# Dynamical Spectral Weight Transfer in the Orbital HK Model

Gaurav Tenkila, Jinchao Zhao, Philip W. Phillips  Department of Physics and
The Anthony J Leggett Institute for Condensed Matter Theory, University of
Illinois at Urbana-Champaign, Urbana, IL 61801, USA

###### Abstract

We compute explicitly the low-energy spectral weight (LESW) in the lower
Hubbard band using the orbital Hatsugai-Kohmoto (OHK) model. We show that the
dynamical mixing enters the LESW with a universal slope independent of the
number of orbitals. This independence on orbital number corroborates the rapid
convergence of the OHK model to the Hubbard model. As a result, OHK is an
effective simulator of Hubbard physics.

The Landau one-to-one correspondence guarantees that anytime a single electron
is removed from a Fermi liquid, just a single empty k-state appears above the
chemical potential. As a result, counting electrons completely exhausts the
spectral weight in a Fermi liquid. However, for doped Mott insulators, this
sum rule fails as shown originally by Harris and LangeHarris and Lange (1967)
in the Hubbard model and confirmed experimentallyChen et al. (1991). The root
cause is the bifurcation of the spectral weight per k-state into lower and
upper Hubbard bands which leads experimentally to a breakdown 1-1
correspondence in Fermi liquid theory. As a result of this bifurcation, every
hole doped into a Mott insulator creates two empty states immediately above
the chemical potential. There are course dynamical corrections arising from
the kinetic mixing between the bands. Consequently, the low-energy spectral
weight in doped Mott insulators increases as

$\displaystyle\Lambda(x)=2x+f(t,x,U)>2x,$ (1)

where $f(t,x,U)$ is determined by the projected kinetic energyHarris and Lange
(1967); Eskes et al. (1991); Meinders et al. (1993) in the lower band. The
inequality stems from the fact that the dynamical correction is inherently
positive as it is mediated by double occupancy which necessarily leads to
empty states in the lower band. The occupied (below the chemical potential)
part of the lower band has a spectral weight of $1-x$. As a consequence, the
total weight of the lower band is $1+x+f(t,U)>1+x$. However, only $1+x$
electrons can fit into the lower band. Consequently, counting electrons does
not exhaust the low-energy spectral weight in a doped Mott insulator, and
degrees of freedom that have no interpretation in terms of standard fermionic
quasiparticles populate the band. This is not surprising as Mottness is
inherently about the breakdown of the single-particle concept.

Typically, perturbative calculationsHarris and Lange (1967); Eskes et al.
(1991); Leigh et al. (2007), phenomenologyLehner et al. (1994) or numerical
methodsMeinders et al. (1993) are employed to determine the mixing function,
$f(t,x,U)$. We show here that it can be determined precisely from a solvable
model for a doped Mott insulator. This constitutes the first time the low-
energy spectral weight in a doped Mott insulator has been calculated exactly
in any soluble model. The mixing contribution between the bands implies that
the spectral weight depends on the parameters in the Hamiltonian, quite
different from a Fermi liquid in which it is strictly the size of the Hilbert
space that determines low-energy degrees of freedom.

##  I Model and Setup

The starting point for our analysis is the orbital versionManning-Coe and
Bradlyn (2023); Mai et al. (2024) of the Hatsugai-KohmotoHatsugai and Kohmoto
(1992) (HK) model, given by the following tight-binding Hamiltonian on a
square lattice

$\displaystyle H_{\text{OHK}}=$ $\displaystyle\sum_{\bf
k,\alpha,\alpha^{\prime},\sigma}g_{\alpha,\alpha^{\prime}}({\bf k})c_{{\bf
k}\alpha\sigma}^{\dagger}c_{{\bf k}\alpha^{\prime}\sigma}-\mu\sum_{{\bf
k},\alpha}n_{{\bf k}\alpha,\sigma}$ (2) $\displaystyle+\sum_{{\bf
k},\alpha,\alpha^{\prime}}U_{\alpha,\alpha^{\prime}}n_{{\bf
k}\alpha\uparrow}n_{{\bf k}\alpha^{\prime}\downarrow},$

with the band index given by $k$ and $\alpha(\beta)$ label the orbital
indices. While the band-HK model lacks DSWT, not so in the orbital extension
which we have shown numericallyMai et al. (2024) contains such a contribution
in excellent agreement with simulationsEskes et al. (1991) on the 1D Hubbard
model. As we also pointed out previouslyMai et al. (2024), the orbital
Hatsugai-Kohmoto (OHK) model converges rapidly to the Hubbard model regardless
of the spatial dimension with quantitative agreement for double occupancy
achieved already at two or four-orbital extensions. For the first section of
this note, we will set the number of orbitals $N_{\alpha}=2$ and employ the
dispersion matrix,

$\displaystyle g=-2t\eta({\bf k})\sigma_{x},$ (3)

where $\sigma_{1}$ is the first Pauli matrix acting on the orbital indices,
$t$ is the hopping amplitude and $\eta(\bf k)$ is the single-particle
dispersion given by

$\eta({\bf k})=2\
\text{cos}\left(\frac{k_{x}}{2}\right)\text{cos}\left(\frac{k_{y}}{2}\right).$
(4)

We will later explore how the results vary with orbital number and with the
lattice dimensionality. It is to be noted that there is a minor difference in
convention for stating the dispersion here. Mai et al. Mai et al. (2024) use a
reduced Brillouin zone convention whereas we use the standard Brillouin zone
with the explicit scaling for $\mathbf{k}$. The model thus stated is diagonal
in $\bf k$ space and thus can be exactly diagonalized by employing a complete
basis in Fock space. We can thus exactly compute observables on each $\bf k$
site and proceed to the continuum limit with the substitution

$\sum_{{\bf k}\in\text{BZ}}\longrightarrow\frac{1}{4\pi^{2}}\int_{{\bf
k}\in\text{BZ}}.$ (5)

With malice aforethought, we assume that we are in the strongly interacting
limit with the bound $U>16\sqrt{3}t$.

The Hilbert space of the system can be decomposed into a direct product over
$\bf k$ states as $\mathcal{H}\longrightarrow\bigotimes_{k}\mathcal{H}_{k}$,
where each $\mathcal{H}_{k}$ has 16 basis states. This can be exactly
diagonalized to yield 8 distinct eigenbands as shown in Fig. 1. Since we are
interested in the low-energy spectral weight and for hole doping, we only need
to consider the lowest 5 eigenbands, the boxed region in Fig. 1.

Figure 1: The 8 eigenbands for 2-orbital HK along a slice of the Brillouin
zone. The box indicates the low energy sector of the model.

The model also conserves particle number, allowing us to decompose the Fock
space at each $\bf k$ state into fixed particle number sectors. Since we will
be considering electron doping from half filling for the 2-orbital HK model,
the relevant sectors are $N=2,3$ and $4$. The states can be labelled by a
binary index, $\ket{n_{\bf{k}1\downarrow}\ n_{\bf{k}1\uparrow}\
n_{\bf{k}2\downarrow}\ n_{\bf{k}2\uparrow}}$. Table. 1 enumerates the basis
states for the different particle sectors, along with the associated
eigenvalues.

Particle Sector | Eigenvalue | Eigenvectors
---|---|---
$N=2$ | $E_{1}=\frac{1}{2}\left(U-4sU-\sqrt{U^{2}+(8t\eta)^{2}}\right)$ |  $\frac{1}{2}\sqrt{1-\frac{U^{2}}{\sqrt{U^{2}+16(2t\eta)^{2}}}}(\ket{0011}+\ket{1100})$ $+\frac{1}{2}\sqrt{1+\frac{U^{2}}{\sqrt{U^{2}+16(2t\eta)^{2}}}}(\ket{1001}+\ket{0110})$
$N=3$ | $E_{2}=U-3sU-2t\eta$ |  $\frac{1}{\sqrt{2}}(\ket{1101}+\ket{0111})$, $\frac{1}{\sqrt{2}}(\ket{1110}+\ket{1011})$
$N=4$ | $E_{3}=2U-4sU$ | $\ket{1111}$
Table 1: Block decomposition and the energy levels of the 2-orbital HK
Hamiltonian

##  II Occupancy

We first need to determine the total occupancy of the system as a function of
the Hamiltonian parameters (in particular, the chemical potential $\mu$). The
deviation from half filling gives us the dopant concentration. The total
ground state occupancy is given by

$n(\mu)=\frac{1}{4\pi^{2}}\int d{\bf
k}\frac{1}{2}\sum_{\alpha,\sigma}\bra{\psi_{0}({\bf k})}c_{{\bf
k}\alpha\sigma}^{\dagger}c_{{\bf k}\alpha\sigma}\ket{\psi_{0}(\bf k)},$ (6)

where $\ket{\psi_{0}(\bf k)}$ is the ground state of the system with momentum
$\bf k$. We proceed by identifying the lowest eigenenergies as a function of
$\bf k$ and noting that the bands partition the Brillouin zone. Fig. 2 depicts
the partitioning between the $N=3$ and $N=2$ bands under electron doping for
two different values of $t/U$. It is important to note that the partitioning
is non-trivial for general values of $t/U$. We obtain a circular partition
that is analytically tractable only under the strongly interacting criterion
$t/U<\frac{1}{16\sqrt{3}}=r^{\ast}$.

We define a dimensionless parameter $s$ as $\mu=sU$. Note that for $0.5\leq
s\leq 1$, the ground state energies for all $\bf k$ are either $E_{1}$ or
$E_{2}$. For $s\geq 1$, the ground states energies are $E_{2}$ or $E_{3}$. We
will restrict ourselves to the former case, since the procedure for the latter
is analogous.

Figure 2: Partitioning of the Brillouin Zone into eigen-regions indexed by the
occupation eigenvalue. The figure on the left represents the partitioning
under electron doping for a system with $t/U>E^{\ast}$, whereas the figure on
the right represents $t/U<E^{\ast}$.

The two bands partition the Brillouin zone into $\Omega_{1}$ and $\Omega_{2}$,
where the ground state is either given by $E_{1}$ or $E_{2}$, respectively.
This will lead to a two-integral contribution,

$\displaystyle 8\pi^{2}n(s)=$ $\displaystyle\int_{{\bf k}\in\Omega_{1}}d{\bf
k}\bra{E_{1}(\bf k)}c_{\bf k}^{\dagger}c_{\bf k}\ket{E_{1}(\bf k)}$ (7)
$\displaystyle+\int_{{\bf k}\in\Omega_{2}}d{\bf k}\bra{E_{2}(\bf k)}c_{\bf
k}^{\dagger}c_{\bf k}\ket{E_{2}(\bf k)},$

to the occupancy. The expectation values in fact do not depend on $\bf k$ and
are simply equal to $2$, $3$ and $4$ for $E_{1},E_{2}$ and $E_{3}$
respectively. Thus, the problem is reduced to finding the areas of the regions
partitioned by the implicitly defined closed curve $E_{1}({\bf k})=E_{2}({\bf
k})$:

$\displaystyle n(s)$
$\displaystyle=\frac{1}{4\pi^{2}}\left(|\Omega_{1}|+\frac{3}{2}|\Omega_{2}|\right)$
$\displaystyle=\frac{4\pi^{2}-|\Omega_{2}|+1.5|\Omega_{2}|}{4\pi^{2}}$
$\displaystyle=1+\frac{1}{8\pi^{2}}|\Omega_{2}|.$ (8)

The area of this region, in general, does not have an analytical solution.
However, if we restrict to the weak doping limit, where the two bands just
begin to intersect, we can expand the cosine around the origin. This gives us
a circular region where the ground state is $E_{2}(\bf k)$, with a radius
given by,

$\displaystyle
R^{2}(s)=\frac{\left(4t-\frac{64t^{2}}{U}+U\left(s-1\right)\right)}{\frac{t}{2}-\frac{16t^{2}}{U}}.$
(9)

Note that the validity of the above formula is only for a small range around
$d$ where $E_{1}(\bf k)$ and $E_{2}(\bf k)$ intersect:

$\displaystyle s\in(s^{*},s^{*}+\epsilon)$ (10) $\displaystyle
s^{*}=\frac{1}{2}-\frac{4t}{U}+\frac{1}{2}\sqrt{1+\frac{256t^{2}}{U^{2}}}.$
(11)

Thus, in the low doping regime,

$n(s)=1+\frac{R^{2}(s)}{8\pi}$ (12)

defines the occupancy.

##  III Spectral Weight

We now relate the occupancy to the spectral function. As is well known, the
imaginary part of the Green function,

$\displaystyle A(\omega)=\sum_{\alpha,\sigma}\int d{\bf
k}\frac{1}{\pi}\text{Im }G_{\alpha,\sigma}({\bf k},\omega)$ (13)
$\displaystyle G_{\alpha,\sigma}({\bf
k},\omega)=\sum_{m}\frac{|\bra{\psi_{m}({\bf k})}c_{{\bf
k\alpha\sigma}}\ket{\psi_{0}({\bf k})}|^{2}}{\omega-E_{n}+E_{0}+i\eta^{+}},$
(14)

defines the spectral function. Taking the continuum limit and summing over the
orbitals yields our working expression,

$A(\omega)=\frac{1}{4\pi^{2}}\int d{\bf k}\sum_{m}|\bra{\psi_{m}({\bf
k})}c_{{\bf k\alpha\sigma}}\ket{\psi_{0}({\bf k})}|^{2}\delta(\omega-
E_{n}+E_{0})\\\ +|\bra{\psi_{0}({\bf k})}c_{{\bf
k\alpha\sigma}}\ket{\psi_{m}({\bf k})}|^{2}\delta(\omega+E_{n}-E_{0})),$ (15)

for the spectral function.

As noted previously, computing $A(\omega)$ simply requires that we find the
locus of intersection of the eigenregions, restricting our analysis to the low
doping limit. The new quantity to be computed here is the integrand with the
ground states given by $\ket{\psi_{0}}=\ket{E_{1}},\ket{E_{2}}$. Since the
relevant interested quantity is the spectral weight transfer, we want to
integrate the spectral weight from $\omega=0$ to a low energy cutoff. To do
so, we truncate the summation to the eigenbands that exceed the bandwidth in
the region $0.5\leq s\leq 2$. We can also ignore the second term in the
summand, since it is non-zero only for $\omega<0$. Thus, for the two cases of
interest, $\ket{\psi_{0}}=\ket{E_{1}},\ket{E_{2}}$, the integrand is given by

$\displaystyle\sum_{n}(\bra{E_{n}}c_{k}\ket{E_{1}}\bra{E_{1}}c_{k}^{\dagger}\ket{E_{n}}=0$
(16)
$\displaystyle\sum_{n}(\bra{E_{n}}c_{k}\ket{E_{1}}\bra{E_{1}}c_{k}^{\dagger}\ket{E_{n}}=1+\frac{2t\eta({{\mathbf{k}}})}{\sqrt{U^{2}+64t^{2}\eta^{2}({\mathbf{k}})}}.$
(17)

Performing the integral over the Brillouin zone as before yields the full
spectral weight transfer,

$\displaystyle\Lambda(s)=$
$\displaystyle\left(1+\frac{4t}{\sqrt{U^{2}+256t^{2}}}\right)\frac{R^{2}}{4\pi}$
(18)
$\displaystyle+\frac{R^{4}tU^{2}}{4\pi(256t^{2}+U^{2})^{3/2}}+\mathcal{O}(R^{6}).$

Rewriting the above in terms of the doping away from half filling, which is
given by $x=n-1$, we obtain the series expansion,

$\displaystyle\Lambda(x)=2x\left(1+\frac{4(t/U)}{\sqrt{1+256(t/U)^{2}}}\right)$
(19) $\displaystyle+16\pi
x^{2}\frac{(t/U)}{\left(1+256(t/U)^{2}\right)^{3/2}}+\mathcal{O}(x^{3}),$ (20)

for the spectral weight transfer as a function of the doping. In particular,
we can write the excess contribution to the spectral weight,

$\displaystyle f(t,x,U)=\frac{8t}{U}\frac{1}{\sqrt{1+256(t/U)^{2}}}x$ (21)
$\displaystyle+16\pi\frac{t}{U}\frac{1}{(1+256(t/U)^{2})^{3/2}}x^{2},$ (22)

laying plain the hopping dependence. Thus the slope of the dynamic component
of the spectral weight transfer is given simply as $4t/U$. This will be
confirmed by subsequent numerical calculations.

The above computation can be generalized to the case of arbitrary parameter
values to obtain an analytic upper bound on the dynamic spectral weight
transfer. Let us assume that the Brillouin zone is partitioned into sectors
$\Omega_{1}$ and $\Omega_{2}$. The contribution to the LESW only comes from
the $E_{2}(\bf k)$ band, corresponding to the $\Omega_{2}$ region. The SWT is
then given by

$\displaystyle\Lambda(s)=\int_{\Omega_{2}}d{\bf
k}\left(1+\frac{2t\eta({{\mathbf{k}}})}{\sqrt{U^{2}+64t^{2}\eta^{2}({\mathbf{k}})}}\right)$
(23)
$\displaystyle\Lambda(s)\leq\left(1+\frac{4t}{\sqrt{U^{2}+64t^{2}}}\right)|\Omega_{2}|.$
(24)

Using Eq. II, we can rewrite the expression,

$\Lambda(x)\leq 2x\left(1+\frac{4t}{\sqrt{U^{2}+64t^{2}}}\right),$ (25)

in terms of the doping. This is in fact a fairly good estimate for the
spectral weight at low doping, since the maximum is achieved at $\bf k=0$ and
the slope of the integrand is heavily damped by the large numerical factor in
the denominator.

##  IV Symmetry about $1/4$ and $3/4$ filling

Figure 3: Plots of the eigenbands at different doping levels. The x-axis is
the a slice along the Brillouin Zone with $k_{y}=0$. a) shows the bands at
half filling when the only occupied band is the $N=2$ band labelled by
$E_{1}$. b) Positions of bands for the symmetry point where $mu=U$ and the
only occupied band has $N=3$ given by the $E_{2}$. c) Positions of bands in
the fully filled system where $E_{3}$ is the only occupied band with $N=4$

The DSWT is symmetric as a function of the doping about the point $x=0.5$,
which corresponds to $3/4$ filling. This is made apparent by the following
argument. Consider $s\gg 1$ and we are deep in the $N=2$ occupation. As $s$ is
decreased, the locus of the intersection of $E_{2}$ and $E_{3}$,

$E_{2}(k)=E_{3}(k),$ (26)

determines the occupation. As before, we limit to low hole doping, allowing us
to expand the cosine to quadratic order, which yields a circle of a different
radius $R_{*}$. Thus, the occupation under hole doping is now given by

$n(s)=2-\frac{R_{*}^{2}}{2\pi}.$ (27)

Similarly, the spectral weight transfer reduces rto

$\Lambda(s)=2-(1+\frac{4t}{U})\frac{R_{*}^{2}}{\pi},$ (28)

or equivalently

$\Lambda(x)=2-2x(1+\frac{4t}{U}),$ (29)

which we obtain by eliminating $R^{\ast}$ through Eq. (27). Note that $x$ here
is the hole doping fraction from the fully occupied state.

The fillings of $1/4$ and $3/4$ occur at $\mu=0$ and $\mu=U$ respectively.
While we have shown that this symmetry obtains for the 2-orbital case, we
anticipate that it obtains for even orbital number systems. To see this, note
that the $1/4$ and $3/4$ filling points are where the dynamic contributions to
the spectral weight are maximal.

##  V Numerical Results

The complexity of $N_{\alpha}$ orbital HK is identical to that of the
$N_{\alpha}$-site Hubbard model. Thus, an analytic solution for large
$N_{\alpha}$ is untenable at present. However, we can numerically diagonalize
the 4-Orbital HK Hamiltonian to compute the spectral weight transfer (SWT).
Subtracting off the static SWT gives us the dynamic part, shown in Fig. 4. As
$t/U$ increases, the dynamical contribution to the DSWT increases, thereby
laying plain the dynamical nature of this correction to the total spectral
weight. The increase of the spectral weight as a function of $t/U$ regardless
of the doping appears to exhibit a universal rise. To extract the slope at low
doping, we perform a spline interpolation on the DSWT and analytically extract
the slope at $x=1$.

The result is shown in Fig. 5. What is evident here is that the slope is
independent of the number of orbitals up to a critical value of $t/U$ that
demarcates the transition to the weakly interacting regime. The critical value
in general depends on the single-particle dispersion used for the model. At
larger values of $t/U$, the band curvature results in a Brillouin Zone
partition that is annular as shown in Fig. 2. Thus, we cannot expand around
the bottom of the band, which is necessary to evaluate the spectral weight
integral. The independence of the slope on the number of orbitals underscores
the fact that the OHK model is governed by an underlying fixed pointZhao et
al. (2023) and hence no observable ultimately depends on the approach to the
Hubbard model.

Figure 4: Dynamical spectral weight transfer $\Lambda(x)-2x$ for 2 orbital HK
as a function of the doping. Clearly shown is the excess of the DSWT beyond
the static value of $2x$ which increases as the hopping increases. Figure 5:
The slope of the DSWT extracted numerically for 2 and 4 Orbital HK in 2D.
Observe that the results from the derived analytic expression diverge at the
threshold $t/U=E^{\ast}$

We observe a good fit to the analytic result for both the 2-orbital and the
4-orbital HK models below the identified threshold $t/U<16\sqrt{3}$. Recall,
the threshold $16\sqrt{3}$ is the value at which the intersection between the
$N=3$ and $N=2$ eigenbands morphs from a circle to an annulus, as depicted in
Fig. 2. Beyond this threshold, a well defined separation between the upper and
lower bands ceases and no clear criterion permits a unique assignment of the
LESW. Consequently, as long as the Hubbard bands are well separated, the slope
of the LESW is independent or orbital number. While numerics beyond 4-orbital
in 2D are infeasible for large systems, we can repeat the same analysis for
the 1D Orbital HK model, where we also have access to analytic results. In the
Appendix, we derive a general result for 2-orbital HK in d-dimensions at low
doping and at lowest order in $t/U$. For the 1D model, the single particle
dispersion is $\eta(k)=\cos(k/2)$ and

$\displaystyle\Lambda(x)=2x\left(1+2\frac{t}{U}\right).$ (30)

Observe that in 1D, this is consistent with the low doping results due to
Eskes et al. Eskes et al. (1994),

$\Lambda(x)=2x+\frac{4t}{\pi U}\text{sin}(\pi x)+\mathcal{O}((t/U)^{2}),$ (31)

which yields a slope of $4t/Ux$ in the low-doping limit. Further, we find that
at small $t/U$, the numerically obtained slope up to 5-orbital HK is
consistent with the analytic result, as shown in Fig. 6. This amplifies that
the orbital index $n$ plays no role in the LESW. Whether this remains the case
for all experimental observable remains to be seen.

Figure 6: Numerically extracted slopes for Orbital HK in 1D. We observe a
similar agreement at low $t/U$ across orbitals and an agreement with the
analytic expression.

##  VI Final Remarks

While DSWT is well documented both experimentallyChen et al. (1991) and
theoreticallyEskes et al. (1991); Meinders et al. (1993); Eskes et al. (1994),
an analytical treatment has been lacking. We have shown that the OHK model is
amenable to an analytical extraction of the DSWT. While the model approaches
the Hubbard model in the infinite orbital limit, we find that the slope of the
DSWT as a function of $t/U$ is independent of the orbital number. The
implication here is two-fold. First, it indicates a rapid convergence to the
Hubbard model. Second, the DSWT is a generic feature of Mott physics
determined entirely by the dimensionless ratio $t/U$ and not the orbital
number. Since, OHK preserves the decoupling of the momenta as the model can
still be written as $\sum_{k}h_{k}$, the dynamics are still governed by the HK
fixed pointZhao et al. (2023). Hence, no experimental observable can depend on
flow parameter that yields the Hubbard model, namely $N_{\alpha}$. Since the
OHK model is exactly solvable for small numbers of orbitals, this platform
affords an exact treatment of spectral weights in strongly correlated matter.
Consequently, the OHK model is an effective simulator of Mott physics
described by the Hubbard model.

##  VII Acknowledgements

We acknowledge NSF DMR-2111379 for partial funding of this project. This
research was supported in part by the Illinois Computes project which is
supported by the University of Illinois Urbana-Champaign and the University of
Illinois System. We would also like to thank Peizhi Mai for useful
discussions.

##  VIII Appendix

###  VIII.1 DSWT in d-dimensions

Our work here generalizes to any dimension. The key to deriving the
d-dimensional result is recognizing that the eigensystem at each $\bf{k}$
point is quite general for 2-orbital HK. The only differences arise from the
dimensionality of the Brillouin Zone and the single particle dispersion
$\eta({\mathbf{k}})$.

To proceed, we note that the general result for the occupation as a function
of the chemical potential is the same as before,

$n(s)=|\Omega^{d}_{1}|+\frac{3}{2}|\Omega^{d}_{2}|,$ (32)

where now $\Omega^{d}_{1}$ and $\Omega^{d}_{2}$ are in general higher
dimensional regions. In the low doping and flat-band limit, $\Omega^{d}_{2}$
is a hypersphere of some radius $R$. We use the fact that the Brillouin Zone
has total volume $(2\pi)^{d}$ to obtain,

$n(s)=1+\frac{V_{d}(R)}{2(2\pi)^{d}},$ (33)

where $V_{d}(R)$ is the volume of a d-dimensional ball with radius $R(s)$.
Next, we use the results in Eq. 16 to compute the spectral weight transfer as,

$\Lambda(s)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{B}_{R}}d^{d}\mathbf{k}\left(1+\frac{2t\eta({\mathbf{k}})}{\sqrt{U^{2}+64t^{2}\eta^{2}({\mathbf{k}})}}\right).$
(34)

To first order in the radius $R$, we obtain the expression,

$\displaystyle\Lambda(s)=\frac{V_{d}(R)}{(2\pi)^{d}}(1+2t\eta(\mathbf{k_{0}}))$
(35)
$\displaystyle\implies\Lambda(x)=2x\left(1+2\eta(\mathbf{k_{0}})\frac{t}{U}\right),$
(36)

as the general $d-$dimensional expression for the spectral weight. Note that
the $d-$dependence is entirely contained in the single particle dispersion
$\eta(\mathbf{k})$ at $\mathbf{k_{0}}=\mathbf{0}$. While this is in general
dependent on how the unit cells are defined, if we systematically construct
the $d-$dimensional system by stacking $(d-1)-$dimensional systems and
shifting each adjacent system by a lattice spacing, we arrive at a general
expression:

$\eta)\mathbf{k})=\sum_{i=1}^{d}\text{cos}\left(\frac{k_{i}}{2}\right).$ (37)

This yields the universal result

$\Lambda(x)=2x+4d\frac{t}{U}x.$ (38)

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# Erbium-based multifuncional compounds as molecular microkelvin-tunable
driving-sensing units.

Jarosław Rybusinski  Institute of Experimental Physics, Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Tomasz Fas  Institute
of Experimental Physics, Faculty of Physics, University of Warsaw, Pasteura 5,
02-093 Warsaw, Poland  Pablo Martin-Ramos  Advanced Materials Laboratory,
ETSIIAA, Universidad de Valladolid, Avenida de Madrid 44, 34004 Palencia,
Spain.  Victor Lavín  Department of Physics and MALTA Consolider Team,
Universidad de La Laguna, E-38206 San Cristóbal de La Laguna, Santa Cruz de
Tenerife, Spain.  Jacek Szczytko  Institute of Experimental Physics, Faculty
of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Jan
Suffczynski  Institute of Experimental Physics, Faculty of Physics, University
of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Inocencio R. Martín  Department
of Physics and MALTA Consolider Team, Universidad de La Laguna, E-38206 San
Cristóbal de La Laguna, Santa Cruz de Tenerife, Spain.  Jesus Martin-Gil
Advanced Materials Laboratory, ETSIIAA, Universidad de Valladolid, Avenida de
Madrid 44, 34004 Palencia, Spain.  Manuela Ramos Silva  CFisUC, Physics
Department, Universidade de Coimbra, Rua Larga, P-3004-516, Coimbra, Portugal.
Bruno Cury Camargo<EMAIL_ADDRESS>Institute of Experimental
Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw,
Poland

(August 30, 2024)

###### Abstract

We demonstrate the selective control of the magnetic response and
photoluminescence properties of $\text{Er}^{3+}$ centers with light, by
associating them with a highly conjugated $\beta$-diketonate
(1,3-di(2-naphtyl)-1,3-propanedione) ligand. We demonstrate this system to be
an optically-pumped molecular compound emitting in infra-red, which can be
employed as a precise heat-driving and detecting unit for low temperatures.

$\beta$-diketonate, erbium(III), magneto-optic effect, NIR, photoluminescence

##  I Introduction

Lanthanide ion coordination compounds possess fascinating physical properties
and important technological applications, which are often governed by their
magnetism and optical responses [1, 2, 3]. Unfortunately, such ions usually
present poor absorption characteristics, causing a severe hindrance to their
optically-pumped luminescence. Luckily, the photo-excitation of magnetically-
relevant emitting levels in these materials can be achieved almost at will by
employing organic ligands as chromophores. These components strongly absorb
light at selected wavelengths, sensitizing the magnetic ions through
intramolecular energy transfer – called the “antenna effect” [4, 5].

$\beta$-diketones containing aromatic groups are prime candidates for such a
role, as they exhibit strong absorption over a wide wavelength range and are
known to provide efficient energy transfer to lanthanide ions [6, 7]. Among
possible choices, the highly conjugated $\beta$-diketonate
1,3-di(2-naphthyl)-1,3-propanedione (Hdnm) is expected to sensitize
coordinated Er(III) and Eu(III) upon excitation in the visible range ($>400$
nm) [8, 9, 10, 11, 12].

However, the crystallization of organic compounds remains a challenging task,
with growth attempts often suffering from low rates, low yields, and
polymorphic specimens with minute fractions of the desired phase [13]. Optical
measurements in such small quantities – tenths or hundredths of micrograms –
pose a challenge in terms of proper collection and amplification of the
scattered light.

An alternative is to explore thermodynamic properties at equilibrium (e.g.
specific heat) of illuminated bulk samples. Conventional calorimetry, however,
involves the same challenges associated with optical experiments in very small
specimens, and dedicated instrumentation is required (see, e.g., ref. [14]).
On the other hand, the strong magnetic response of rare-earth-based compounds
makes such materials excellent candidates for magnetic measurements. Indeed,
conventional commercial SQUID magnetometry is capable of detecting the
magnetic response of as little as $10^{13}$ spins, which for the typical Er-
based composites considered here, corresponds to $10^{12}$ molecules ($\sim$ 4
ng).

This precision allows one to observe, a priori, variations in the magnetic
response or the magnetic ion when illuminating the molecular compound. In this
work, we explore this possibility by studying the magnetism of
$[\text{Er(dnm)}_{3}\text{(bipy)}]$ subjected to light excitation. We
demonstrate that SQUID magnetometry in this strong paramagnet acts as a viable
thermodynamical alternative to probe the optical and thermal properties of
minute sample quantities. Possible applications of such a system in
thermometry are discussed.

##  II Samples and experimental setups

The sample considered in this study is
tris(1,3-di(2-naphthyl)-1,3-propanedionate)mono(2,2’-bipyridine)erbium(III),
[$\text{Er(dnm)}_{3}\text{(bipy)}$,
$\text{C}_{79}\text{H}_{53}\text{ErN}_{2}\text{O}_{6}$]. This novel compound,
in a form of a yellow powder, was synthesized in-house following the procedure
outlined in the supplementary information (SI) [15]. Its structure consists of
a central rare-earth element ($\text{Er}^{3+}$), surrounded by six oxygen
atoms from $\beta$-diketonate ligands and two nitrogen atoms from a
2,2’-bipyridine neutral molecule (see Fig. 1), prompting a distorted square
antiprism chemical environment for the lanthanide. Chemical and structural
analysis confirmed a pure, yet disordered, solid (see the SI for the full
characterization [15]).

Figure 1:  Chemical structure of $\text{Er(dnm)}_{3}\text{(bipy)}$.

After the synthesis, the physical properties of the material were probed
through optical and magnetic measurements. Optical characterization was
achieved through absorption and luminescence measurements in the 5.0 K $\leq$
T $\leq$ 300.0 K temperature interval. They were performed by illuminating the
sample with the laser light in a wavelength range between 400 nm and 800 nm,
with a linewidth of 2 nm, using a LLTF Contrast powered by NKT 8 W
supercontinuum laser. The scattered and emitted light responses were captured
by an Andor SR-500i spectrometer with an IDus InGaAs array. Magnetic
measurements were carried out on a QuantumDesign MPMS 7T platform, in the 2 K
$\leq$ T $\leq$ 300 K temperature range, under magnetic fields up to 7 T [16].
The setup was adapted with an optic fiber window, to allow illumination of the
sample during magnetic measurements. For this purpose, a spectrometer-filtered
Xenon arc discharge light bulb was employed.

##  III Results

Figure 2:  $\chi$T vs. temperature for the sample considered herein, obtained
at $\mu_{0}H=0.2$ T. The inset shows the magnetic response per molecule
measured at (from top to bottom) T = 2 K, 5 K, 10 K, 50 K, 100 K, 200 K and
300 K. The red line in the main panel and the dashed lines in the inset
represent magnetization curves obtained with Phi [17] for an isolated $J=15/2$
$\text{Er}^{3+}$ magnetic ion in a distorted $D_{4d}$ crystallographic
environment. The y-axis experimental uncertainty is not visible in the scale
shown.

Temperature-scaled magnetic susceptibility vs. temperature
($\chi\text{T}\times\text{T}$) and magnetization vs. magnetic field
($\text{M}\times\text{H}$) for the compound under study are shown in Fig. 2.
Measurements revealed a clear paramagnetic-like behavior, which was, however,
not well-described by the conventional Curie-law ($\chi\propto\text{T}^{-1}$)
above 6 K. This can be attributed to the chemical environment for the
$\text{Er}^{3+}$ ion in $[\text{Er(dnm)}_{3}\text{(bipy)}]$. Indeed,
simulations using PHI software [17] reproduced well the
$\text{M}\times\text{H}$ and $\chi\text{T}\times\text{T}$ sample behavior by
assuming isolated $J=15/2$ magnetic centers in a $D_{4}$ crystalline
environment, in agreement with XRD data (see the SI [15]).

Strikingly, upon illumination, a remarkable variation of the sample magnetic
response was observed. Its magnitude, denoted
$|\Delta\text{M}|\equiv|\text{M}_{\text{dark}}-\text{M}_{\text{lit}}|$, was
determined by performing consecutive magnetization measurements for the sample
without ($\text{M}_{\text{dark}}$) and under ($\text{M}_{\text{lit}}$)
irradiation at fixed temperatures. The amplitude of $\Delta\text{M}$ closely
followed the intensity profile of the diffuse reflectance spectrum of the
material, and was most pronounced at low temperatures (see Fig. 3).

Among the features visible both in magnetic and optical measurements, a broad
absorption band in the 200-500 nm range can be mainly attributed to the
$\pi$-$\pi^{*}$ transitions of the dnm $\beta$-diketonate [18], with
overlapping bands from the 2,2’-bipyridine organic ligand (240-290 nm region)
[19] and from the
${}^{4}{\text{I}}_{15/2}\rightarrow({}^{2}\text{G},{}^{4}\text{F},{}^{2}\text{H})^{9/2}$
$\text{Er}^{3+}$ transition (407 nm). Above 490 nm, sharp peaks are associated
with intra-configurational ${}^{4}f^{11}-{}^{4}f^{11}$ electronic transitions
starting from the ${}^{4}{\text{I}}_{15/2}$ ground state of the
$\text{Er}^{3+}$ magnetic center, superimposed to the ligand’s absorption tail
[20].

The excitation spectrum of the transition at $\lambda\approx 1550$ nm
(${}^{4}{\text{I}}_{13/2}\rightarrow{}^{4}{\text{I}}_{15/2}$, see the inset in
Fig. 3) featured a broad peak when the sample was pumped with a wavelength of
about 450 nm. Such a feature is significantly red shifted compared to other
rare-earth-based coordination compounds [10, 18], including those based on
$\beta$-diketonate complexes [19]. Its occurrence is closely related to the
absorption maximum of the organic ligand used here, which occurs in the UV-Vis
range (see Fig. 3 and the SI [15]). Such a result strongly indicates the
sensitization of the emissive metal center in our compound by the antenna
effect.

Figure 3:  Diffuse reflectance (black line, left axis) and variation of
magnetic response measured at T = 2 K and $\mu_{0}H=0.2$ T
($|\Delta\text{M}|\equiv|\text{M}_{\text{dark}}-\text{M}_{\text{lit}}|$, red
points, right axis) as a function of wavelength. The labeled absorption lines
correspond to transitions from the ${}^{4}{\text{I}}_{15/2}$ ground state of
the $\text{Er}^{3+}$ ion, while the smooth background is associated with the
organic ligand. The inset shows the photoluminescence measured at
$\lambda=1550$ nm (pointed by an arrow in the main panel). The labelled peaks
in the inset are associated with the corresponding Er absorption lines of the
main panel, while the broad maximum centered at $\lambda\approx 450$ nm is due
to the energy transfer from the organic ligand to the metallic ion.

To better understand how absorption through the ligand influenced on the
magnetic center of the molecule, we measured the temporal evolution of the
sample’s photoluminescence at excitation pulses with wavelength
$\lambda_{\text{exc}}=375$ nm. This value is centered around the broad UV
absorption band of the $\text{(dnm)}_{3}$ ligand (see Fig. 3). The results
revealed that an initially strong fluorescent emission at $\lambda\approx 480$
nm quickly gave way to an emission line centered at $\lambda\approx 620$ nm
(see Fig. 4). The latter is ascribed to the triplet state of the organic
ligand, and largely overlaps with the ${}^{4}\text{F}_{9/2}$ absorption line
of $\text{Er}^{3+}$ (for a full description, see the SI [15]). This results in
the pumping of the rare earth ion through a resonant energy transfer process
[20], followed by relaxation and radiative decay (see the diagram provided in
the SI [15], Fig. S7), thus yielding the characteristic luminescence spectra
at around 1550 nm showcased in the inset of Fig. 3.

Figure 4:  Time-resolved evolution of the PL emission in the visible range of
$\text{Er(dnm)}_{3}\text{(bipy)}$.

The triplet state at $\lambda\approx 620$ nm exhibited a non-exponential
decay, with characteristic lifetimes of a few nanoseconds (see the SI [15],
Sec. V). These relatively small values suggest an efficient ligand-to-metal
energy transfer in the system [21]. The PL decay of the $\text{Er}^{3+}$
${}^{4}\text{I}_{13/2}$ multiplet ($\lambda\approx 1550$ nm), however,
exhibited a single exponential behavior, which indicates a consistent
coordination environment around the lanthanide ion (see the SI [15], Fig. S9).
The characteristic lifetime extracted for this transition ranged around
$\tau\approx 1.3$ $\mu\text{s}$, comparable to other $\text{Er}^{3+}$
compounds found in the literature [19, 22, 23]. Nevertheless, considering the
radiative lifetime of $\text{Er}^{3+}$ at approx. 1-2 ms allows the estimation
of the quantum efficiency of the transition at $\sim 0.1$% [24]. This suggests
that most energy captured by the organic ligands is absorbed by the material,
rather than being re-emitted by the rare earth center.

Such a small quantum efficiency does not allow $\Delta\text{M}(\lambda)$,
shown in Fig. 3, to be mainly attributed to a variation of the magnetic state
of the $\text{Er}^{3+}$ ion. Indeed, assuming that each incident photon is
absorbed by a molecule, leasing to a change of the magnetic state in the Er
ion, the required power delivery at $\lambda=520$ nm to induce
$\Delta\text{M}\approx 10^{-5}$ emu at T = 2 K would be of $\approx 4$ W. This
a value is unrealistic in our setup. Instead, at the highest applied power,
the population of excited Er ions in dynamical equilibrium is not larger than
$10^{8}$ (assuming a long, 10 $\mu$s relaxation time). This is far below the
detection threshold of $10^{18}$ spins for the SQUID magnetometer.

Instead, the change in the magnetic response upon illumination was directly
correlated with the change of M with T. This is shown in Fig. 5 for the
brightest excitation available in our experimental setup, which was of
$P_{100}=1170$ $\mu\text{W}$ at the sample location. In the figure, a
remarkable overlap is observed between $d\text{M}/d\text{T}$ and
$|\Delta\text{M}|/\Delta\text{T}$, with $\Delta\text{T}$ a constant
representing a temperature change. This result strongly suggests heat as the
driving mechanism behind the modulation of the sample magnetization, following
$\Delta\text{M}=d\text{M}/d\text{T}\times\Delta\text{T}$. For the case shown
in the figure, the variation in sample temperature is estimated at
$\Delta\text{T}\approx 0.25$ K. Such a value could be indeed controlled by
changing the intensity of the incident light. This is illustrated in the inset
of Fig. 5, through the demonstration that $|\Delta\text{M}|$ was well-
described by the phenomenological relation
$\Delta\text{M}\propto(P/P_{100})^{0.8}$ over the $2.1\text{ }\mu\text{W }\leq
P\leq 1170$ $\mu\text{W}$ interval.

Nonetheless, it should be noted that the power delivered to the sample can be
continuously tuned by controlling the excitation wavelength, and not only its
strength. This occurs because the absorption background of the molecule is
non-monotonic (see Fig. 3). That is, the compound is heated more intensely for
UV wavelengths and around absorption lines. For $\lambda\approx 520$ nm, the
relation reads $\Delta\text{T}=0.25\times(P/P_{100})^{0.8}$ K. It should be
stressed that this possibility results from the choice of a ligand with a
broad-band energy-absorption, bound to a rare-earth magnetic ion. The former
acts as the heating element of the molecule, whereas the latter provides a
mensurable temperature-dependent quantity.

Figure 5:  $d\text{M}/d\text{T}$ vs. temperature (filled black squares),
plotted together with $|\Delta\text{M}|/\Delta\text{T}$ (empty blue circles).
In it, $|\Delta\text{M}|$ is defined as
$|\Delta\text{M}|\equiv|\text{M}_{\text{dark}}-\text{M}_{\text{lit}}|$, where
$\text{M}_{\text{lit}}$ was measured for a sample illuminated with
$P_{100}=1170$ $\mu\text{W}$ of light at $\lambda=520$ nm. The term
$\Delta\text{T}$ is a scaling parameter set at $\Delta\text{T}=0.25$ K, chosen
so that both curves would overlap. The red line is a function of the type
$\Delta\text{M}^{\prime}=\text{M}^{\prime}_{0}(\text{T}/\text{T}_{0})^{-1.8}$,
$\text{T}_{0}=1$ K and $\text{M}^{\prime}_{0}=18\times 10^{-3}$ emu/K. Error
bars are not visible in the scale. The inset shows the variation rate
$\Delta\text{M}/\text{M}_{dark}$ measured at $\text{T}=2$ K as a function of
the relative light intensity $P/P_{100}$ (region indicated by a black arrow
and a dashed line in the main panel). The red line in the inset follows the
relation $|\Delta\text{M}|/\text{M}_{dark}\propto(P/P_{100})^{0.8}$.

Knowing the $\text{M}_{\text{dark}}(\text{T})$ behavior of the sample thus
allows a fine control of the system’s temperature through tracking the
magnetic response of the material in the presence of light. Considering a
realistic resolution of the SQUID magnetometer utilized here at $10^{-7}$ emu,
allows a precision of the temperature driving system at
$\Delta\text{T}_{\text{min}}\approx 10^{-7}\text{
emu}\times(d\text{M}/d\text{T})^{-1}$ . For the compound chosen,
$d\text{M}/d\text{T}\propto\text{T}^{-1.8}$ (see Fig. 5), which translates to
an enhanced resolution at lower temperatures. In the present work, the value
attained in $\Delta\text{M}(\lambda,\text{T}=2\text{ K})$ allows to infer a
resolution $\Delta\text{T}_{\text{min}}\approx 20$ $\mu\text{K}$. The mass of
the sample (0.2 mg) sets the sensitivity of our sample/heating device at 100
$\mu\text{K}$/mg. However, this sensitivity is likely to be at least two
orders of magnitude better, as absorption occurs in the sample surface region
over a depth of $\sim$100 nm, which may be estimated at not more than 10% of
the total volumetric content.

Considering the scenario outlined above, it is feasible to embed this magnetic
compound in a matrix of interest, and track the temperature of the latter by
measuring the magnetic properties of the Er complex. Provided that the
molecular compound remains in a dilute regime, is chemically passive in
relation to the matrix, and that crystalline fields introduced by the matrix
(if any) remain weak in comparison with local bonds, no major variations on
the magnetic response are expected. This occurs because the main factor
dictating magnetic properties of weakly interacting molecular compounds is the
ligand field around the magnetic center, which is sensitive to the molecular
configuration [25].

Similar approaches to temperature sensing have been employed in the past, e.g.
through the use of the magnetic response of magnetite [26] and paramagnetic
salts [27], or through tracking the photoluminescence response of organic-
rare-earth compounds [3]. Other applications have also been suggested for Er-
based materials (see e.g. [24]), relying on their spectroscopic properties.
However, here we combine both the optical and magnetic properties, resulting
in a sensing-driving device for fine temperature control. We stress that the
choice of a hybrid Er-organic compound poses a decisive advantage over the
simple inclusion of magnetic centers as thermometers (as in the case of
magnetite [26], paramagnetic salts [27], or single Er ions [24]). In addition
to the possibility of passivizing the magnetic ion with respect to the matrix
of interest, the organic “antenna” of these molecules can also have absorption
lines engineered at wavelengths for which the matrix is transparent [7, 6].
This ensures the delivery of thermal power to the system at the detecting
centers (rather than at the sample surface), which can be (for example)
diluted across the material to ensure homogeneous heating/sensing. We also
note that the resolution reported by us depends on the magnetic response of
the material under consideration, and may be altered by engineering a desired
MxT slope.

Taking into consideration the relative flexible operational temperature range,
the possibility of a tunable optical excitation, and the relatively simple
synthesis method for the material shown here, our reported resolution of 100
$\mu$K/mg is competitive in relation with other existing alternatives with
detection limits in the $\mu$K range. Among them, state-of-art low-temperature
thermometry using on-chip Coulomb blockade sensors yields a precision of 1%
for temperatures around 10 mK [28]. Albeit very powerful, such a technique
comes at a much larger cost, requires electrodes, is confined to low
temperatures, and demands complex microfabrication processes. Another
interesting approach makes use of an infra-red pyrometer operating in a
differential configuration on thermally-stable surroundings. Such an
instrument is reported to reach a remarkable precision around 30 $\mu$K at
room temperature, but is a passive technique fit for chemical and biological
processes [29]. Finally, opto-mechanical measurements probing vibrating modes
of nanometric membranes yield resolutions around 15 $\mu$K [30]. Such
impressive values, however, are suitable for on-chip detection, as they are
inherently sensitive to the environment surrounding the membranes. The method
outlined here by us, in contrast, requires a small magnetic field to probe
temperature, but is contact-free and is based on a compound that may be
dispersed in the material being probed. This approach eliminates electronic
heat contribution that may disrupt measurements otherwise (see e.g. [31]) and
ensures thermalization with the sample. Such an approach is promising for
systems in which electrodes are unnecessary, as it minimizes conductive heat
transfer while allowing selective heating by tuning light excitation
wavelength at a fixed output power.

##  IV Conclusions

In summary, we have established the magnetic and optical properties of a
newly-synthesized magnetic molecule based on a $\beta$-diketonate ligand
attached to a rare-earth $\text{Er}^{3+}$ ion. The presence of the organic
ligand in this compound acts as an absorption center (or “antenna”) for the
incoming radiation. Part of the absorbed energy is transferred towards the
$\text{Er}^{3+}$, which causes the material to act as an optically-pumped IR-
emitting compound. The remaining energy is absorbed by the material, which
then acts as a wavelength-tunable heating source. Its temperature can be
minutely tracked through the strong paramagnetic response of the embedded
magnetic ion. The resolution of the temperature tracking using this method
reciprocally increases with the temperature, reaching the value of 20
$\mu\text{K}$ at $\text{T}=2$ K for the compound shown here. By properly
choosing the organic antenna attached to the $\text{Er}^{3+}$ ion, the
engineering of local heater/thermometers embedded in a sample of interest may
be envisaged. The system can be homogeneously driven at wavelengths for which
the sample is transparent, while simultaneously having its temperature
magnetically probed.

## Acknowledgements

This work was carried out with the support of FCT (Fundação para a Ciência e a
Tecnologia) UIDB/04564/2020, and UIDP/04564/2020.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# On universal sign patterns

Vladimir Petrov Kostov  Université Côte d’Azur, CNRS, LJAD, France
<EMAIL_ADDRESS>
###### Abstract.

We consider polynomials $Q:=\sum_{j=0}^{d}a_{j}x^{j}$,
$a_{j}\in\mathbb{R}^{*}$, with all roots real. When the sign pattern
$\sigma(Q):=({\rm sgn}(a_{d}),{\rm sgn}(a_{d-1})$, $\ldots$, ${\rm
sgn}(a_{0}))$ has $\tilde{c}$ sign changes, the polynomial $Q$ has $\tilde{c}$
positive and $d-\tilde{c}$ negative roots. We suppose the moduli of these
roots distinct. The order of these moduli is defined when in their string as
points of the positive half-axis one marks the places of the moduli of
negative roots. A sign pattern $\sigma^{0}$ is universal when for any possible
order of the moduli there exists a polynomial $Q$ with $\sigma(Q)=\sigma^{0}$
and with this order of the moduli of its roots. We show that when the
polynomial $P_{m,n}:=(x-1)^{m}(x+1)^{n}$ has no vanishing coefficients, the
sign pattern $\sigma(P_{m,n})$ is universal. We also study the question when
$P_{m,n}$ can have vanishing coefficients.

Key words: real polynomial in one variable; hyperbolic polynomial; sign
pattern; Descartes’ rule of signs

AMS classification: 26C10; 30C15

##  1\. Introduction

The present paper deals with univariate real polynomials
$Q:=\sum_{j=0}^{d}a_{j}x^{j}$, $a_{j}\in\mathbb{R}$, $a_{d}\neq 0$. The
classical Descartes’ rule of signs states that the number $\ell_{+}$ of
positive roots of such a polynomial (counted with multiplicity) is majorized
by the number $\tilde{c}$ of sign changes in the sequence of its coefficients
and the difference $\tilde{c}-\ell_{+}$ is even, see [1], [2], [3], [4], [5],
[6], [7], [8], [15] or [16]. (When defining the number $\tilde{c}$ one has to
erase the zero coefficients.) We are mainly concerned with hyperbolic
polynomials, i. e. ones with all roots real. If all coefficients of a
hyperbolic polynomial are non-vanishing, then $\ell_{+}=\tilde{c}$ and
$\ell_{-}=\tilde{p}$, where $\ell_{-}$ is the number of negative roots and
$\tilde{p}$ is the number of sign preservations; clearly
$\ell_{+}+\ell_{-}=\tilde{c}+\tilde{p}=d$.

We are interested in a problem closely related to Descartes’ rule of signs.

######  Definition 1.

A sign pattern of length $d+1$ is a string of $d+1$ signs $+$ and/or $-$. If
all coefficients $a_{j}$ are non-zero, then we say that the polynomial $Q$
defines the sign pattern $\sigma(Q):=({\rm sgn}(a_{d}),{\rm sgn}(a_{d-1})$,
$\ldots$, ${\rm sgn}(a_{0}))$. If there is at least one vanishing coefficient,
then we say that $\sigma(Q)$ is a generalized sign pattern and its components
can be equal to $+$, $-$ or $0$. For a given sign pattern, we define its
corresponding change-preservation pattern by writing the letter $c$ when there
are two consecutive signs $(+,-)$ or $(-,+)$ and the letter $p$ if there are
signs $(+,+)$ or $(-,-)$. Thus the change-preservation pattern corresponding
to the sign pattern $(+,+,-,-,-,+,-,+)$ is $pcppccc$. We consider mainly monic
polynomials, so the correspondence between sign and change-preservation
patterns is bijective.

One can consider the moduli of the roots of $Q$. We limit ourselves to the
case when all these moduli are distinct. The definition of the order of these
moduli should be clear from the following example.

######  Example 1.

Suppose that $d=8$ and the polynomial $Q$ has positive roots
$\alpha_{1}<\alpha_{2}<\alpha_{3}$ and negative roots $-\gamma_{j}$, $j=1$,
$\ldots$, $5$, such that

$\gamma_{1}<\alpha_{1}<\gamma_{2}<\gamma_{3}<\gamma_{4}<\alpha_{2}<\gamma_{5}<\alpha_{3}~{}.$

Then we say that the order of moduli defined by the roots of $Q$ is
$NPNNNPNP$, i. e. the letters $P$ and $N$ indicate the relative positions of
the moduli of positive and negative roots.

######  Definition 2.

(1) A couple (sign pattern, order of moduli) is compatible if the number of
sign changes (resp. sign preservations) of the sign pattern is equal to the
number of letters $P$ (resp. $N$) of the order of moduli. In what follows we
consider only compatible couples and we write “couples” for short.

(2) A couple is realizable if there exists a hyperbolic polynomial which
defines the sign pattern of the couple and the moduli of whose roots define
the given order. We can also say that the given sign pattern is realizable
with the given order of moduli or vice versa.

In the present paper we consider the following problem:

######  Problem 1.

For a given degree $d$, which couples are realizable?

The problem is settled for $d\leq 6$, see [11] and [12]. We remind now some
results towards the resolution of Problem 1.

######  Definition 3.

Given a sign pattern $\sigma^{*}$ one defines its corresponding canonical
order of moduli as follows: one reads $\sigma^{*}$ from the right and to each
sign change (resp. sign preservation) one puts into correspondence the letter
$P$ (resp. $N$). Example: the canonical order corresponding to the sign
pattern $(+,-,-,-,+,+,-,+,-,-,+)$ is $PNPPPNPNNP$.

######  Remarks 1.

(1) Each couple (sign pattern, corresponding canonical order) is realizable,
see [13, Proposition 1]. However there exist sign patterns (called also
canonical, see [10]) which are realizable only with their corresponding
canonical orders. They are characterized by the property not to contain four
consecutive signs $(+,+,-,-)$, $(-,-,+,+)$, $(+,-,-,+)$ or $(-,+,+,-)$, so the
corresponding change-preservation patterns have no triple $cpc$ or $pcp$, i.
e. no isolated sign change and no isolated sign preservation except at the two
ends.

(2) There exist also orders of moduli (called rigid) which are realizable with
a single sign pattern. Except the trivial case when all roots are of the same
sign and the change-preservation pattern is $cc\ldots c$ or $pp\ldots p$,
rigid are the orders $\ldots NPNPNP$ and $\ldots PNPNPN$, realizable with the
sign patterns $(+,-,-,+,+,-,-,+,+,\ldots)$ and $(+,+,-,-,+,+,-,-,+,+,\ldots)$
respectively, see [14].

The present paper explores the inverse phenomenon, i. e. sign patterns
realizable with all compatible orders of moduli.

######  Remark 1.

Excluding the trivial case, orders of moduli realizable with all compatible
sign patterns do not exist. Indeed, if such an order $\Omega$ exists (with
$\tilde{c}>0$ letters $P$ and $\tilde{p}>0$ letters $N$), then it should be
realizable with each of the two change-preservation patterns, $cc\ldots
cpp\ldots p$ and $pp\ldots pcc\ldots c$, each with $\tilde{c}$ letters $c$ and
$\tilde{p}$ letters $p$. However these sign patterns are canonical, see part
(1) of Remarks 1, and their corresponding canonical orders of moduli are
different, see Definition 3, so $\Omega$ cannot be equal to each of them.

######  Definition 4.

A sign pattern is called universal if it is realizable with all compatible
orders of moduli.

There exist natural candidates for the role of universal sign patterns.

######  Notation 1.

(1) We set $P_{m,n}:=(x-1)^{m}(x+1)^{n}$, $m$, $n\in\mathbb{N}$. We denote by
$\sigma_{j}(P_{m,n})$ the sign $+$, $-$ or $0$ of the coefficient of $x^{j}$
of the polynomial $P_{m,n}$. It is clear that the polynomial $P_{m,n}$ is
self-reciprocal, i. e. $x^{m+n}P_{m,n}(1/x)=(-1)^{m}P_{m,n}(x)$. Hence the
sign changes and sign preservations in $\sigma(P_{m,n})$ are situated
symmetrically w.r.t. the middle of $\sigma(P_{m,n})$. As
$P_{m,n}(-x)=(-1)^{m+n}P_{n,m}(x)$, it suffices to study the polynomials
$P_{m,n}$ for $m\leq n$.

(2) We denote by $\Sigma_{k_{1},k_{2},k_{3},\ldots}$ a sign pattern beginning
with $k_{1}$ signs $+$ followed by $k_{2}$ signs $-$ followed by $k_{3}$ signs
$+$ etc. For a generalized sign pattern we use a similar notation including
zeros at the places between two numbers $k_{i}$ where zero coefficients occur,
see part (1) of Lemma 1. Thus the generalized sign pattern
$(+,+,0,-,-,-,+,0,-,-)$ will be denoted also by $\Sigma_{2,0,3,1,0,2}$.

######  Remark 2.

For $d\geq 4$, a sign pattern $\Sigma_{k_{1},k_{2},\ldots k_{s}}$ is canonical
if and only if $k_{i}\neq 2$ for $i=2$, $\ldots$, $s-1$. For $d=3$, not
canonical are exactly $\Sigma_{1,2,1}$ and $\Sigma_{2,2}$.

######  Theorem 1.

(1) When the polynomial $P_{m,n}$ has no vanishing coefficients, the sign
pattern $\sigma(P_{m,n})$ is universal.

(2) For $d=2m+1$, a sign pattern with $m$ sign changes and $m+1$ sign
preservations is universal if and only if this is the sign pattern
$\sigma(P_{m,m+1})=\Sigma_{2,2,\ldots,2}$.

(3) For $m$ even, one has $\sigma(P_{m,m+2})=\Sigma_{2,\ldots,2,3,2,\ldots,2}$
($m/2$ times $2$, one $3$ and again $m/2$ times $2$). For $m$ odd,
$\sigma(P_{m,m+2})=\Sigma_{2,2,\ldots,2,0,2,2,\ldots,2}$ is a generalized sign
pattern consisting of $(m+1)/2$ couples of equal signs (the signs of two
consecutive couples being opposite), one $0$ and again $(m+1)/2$ couples of
equal signs.

We give comments on the theorem in the next section and its proof in Section
4. While the proof of part (1) is short and needs no preliminary results, the
proofs of parts (2) and (3) rely on Section 3 in which we study the question
when the polynomial $P_{m,n}$ can have zero coefficients.

##  2\. Comments

If a hyperbolic polynomial $Q$ has at least one vanishing coefficient, then it
is not true that $\ell_{-}=\tilde{p}$, but only that $\ell_{-}$ is equal to
the number of sign changes in the sequence of coefficients of the polynomial
$Q(-x)$. Example: for $Q:=x^{2}-1$, one has
$\ell_{+}=\ell_{-}=\tilde{c}=1\neq\tilde{p}=0$; see also Remark 4. Bearing
this difference in mind, one can define realizability of couples (generalized
sign pattern, order of moduli) as this is done for sign patterns in part (2)
of Definition 2, and then define universal generalized sign patterns by
analogy with Definition 4.

For $d\leq 6$, the following two statements hold true (this can be directly
checked using the results in [11] and [12]):

  * •

All universal sign patterns are of the form $\sigma(P_{m,n})$ for some $m$ and
$n$.

  * •

When $\sigma(P_{m,n})$ is a generalized sign pattern, not a sign pattern, then
there exists no universal sign pattern with $\tilde{c}=m$ and $\tilde{p}=n$.
Hence for $d\leq 6$, no generalized sign pattern is universal, otherwise all
sign patterns obtained from it by replacing its zeros by $+$ or $-$ should
also be universal. (One can perturb the coefficients of $P_{m,n}$ to make them
all non-zero without changing the order of moduli.)

It would be interesting to prove or disprove that these two statements hold
true for any $d\in\mathbb{N}$.

In the next section we deal with the situation when $\sigma(P_{m,n})$ is not a
sign pattern, but a generalized sign pattern, so part (1) of Theorem 1 cannot
be applied. For $n>m$ and as $m\rightarrow\infty$, in asymptotically $1/4$ of
the cases there is a vanishing coefficient in $P_{m,n}$. These are the cases
when $m$ and $n$ are odd and this is the middle coefficient, see part (1) of
Proposition 1. Zero coefficients not in the middle also occur, but they seem
to appear less and less frequently as $m\rightarrow\infty$, see Propositions 1
and 3, Remarks 2 and Remarks 3.

##  3\. Vanishing coefficients of the polynomials $P_{m,n}$

###  3.1. Generalized sign patterns of the polynomials $P_{m,n}$

We begin this subsection by a result about the vanishing coefficients of the
polynomials $P_{m,n}$, in particular for small values of $m$.

######  Proposition 1.

(1) For $m$ and $n$ odd, the middle coefficient of the polynomial $P_{m,n}$
vanishes. For any $m$ and $n$, if the coefficient of $x^{j}$ in $P_{m,n}$
vanishes, then the coefficient of $x^{m+n-j}$ also vanishes. For $m=1$ and $n$
even, the polynomial $P_{m,n}$ has no vanishing coefficient. For $m=1$ and $n$
odd, the middle coefficient of $P_{m,n}$ is its only vanishing coefficient.
For $m$ and $n$ even, the middle coefficient of $P_{m,n}$ does not vanish.

(2) For $m=2$ and $n\geq 2$, the polynomial $P_{m,n}$ has vanishing
coefficients exactly when $n=\nu^{2}-2$, $\nu\in\mathbb{N}^{*}$, $\nu\geq 2$.
These are the coefficients of $x^{j}$, $j=(\nu\pm 1)\nu/2$, and the
generalized sign pattern $\sigma(P_{2,n})$ equals
$\Sigma_{(\nu-1)\nu/2,0,\nu-1,0,(\nu-1)\nu/2}$.

For $\nu^{2}-2<n<(\nu+1)^{2}-2$, one has

$\left\\{\begin{array}[]{ll}\sigma(P_{2,n})=\Sigma_{[(n-\nu)/2]+1,~{}\nu+1,~{}[(n-\nu)/2]+1}&{\rm
for}~{}~{}n-\nu~{}~{}{\rm even~{}~{}and}\\\ \\\
\sigma(P_{2,n})=\Sigma_{[(n-\nu)/2]+2,~{}\nu,~{}[(n-\nu)/2]+2}&{\rm
for}~{}~{}n-\nu~{}~{}{\rm odd.}\end{array}\right.$

(3) For $m=3$ and $n\geq 3$, the polynomial $P_{m,n}$ has vanishing
coefficients exactly in the following two cases:

(a) when $n$ is odd and the coefficient of $x^{(n+3)/2}$ vanishes (see part
(1));

(b) when $n=(\mu^{2}-7)/3$, where $\mu\in\mathbb{N}$, $\mu\geq 4$, is not
divisible by $3$; in this case vanishing are the coefficients of $x^{j}$,
$j=(\mu\pm 1)(\mu\pm 2)/6$.

Set $\gamma:=(\mu-1)(\mu-2)/6$. For $n=(\mu^{2}-7)/3$, one obtains

$\sigma(P_{3,n})=\left\\{\begin{array}[]{ll}\Sigma_{\gamma,0,(\mu-1)/2,(\mu-1)/2,0,\gamma}&{\rm
for~{}}\mu\equiv 1~{}{\rm or~{}}5~{}\mod 6~{}{\rm and}\\\ \\\
\Sigma_{\gamma,0,\mu/2-1,0,\mu/2-1,0,\gamma}&{\rm for~{}}\mu\equiv 2~{}{\rm
or~{}}4~{}\mod 6~{}.\end{array}\right.$

For $(\mu^{2}-7)/3<n<((\mu+1)^{2}-7)/3$, one has

$\sigma(P_{3,n})=\left\\{\begin{array}[]{ll}\Sigma_{\rho,\kappa,0,\kappa,\rho}&{\rm
for~{}}n~{}{\rm odd,~{}with}\\\ \\\
\rho=\frac{n+3-\mu}{2},~{}\kappa=\frac{\mu}{2}~{}{\rm
for~{}}\mu{\rm~{}even,}&\rho=\frac{n+4-\mu}{2},~{}\kappa=\frac{\mu-1}{2}~{}{\rm
for~{}}\mu{\rm~{}odd,}\\\ \\\ \Sigma_{\rho,\kappa,\kappa,\rho}&{\rm
for~{}}n{\rm~{}even,~{}with}\\\ \\\
\rho=\frac{n+4-\mu}{2},~{}\kappa=\frac{\mu}{2}~{}{\rm
for~{}}\mu{\rm~{}even,}&\rho=\frac{n+3-\mu}{2},~{}\kappa=\frac{\mu+1}{2}~{}{\rm
for~{}}\mu{\rm~{}odd.}\end{array}\right.$

(4) For $m\in\mathbb{N}^{*}$, the coefficients of all odd powers of $x$ in
$P_{m,m}$ and only they are $0$ while $P_{m,m+1}$ and, for even $m$,
$P_{m,m+2}$ have no vanishing coefficients. The only vanishing coefficient of
$P_{m,m+2}$ for $m$ odd is the one of $x^{m+1}$.

######  Remark 3.

The values of $n$ as defined in part (3)(b) of Proposition 1 form the
following sequence $\mathcal{S}$ (see [17, A369304]): $3$, $6$, $14$, $19$,
$31$, $38$, $54$, $63$, $83$, $94$, $118$, $131$, $\ldots$, in which pairs of
even numbers alternate with pairs of odd numbers. For an even $n$ of this
sequence, the polynomial $P_{3,n}$ has exactly two and for an odd $n$, it has
exactly three zero coefficients, see part (3) of Proposition 1. Consider the
sequence of differences between two consecutive terms of the sequence
$\mathcal{S}$: $3$, $8$, $5$, $12$, $7$, $16$, $9$, $20$, $11$, $24$, $13$,
$\ldots$. It takes alternatively the values of the two arithmetic progressions
$3$, $5$, $7$, $\ldots$ and $8$, $12$, $16$, $\ldots$.

###### Proof of Proposition 1.

The first two claims of part (1) follow from the self-reciprocity of
$P_{m,n}$. In particular, when $m$ and $n$ are odd, one has
$x^{m+n}P_{m,n}(1/x)=-P_{m,n}(x)$ and the middle coefficient of $P_{m,n}$ is
equal to its opposite, so it is $0$. For $m=1$, one has
$P_{m,n}=\sum_{j=0}^{n}({n\choose j-1}-{n\choose j})x^{j}$, so $P_{m,n}$ has a
zero coefficient if and only if $n$ is odd and $j=(n+1)/2$. If $m$ and $n$ are
even and the middle coefficient vanishes, then by part (2) of Lemma 1 there is
a sign change in the middle of the sign pattern $\sigma(P_{m,n})$. By self-
reciprocity the remaining sign changes are symmetrically situated w.r.t. the
middle of $\sigma(P_{m,n})$, so the total number of sign changes is odd which
contradicts the parity of $m$.

Part (2). For $m=2$, one has

$\begin{array}[]{cccccc}P_{2,n}&=&(x^{2}-2x+1)\sum_{j=0}^{n}{n\choose
j}x^{j}&=&\sum_{j=0}^{n+2}b_{n,j}x^{j}~{},&{\rm where}\\\ \\\
b_{n,j}&:=&{n\choose j}-2{n\choose j-1}+{n\choose j-2}~{}.&&&\end{array}$

The condition $b_{n,j}=0$ reads

$\begin{array}[]{lccc}(n-j+2)(n-j+1)-2(n-j+2)j+j(j-1)&=&0~{},&{\rm i.~{}e.}\\\
\\\ 4j^{2}-(4n+8)j+(n+2)(n+1)&=&0~{},&\end{array}$

so one obtains $j=\tilde{j}_{\pm}:=(n+2\pm\sqrt{n+2})/2$. These values of $j$
are integer exactly when $n=\nu^{2}-2$, $\nu\in\mathbb{N}^{*}$, $\nu\geq 2$,
from which the first two claims of part (2) follow.

Suppose that $\nu^{2}-2<n<(\nu+1)^{2}-2$, i. e.

$\nu^{2}<n+2<(\nu+1)^{2}~{}~{}~{}\,{\rm
and}~{}~{}~{}\,(n+1-\nu)/2<\tilde{j}_{-}<(n+2-\nu)/2~{}.$

If $n-\nu$ is even, then

$[(n-\nu)/2]<[(n-\nu)/2]+1/2<\tilde{j}_{-}<[(n-\nu)/2]+1$

(where $[.]$ stands for the integer part). If $n-\nu$ is odd, then

$[(n-\nu)/2]+1<\tilde{j}_{-}<[(n-\nu)/2]+3/2<[(n-\nu)/2]+2$

from which the rest of part (2) follows.

Part (3). For $m=3$, one has

$\begin{array}[]{l}P_{3,n}=(x^{3}-3x^{2}+3x-1)\sum_{j=0}^{n}{n\choose
j}x^{j}=\sum_{j=0}^{n+2}c_{n,j}x^{j}~{},~{}~{}~{}\,{\rm where}\\\ \\\
c_{n,j}:={n\choose j}-3{n\choose j-1}+3{n\choose j-2}-{n\choose
j-3}~{}.\end{array}$

The condition $c_{n,j}=0$ yields

$\begin{array}[]{l}(n-j+3)(n-j+2)(n-j+1)-3(n-j+3)(n-j+2)j\\\ \\\
+3(n-j+3)j(j-1)-j(j-1)(j-2)=0~{},~{}~{}~{}\,{\rm i.~{}e.}\\\ \\\
2(j-(n+3)/2)(4j^{2}-(4n+12)j+(n+2)(n+1))=0~{}.\end{array}$

The first factor defines an integer solution $j=(n+3)/2$ exactly when $n$ is
odd. This proves part (3)(a).

The second factor is $0$ for $j=j^{*}_{\pm}:=(n+3\pm\sqrt{3n+7})/2$ and
$j^{*}_{\pm}$ are integer only if $n=(\mu^{2}-7)/3$, $\mu\in\mathbb{N}$, in
which case $n$ and $\mu$ are of opposite parity and $j^{*}_{\pm}=(\mu\pm
1)(\mu\pm 2)/6$. Both these values are integer exactly when $\mu$ is not
divisible by $3$. This proves the first displayed formula of part (3)(b).

It is clear that if $j^{*}_{\pm}$ are not integer, then $\rho=[j^{*}_{-}]+1$.
The condition

$(\mu^{2}-7)/3<n<((\mu+1)^{2}-7)/3$

can be given the equivalent form

$-\mu-1<-\sqrt{3n+7}<-\mu~{},$

therefore

$(n+2-\mu)/2<j^{*}_{-}<(n+3-\mu)/2~{},~{}~{}~{}\,{\rm i.~{}e.}$

$\begin{array}[]{cl}(n-\mu)/2+1<j^{*}_{-}<(n-\mu)/2+3/2<(n-\mu+4)/2&{\rm
for}~{}~{}n-\mu~{}~{}{\rm even}~{},\\\ \\\
(n-\mu-1)/2+3/2<j^{*}_{-}<(n-\mu+3)/2&{\rm for}~{}~{}n-\mu~{}~{}{\rm
odd}~{}.\end{array}$

When $n$ and $\mu$ are of the same parity, then $n-\mu$ is even and
$\rho=(n-\mu+4)/2$. When they are of different parity, then $n-\mu$ is odd and
$\rho=(n-\mu+3)/2$.

It remains to find $\kappa$. To this end one has to notice that when $n$ is
odd, then the length of the sign pattern is odd and $\kappa+\rho=(n+3)/2$, so
$\kappa=\mu/2$ for $\mu$ even and $\kappa=(\mu-1)/2$ for $\mu$ odd. When $n$
is even, then the length of the sign pattern is even and
$\kappa+\rho=(n+4)/2$, thus $\kappa=\mu/2$ for $\mu$ even and
$\kappa=(\mu+1)/2$ for $\mu$ odd. This proves the second displayed formula of
part (3)(b).

Part (4). The first claim of part (4) results from $P_{m,m}=(x^{2}-1)^{m}$. As

(3.1) $P_{m,m+1}=(x+1)P_{m,m}=x\sum_{j=0}^{m}{m\choose
j}(-1)^{j}x^{2j}+\sum_{j=0}^{m}{m\choose j}(-1)^{j}x^{2j}~{},$

any coefficient of $P_{m,m+1}$ up to a sign is a binomial coefficient hence
non-zero. As in equality (3.1) one finds that any coefficient of

$\begin{array}[]{ccl}P_{m,m+2}&=&(x^{2}+2x+1)P_{m,m}\\\ \\\
&=&2x\sum_{j=0}^{m}(-1)^{j}{m\choose
j}x^{2j}+\sum_{j=0}^{m+1}\left((-1)^{j-1}{m\choose j-1}+(-1)^{j}{m\choose
j}\right)x^{2j}\end{array}$

is either of the form $\pm 2{m\choose j}\neq 0$ or of the form $\pm({m\choose
j}-{m\choose j-1})$. If $m$ is even, then the latter difference is non-zero
for any $j$, $1\leq j\leq m$. If $m$ is odd, then this difference vanishes
exactly when $j=(m+1)/2$ which gives the coefficient of $x^{m+1}$. ∎

######  Remarks 2.

(1) It would be interesting to show that for any $m\geq 1$ fixed, there are
infinitely many values of $n$ for which no coefficient of $P_{m,n}$ vanishes
and infinitely many values for which at least one coefficient is $0$.

(2) In the proofs of parts (2) and (3) of Proposition 1, for $m=2$ and $3$, we
obtained formulas for the values of $j$ expressed as functions of $n$ such
that when $j\in\mathbb{N}$, then the coefficient of $x^{j}$ in $P_{m,n}$
vanishes. Part (1) of the proposition implies that for $m=1$, one obtains
$j=(n+1)/2$. Similar formulas exist also for $m=4$ and $m=5$. For $m=4$, one
obtains four values of $j$:

$(n+4\pm\sqrt{\pm\theta+3n+8})/2~{},~{}~{}~{}\,\theta:=\sqrt{6n^{2}+30n+40}~{}.$

For $n=13$ and for $-\theta$, one obtains $j=(17\pm 3)/2$, i. e. $j=7$ and
$10$; for $n=13$ and for $+\theta$, one gets $j=3.89\ldots$ and $13.10\ldots$,
i. e. one does not obtain integer values for $j$. Thus
$\sigma(P_{4,13})=\Sigma_{4,3,0,2,0,3,4}$. For $n=62$ and for $-\theta$, one
obtains $j=(66\pm 6)/2$, i. e. $j=30$ and $36$; for $n=62$ and for $+\theta$,
one does not obtain integer values for $j$ and in a similar way one concludes
that $\sigma(P_{4,62})=\Sigma_{24,6,0,5,0,6,24}$. For $m=4$ and $5\leq n\leq
200$, these are the only cases in which one obtains vanishing coefficients in
$P_{m,n}$. For $n=151$, one has $\theta=2^{3}47$, but neither of the two
quantities $\pm\theta+3n+8$ is an exact square.

For $m=5$, the five values of $j$ are $(n+5)/2$ and

$(n+5\pm\sqrt{\pm\chi+5n+15})/2~{},~{}~{}~{}\,\chi:=\sqrt{10n^{2}+50n+76}~{}.$

For $n=12$ and for $+\chi$, one obtains $j=(17\pm 11)/2$, i. e. $j=3$ and
$14$; for $n=12$ and for $-\chi$, one does not obtain integer values for $j$;
$(n+5)/2$ is not integer for $n=12$. One has
$\sigma(P_{5,12})=\Sigma_{3,0,2,3,3,2,0,3}$. For $n=31$, one obtains
$\chi=106$; for $-\chi$, one gets $j=(36\pm 8)/2$, i. e. $j=14$ and $22$; for
$+\chi$, one does not obtain integer values for $j$, so
$\sigma(P_{5,31})=\Sigma_{10,4,0,3,0,3,0,4,10}$.

Set $\tilde{\chi}_{\pm}:=\pm\chi+5n+15$. For $n=62$, one obtains
$\chi=2^{2}\times 3\times 17=204$, $\tilde{\chi}_{+}=23^{2}$,
$\tilde{\chi}_{-}=11^{2}$ and

$\begin{array}[]{ccccc}(62+5\pm\sqrt{\tilde{\chi}_{+}})/2&=&(67\pm
23)/2&=&45~{}~{}~{}\,{\rm and}~{}~{}~{}\,22~{},\\\ \\\
(62+5\pm\sqrt{\tilde{\chi}_{-}})/2&=&(67\pm 11)/2&=&39~{}~{}~{}\,{\rm
and}~{}~{}~{}\,28~{},\end{array}$

so $\sigma(P_{5,62})=\Sigma_{22,0,5,0,5,5,0,5,0,22}$. These are all cases with
$6\leq n\leq 200$ for which there is more than one vanishing coefficient in
$P_{5,n}$.

###  3.2. Which triples of consecutive signs are possible for
$\sigma(P_{m,n})$?

The following lemma (see [9, Lemma 7]) exhibits some constraints about the
presence of vanishing coefficients in hyperbolic polynomials.

######  Lemma 1.

Suppose that $V$ is a hyperbolic polynomial of degree $d\geq 2$ with no root
at $0$. Then:

(1) $V$ does not have two or more consecutive vanishing coefficients.

(2) If $V$ has a vanishing coefficient, then the signs of its surrounding two
coefficients are opposite.

######  Remark 4.

For a hyperbolic degree $d$ polynomial $Q$ with non-zero constant term it is
true that $\ell_{+}=\tilde{c}$. Indeed, denote by $\lambda$ the number of
vanishing coefficients of $Q$. By Lemma 1 the triples of signs of coefficients
with a vanishing coefficient in the middle are of the form $(+,0,-)$ or
$(-,0,+)$. Denote by $\tilde{c}^{*}$ the number of sign changes $(+,-)$ or
$(-,+)$ and by $\tilde{p}^{*}$ the number of sign preservations of the form
$(+,+)$ or $(-,-)$ in $\sigma(Q)$. Then $\tilde{c}=\tilde{c}^{*}+\lambda$ and
the number of sign changes in $\sigma(Q(-x))$ equals $\tilde{p}^{*}+\lambda$.
But by Descartes’ rule of signs

$\begin{array}[]{cc}\ell_{+}\leq\tilde{c}=\tilde{c}^{*}+\lambda~{},&\ell_{-}\leq\tilde{p}^{*}+\lambda\\\
\\\ {\rm
and}&\tilde{c}^{*}+\tilde{p}^{*}+2\lambda=d=\ell_{+}+\ell_{-}~{},\end{array}$

so $\ell_{+}=\tilde{c}^{*}+\lambda=\tilde{c}$ and
$\ell_{-}=\tilde{p}^{*}+\lambda$. Observe that $\tilde{p}=\tilde{p}^{*}$.
Hence if $\lambda>0$, then $\ell_{-}>\tilde{p}$.

With the help of Lemma 1 we prove another proposition about the polynomials
$P_{m,n}$:

######  Proposition 2.

(1) For $m<n$, if a component of $\sigma(P_{m,n})$ equals $+$ (resp. equals
$-$), then at least one of the two surrounding components equals $+$ (resp.
$-$). In particular, $\sigma_{m+n-1}(P_{m,n})=+$ and
$\sigma_{1}(P_{m,n})=\sigma_{0}(P_{m,n})={\rm sign}((-1)^{m})$.

(2) There are $20$ triples of consecutive components of a generalized sign
pattern compatible with Lemma 1. Out of these the following $12$ can be
encountered in some polynomial $P_{m,n}$:

(3.2)
$(\pm,\pm,\pm)~{},~{}~{}~{}\,(\pm,\mp,\mp)~{},~{}~{}~{}\,(\pm,\pm,\mp)~{},~{}~{}~{}\,(0,\pm,\pm)~{},~{}~{}~{}\,(\pm,\pm,0)~{}~{}~{}\,{\rm
and}~{}~{}~{}\,(\pm,0,\mp)~{},$

while the following $8$ cannot be encountered:

(3.3)
$(\pm,\mp,\pm)~{},~{}~{}~{}\,(0,\pm,0)~{},~{}~{}~{}\,(0,\pm,\mp)~{}~{}~{}\,{\rm
and}~{}~{}~{}\,(\pm,\mp,0)~{}.$

######  Corollary 1.

For $1\leq m<n$, the (generalized) sign pattern $\sigma(P_{m,n})$ contains
strings of signs $+$ and strings of signs $-$. All these strings are of length
$\geq 2$, where two consecutive strings consist of opposite signs and they
either follow one another immediately or are separated by one $0$. Hence in
the (generalized) sign patterns $\sigma(P_{m,n})$ all sign changes are
isolated.

###### Proof of Proposition 2.

There are $27$ a priori possible triples of consecutive components of
generalized sign patterns – each component equals $+$, $-$ or $0$. The 7
triples $(0,0,0)$, $(\pm,0,0)$, $(0,0,\pm)$ and $(\pm,0,\pm)$, and only they,
are not compatible with Lemma 1. This proves the first claim of part (2).

As $P_{2,6}=x^{8}+4x^{7}+4x^{6}-4x^{5}-10x^{4}-4x^{3}+4x^{2}+4x+1$, one
encounters the triples $(\pm,\pm,\pm)$, $(\pm,\mp,\mp)$ and $(\pm,\pm,\mp)$ in
$\sigma(P_{2,6})$ and the remaining $6$ triples of (3.2) in
$\sigma(P_{2,14})$, where

$\sigma(P_{2,14})=(+,+,+,+,+,+,0,-,-,-,0,+,+,+,+,+,+)~{}.$

This proves the second claim of part (2).

The rest of the proposition is proved by induction on $m$. The induction base
is the case of $P_{1,n}$, $n\in\mathbb{N}$, $n>1$, with
$\sigma(P_{1,n})=(+,\ldots,+,0,-,\ldots,-)$ for $n$ odd ($(n+1)/2$ times $+$
and $(n+1)/2$ times $-$) and $\sigma(P_{1,n})=\Sigma_{n/2+1,n/2+1}$ for $n$
even; in this case all claims of the proposition are readily checked.

Suppose that the proposition holds true for $P_{m,n}$. Consider the polynomial
$P_{m+1,n+1}=(x^{2}-1)P_{m,n}$. Suppose that $\sigma(P_{m,n})$ contains the
following sign change:

$\begin{array}[]{cccc}{\rm
either}&(\sigma_{k}(P_{m,n}),\ldots,\sigma_{k-3}(P_{m,n}))&=&(\pm,\pm,\mp,\mp)\\\
\\\ {\rm
or}&(\sigma_{k}(P_{m,n}),\ldots,\sigma_{k-4}(P_{m,n}))&=&(\pm,\pm,0,\mp,\mp)~{}.\\\
\\\ {\rm
Then}&(\sigma_{k}(P_{m+1,n+1}),\sigma_{k-1}(P_{m+1,n+1}))&=&(\mp,\mp)\\\ \\\
{\rm
or}&(\sigma_{k}(P_{m+1,n+1}),\ldots,\sigma_{k-2}(P_{m+1,n+1}))&=&(\mp,\mp,\mp)\end{array}$

respectively. Thus in $\sigma(P_{m+1,n+1})$ there are $m$ strings of
consecutive equal signs $+$ or $-$ corresponding to the $m$ sign changes in
$\sigma(P_{m,n})$. It is clear that

$\begin{array}[]{cccccc}\sigma_{m+n+2}(P_{m+1,n+1})&=&\sigma_{m+n+1}(P_{m+1,n+1})&=&+&{\rm
and}\\\ \\\ \sigma_{1}(P_{m+1,n+1})&=&\sigma_{0}(P_{m+1,n+1})&=&{\rm
sgn}((-1)^{m+1})\end{array}$

which defines another $2$ strings. The signs $\pm$ of these $m+2$ strings
alternate; to prove this one can use the fact that
$\sigma_{k}(P_{m,n})=-\sigma_{k}(P_{m+1,n+1})$. Hence there are $\geq m+1$
sign changes in $\sigma(P_{m+1,n+1})$.

But by Descartes’ rule of signs these changes are exactly $m+1$, see Remark 4.
We show that then $\sigma(P_{m+1,n+1})$ cannot contain a triple of consecutive
components $(\pm,\mp,\pm)$, $(0,\pm,\mp)$, $(\pm,\mp,0)$ or $(0,\pm,0)$.
Indeed, consider two consecutive strings of equal signs in
$\sigma(P_{m+1,n+1})$ corresponding to sign changes in $\sigma(P_{m,n})$:
$\omega_{L}:=(\pm,\ldots,\pm)$ and $\omega_{R}:=(\mp,\ldots,\mp)$. We consider
the case when $\omega_{L}:=(+,\ldots,+)$ and $\omega_{R}:=(-,\ldots,-)$, the
opposite case is treated by analogy. We assume that $\omega_{L}$ and
$\omega_{R}$ are maximal, i. e. they cannot be further extended.

Denote by $\omega_{C}$ the possible string of components of
$\sigma(P_{m+1,n+1})$ between them. Suppose that $\omega_{C}$ has exactly two
components. Then they cannot be $(0,0)$, $(0,+)$ or $(-,0)$ (contradiction
with Lemma 1), neither $(+,0)$, $(0,-)$, $(+,+)$, $(+,-)$ or $(-,-)$
(contradiction with the maximality of $\omega_{L}$ and $\omega_{R}$). If they
are $(-,+)$, then there will be more that $m+1$ sign changes in
$\sigma(P_{m+1,n+1})$. Hence $\omega_{C}$ cannot have exactly two components.

Suppose that $\omega_{C}$ has $w\geq 3$ components. Set
$\omega_{C}:=(\alpha_{1},\ldots,\alpha_{w})$. Then in the same way
$\alpha_{1}=0$ or $\alpha_{1}=-$ while $\alpha_{w}=0$ or $\alpha_{w}=+$. If
$\alpha_{1}=0$ (resp. $\alpha_{w}=0$), then by Lemma 1, $\alpha_{2}=-$ (resp.
$\alpha_{w-1}=+$). If $(\alpha_{1},\alpha_{2})=(0,-)$, then between
$\alpha_{2}$ and $\omega_{R}$ there is no $+$, otherwise $\sigma(P_{m+1,n+1})$
has more than $m+1$ sign changes. If all components between $\alpha_{2}$ and
$\omega_{R}$ equal $-$, then this contradicts the maximality of $\omega_{C}$.
If there is at least one $0$, this contradicts Lemma 1. So
$(\alpha_{1},\alpha_{2})=(0,-)$ is impossible and similarly one cannot have
$(\alpha_{w-1},\alpha_{w})=(+,0)$. But then $\alpha_{1}=-$ and $\alpha_{w}=+$
which means that $\sigma(P_{m+1,n+1})$ has more than $m+1$ sign changes.

Thus either $\omega=\emptyset$ or $\omega$ consists of just one component. As
$\omega_{L}$ and $\omega_{R}$ are maximal, this component equals $0$. Thus in
the string $\omega_{L}\omega_{C}\omega_{R}$ (and hence in
$\sigma(P_{m+1,n+1})$) one does not encounter any of the triples (3.3). ∎

###  3.3. When does the coefficient of $x^{k}$ of $P_{m,n}$ vanish?

In this subsection we ask the question when for $k=1$, $\ldots$, $5$ does the
coefficient of $x^{k}$ of the polynomial $P_{m,n}$ vanish.

######  Remark 5.

The coefficient of $x$ of the polynomial $P_{m,n}$ vanishes if and only if
$m=n$. Indeed, this coefficient equals ${n\choose 1}-{m\choose 1}$. Hence this
is the (even) polynomial $(x^{2}-1)^{m}$.

######  Proposition 3.

(1) For $m<n$, the coefficient of $x^{2}$ of the polynomial $P_{m,n}$ vanishes
if and only if $m=s(s-1)/2$, $n=s(s+1)/2$, $s\in\mathbb{N}^{*}$.

(2) For $m<n$, the coefficient of $x^{3}$ of the polynomial $P_{m,n}$ vanishes
if and only if $m=(s-1)(s-2)/6$, $n=(s+1)(s+2)/6$, where $s\in\mathbb{N}^{*}$
is not divisible by $3$.

Hence there exist infinitely many values of $m$ for which there exists $n>m$
such that the polynomial $P_{m,n}$ has more than one vanishing coefficient.

###### Proof.

Part (1). The coefficient of $x^{2}$ of $P_{m,n}$ equals

$m(m-1)/2-mn+n(n-1)/2=((n-m)^{2}-(m+n))/2~{}.$

It vanishes exactly when $(n-m)^{2}=m+n$. Set $n-m=:s$, $m+n=:d$. Hence
$d=s^{2}$, $m=s(s-1)/2$, $n=s(s+1)/2$, $s\in\mathbb{N}^{*}$.

Part (2). The coefficient of $x^{3}$ of $P_{m,n}$ equals

${n\choose 3}-{n\choose 2}m+n{m\choose 2}-{m\choose
3}=(n-m)((n-m)^{2}-3(n+m)+2)/6~{}.$

It vanishes exactly when $d:=n+m=((n-m)^{2}+2)/3$, so one sets $s:=n-m$ hence
$n+m=(s^{2}+2)/3$, $n=(s+1)(s+2)/6$ and $m=(s-1)(s-2)/6$. For
$s\in\mathbb{N}^{*}$, the numbers $m$ and $n$ are integer if and only if $s$
is not divisible by 3. ∎

######  Remarks 3.

(1) One can try to find in the same way values of the couple $(m,n)$ for which
the coefficient of $x^{4}$ of the polynomial $P_{m,n}$ vanishes. This
coefficient equals $\kappa:=(s^{4}-6s^{2}d+8s^{2}+3d^{2}-6d)/24$, where
$d=m+n$ and $s=n-m$. The condition $\kappa=0$ yields

(3.4) $d=s^{2}+1\pm\sqrt{1+2(s^{4}-s^{2})/3}~{}.$

For $s\leq 20$, one gets integer values for $m=(d-s)/2$ and $n=(d+s)/2$ only
for $s=1$, $2$, $3$ and $6$. One can exclude the trivial cases with $m+n<4$.
The remaining cases are:

$(m,n)~{}=~{}(3,5),~{}(7,10),~{}(30,36)~{}~{}~{}\,{\rm
and}~{}~{}~{}\,(1,7)~{}.$

The latter couple is the only one in which one chooses the sign “$-$” in
formula (3.4). We list the signs of the last $6$ coefficients of the
polynomials $P_{m,n}$ corresponding to these $4$ cases:

$(-,0,+,+,-,-),~{}(-,0,+,+,-,-),~{}(+,0,-,-,+,+)~{}~{}~{}\,{\rm
and}~{}~{}~{}\,(+,0,-,-,-,-)~{}.$

For $21\leq s\leq 200$, only for $s=91$ does one find integer couples $(m,n)$,
namely $(715,806)$ with the sign “$-$” and $(7476,7567)$ with the sign “$+$”.
The signs of their $6$ last coefficients are $(+,0,-,-,-,-)$ and
$(+,0,-,-,+,+)$ respectively.

(2) Consider the couples $(m,n)$ for which the coefficient of $x^{5}$ of
$P_{m,n}$ equals 0. With the notation of part (1) this coefficient is equal to
$d(s^{4}-10s^{2}d+20s^{2}+15d^{2}-50d+24)/120$. It vanishes for

(3.5) $d=(1/15)(5s^{2}+25\pm\sqrt{10s^{4}-50s^{2}+265})~{}.$

For $s\leq 20$, integer values for $m$ and $n$ are obtained for $s=1$, $2$,
$3$, $4$, $8$ and $11$. Eliminating the couples $(m,n)$ with $m+n<5$ leaves
$5$ couples:

$(m,n)~{}=~{}(3,7),~{}(14,22),~{}(1,9),~{}(28,39)~{}~{}~{}\,{\rm
and}~{}~{}~{}\,(3,14)~{}.$

The signs of their last $7$ coefficients equal respectively

$\begin{array}[]{lcl}(-,0,+,+,-,-,-)~{},&(+,0,-,-,+,+,+)~{},&(+,0,-,-,-,-,-)~{},\\\
\\\ (+,0,-,-,+,+,+)&{\rm and}&(+,0,-,-,-,-,-)~{}.\end{array}$

For the third and fifth case one chooses the sign “$-$” (see (3.5)), for the
rest the sign “$+$”. For $21\leq s\leq 200$, one obtains integer couples
$(m,n)$ only for $s=23$. These are $(22,45)$ with the sign “$-$” and
$(133,156)$ with the sign “$+$”. The signs of their $7$ last coefficients are
$(-,0,+,+,+,+,+)$ and $(-,0,+,+,-,-,-)$ respectively.

##  4\. Proof of Theorem 1

Part (1). For any order $\Omega$ of $m$ moduli of positive and $n$ moduli of
negative roots, one can find $m$ positive numbers $\alpha_{i}$ and $n$
negative numbers $-\beta_{j}$ such that their moduli are all distinct and
realize the order $\Omega$. Then for all $\varepsilon>0$ sufficiently small,
the moduli of the numbers $1+\varepsilon\alpha_{i}$ and
$-1-\varepsilon\beta_{j}$ are also distinct and realize the order $\Omega$.

On the other hand for $\varepsilon>0$ sufficiently small, the coefficients of
the polynomial

$\tilde{P}:=\prod_{i=1}^{m}(x-1-\varepsilon\alpha_{i})\prod_{j=1}^{n}(x+1+\varepsilon\beta_{j})$

are close to the ones of the polynomial $P_{m,n}$, so
$\sigma(\tilde{P})=\sigma(P_{m,n})$. Hence $\sigma(P_{m,n})$ is a universal
sign pattern.

Part (2). A universal sign pattern as in the theorem is realizable with the
order of moduli $NPNPN\ldots N$. This is a rigid order, see part (2) of
Remarks 1. It is realizable only with the sign pattern
$\Sigma_{2,2,\ldots,2}$. On the other hand by part (4) of Proposition 1 the
polynomial $P_{m,m+1}$ has no vanishing coefficients. This polynomial has
$2m+2$ coefficients and $m$ sign changes. By part (1) of Proposition 2,
$\sigma(P_{m,m+1})$ contains $m+1$ sequences of equal signs, each of length
$\geq 2$, so $\sigma(P_{m,m+1})=\Sigma_{2,2,\ldots,2}$.

Part (3). Suppose that $m$ is even. The polynomial $P_{m,m+2}$ has $2m+3$
coefficients. By part (1) of Proposition 2, there must be $m+1$ sequences of
at least two consecutive coefficients of equal signs. Hence there are $m$
sequences of two and one sequence of three such coefficients. The polynomial
is self-reciprocal, so the sequence of three coefficients is in the middle.

Suppose that $m$ is odd. The middle coefficient of $P_{m,m+2}$ is $0$. By part
(1) of Proposition 2, the rest of its $2m+3$ coefficients form $m+1$ couples
of two consecutive coefficients of equal signs, $(m+1)/2$ to the left and
$(m+1)/2$ to the right of the $0$ in the middle.

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