ is
the electron–electron interaction, $H_{\text{el-ph}}$ is the electron–phonon
coupling, and $H_{\text{ph}}$ is the phonon Hamiltonian. This approach is
analogous to that used in previous theoretical works on Ta2NiSe5 [(43; 8;
37)], and like there we use a form of $H_{\text{el-ph}}$ that explicitly
breaks the symmetry of $H$ (see Supplementary Materials Sec. S7 and [(44)] for
more details, calculations at various temperatures, and for an analysis of the
opposite case with no explicit symmetry breaking). The microscopic parameters
corresponding to the band structure and the electron–electron interaction were
estimated by fitting previous experimental ARPES results [(8)] (see
Supplemental Materials Sec. S7B). In Fig. 4A, the calculated linear response
function shows the dispersion of the massless phase mode in the absence of
electron–phonon coupling. Using this dispersion, the estimated group velocity
is $v_{\text{PM}}$= 1.0 x 105 m/s for this mode, which is of the same order of
magnitude of the velocity observed in the experiments. In Fig. 4B, we show the
result in the presence of the electron–phonon coupling term where the phase
mode becomes massive and a hybridization between the phase and phonon mode
occurs.

In order to determine whether the phonon oscillations induced by the
excitation can propagate at velocities of the order of $v_{\text{PM}}$ (an
electronic-like velocity), we perform a real-space time-dependent mean-field
simulation (see Supplementary Materials Sec. 7A). Here, we use a larger phonon
frequency (indicated by the red, dashed line in Fig. 4B, D-F) than those
observed in the experimental data and we correspondingly consider a smaller
excitation area. These assumptions do not affect the conclusions that we can
draw from these calculations and they are adopted only for the sake of
simplifying the numerical simulation. In Fig. 4C, we show the propagation of
the phonon displacement after a spatially confined excitation. The signal
propagates with a velocity of about 0.6$\times$105 m/s. We can visualize the
time evolution of the phonon displacement at different distances from the
excitation by performing a spatially windowed Fourier Transform, analogous to
a CWT analysis. The results are shown in Fig. 4D-F, where the orange dashed
line indicates the frequency associated with the gap in the phase mode
dispersion for the parameters used in our simulations. These plots illustrate
the spreading of the hybrid mode at the phonon frequency with a velocity
comparable to the phase mode velocity and further support the above scenario
that mixing between the phase mode and the phonon mode leads to the fast
propagation of the phonon oscillations.

We note that in Fig. 4D-F, oscillations in the phonon displacement as a
function of time contain information about the gap of the phase mode at
$k\simeq 0$ (see the Supplementary Materials Sec. S7B for details). While the
data in Fig. 3A shows analogous oscillations, we presently cannot determine if
their origin is connected to the gapped phase mode or to a more trivial
frequency beating between the 2.9 and 3.6 THz modes.

##  IV Conclusions

Using ultrafast pump-probe microscopy, we image the spatial dynamics of
oscillatory modes in the EI candidate Ta2NiSe5. Below 328 K, we observe the
anomalous propagation of these modes, which remain coherent up to $\simeq$ 1
$\mu$m, at a velocity of $1.5\times 10^{5}$ m/s. After carefully considering
various scenarios, we attribute this propagation to the hybridization of
dispersionless phonon modes with the dispersive, low-lying phase mode of an
excitonic condensate, which are excited through the abrupt change in the
quasi-particle density in the excitation region [(45)]. Further studies are
required to establish the precise processes by which these collective modes
are excited. Our results and interpretations have several implications.
Firstly, the experimental observation of the signature of a phase mode
provides us with important insight into the nature of the excitonic order in
Ta2NiSe5. Namely, if the gap of the phase mode was much larger than the phonon
frequency, the mixing between the modes would be suppressed. Hence, the
effects of the electron–phonon coupling and of possible electronic terms that
explicitly break the continuous symmetry [(27; 28; 29; 43)] should be weak,
indicating that the ordered phase in Ta2NiSe5 is an excitonic insulator phase
primarily driven by interband Coulomb interactions (see Supplementary
Materials Sec. S7). Second, a notable experimental observation is the small
anisotropy in the mode propagation. This may seem counterintuitive since
Ta2NiSe5 is a quasi-one-dimensional system and previous DFT calculations found
that the bandwidth of the valence band along the direction perpendicular to
the atomic chains should be almost five times smaller than in the direction
along the chains [(13)]. However, we show theoretically that even in the
presence of anisotropic hopping between the lattice sites, when the system
approaches the BCS-BEC crossover regime, the anisotropy of the phase-mode
velocity becomes small (see Supplementary Materials Sec. S7B3). Hence, the
small experimentally observed anisotropy indicates that the system is close to
the BCS-BEC crossover regime, as suggested in previous works [(14)]. Finally,
our results showcase the possibility of using spatially–resolved, femtosecond
pump-probe measurements to gain crucial insights into the properties of
condensed matter systems manifesting emergent many-body phenomena. By exciting
and detecting coherent collective modes and by studying their propagation
properties, important information can be gathered on the microscopic origin of
these effects. Femtosecond pulses could thus provide a means to control and
read out the properties of these systems in future quantum information
applications.

## Materials and Methods

Sample preparation

Ta2NiSe5 crystals were grown using the procedure outlined in reference [(22)].

To prepare samples for optical measurements, flakes were exfoliated using
gold-assisted exfoliation [desai2016gold ] onto glass coverslips and
subsequently encapsulated with a second glass slide while inside a nitrogen
glovebox.

Pump-probe microscopy A Yb:KGW laser system (Pharos, Light Conversion),
delivers 1030 nm light with pulses of 200 fs duration and 30 $\mu$J power at a
repetition rate of 200 kHz. The output beam is split to seed two broadband
white-light continuum generation stages, used to create pump and probe pulses.
The broadband probe beam is generated using a 3 mm YAG crystal, and a fused-
silica prism is then used to select the spectral range of interest (from 650
nm to 950 nm). The white-light continuum employed for pump pulses is instead
generated using a 3 mm sapphire crystal, delivering pulses covering the
wavelength range from 500 nm to 650 nm (a long wavelength cut off is achieved
using a short-pass filter (FESH650, Thorlabs)). The pulses are temporally
compressed using chirped mirrors and fused silica wedges, to achieve ¡12 fs
pulses. In particular, two sets of Layertec (109811) mirrors are used on the
pump path to also compensate for the glass elements contained in the
objective; Venteon (DCM9) (Venteon) mirrors are instead used in the probe
path. The pulse length is verified using a frequency resolved optical gating
technique [(47)]. A mechanical chopper (Thorlabs, MC2000B) is used to modulate
the pump pulse stream at a frequency of 30 Hz, and a 40 $\mu$m pinhole is used
to spatially clean the laser mode. The pump beam is finally expanded to
achieve proper filling of the back aperture of an oil-immersion, 1.1 NA
objective, which finally focuses the pulses onto the sampleto a spot size of
$\sim$400 nm full-width-half-maximum (FWHM). Reflected components of the pump
are removed from the collection path using a longpass filter. The probe pulses
are mechanically delayed with respect to the pump excitation using a closed-
loop piezo translation stage (P-625.1CL, Physik Instrumente), and then focused
onto the sample to a size of $\sim$15 $\mu$m FWHM using a spherical mirror.
The transmitted probe pulse is collected by the objective, sent through a 10
nm bandpass filter (Thorlabs) to select the wavelength range of interest, and
is finally imaged onto an EMCCD camera (Rolera Thunder, QImaging). For further
details on the experimental apparatus, see reference (32).

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Acknowledgements and funding

The calculations were run on the Beo05 cluster at the University of Fribourg.
We acknowledge the CECAM workshop “Excitonic insulator: New perspectives in
long-range interacting systems” at EPFL Lausanne for insights and discussions
and for providing the opportunity to start this collaboration. We also thank
A. Boris (Max Planck Institute for Solid State Research) and A. Musser
(Cornell University) for helpful discussions. Funding: H.M.B., P.A., and A.R.
acknowledge support from the Winton Programme for the Physics of
Sustainability, the Engineering and Physical Sciences Research Council, and
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement 758826). N.R.C.
acknowledges funding from the Engineering and Physical Sciences Research
Council (EPSRC grant no. EP/P034616/1). Y.M. thanks the Japan Society for the
Promotion of Science and the Japan Science and Technology Agency for funding
through KAKENHI grant no. JP19K23425 and JST CREST grant no. JPMJCR1901. P.W.
acknowledges funding from the ERC Consolidator grant no. 724103 and from the
Swiss National Science Foundation via NCCR Marvel. A.K.S. thanks the
Department of Science and Technology, India for support under Nanomission and
Year of Science Professorship. L.H. acknowledges the financial support from
the Department of Science and Technology (DST), India [grant no.
SR/WOS-A/PM-33/2018 (G)] and IISER Pune for providing the facilities for
crystal growth and characterization. B.R. acknowledges support from the
Cambridge International Trust and Wolfson College, Cambridge. D.G. is
supported by the Slovenian Research Agency (ARRS) under program nos. P1-0044
and J1-2455. N.R.C., A.J.M., and D.G. acknowledge support from the Simons
Foundation.

Author Contributions
HMB and PA performed measurements and analyzed the data. PT, AS, and LH worked
on the growth and characterization of the material. PA, AKS and AR conceived
the experiments. YM, DG, and BR performed theoretical modelling with input
from AJM, PW and NRC. HMB, PA, and AR wrote the paper with input from all
authors.

Competing Interests
The authors declare that they have no competing financial interests.

Data availability
The data underlying all figures in the main text is publicly available at
https://doi.org/10.17863/CAM.64033

Figure 1: Pump-probe microscopy of Ta2NiSe5. (A, top) Crystal structure of
Ta2NiSe5. The alternating chains of Ta and Ni atoms confer a quasi-1D nature
to the material and independently host conduction and valence band states
[[8]]. (A, bottom) Schematic electronic band structure for a prototypical
excitonic insulator. Above the critical temperature the material is close to
the semiconductor-semimetal transition (portrayed here as a semiconductor of
energy gap Eg). Below the critical temperature, the exciton binding energy
(Eb) exceeds the single-particle gap (Eg), resulting in a macroscopic coherent
state. (B) Schematic of the measurement setup at the location of the sample
with the photoinduced transmissivity change ($\Delta$T/T) signal. The white
dot and dashed circle are examples of pixel regions over which we average the
signal in our analysis. (C) Schematic diagram of a possible low-energy
excitation structure for an excitonic insulator in the presence of electron-
phonon coupling. The phonon modes of the material hybridize with the phase
mode resulting in mixed phonon-phase modes as long as the energy gap in the
phase mode dispersion at k = 0 is smaller than the phonon energy Eph,i. The
phonon and phase content of the modes is represented as a color gradient from
green (pure phase mode) to red (pure phonon mode). (D) Photoinduced
transmissivity change ($\Delta$T/T) as a function of the pump-probe delay time
collected at the center of the pump region (inset) and its Fourier Transform
power density. Figure 2: Temperature and fluence dependent propagation. (A)
Fourier Transform (FT) power density as a function of the distance of the ring
over which we average the signal from the center of the pump region. This data
was collected using a pump fluence of 0.3 mJ/cm2. The data set for the 1 THz
mode is magnified by a factor of 15 to improve its visibility. (B,D) Spatial
dependence of the normalized FT power density for the 2.9 THz mode as a
function of pump fluence (measured at room temperature) (B) and of temperature
(each measured at a comparable fluence of $\simeq$ 1.2 mJ/cm2) (D). The dashed
line indicates the profile of the pump pulse. (C) Half-width half-maximum
(HWHM) of the phonon spatial extension as a function of the pump fluence for
the 1 THz and 2.9 THz modes. Figure 3: Temporal characterization of the
oscillatory modes’ propagation. (A) Continuous wavelet transform (CWT) of the
oscillatory component of the $\Delta$T/T signal at three different distances
(333, 555, and 777 nm) from the center of the pump region and measured at a
fluence of $\simeq$ 1.6 mJ/cm2. The color map represents the magnitude of the
CWT. The white dotted lines approximately mark the location of the strongest
phonon oscillations. The periodic oscillations in the CWT are clearly resolved
and their possible origins are discussed in the main text. (B) Time delay of
the region of strong oscillations as a function of the ring distance. The
orange dashed line represents the linear fit of the data in the region outside
the pump area and has an inverse slope 1.51$\leavevmode\nobreak\ \times$ 105
m/s. Figure 4: Time-dependent Mean-Field Model. (A)(B) Response function of
the order parameter for the microscopic two-band model on the one-dimensional
lattice with purely excitonic behavior (A) and in the presence of
electron–phonon coupling (B). The coupling is small with respect to the
Coulomb interaction and explicitly breaks the continuous symmetry of the
system. (C) Spatial and temporal evolution of the phonon displacement, $\delta
X$ for the case of non-zero electron–phonon coupling following an excitation
at the center of the system at time t = 0, calculated within the mean-field
description. (D-F) Windowed Fourier Transform of the results in (C) at
specified lattice sites, where $\Delta r$ indicates the distance from the
center of the excitation area. The red and orange dashed lines indicate the
bare phonon frequency used in these calculations and the size of the phase
mode gap, respectively. The bandwidth of the conduction and valence bands is
set to 1.6 eV and the Coulomb interaction to 0.84 eV. The corresponding band
gap is about 0.32 eV.


#  $m$-quasi Einstein manifolds with convex potential

Absos Ali Shaikh1∗, Prosenjit Mandal2, Chandan Kumar Mondal3†, Akram Ali 4
1,2,3Department of Mathematics,
The University of Burdwan,Golapbag,
Purba Bardhaman-713101,
West Bengal, India
†School of Sciences,
Netaji Subhas Open University,
Durgapur Regional Center, Durgapur-713214
Paschim Bardhaman,
West Bengal, India
4Department of Mathematics,
College of Science,
King Khalid University,
9004 Abha, Saudi Arabia<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>4<EMAIL_ADDRESS>
###### Abstract.

The main objective of this paper is to investigate the $m$-quasi Einstein
manifold when the potential function becomes convex. In this article, it is
proved that an $m$-quasi Einstein manifold satisfying some integral conditions
with vanishing Ricci curvature along the direction of potential vector field
has constant scalar curvature and hence the manifold turns out to be an
Einstein manifold. It is also shown that in an $m$-quasi Einstein manifold the
potential function agrees with Hodge-de Rham potential up to a constant.
Finally, it is proved that if a complete non-compact and non-expanding
$m$-quasi Einstein manifold has bounded scalar curvature and the potential
vector field has global finite norm, then the scalar curvature vanishes.

††footnotetext: $\mathbf{2020}$ Mathematics Subject Classification: 53C20;
53C21; 53C44.
Key words and phrases: Quasi Einstein Manifold; scalar curvature; convex
function; Einstein Manifold.

##  1\. Introduction and preliminaries

A Riemannian manifold $(M,g)$ of dimension $n(>2)$ is called an $m$-quasi
Einstein manifold (see e.g. [3, 4, 17]), if its Ricci tensor $Ric$ satisfies
the following relation:

(1) $Ric+\frac{1}{2}\pounds_{X}g-\frac{1}{m}X^{\flat}\otimes X^{\flat}=\lambda
g,$

where $X$ is a smooth vector field on $M$, $X^{\flat}$ is the dual 1-form of
$X$ with respect to the metric $g$ and $m,\lambda$ are scalar such that
$0<m\leq\infty$. If we omit the lie derivative term in (1), then we get the
notion of quasi Einstein manifold (see e.g. [8, 21, 20] and also references
therein) which appeared in the literature while considering the investigation
of exact solution of Einstein field equation and also during the consideration
of quasiumbilical hypersurfaces, studied by Cartan. In [9] it is shown that
$3$-dimensional Cartan hypersurfaces are quasi-Einstein manifolds. Throughout
the paper we will consider the $m$-quasi Einstein manifolds ([4]), which is a
generalization of Ricci soliton in case of $m$-Bakry-Emery tensor. An
$m$-quasi Einstein manifold is said to be expanding (resp., steady or
shrinking), if $\lambda<0$ (resp., $\lambda=0$ or $\lambda>0$). If $X$ is the
gradient of a smooth function $f$, then (1) reduces to the following form:

(2) $Ric+\nabla^{2}f-\frac{1}{m}df\otimes df=\lambda g.$

The function $f$ is called potential function of the $m$-quasi Einstein
manifold. The tensor of the left hand side of (2) is called the $m$-Bakry-
Emery tensor. Further, if $m=\infty$, then (2) reduces to the equation of
gradient Ricci soliton (see e.g. [7, 11, 15, 16, 19]) with the potential
function $f$. If $m$ and $\lambda$ are smooth functions on $M$, then (2) is
called the generalized $m$-quasi Einstein manifold [5]. Moreover, taking the
trace of (2), we deduce

(3) $\Delta f=\lambda n-R+\frac{1}{m}|\nabla f|^{2},$

where $R$ denotes the scalar curvature of the manifold. Case et al. [4] proved
that a compact $m$-quasi Einstein metric with constant scalar curvature is
trivial. They also showed that all $2$-dimensional $m$-quasi Einstein metrics
on compact manifolds are trivial. In 2012, Barros and Ribeiro [3] proved that
in a complete non-compact $m$-quasi Einstein manifold if $\lambda n\geq R$ and
$|\nabla f|\in L^{1}(M)$, then the manifold becomes Einstein. In 2015, Hu et
al. [12] classified $m$-quasi Einstein manifolds with parallel Ricci tensor,
and also in [13], they classified $m$-quasi Einstein manifolds with all
eigenvalues of the Ricci tensor as constant. Kim and Shin [14] also classified
3-dimensional $m$-quasi Einstein manifolds. Barros and Gomes [2] studied
compact $m$-quasi Einstein manifolds and also showed that if such a manifold
is Einstein, then its potential vector field vanishes. We note that the
investigation of various structures of Riemannian manifolds through the
convexity revel several geometrical and topological properties of such
manifolds. Hence by motivating the above studies as well as the study of [1],
in the present paper, we prove the following:

######  Theorem 1.1.

Suppose $(M,g)$ is an $m$-quasi Einstein manifold which is complete and non-
compact and endowed with a positive convex potential function $f$ and
$Ric\geq-(n-1)K$ for some positive constant $K$. If the Ricci curvature of the
manifold vanishes along $\nabla f$ and $f$ satisfies

(4) $\quad\int_{M-B(p,r)}\frac{f}{d(x,p)^{2}}<\infty,$

where $B(p,r)$ is an open ball with center $p$ and radius $r$, and $d(x,p)$ is
the shortest distance between $x$ and $p$ in $M$, then the following holds:
(i) The manifold is Einstein,
(ii) The scalar curvature of the manifold is a non-positive constant,
(iii) The Ricci curvature of the manifold is non-positive everywhere in $M$.

######  Corollary 1.1.1.

Let $(M,g)$ be a complete non-compact $m$-quasi Einstein manifold with
positive convex potential function $f$. If the manifold is radially Ricci flat
and satisfies the condition (4), then the manifold is Einstein with non-
positive Ricci curvature.

######  Theorem 1.2.

If $(M,g)$ is a compact oriented $m$-quasi Einstein manifold such that $f$ is
a potential function on $M$, then $f$ agrees, upto a constant, with the Hodge-
de Rham potential.

######  Theorem 1.3.

Let $(M,g)$ be a complete non-compact and non-expanding $m$-quasi Einstein
manifold with finite volume. If the potential vector field $X$ is of finite
global norm, then there exists an open ball where the scalar curvature
$R\geq\lambda n$.

######  Corollary 1.3.1.

Let $(M,g)$ be a complete non-compact and non-expanding $m$-quasi Einstein
manifold with finite volume. If the potential vector field $X$ is of finite
global norm and the scalar curvature is bounded by $\lambda n$, then the
scalar curvature vanishes in $M$.

##  2\. Proof of the results

To prove Theorem 1.1 we need the following result of Hu et al. [12], which we
state at first.

######  Theorem 2.1.

[12] In an $m$-quasi Einstein manifold $(M,g)$ with constant scalar curvature,
$\lambda\leq 0$.

###### Proof of Theorem 1.1.

Since $f\in C^{\infty}(M)$, ring of smooth functions on $M$, is a non-constant
convex function on $M$, it follows that [23], $M$ is non-compact. Let us
consider the cut-off function, studied in [6], $\varphi_{r}\in
C^{2}_{0}(B(p,2r))$ for $r>0$, where $C^{2}_{0}(B(p,2r))$ is a class of second
order continuously differentiable functions with compact support and $C$ being
a constant, $(B(p,2r))\subseteq M$, $p\in M$ such that

$\begin{cases}\varphi_{r}=1&\text{ in }B(p,r)\\\ 0\leq\varphi_{r}\leq 1&\text{
in }B(p,2r)\\\ \Delta\varphi_{r}\leq\frac{C}{r^{2}}&\text{ in
}B(p,2r).\end{cases}$

Then for $r\rightarrow\infty$, we have $\Delta\varphi^{2}_{r}\rightarrow 0$ as
$\Delta\varphi^{2}_{r}\leq\frac{C}{r^{2}}$. The convexity of $f$ implies that
$f$ is also subharmonic [10], i.e., $\Delta f\geq 0$. In view of integration
by parts, we obtain

(5) $\int_{M}\Delta f\varphi^{2}_{r}=\int_{M}f\Delta\varphi^{2}_{r}.$

Since $\varphi_{r}\equiv 1$ in $B(p,r)$, using (5), we get

$\int_{B(p,r)}\Delta f=0.$

Again, in view of integration by parts and also by our assumption, we obtain

$0\leq\int_{B(p,2r)}\varphi_{r}^{2}\Delta
f=\int_{B(p,2r)-B(p,r)}f\Delta\varphi_{r}^{2}\leq\int_{B(p,2r)-B(p,r)}f\frac{C}{r^{2}}.$

But the right hand side tends to zero as $r\rightarrow\infty$. Hence we get

$\int_{M}\Delta f=0.$

Thus, the subharmonocity of $f$ implies that $\Delta f=0$ in $M$. Therefore,
(3) entails that

(6) $R-\lambda n=\frac{1}{m}|\nabla f|^{2},$

which implies that $R\geq\lambda n$. Taking $u=\log f$ and then simplifying we
obtain

$-\Delta u=|\nabla u|^{2}=\frac{|\nabla f|^{2}}{f^{2}}.$

Let $l(x)$ be the distance of $x\in M$ from the fixed point $p$. For any
$r>0$, consider the function $\eta:[0,\infty)\rightarrow[0,1]$ satisfying the
following properties:

$\begin{cases}\eta(t)=1\text{ for }t\leq r\\\ \eta(t)=0\text{ for }t\geq 2r\\\
\eta^{\prime}\leq 0\\\
(\eta^{\prime})^{2}\leq\frac{(\eta^{\prime})^{2}}{\eta}\leq\frac{C}{r^{2}}\\\
|\eta^{\prime\prime}|\leq\frac{C}{r^{2}},\end{cases}$

for some constant $C<\infty$. Now define the function $\eta$ on $M$ by
$\eta(x)=\eta(l(x))$ for $x\in M$. Then the function $u$ satisfies the
following inequality:

$|\nabla\nabla u|^{2}\geq\frac{1}{n}(\Delta u)^{2}=\frac{1}{n}|\nabla u|^{4}.$

By virtue of above inequality and the Bochner formula, we obtain

$\displaystyle\frac{1}{2}\Delta(\eta|\nabla u|^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Delta\eta|\nabla u|^{2}+\nabla\eta\nabla|\nabla
u|^{2}+\frac{1}{2}\eta\Delta|\nabla u|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Delta\eta|\nabla u|^{2}+\nabla\eta\nabla|\nabla
u|^{2}+\eta|\nabla\nabla u|^{2}+\eta\nabla u\nabla(\Delta u)+\eta Ric(\nabla
u,\nabla u)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\Delta\eta|\nabla
u|^{2}+\eta\nabla\eta\nabla|\nabla u|^{2}+\eta|\nabla\nabla u|^{2}-\eta\nabla
u\nabla|\nabla u|^{2}+\eta Ric(\nabla u,\nabla u)$ $\displaystyle\geq$
$\displaystyle\Big{(}\frac{1}{2\eta}\Delta\eta-\frac{1}{\eta^{2}}|\nabla\eta|^{2}+\frac{1}{\eta}\nabla\eta\nabla
u\Big{)}\eta|\nabla u|^{2}+\Big{(}\frac{1}{\eta}\nabla\eta-\nabla
u\Big{)}\nabla(\eta|\nabla u|^{2})$ $\displaystyle+\frac{1}{n}\eta|\nabla
u|^{4}+\eta Ric(\nabla u,\nabla u).$

Again, calculation shows that $\eta$ satisfies the following inequality:

(7) $|\nabla\eta\nabla u|\leq\frac{\eta|\nabla
u|^{2}}{2n}+\frac{n}{2\eta}|\nabla\eta|^{2}.$

Since $|\nabla u|^{2}$ is non-zero, there is a point at which $\eta|\nabla
u|^{2}$ is maximum where $\eta$ is smooth, and hence $\Delta(\eta|\nabla
u|^{2})\leq 0$ and $\nabla(\eta|\nabla u|^{2})=0$ such that

$0\geq\frac{1}{2}\Delta\eta-\frac{1}{\eta}|\nabla\eta|^{2}+\nabla\eta\nabla
u+\frac{1}{n}\eta|\nabla u|^{2}+\eta^{2}Ric(\nabla u,\nabla u).$

By using (7), we have

(8)
$\displaystyle-\Delta\eta+\frac{n}{\eta}|\nabla\eta|^{2}+\frac{2}{\eta}|\nabla\eta|^{2}$
$\displaystyle=$
$\displaystyle-\Delta\eta+\frac{n}{\eta}|\nabla\eta|^{2}+\frac{2}{\eta}|\nabla\eta|^{2}+2\nabla\eta\nabla
u-2\nabla\eta\nabla u$ (9) $\displaystyle\geq$
$\displaystyle\frac{n}{\eta}|\nabla\eta|^{2}+2\nabla\eta\nabla
u+\frac{2}{n}\eta|\nabla u|^{2}+2\eta^{2}Ric(\nabla u,\nabla u)$ (10)
$\displaystyle\geq$ $\displaystyle\frac{1}{n}\eta|\nabla
u|^{2}+2\eta^{2}Ric(\nabla u,\nabla u).$

Therefore, we get

$-\eta^{\prime}\Delta
d-\eta^{\prime\prime}+\frac{n+2}{\eta}(\eta^{\prime})^{2}\geq\frac{1}{n}\eta|\nabla
u|^{2}+2\eta^{2}Ric(\nabla u,\nabla u).$

Since $Ric\geq-(n-1)K$, the Laplace comparison theorem implies that

$\Delta d\leq\frac{n-1}{d}(1+\sqrt{K}d)\leq\frac{n-1}{r}+(n-1)\sqrt{K},$

where $\eta^{\prime}\neq 0$. Hence, we obtain

$\frac{1}{n}\eta|\nabla u|^{2}+2\eta^{2}Ric(\nabla u,\nabla
u)\leq\frac{C^{\prime}(n)}{r^{2}}+\frac{C^{\prime\prime}(n)}{r}\sqrt{K},$

where $C^{\prime}(n)$ and $C^{\prime\prime}(n)$ are positive constants depend
only on $n$. Consequently, we get

$\frac{1}{n}\eta|\nabla
u|^{2}\leq\frac{C^{\prime}(n)}{r^{2}}+\frac{C^{\prime\prime}(n)}{r}\sqrt{K}-2\eta^{2}Ric(\nabla
u,\nabla u).$

Taking limit as $r\rightarrow\infty$ in both sides, yields for all $x\in M$,

(11) $\displaystyle|\nabla u|^{2}_{x}$ $\displaystyle\leq$
$\displaystyle-2nRic_{x}(\nabla u,\nabla u)$ $\displaystyle=$
$\displaystyle\frac{-2n}{f^{2}}Ric_{x}(\nabla f,\nabla f).$

But Ricci curvature vanishes along $\nabla f$, i.e., $Ric(\nabla f,\nabla
f)=0$, hence, we have $|\nabla u|=0$, which shows that $f$ is constant.
Therefore, (2) implies that the manifold becomes Einstein and $R$ is equal to
$\lambda n$ which is a constant. Again, Theorem 2.1 implies that in an
$m$-quasi Einstein manifold $\lambda\leq 0$. Hence, $R=\lambda n\leq 0$ and it
also implies that Ricci curvature is non-positive. ∎

###### Proof of Theorem 1.2.

Let $(M,g)$ is a Riemannian manifold which is compact and oriented. If $X$ is
a vector field on $M$, the by virtue of Hodge-de Rham decomposition theorem,
(see e.g. [22]), $X$ can be written as

(12) $X=W+\nabla\xi,$

where $div\ W=0$ and $\xi$ is a smooth function called the Hodge-de Rham
potential. We consider an $m$-quasi Einstein manifold $(M,g)$ such that
equation (1) yields

(13) $R+divX-\frac{1}{m}|X|^{2}=\lambda n.$

Hence (12) entails that $divX=\Delta\xi$ and consequently $(\ref{qe1})$ takes
the form

(14) $R+\Delta\xi-\frac{1}{m}|X|^{2}=\lambda n.$

Again from $(\ref{e0})$ we have

$R+\Delta f-\frac{1}{m}|\nabla f|^{2}=\lambda n.$

From $(\ref{qe2})$, it follows that

(15) $\Delta(f-\xi)=0,$

which implies that $f=\xi+C_{1}$, for some constant $C_{1}$, this proves the
result. ∎

Let $(M,g)$ be an oriented Riemannian manifold and $\Lambda^{k}(M)$ be the set
of all $k$-th differential forms in $M$. Now for any integer $k\geq 0$, the
global inner product in $\Lambda^{k}(M)$ is defined by

$\langle\eta,\omega\rangle=\int_{M}\eta\wedge*\omega,$

for $\eta,\omega\in\Lambda^{k}(M)$. Here $`*^{\prime}$ is the the Hodge star
operator. Hence, we define the global norm of $\eta\in\Lambda^{k}(M)$ by
$\|\eta\|^{2}=\langle\eta,\eta\rangle$ and remark that
$\|\eta\|^{2}\leq\infty$.

###### Proof of Theorem 1.3.

For any $r>0$ we have

$\displaystyle\frac{1}{r}\int_{B(p,2r)}|X|dV$ $\displaystyle\leq$
$\displaystyle\Big{(}\int_{B(p,2r)}\langle X,X\rangle
dV\Big{)}^{1/2}\Big{(}\int_{B(p,2r)}\Big{(}\frac{1}{r}\Big{)}^{2}dV\Big{)}^{1/2}$
$\displaystyle\leq$
$\displaystyle\|X\|_{B(p,2r)}\frac{1}{r}\Big{(}Vol(M)\Big{)}^{1/2},$

where $Vol(M)$ denotes the volume of $M$. Thus we obtain

$\liminf_{\ r\rightarrow\infty}\frac{1}{r}\int_{B(p,2r)}|X|dV=0.$

Again, there exists a Lipschitz continuous function $\omega_{r}$ such that for
some constant $K>0$, (see [24]),

$\displaystyle|d\omega_{r}|\leq\frac{K}{r}\qquad\text{almost everywhere on }M$
$\displaystyle 0\leq\omega_{r}(x)\leq 1\quad\forall x\in M$
$\displaystyle\omega_{r}(x)=1\quad\forall x\in B(p,r)$ $\displaystyle\text{
supp }\omega_{r}\subset B(p,2r).$

Then taking limit, we get $\lim\limits_{r\rightarrow\infty}\omega_{r}=1$.
Therefore, by using the function $\omega_{r}$, we have

$\Big{|}\int_{B(p,2r)}\omega_{r}divXdV\Big{|}\leq\frac{C}{r}\int_{B(p,2r)}|X|dV.$

In view of the $m$-quasi Einstein manifold equation $(\ref{q1})$, we get

$\int_{M}\\{\lambda n-R+\frac{1}{m}|X|^{2}\\}dV=0,$

which yields

(16) $\int_{M}(\lambda n-R)dV\leq 0.$

Since $R$ is continuous, there is an open ball where $R\geq\lambda n$. ∎

###### Proof of Corollary 1.3.1.

Our assumption and (16) together imply that $R=\lambda n$ in $M$. Again, using
Theorem 2.1, we get $\lambda\leq 0$. But the manifold is non-expanding
$m$-quasi Einstein. Therefore, $R=\lambda n=0$ in $M$. ∎

## Acknowledgment

The authors extend their appreciation to the deanship of scientific research
at King Khalid University for funding this work through research groups
program under grant number R.G.P.1/50/42.

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