# Quasi-one-dimensional magnetism in the spin-$\frac{1}{2}$ antiferromagnet
BaNa2Cu(VO4)2

Sebin J. Sebastian  K. Somesh  School of Physics, Indian Institute of Science
Education and Research Thiruvananthapuram-695551, India  M. Nandi  Ames
Laboratory and Department of Physics and Astronomy, Iowa State University,
Ames, Iowa 50011, USA  N. Ahmed  P. Bag  School of Physics, Indian Institute
of Science Education and Research Thiruvananthapuram-695551, India  M. Baenitz
Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40,
01187 Dresden, Germany  B. Koo  Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany  J. Sichelschmidt  Max
Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40, 01187
Dresden, Germany  A. A. Tsirlin<EMAIL_ADDRESS>Experimental Physics VI,
Center for Electronic Correlations and Magnetism, University of Augsburg,
86135 Augsburg, Germany  Y. Furukawa  Ames Laboratory and Department of
Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA  R. Nath
<EMAIL_ADDRESS>School of Physics, Indian Institute of Science Education
and Research Thiruvananthapuram-695551, India

###### Abstract

We report synthesis and magnetic properties of quasi-one-dimensional
spin-$\frac{1}{2}$ Heisenberg antiferromagnetic chain compound BaNa2Cu(VO4)2.
This orthovanadate has a centrosymmetric crystal structure, $C2/c$, where the
magnetic Cu2+ ions form spin chains. These chains are arranged in layers, with
the chain direction changing by 62$\degree$ between the two successive layers.
Alternatively, the spin lattice can be viewed as anisotropic triangular layers
upon taking the inter-chain interactions into consideration. Despite this
potential structural complexity, temperature-dependent magnetic
susceptibility, heat capacity, ESR intensity, and NMR shift agree well with
the uniform spin-$1/2$ Heisenberg chain model with an intrachain coupling of
$J/k_{\rm B}\simeq 5.6$ K. The saturation field obtained from the magnetic
isotherm measurement consistently reproduces the value of $J/k_{\rm B}$.
Further, the 51V NMR spin-lattice relaxation rate mimics the 1D character in
the intermediate temperature range, whereas magnetic long-range order sets in
below $T_{\rm N}\simeq 0.25$ K. The effective interchain coupling is estimated
to be $J_{\perp}/k_{\rm B}\simeq 0.1$ K. The theoretical estimation of
exchange couplings using band-structure calculations reciprocate our
experimental findings and unambiguously establish the 1D character of the
compound. Finally, the spin lattice of BaNa2Cu(VO4)2 is compared with the
chemically similar but not isostructural compound BaAg2Cu(VO${}_{4})_{2}$.

##  I Introduction

The studies of low-dimensional and frustrated spin systems have contributed
substantially in understanding the quantum phase transitions at low
temperatures S. Sachdev (2007); Ramirez (1994). In one-dimensional (1D)
antiferromagnetic (AFM) spin-$1/2$ uniform Heisenberg chains, magnetic long-
range-order (LRO) is forbidden at zero temperature as a result of enhanced
quantum fluctuations, thereby exhibiting a gapless excitation spectrum and
power-law decay of spin-spin correlations Mermin and Wagner (1966). However,
non-zero inter-chain interactions, inherent to real materials, lead to the
formation of magnetic LRO at finite temperatures Yasuda  _et al._ (2005);
Schulz (1996). On the other hand, the inter-chain interactions often create
frustrated network between the chains that eventually prevents the system from
achieving the conventional LRO but stabilizes different exotic states instead
Ramirez (1994); Greedan, John E. (2001); Kojima  _et al._ (1997); Lancaster
_et al._ (2006). Further, competing interactions as realized in a set of
compounds, add magnetic frustration in spin chains which along with quantum
fluctuations host a multitude of intriguing magnetic ground states Furukawa
_et al._ (2010); Hase  _et al._ (1993); Drechsler  _et al._ (2007). The
transition-metal oxides offer nearly endless opportunities for realizing 1D
spin chains with different types of exchange couplings, and may harbor wide
varieties of exotic phases of matter.

Figure 1:  Left panel: crystal structure of BaNa2Cu(VO${}_{4})_{2}$ showing
corner-shared CuO4 plaquettes and VO4 tetrahedra forming layers of spin
chains. The coupling of Na1+ ions with the magnetic Cu2+ ions is also shown.
Middle panel: crystal structure of BaNa2Cu(VO${}_{4})_{2}$ shown in a
different orientation to visualize the spin chains running along the $[110]$
and $[1\bar{1}0]$ directions; black spheres show the Ba atoms, the Na atoms
are omitted for clarity. Right panel: the structure of the single spin chain
with the geometrical parameters $\varphi$ and $r$ that control the sign and
strength of superexchange through the double bridges of the VO4 tetrahedra.

Recently, synthesis and magnetic properties of a series of compounds
$AA^{\prime}M$(VO4)2 ($A=$ Ba and Sr, $A^{\prime}=$ Na2 and Ag2, and $M=$ Mn,
Ni, Co, Fe, and Cu) were reported. Despite some variations in their crystal
structures, the magnetic model of anisotropic triangular lattice has been
generally used to understand their magnetism Amuneke  _et al._ (2011); Möller
_et al._ (2012); Nakayama  _et al._ (2013); Reuß  _et al._ (2018); Sanjeewa
_et al._ (2019); Amuneke  _et al._ (2014); Lee  _et al._ (2020).
BaAg2Cu(VO${}_{4})_{2}$ stands as an exception in this series, because its
crystal structure is triclinic (space group: $P\overline{1}$) Amuneke  _et
al._ (2011), and indeed microscopic analysis via density-functional band-
structure calculations Tsirlin  _et al._ (2012) combined with resonance
spectroscopy Krupskaya  _et al._ (2017) revealed 1D magnetism with two
dissimilar types of spin chains, one ferromagnetic and one antiferromagnetic,
coexisting in the structure.

Here, we present for the first time the magnetic properties of
BaNa2Cu(VO${}_{4})_{2}$, another Cu2+ member of the series von Postel and
Müller-Buschbaum (1992). Its structure features four equal Cu–Cu distances of
5.507 Å as well as two slightly longer distances of 5.686 Å, all in the $ab$
plane. This interaction geometry is a pre-requisite of the triangular-lattice
scenario previously established for other members of the $AA^{\prime}M$(VO4)2
series. On the other hand, the square-planar oxygen coordination of Cu2+ and
the VO4 bridges between such CuO4 plaquette units may lead to one preferred
direction for magnetic couplings in the $ab$ plane (Fig. 1). Interestingly,
this preferred direction changes from $\mathbf{a}+\mathbf{b}$ in one plane to
$\mathbf{a}-\mathbf{b}$ in the adjacent plane, thus leading to the formation
of crossed spin chains arranged at $62^{\circ}$ relative to each other. This
geometry resembles the crossed-chain magnetic model, where exotic ground
states and potential spin-liquid behavior have been proposed theoretically
Starykh  _et al._ (2002); Sindzingre  _et al._ (2002); Brenig and Grzeschik
(2004); Starykh  _et al._ (2005); Bishop  _et al._ (2012).

Here, we use magnetization, heat capacity, electron spin resonance (ESR), and
nuclear magnetic resonance (NMR) measurements, as well as complementary band-
structure calculations to uncover magnetic interactions in
BaNa2Cu(VO${}_{4})_{2}$ and establish its microscopic magnetic model. Our data
suggest the formation of uniform AFM spin chains with the exchange coupling
$J/k_{\rm B}\simeq 5.6$ K, and the subsequent onset of magnetic LRO below
$T_{\rm N}\simeq 0.25$ K. We suggest that this magnetic order can be driven by
residual inter-chain couplings of $J_{\perp}/k_{\rm B}\simeq 0.1$ K that
remain non-frustrated despite the crossed-chain structural geometry. Our
results establish that the mere presence of spin chains arranged along two
different directions is insufficient to reach the interesting physics of the
crossed-chain model, and an additional condition for the lateral displacement
of these chains has to be met experimentally.

##  II Methods

Figure 2:  Powder XRD pattern of BaNa2Cu(VO4)2 measured at $T=300$ K. The
circles are experimental data and the solid black line is the Le-Bail fit. The
Bragg positions are indicated by green vertical lines and the bottom solid
blue line indicates the difference between the experimental and calculated
intensities.

Polycrystalline sample of BaNa2Cu(VO4)2 was prepared by the usual solid-state
reaction method. Initially, the reactants Na2CO3 (Aldrich, 99.995%), BaCO3
(Aldrich, 99.995%), CuO (Aldrich, 99.999%), and V2O5 (Aldrich, 99.995%) were
mixed in proper molar ratios, thoroughly ground, and then pressed into
pellets. The pellets were sintered in an alumina crucible at 500 0C for three
days in air with several intermediate grindings. The phase purity of the
sample was confirmed from the powder x-ray diffraction (XRD) performed at room
temperature. For the powder XRD experiment, a PANalytical powder
diffractometer with CuKα radiation ($\lambda_{\rm avg}\simeq 1.54182$ Å) was
used. Le-Bail analysis of the powder XRD pattern was performed using the
`FullProf` software package J. Rodríguez-Carvajal (1993). Figure 2 displays
the room-temperature powder XRD data along with the fit. The structural
parameters given in Ref. von Postel and Müller-Buschbaum (1992) were used as
the initial parameters. The goodness-of-fit was found to be $\chi^{2}\simeq
3.57$. The obtained lattice parameters are $a=9.4379(1)$ Å, $b=5.6926(1)$ Å,
$c=14.0519(1)$ Å, and $\beta=92.3434(8)^{\circ}$ and the unit cell volume
$V_{cell}\simeq 754.34$ Å3, which are in close agreement with the previous
report von Postel and Müller-Buschbaum (1992).

Magnetization ($M$) measurements were performed as a function of temperature
(0.48 K $\leq T\leq 380$ K) and magnetic field ($0\leq H\leq 14$ T) using a
Superconducting Quantum Interference Device (SQUID, Quantum Design)
magnetometer and a Physical Property Measurement System (PPMS, Quantum
Design). The SQUID enabled us to measure magnetization down to 0.48 K with a
3He attachment. High-field magnetization up to 14 T were measured using PPMS.
Heat capacity ($C_{\rm p}$) was measured as a function of $T$ (0.4 K $\leq
T\leq 200$ K) on a sintered pellet using the thermal relaxation method in
PPMS. The temperature down to 0.4 K was achieved using a 3He attachment to the
PPMS.

The ESR experiments were performed on the powder sample with a standard
continuous-wave spectrometer in the temperature range 2.5 K$\leq T\leq 300$ K.
As a function of external magnetic field $B$, the resonance shows up as an
absorbed power $P$ of a transversal magnetic microwave field ($\nu\simeq 9.4$
GHz, X-band). In order to improve the signal-to-noise ratio, a lock-in
technique was used by modulating the applied field, which yields the
derivative of power absorption $dP/dB$ as a function of $B$. By using the
resonance condition $g=\frac{h\nu}{\mu_{\rm B}H_{\rm res}}$, where $h$ is the
Planck’s constant, $\mu_{\rm B}$ is the Bohr magneton, $\nu$ is the resonance
frequency, and $H_{\rm res}$ is the corresponding resonance field, the
$g$-value was obtained.

The pulsed NMR measurements were performed on both 23Na (nuclear spin $I=3/2$
and gyromagnetic ratio $\gamma=11.26$ MHz/T) and 51V ($I=7/2$ and
$\gamma=11.19$ MHz/T) nuclei in the temperature range 0.044 K$\leq T\leq 200$
K. For measurements above 2 K a 4He cryostat (Oxford Instrument) with a field-
sweep superconducting magnet was used, while for measurements in the low-
temperature range (0.044 K$\leq T\leq 2$ K), a 3He/4He dilution refrigerator
(Kelvinox, Oxford Instruments) with a field sweep magnet was used. All the NMR
measurements were carried out in a radio frequency of 77 MHz. The NMR spectra
were measured as a function of temperature $T$ by sweeping the magnetic field
at a constant radio frequency of 77 MHz. The NMR shift was calculated for both
23Na and 51V nuclei as $K(T)$ = [$H_{\rm ref}-H(T)$]/$H(T)$, where $H$ is the
resonance field for 23Na and 51V and $H_{\rm ref}$ is the resonance field of
the non-magnetic reference sample. The spin-lattice relaxation rate $1/T_{1}$
was measured by the conventional single saturation pulse method.

Density-functional (DFT) band-structure calculations were performed in the
FPLO code Koepernik and Eschrig (1999) using the structural parameters from
Ref. von Postel and Müller-Buschbaum (1992) and local-density approximation
(LDA) for the exchange-correlation potential Perdew and Wang (1992). Exchange
parameters of the spin Hamiltonian

$\mathcal{H}=\sum_{\langle ij\rangle}J_{ij}\mathbf{S}_{i}\mathbf{S}_{j}$ (1)

with $S=\frac{1}{2}$ and the summation over atomic pairs $\langle ij\rangle$,
were extracted via two complementary procedures. First, band structure
obtained on the LDA level was mapped onto a tight-binding model for the half-
filled $d_{x^{2}-y^{2}}$ orbitals of Cu2+ as the magnetic ion. Squared hopping
parameters $t_{i}$ of this tight-binding model are proportional to AFM
contributions to the exchange, $J_{i}^{\rm AFM}=4t_{i}^{2}/U_{\rm eff}$, where
$U_{\rm eff}$ is the effective on-site Coulomb repulsion. Alternatively, full
exchange couplings $J_{i}$ comprising both FM and AFM contributions are
extracted by a mapping procedure Xiang  _et al._ (2011) from total energies of
magnetically ordered states calculated on the DFT+$U$ level, with correlation
effects in the Cu $3d$ shell modeled by the on-site Coulomb repulsion
$U_{d}=6$ eV, Hund’s exchange $J_{d}=1$ eV, and around-mean-field flavor of
the double-counting correction Janson  _et al._ (2011); Tsirlin  _et al._
(2012). The $k$ mesh with up to 150 points in the symmetry-irreducible part of
the first Brillouin zone was used.

Field-dependent magnetization and magnetic specific heat of a uniform
spin-$\frac{1}{2}$ chain were obtained from quantum Monte-Carlo simulations
for $L=32$ finite lattices with periodic boundary conditions. The loop Todo
and Kato (2001) and dirloop_sse Alet  _et al._ (2005) algorithms of the ALPS
simulation package Albuquerque  _et al._ (2007) were used.

##  III Results and Discussion

###  III.1 Magnetization

Figure 3:  $\chi$ of polycrystalline BaNa2Cu(VO4)2 sample as a function of
temperature in an applied field $\mu_{0}H=1$ T. The solid line is the fit
using Bonner-Fisher model [Eq. (2)] for uniform Heisenberg spin-$1/2$ chain.
Upper inset: inverse susceptibility $1/\chi$ vs $T$ and the solid line
represents the CW fit, as discussed in the text. Lower inset: the low-
temperature $\chi(T)$ measured in two different fields $\mu_{0}H=1$ T and 3 T.

Temperature-dependent magnetic susceptibility $\chi(T)(=M/H$) of the
polycrystalline Na2BaCu(VO4)2 sample measured in two different applied fields
$H=1$ T and 3 T is depicted in the Fig. 3. The most significant feature in the
$\chi(T)$ curve is the presence of a broad maximum at 3 K, signaling a
crossover to an AFM short-range ordered state, typical for low-dimensional
spin systems Bonner and Fisher (1964); Eggert  _et al._ (1994). This broad
maximum is more pronounced in the 3 T data shown in the lower inset of Fig. 3.
No anomaly indicative of the potential LRO could be seen down to 0.48 K.

The preliminary analysis was done by fitting the $\chi(T)$ data using the
Curie-Weiss (CW) law, $\chi(T)=\chi_{0}+C/(T+\theta_{\rm CW}$), where
$\chi_{0}$ is the temperature-independent susceptibility, $C$ is the Curie
constant, and $\theta_{\rm CW}$ is the characteristic CW temperature. The fit
shown in the upper inset of Fig. 3 in the high-temperature regime ($T\geq 16$
K) yields the following parameters: $\chi_{0}\simeq 7.9288\times 10^{-5}$
cm3/mol, $C\simeq 0.445$ cm3K/mol, and $\theta_{\rm CW}\simeq 3$ K. In order
to estimate the Van-Vleck paramagnetic susceptibility ($\chi_{\rm VV}$), which
arises from the second-order contribution to free energy in the presence of
magnetic field, core diamagnetic susceptibility $\chi_{\rm core}$ of
Na2BaCu(VO4)2 was calculated to be $-1.57\times 10^{-4}$ cm3/mol by summing
the core diamagnetic susceptibilities of individual ions Na+, Ba2+, Cu2+, V5+,
and O2- P. W. Selwood (2013); Mendelsohn  _et al._ (1970). Subsequently,
$\chi_{\rm VV}$ was obtained by subtracting $\chi_{\rm core}$ from $\chi_{0}$
to be $\sim 2.36\times 10^{-4}$ cm3/mol, which is close to the values reported
for other cuprates Motoyama  _et al._ (1996); Nath  _et al._ (2005); Ahmed
_et al._ (2015) and consistent with tetragonal crystal-field splitting at the
Cu2+ site with the square-planar oxygen coordination Takigawa  _et al._
(1989).

From the Curie constant $C$, the effective moment is calculated using the
relation $\mu_{\rm eff}=\sqrt{3k_{\rm B}C/N_{\rm A}}$ to be $\simeq 1.88$
$\mu_{\rm B}$, where $k_{\rm B}$ is the Boltzmann constant, $\mu_{\rm B}$ is
the Bohr magneton, and $N_{\rm A}$ is the Avogadro’s number. For a
spin-$\frac{1}{2}$ system, the spin-only effective moment is expected to be
$\mu_{\rm eff}=g\sqrt{S(S+1)}\mu_{\rm B}\simeq 1.73$ $\mu_{\rm B}$, assuming
Landé $g$-factor $g=2$. However, our experimental value of $\mu_{\rm
eff}\simeq 1.88$ $\mu_{\rm B}$ corresponds to a $g$-factor of $g\simeq 2.17$,
which is consistent with the ESR experiments discussed later. The positive
value of $\theta_{\rm CW}$ suggests that the dominant exchange interactions
between the Cu2+ ions are AFM in nature.

In order to estimate the exchange coupling between the Cu2+ ions, we
decomposed $\chi(T)$ into three components,

$\chi(T)=\chi_{0}+\frac{C_{\rm imp}}{T}+\chi_{\rm spin}(T).$ (2)

Here, the second term is the Curie law, which accounts for the paramagnetic
contributions from impurity spins and/or defects, and $\chi_{\rm spin}(T)$ is
the intrinsic spin susceptibility. This last term can be chosen in different
forms depending on the underlying magnetic model. The best fit was achieved
with the spin-chain model, which is further supported by the specific-heat
data (Sec. III.3) and ab initio calculations (Sec. III.5).

The susceptibility of a spin-$\frac{1}{2}$ uniform Heisenberg AFM chain takes
the form

$\chi_{\rm
spin}=\frac{N_{A}\mu_{B}^{2}g^{2}}{k_{B}T}\frac{0.25+0.0775x+0.0752x^{2}}{1+0.993x+0.1721x^{2}+0.7578x^{3}},$
(3)

with $x=\lvert J\rvert/k_{\rm B}T$ Bonner and Fisher (1964). This is simply a
high-temperature series expansion (HTSE) valid in the regime $k_{\rm B}T/J\geq
0.5$. The solid line in Fig. 3 represents the best fit of the $\chi(T)$ data
above 4 K by Eq. (2). The following parameters were obtained: $\chi_{0}\simeq
1.44\times 10^{-4}$ cm3/mol, $C_{\rm imp}\simeq 0.0258$ cm3K/mol, $g\simeq
2.13$, and the dominant intra-chain AFM exchange coupling $J/k_{\rm B}\simeq
5.6$ K. From the value of $C_{\rm imp}$, the sample was found to contain $\sim
6$% spin-$\frac{1}{2}$ impurities/defects. At temperatures below 1 K, this
impurity contribution becomes dominant and causes the reduction in the
susceptibility with the applied field, even though $\chi_{\rm spin}(T)$ should
increase when the field is applied Klümper (1998).

Figure 4:  Magnetization ($M$) vs $H$ measured at $T=2$ K. Inset: $dM/dH$ vs
$H$ to highlight the saturation field $H_{\rm s}$.

The magnetic isotherm at $T=2$ K upto 14 T is shown in Fig. 4. $M$ increases
almost linearly with $H$ but with a small curvature. It develops a tendency of
saturation above 9 T. A more accurate value of the saturation field $H_{\rm
s}\simeq 9$ T was found by drawing tangential at the curvature (see Fig. 4).
The field derivative of the $M$ vs $H$ plot also implies $H_{\rm s}\simeq 9$ T
(see the inset of Fig. 4). For a spin-$1/2$ Heisenberg AFM chain, the
saturation field is directly proportional to the intra-chain exchange coupling
as $H_{\rm s}=2J_{\rm 1D}(k_{B}/g\mu_{B})$ Lebernegg  _et al._ (2011). Using
the value of $J/k_{\rm B}\simeq 5.6$ K, the saturation field is calculated to
be $H_{\rm s}\simeq 8.34$ T, which matches well with the experimental value,
confirming the dominant 1D character of the compound.

###  III.2 ESR

Figure 5: (a) Integrated ESR intensity vs temperature and the solid line
represents the fit as described in the text. Inset: ESR spectrum at room
temperature measured at a microwave frequency of 9.4 GHz together with the
powder-averaged Lorentzian fit (solid line). (b) Temperature variation of the
$g$ values (both perpendicular and parallel components) obtained from the
Lorentzian fit. (c) Temperature-dependent ESR linewidth $\Delta B$ (both
perpendicular and parallel components).

ESR experiment was performed on the powder sample and the results are shown in
Fig. 5. The inset of Fig. 5(a) depicts a typical ESR powder spectrum at 300 K.
The uniaxial $g$ factor anisotropy was obtained by fitting the spectra using
the powder-averaged Lorentzian line. The fit of the spectral at room
temperature yields the anisotropic $g$-tensor components
$g_{\parallel}$$\simeq 2.315$ and $g_{\perp}$$\simeq 2.098$. From these
values, the average $g$-value was calculated as
$g=[(g_{\parallel}+2g_{\perp})/3]\simeq 2.17$ Abragam and Bleaney (2012). This
value is slightly larger ($\Delta g/g\simeq 0.085$) compared to the free
electron value ($g=2$), typical for Cu2+ based oxides Kochelaev  _et al._
(1997); Nath  _et al._ (2014). The integrated ESR intensity ($I_{\rm ESR}$)
obtained from the above fit is plotted as a function of temperature in Fig.
5(a). It shows similitude with the $\chi(T)$ behavior, which traces a broad
maximum at around $T^{\max}_{\rm ESR}$ $\simeq 3.7$ K. Indeed, the $I_{\rm
ESR}$ vs $\chi$ plot with temperature as an implicit parameter follows a
straight line down to $\sim 5$ K (not shown). The variation of $g$ with
respect to temperature is shown in Fig. 5(b). Both the components of $g$ were
found to be almost temperature-independent at high temperatures ($T\geq 20$
K). However, below 20 K a weak deviation from the room-temperature values is
observed.

In order to estimate the exchange coupling, $I_{\rm ESR}(T)$ was fitted by

$I_{\rm ESR}(T)=A+B\chi_{\rm spin}(T).$ (4)

Here, $A$ and $B$ are arbitrary constants, and $\chi_{\rm spin}$ is given by
Eq. (3). Our fit (see Fig. 5) in the high-temperature regime ($T\geq 5$ K)
produced $J/k_{\rm B}\simeq 5.55~{}K$. This value of $J/k_{\rm B}$ is close to
the one obtained from the $\chi(T)$ analysis. During the fit, the value of $g$
was kept constant to 2.17, as obtained above. We have also fitted the
$1/I_{\rm ESR}$ data in the high-temperature regime ($T\geq 10$ K) using the
relation $I_{\rm ESR}=M+N/(T+\theta_{\rm CW})$ where $M$ and $N$ are arbitrary
constants. As shown in the lower inset of Fig. 5(a), the fit returns
$\theta_{\rm CW}\simeq 3.9~{}K$, which is in good agreement with the value
obtained from the $\chi^{-1}(T)$ analysis.

The temperature-dependent ESR linewidth, or equivalently the half-width at
half maximum of the ESR absorption signal, is presented in Fig. 5(c). Both the
parallel ($\Delta B_{\parallel}$) and perpendicular ($\Delta B_{\perp}$)
components of the ESR line width follow the general trend, commonly observed
in most of the low-dimensional spin systems Ivanshin  _et al._ (2003);
Sichelschmidt  _et al._ (2002). The rapid increase/divergence below $\sim 25$
K indicates the growth of strong spin correlations at low temperatures as the
system approaches the magnetic LRO state.

###  III.3 Heat Capacity

Figure 6:  Upper panel: Heat capacity ($C_{\rm p}$) vs $T$ in zero applied
field. The solid line denotes the phonon contribution to the heat capacity
$C_{\rm ph}$ using the Debye-Einstein fit. The blue solid spheres indicate the
magnetic contribution to the heat capacity $C_{\rm mag}$. Inset: $C_{\rm p}$
vs $T$ in the whole measured temperature range along with the Debye-Einstein
fit. Lower panel: The left $y$-axis shows $C_{\rm mag}/T$ and the right
$y$-axis shows the magnetic entropy $S_{\rm mag}$ vs $T$. Inset: $C_{\rm
mag}/R$ vs $T$.

Temperature-dependent heat capacity $C_{\rm p}$ of the polycrystalline sample
is shown in the upper panel of Fig. 6. In magnetic insulators, the two major
contributions to $C_{\rm p}$ are from phonon and magnetic parts. At high
temperatures, $C_{\rm p}(T)$ is dominated by the phonon part, while at low
temperatures it is dominated by the magnetic part. Our experimental $C_{\rm
p}$ data exhibit a pronounced broad maximum at $T\simeq 1.52$ K indicative of
the low-dimensional short-range order and also reflects the dominant magnetic
contribution at low temperatures. In order to estimate the magnetic
contribution to the heat capacity $C_{\rm mag}$, we proceed as follows. First
we approximate the lattice contribution $C_{\rm ph}$ by fitting the high-
temperature data by a linear combination of one Debye and two Einstein terms
(Debye-Einstein model) as Kittel (c2005); Caslin  _et al._ (2014)

$C_{\rm ph}(T)=f_{\rm D}\,C_{\rm D}(\theta_{\rm
D},T)+\sum_{i=1}^{2}g_{i}\,C_{{\rm E}_{i}}(\theta_{{\rm E}_{i}},T).$ (5)

The first term in Eq. (5) is the Debye term,

$C_{\rm D}(\theta_{\rm D},T)=9nR\left(\frac{T}{\theta_{\rm
D}}\right)^{3}\int_{0}^{\frac{\theta_{\rm
D}}{T}}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx.$ (6)

Here, $x=\frac{\hbar\omega}{k_{\rm B}T}$, $\omega$ is the vibration frequency,
$R$ is the universal gas constant, and $\theta_{\rm D}$ is the characteristic
Debye temperature. The second term in Eq. (5) is a combination of the Einstein
terms that are usually responsible for flat optical modes in the phonon
spectrum,

$C_{\rm E}(\theta_{\rm E},T)=3nR\left(\frac{\theta_{\rm
E}}{T}\right)^{2}\frac{e^{\,\theta_{\rm E}/T}}{\left(e^{\,\theta_{\rm
E}/T}-1\right)^{2}}.$ (7)

Here, $\theta_{\rm E}$ is the characteristic Einstein temperature. The
coefficients $f_{\rm D}$, $g_{1}$, and $g_{2}$ are the weight factors, which
take into account the number of atoms per formula unit ($n$) and are
conditioned such that at high temperatures the Dulong-Petit value of $3nR$ is
satisfied. The $C_{\rm p}(T)$ data above $\sim 15$ K were fitted by Eq. (5)
and the obtained parameters are $f_{\rm D}\simeq 0.34$, $g_{1}\simeq 0.35$,
and $g_{2}\simeq 0.31$, $\theta_{\rm D}\simeq 214$ K, $\theta_{{\rm
E}_{1}}\simeq 356$ K, and $\theta_{{\rm E}_{2}}\simeq 897$ K. Further Einstein
terms beyond $\theta_{{\rm E}_{2}}$ rendered the fit unstable. The fit itself
is phenomenological in nature, although one may tentatively associate
$\theta_{\rm D}$ with low-energy vibrations of heavier atoms (Ba, Cu, and V)
that constitute 28.5 %, about $\frac{1}{3}$ of the atomic species in
BaNa2Cu(VO4)2. The lower Einstein temperature $\theta_{{\rm E}_{1}}$ may
correspond to Na atoms and two apical oxygens of the VO4 tetrahedra
(altogether 6 atoms per formula unit), whereas $\theta_{{\rm E}_{2}}$ reflects
higher-energy vibrations of the remaining four oxygens that are bound to V and
Cu at the same time.

The high-$T$ fit was extrapolated down to low temperatures and $C_{\rm
mag}(T)$ was estimated by subtracting $C_{\rm{ph}}(T)$ from $C_{\rm p}(T)$
[see Fig. 6 (upper panel)]. $C_{\rm mag}(T)/T$ was plotted as a function of
temperature in the lower panel of Fig. 6. The broad maximum corresponding to
the short-range order is apparent at $T\simeq 1.52$ K. At low temperatures,
$C_{\rm mag}(T)/T$ shows a rapid increase, which could be related to the onset
of magnetic LRO below 0.4 K. The magnetic entropy was calculated as
$S_{\rm{mag}}(T)=\int_{\rm
2\,K}^{T}\frac{C_{\rm{mag}}(T^{\prime})}{T^{\prime}}dT^{\prime}$, which yields
$S_{\rm mag}\simeq 5.83$ J/mol K at 20 K (see the lower panel of Fig. 6). This
value is close to the expected magnetic entropy for spin-$\frac{1}{2}$:
$S_{\rm mag}=R\ln 2=5.76$ J/mol K.

In the inset of the lower panel of Fig. 6, $C_{\rm mag}/R$ is plotted against
$T$. The peak of $C_{\rm mag}/R$ can be used to discriminate between different
microscopic scenarios. Its height depends on the nature of the underlying spin
lattice Bernu and Misguich (2001). Our experimental peak value of $C_{\rm
mag}/R\simeq 0.323$ fits well to the aforementioned 1D scenario, suggesting
that VO4 bridges choose the direction of spin chains. Alternatively, four
shortest Cu–Cu contacts of 5.507 Å could cause interactions of equal strength
and form a 2D square-lattice interaction topology that should manifest itself
by a much higher peak with $C_{\rm mag}/R\simeq 0.47$. On the other hand, the
triangular-lattice scenario would reduce the peak value to $C_{\rm
mag}/R\simeq 0.22$, lower than seen experimentally. We thus conclude that our
specific-heat data favor the spin-chain scenario for BaNa2Cu(VO${}_{4})_{2}$.

###  III.4 23Na and 51V NMR

NMR is a potent tool to study the static and dynamic properties of spin
systems. In Na2BaCu(VO4)2, the 23Na and 51V nuclei are hyperfine-coupled to
the magnetic Cu2+ ions along the spin chains. Therefore, the low-lying
excitations of Cu2+ spins can be probed by means of 23Na and 51V NMR
measurements.

Figure 7:  Field-sweep NMR spectra of the polycrystalline Na2BaCu(VO4)2
sample, measured at 77 MHz as a function of temperature. The spectral lines
corresponding to 23Na and 51V nuclei for $T=80$ K are marked by arrows. The
solid line is the simulated spectrum.

The quadrupole nuclei 23Na ($I=3/2$) and 51V ($I=7/2$) are in a non-cubic
symmetry that may produce an asymmetric charge distribution and hence electric
field gradient (EFG). Therefore, the four-fold and eight-fold degeneracies of
the $I=3/2$ and $I=7/2$ spins, respectively, are lifted partially due to the
interaction between the nuclear quadrupole moment ($Q$) and the surrounding
EFG. In this case, the nuclear spin Hamiltonian is a sum of the Zeeman and
quadrupolar interaction terms Curro (2009); Slichter (1992),

$\mathcal{H}=-\gamma\hbar{\hat{I}}H(1+K)+\frac{h\nu_{Q}}{6}[(3\hat{I}_{z}^{2}-\hat{I}^{2})+\eta(\hat{I}_{x}^{2}-\hat{I}_{y}^{2})].$
(8)

Here, the nuclear quadrupole resonance (NQR) frequency is defined as $\nu_{\rm
Q}=\frac{3e^{2}qQ}{2I(2I-1)h}$, $e$ is the electron charge,
$\hbar~{}(=h/2\pi)$ is the Planck’s constant, $H$ is the applied field along
$\hat{z}$, $K$ is the magnetic shift due to hyperfine field at the nuclear
site, $V_{\alpha\beta}$ are the components of the EFG tensor, $eq=V_{zz}$ is
the largest eigenvalue or principal component of the EFG, and
$\eta=|V_{xx}-V_{yy}|/V_{zz}$ is the EFG asymmetry (here, the principal axes
of EFG are chosen such that $|V_{zz}|\geq|V_{yy}|\geq|V_{xx}|$.).
Experimentally, the transitions can be observed at the frequency
$\nu_{z}=\nu_{Q}\sqrt{1+\eta^{2}/3}$.

The principal axes $\\{x,y,z\\}$ of the EFG tensor are defined by the local
symmetry of the crystal structure. Consequently, the corresponding resonance
frequency to any nuclear transition will have strong dependence on the
direction of the applied field with respect to the crystallographic axes. For
a site with axial symmetry ($\eta=0$), there will be $2I-1$ quadrupolar
resonances at frequencies $n\nu_{\rm Q}$, where $n=$ 1, ….$2I-1$. When
$\eta>0$, the resonances are not equally spaced. The EFG is fully
characterized by the parameters $\nu_{\rm z}$, $\eta$, and $\hat{z}$, where
$\hat{z}$ is the unit vector in the direction of the principal axis of the EFG
with the largest eigenvalue. When the Zeeman term dominates over the
quadrupole term, first-order perturbation theory is enough for describing the
system. In such a scenario, for a quadrupole nucleus, equally spaced satellite
peaks should appear on either side of the central peak separated by $\nu_{Q}$
Lang  _et al._ (2005).

The NMR spectra as a function of temperature measured by sweeping the magnetic
field at 77 MHz are presented in Fig. 7. Since 23Na and 51V nuclei have nearly
the same $\gamma$ values, one expects their spectral lines to appear very
close to each other. Further, 23Na and 51V are quadrupolar nuclei with nuclear
spins $I=3/2$ and $7/2$, respectively and the transitions with $\Delta m=\pm
1$ are expected between the energy levels. Therefore, one would anticipate
three NMR lines for 23Na: one central line corresponding to
$I_{z}=+1/2\longleftrightarrow-1/2$ and two equally spaced satellite lines
corresponding to $I_{z}=\pm 3/2\longleftrightarrow\pm 1/2$ and seven NMR lines
for 51V: the central line being $I_{z}=+1/2\longleftrightarrow-1/2$ and the
satellite lines $I_{z}=\pm 1/2\longleftrightarrow\pm 3/2\longleftrightarrow\pm
5/2\longleftrightarrow\pm 7/2$. Indeed, at high temperatures, we observed two
sharp and prominent peaks at the resonance field position and two satellite
peaks on either side of those. The central peak towards the low-field side is
identified to be the signal coming from the 23Na nuclei, while the one towards
the high-field side appears to be the 51V peak. In addition to the central
peaks, two satellite peaks correspond to the 23Na line. At high temperatures,
the NMR spectra are found to be narrow and one can distinguish the 23Na and
51V signals. As the temperature is lowered, the line broadens asymmetrically
and the central lines shift weakly with temperature. No abrupt line broadening
was noticed down to 44 mK, which may signal the absence of magnetic LRO
Ranjith, K. M. and Nath, R. and Majumder, M. and Kasinathan, D. and Skoulatos,
M. and Keller, L. and Skourski, Y. and Baenitz, M. and Tsirlin, A. A. (2016).

Figure 8:  Upper panel: NMR spectra at $T=125$ K showing the 23Na and 51V
central lines, with the downward arrows pointing to the 23Na satellites. The
solid line is the simulation of the spectra assuming the superposition of the
23Na and 51V signals. Lower panel: temperature-dependent NMR shift $K$ as a
function of temperature for 23Na and 51V, measured at 77 MHz. Solid line is
the fit using Eq. (9). Inset: NMR shift vs $\chi$ measured at 3 T. Solid lines
are the linear fits.

The spectra were fitted assuming the superposition of 23Na and 51V signals.
The spectral fit at $T=125$ K is presented in the upper panel of Fig. 8, where
23Na and 51V lines and their satellites are marked by arrows. The obtained
fitting parameters are $K\simeq 0.0345$% (isotropic shift), $\eta=0$
(asymmetry parameter), and $\nu_{Q}\simeq 0.92$ MHz (NQR frequency) for 23Na
and $K\simeq 0.627$%, $\eta=0$, and $\nu_{Q}\simeq 0.234$ MHz for 51V. The
quadrupole frequency is found to be almost constant with temperature down to
1.5 K, which essentially excludes the possibility of any structural distortion
in the studied compound.

The NMR shift $K(T)$ for both 23Na and 51V lines obtained from the spectral
fits is plotted in the lower panel of Fig. 8. The temperature-dependent 23Na
shift [${}^{\rm 23}K(T)$] is found to have a broad maximum at around 3 K,
similar to the $\chi(T)$ data. As $K(T)$ in an intrinsic measure of the spin
susceptibility $\chi_{\rm spin}$, one can write the linear relation

$K(T)=K_{0}+\frac{A_{\rm hf}}{N_{A}\mu_{B}}\chi_{\rm spin},$ (9)

where $K_{0}$ is the temperature-independent chemical shift and the
proportionality constant $A_{\rm hf}$ is the hyperfine coupling between the
probed nuclei and the electron spins.

From Eq. (9), $A_{\rm hf}$ can be calculated by taking the slope of the linear
$K$ vs $\chi$ plot (inset of Fig. 8) with temperature as an implicit
parameter. In the case of 23Na, the data for $T\geq 5$ K were fitted well by a
linear function, and the slope of the fit yields ${}^{23}A_{\rm hf}\simeq
0.021$ T/$\mu_{\rm B}$. Similarly, for 51V the linearity is found over a large
temperature range down to 10 K, and the linear fit returns ${}^{51}A_{\rm
hf}\simeq-0.016$ T/$\mu_{\rm B}$. To estimate the exchange coupling,
${}^{23}K(T)$ above 2.5 K was fitted by Eq. (9) taking $\chi_{\rm spin}$ for
the 1D $S=1/2$ Heisenberg chain [Eq. (3)]. The fit returns $J/k_{\rm B}\simeq
4.22$ K and ${}^{23}A_{\rm hf}\simeq 0.0194$ T/$\mu_{\rm B}$. The value of $g$
was fixed to $g=2.17$ during the fitting procedure. This value of $J/k_{\rm
B}$ is close to the one obtained from the $\chi(T)$ analysis, whereas
${}^{23}A_{\rm hf}$ is also in good agreement with the value obtained from the
$K$ vs $\chi$ analysis. An anomaly at $\sim 0.3$ K in ${}^{\rm 23}K(T)$ could
be due to a magnetic transition.

Figure 9:  $1/T_{1}$ as a function of temperature measured on the 51V nuclei
down to 0.044 K. Inset: $1/T_{1}$ above 2 K is shown in order to highlight the
features around 10 K.

To study the spin dynamics, spin-lattice relaxation rate ($1/T_{1}$) was
measured by irradiating the central position of the 51V spectra corresponding
to the $1/2\longleftrightarrow-1/2$ transition, choosing an appropriate pulse
width. The recovery of the longitudinal magnetization was fitted by the
following exponential function relevant for a quadrupole ($I=7/2$) nuclei M.
I. Gordon and M. J. R. Hoch (1978); Simmons  _et al._ (1962)

$\displaystyle 1-\frac{M(t)}{M(\infty)}$ $\displaystyle=$ $\displaystyle
0.0119\times e^{(-t/T_{1})}+0.068\times e^{(-6t/T_{1})}$ (10) $\displaystyle+$
$\displaystyle 0.21\times e^{(-15t/T_{1})}+0.71\times e^{(-28t/T_{1})}$

Here, $M(t)$ and $M(\infty)$ are the nuclear magnetizations at a time $t$ and
$t\longrightarrow\infty$, respectively, after the saturation pulse.
Temperature dependence of 51V $1/T_{1}$ obtained from the above fit is shown
in Fig. 9. Our measurements were carried out down to 0.04 K. At high
temperatures, $1/T_{1}$ is almost temperature-independent as expected in the
paramagnetic regime Moriya (1956). At low temperatures, it exhibits a sharp
peak at $T\simeq 0.25$ K due to slowing down of the fluctuating moments and is
a direct evidence of the onset of magnetic LRO. In order to highlight the
behavior in the intermediate temperature range, $1/T_{1}$ above 2 K is
magnified in the inset of Fig. 9. As the temperature is lowered, $1/T_{1}$
decreases linearly below about 25 K, remains almost temperature-independent
for 4 K$\leq T\leq 10$ K, and then starts increasing for $T\leq 4$ K. This
increase below 4 K can be attributed to the growth of AFM correlations as the
system approaches the magnetic LRO state.

Further, $1/T_{1}T$ is directly proportional to the imaginary part of the
dynamic susceptibility $\chi_{M}(\vec{q},\omega_{0})$ at the nuclear Larmor
frequency $\omega_{0}$, which is $q$-dependent Moriya (1956). In low-
dimensional spin systems, temperature-dependent $1/T_{1}$ often reflects
dominant contributions from different $q$ values in different temperature
regimes. For instance, for spin-$1/2$ Heisenberg AFM spin chains, it is
theoretically predicted that with the dominant staggered contribution
($q=\pm\pi/a$) the spin-lattice relaxation rate behaves as $1/T_{1}\sim
T^{0}$, while the dominant contribution of the uniform component ($q=0$)
results in $1/T_{1}\sim T$ Sandvik (1995); Sachdev (1994). The dominant
contributions of $q=\pm\pi/a$ and $q=0$ are typically observed in the low-
temperature ($T<J$) and high-temperature ($T\sim J$) regimes, respectively
Nath  _et al._ (2005, 2008). Thus, our experimentally observed constant and
linear behaviors of $1/T_{1}$ with temperature over 4 K $\leq T\leq 10$ K and
10 K$\leq T\leq 25$ K, respectively (inset of Fig. 9), are compatible with the
1D physics.

In real spin-chain systems, the non-vanishing interchain couplings often leads
to the onset of magnetic LRO at very low temperatures. The interchain coupling
can be calculated using the expression proposed by Schulz Schulz (1996)

$|J_{\perp}|\simeq\frac{T_{\rm N}}{1.28\sqrt{\ln(5.8J/T_{\rm N})}},$ (11)

where $J_{\perp}$ is an effective interchain coupling. Taking $T_{\rm N}\simeq
0.25$ K and $J/k_{\rm B}\simeq 5.6$ K, we arrive at the possible value of
$J_{\perp}/k_{\rm B}\simeq 0.1$ K, which is indeed consistent with the value
estimated from the band-structure calculations, as discussed in the following.

###  III.5 Microscopic magnetic model

Table 1:  Exchange parameters of BaNa2Cu(VO${}_{4})_{2}$ obtained from DFT calculations: Cu–Cu distances $d$ (in Å), electron hoppings $t_{i}$ (in meV), AFM contributions to the exchange $J_{i}^{\rm AFM}=4t_{i}^{2}/U_{\rm eff}$ (in K), and total exchange couplings $J_{i}$ (in K) from the DFT+$U$ mapping procedure.  |  $d_{\rm Cu-Cu}$ | $t_{i}$ |  $J_{i}^{\rm AFM}$ | $J_{i}$
---|---|---|---|---
$J$ | 5.507 | $-40$ | 14.9 | 6.8
$J_{ab}$ | 5.507 | $-5$ | 0.2 | $<\\!0.2$
$J_{ab}^{\prime}$ | 5.686 | $-1$ | 0.01 | $<\\!0.2$
$J_{c}$ | 7.024 | 3 | 0.08 | $<\\!0.2$

LDA band structure of BaNa2Cu(VO${}_{4})_{2}$ (Fig. 10) features Cu $3d$
states below the Fermi level and V $3d$ states above 2 eV, confirming the non-
magnetic state of vanadium. The overall energy spectrum is metallic, as
typical for a transition-metal compound when correlation effects in the $3d$
shell were not taken into account. Nevertheless, this band structure gives an
overview of possible exchange interactions, as the hopping parameters $t_{i}$
are proportional to the LDA bandwidth, whereas $J_{i}^{\rm
AFM}=4t_{i}^{2}/U_{\rm eff}$. The Fermi level is crossed by two narrow bands
formed by the half-filled $d_{x^{2}-y^{2}}$ orbitals of Cu2+. The width of
these bands is less than 0.2 eV, one of the smallest in cuprates, and
indicates very weak exchange couplings in BaNa2Cu(VO${}_{4})_{2}$.

Figure 10:  LDA density of states for BaNa2Cu(VO${}_{4})_{2}$. Note the very
narrow Cu $d_{x^{2}-y^{2}}$ band around 0 eV (Fermi level) that indicates
small electron hoppings and correspondingly weak exchange couplings.

DFT results for the exchange couplings are summarized in Table 1. Only one
sizable coupling, $J/k_{\rm B}\simeq 6.8$ K is found. It corresponds to spin
chains running along $[110]$ in one layer and along $[1\bar{1}0]$ in the
adjacent layer, the direction being chosen by the position of the double VO4
bridges that connect the CuO4 plaquette units (Fig. 1). Such a coupling
mechanism is fairly common among the Cu2+ compounds and can give rise to both
FM and AFM superexchange depending on the orientation of the VO4 tetrahedra
relative to the CuO4 planes Tsirlin  _et al._ (2012). Larger rotations of the
tetrahedra favor FM couplings.

In BaNa2Cu(VO${}_{4})_{2}$, we find $\varphi=99.0^{\circ}$, which is similar
to $\varphi^{(2)}=102.2^{\circ}$ for the AFM coupling $J_{a}^{(2)}/k_{\rm
B}=9.5$ K in BaAg2Cu(VO${}_{4})_{2}$ and very different from
$\varphi^{(1)}=123.7^{\circ}$ for the FM coupling $J_{a}^{(1)}/k_{\rm B}=-19$
K in the same compound Tsirlin  _et al._ (2012). Here, $\varphi$ is the angle
between the face of the VO4 tetrahedron and the plane connecting the adjacent
CuO4 plaquettes, as shown in Fig. 1. Compared to BaAg2Cu(VO${}_{4})_{2}$, the
AFM coupling weakens from 9.5 K to $\sim 6$ K, likely because of the longer
Cu–Cu distance (5.507 Å vs. 5.448 Å) and the increased lateral displacement
$r$ of the CuO4 plaquettes (0.895 Å vs. 0.860 Å).

All couplings beyond the aforementioned spin chains appear to be very weak,
below 0.2 K, and unfeasible for the DFT+$U$ mapping analysis. Their relative
strengths can be assessed from the hopping parameters that suggest the
dominant interchain couplings $J_{ab}$ in the $ab$ plane (along $[1\bar{1}0]$
for the spin chains along $[110]$, and vice versa) and $J_{c}$ along the $c$
direction. The in-plane coupling $J_{ab}^{\prime}$ is negligible. The two
stronger interchain couplings, $J_{ab}$ and $J_{c}$, form a non-frustrated 3D
network. From $4t_{i}^{2}/U_{\rm eff}$ with $U_{\rm eff}=5$ eV Lebernegg  _et
al._ (2013); Ahmed  _et al._ (2015), one expects the coupling strength of 0.2
K or lower, in agreement with the DFT+$U$ results. Altogether, our modeling
results establish weak and non-frustrated interchain couplings in
BaNa2Cu(VO${}_{4})_{2}$, with $J_{\perp}/J\simeq 0.02$. The average interchain
coupling of $J_{\perp}/k_{\rm B}\simeq 0.1$ K leads to $T_{N}/J\simeq 0.22$ K
Yasuda  _et al._ (2005) in good agreement with 0.25 K found experimentally.
Therefore, we argue that long-range magnetic order in BaNa2Cu(VO${}_{4})_{2}$
should be driven by weak interchain couplings, and the Néel temperature
$T_{\rm N}/J$ is determined by the $J_{\perp}/J$ ratio.

Figure 11:  Magnetization normalized to the saturation value (main figure) and
magnetic specific heat (inset) of BaNa2Cu(VO${}_{4})_{2}$. Predictions of the
spin-chain model with $J/k_{\rm B}=5.5$ K and $g=2.17$ are shown with lines.
In magnetization curves, an additional 5 % paramagnetic contribution described
by the Brillouin function was included in order to reproduce the weak bend in
low magnetic fields.

Above $T_{\rm N}$, a purely one-dimensional description should hold. Indeed,
we were able to fit magnetization curves down to 0.49 K using the spin-chain
model with $J/k_{\rm B}=5.5$ K and $g=2.17$ in excellent agreement with 5.6 K
from the fit to the magnetic susceptibility and $g=2.17$ from the ESR
experiment (Fig. 11). This confirms that the interchain couplings are very
weak and play only marginal role even at $T<J$. Magnetic specific heat is also
well described by the spin-chain model showing small deviations below 1 K
only. These deviations correspond to the upturn in $C_{\rm mag}/T$ upon
approaching $T_{\rm N}$ (Fig. 6).

##  IV Conclusions

We have shown that BaNa2Cu(VO${}_{4})_{2}$ strongly deviates from all of its
structural siblings in terms of the magnetic behavior. The majority of these
compounds are triangular magnets, while the only Cu2+ member studied to date,
BaAg2Cu(VO${}_{4})_{2}$, revealed a very unusual coexistence of different spin
chains, one ferromagnetic and one antiferromagnetic Tsirlin  _et al._ (2012);
Krupskaya  _et al._ (2017). Our present results for BaNa2Cu(VO${}_{4})_{2}$
corroborate non-trivial magnetostructural correlations in Cu2+ vanadates,
where the sign of a magnetic coupling strongly depends on the spatial
orientation of the VO4 tetrahedra relative to the spin chains and CuO4
plaquette units.

The disparity of spin chains is absent in BaNa2Cu(VO${}_{4})_{2}$, but now the
chains adopt two different directions and form an unusual crossed pattern.
Interestingly, this crossed pattern does not cause any magnetic frustration,
because the Cu2+ ion of one chain sits exactly on top of the Cu2+ ion of the
adjacent chain (Fig. 1). Then, each magnetic site has only one coupling to a
spin chain of another direction, and not two couplings, as expected
theoretically Starykh  _et al._ (2002). This fact highlights the importance of
lateral displacements between the Cu2+ ions of the crossed chains to induce
the frustration. Such displacements do not occur in BaNa2Cu(VO${}_{4})_{2}$,
but they may potentially appear in sister compounds, because even the
substitution of Na+ by Ag+ causes significant structural changes, although the
two ions are very similar in size. Alternatively, one may consider structure
types with a weaker spatial separation between the crossed chains that, in
turn, allows several non-equivalent interactions to form a frustrated topology
even in the absence of lateral displacements Tsirlin  _et al._ (2011);
Mukharjee  _et al._ (2019); Weickert  _et al._ (2019).

###### Acknowledgements.

We would like to acknowledge SERB, India, for financial support bearing
sanction Grant No. CRG/2019/000960. Work at the Ames Laboratory was supported
by the U.S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division. The Ames Laboratory is operated
for the U.S. Department of Energy by Iowa State University under Contract No.
DEAC02-07CH11358. AT was funded by the Federal Ministry for Education and
Research through the Sofja Kovalevskaya Award of Alexander von Humboldt
Foundation.

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# The volume polynomial of lattice polygons

Ivan Soprunov  Department of Mathematics and Statistics
Cleveland State University
Cleveland, OH USA<EMAIL_ADDRESS>and  Jenya Soprunova  Department of
Mathematical Sciences
Kent State University
Kent, OH USA<EMAIL_ADDRESS>
###### Abstract.

We prove that every indefinite quadratic form with non-negative integer
coefficients is the volume polynomial of a pair of lattice polygons. This
solves the discrete version of the Heine–Shephard problem for two bodies in
the plane. As an application, we show how to construct a pair of planar
tropical curves (or a pair of divisors on a toric surface) with given
intersection number and self-intersection numbers.

###### Key words and phrases:

volume polynomial, lattice polytope, mixed volume, integer quadratic form,
intersection number, tropical curve, toric surface

###### 2020 Mathematics Subject Classification:

Primary 52B20, 52A39; Secondary 11H55, 14T10, 14M25

##  1\. Introduction

With any collection $K_{1},\dots,K_{n}$ of convex bodies in $\mathbb{R}^{d}$
and non-negative scalars $x_{1},\dots,x_{n}\in\mathbb{R}_{\geq 0}$ one can
associate a convex body

$x_{1}K_{1}+\dots+x_{n}K_{n}=\\{x_{1}a_{1}+\dots+x_{n}a_{n}:a_{i}\in
K_{i},1\leq i\leq n\\}\subset\mathbb{R}^{d}.$

In [Min03] Minkowski showed that the $d$-dimensional volume of this body
depends polynomially on the scalars, that is,
$\operatorname{Vol}_{d}(x_{1}K_{1}+\dots+x_{n}K_{n})$ is a homogeneous degree
$d$ polynomial in $x_{1},\dots,x_{n}$. This polynomial is called the volume
polynomial of $K_{1},\dots,K_{n}$ and its coefficients are the mixed volumes
(up to multinomial coefficients). In the case of two planar convex bodies
$K,L$ the volume polynomial is a quadratic form

(1.1)
$\operatorname{Vol}_{2}(xK+yL)=\operatorname{Vol}_{2}(K)x^{2}+2\operatorname{V}(K,L)xy+\operatorname{Vol}_{2}(L)y^{2},$

where the middle coefficient $\operatorname{V}(K,L)$ is called the mixed
volume of $K$ and $L$. It can be expressed by evaluating (1.1) at $x=y=1$ as

(1.2)
$\operatorname{V}(K,L)=\frac{1}{2}\left(\operatorname{Vol}_{2}(K+L)-\operatorname{Vol}_{2}(K)-\operatorname{Vol}_{2}(L)\right).$

The classical Minkowski inequality provides a relation between the
coefficients of (1.1):

(1.3)
$\operatorname{Vol}_{2}(K)\operatorname{Vol}_{2}(L)\leq\operatorname{V}(K,L)^{2}.$

In other words, (1.1) is an indefinite quadratic form. It is not hard to show
that every indefinite quadratic form with non-negative coefficients is the
volume polynomial of some planar convex bodies $K,L$, see Proposition 4.1.

In general, the study of polynomial inequalities between the coefficients of
the volume polynomial is the core of the Brunn-Minkowski theory of convex
bodies. We refer to the book of Schneider [Sch14] for the most complete
account of this theory. Still the problem of giving an explicit description
for the space of volume polynomials in terms of coefficient inequalities is
wide open. Besides the case of two planar bodies described above, such a
description is only known for three planar bodies ($n=3,d=2$) provided by
Heine [Hei38], and two bodies in any dimension ($n=2,d\in\mathbb{N}$) provided
by Shephard [She60]. Inequalities such as the Aleksandrov-Fenchel inequality
and Shephard’s determinantal inequalities uncover deep connections between
mixed volumes, but yet do not provide a complete description for the space of
volume polynomials, in general, see [She60, ABS20]. Some new inequalities
describing the square-free part of the volume polynomial for $n=4$ and $d=2$
have been recently found in [AS23].

In this note we consider a discrete version of the Heine–Shephard problem.
Namely, we are interested in describing the space of volume polynomials of
lattice polytopes. Recall that a convex polytope $P\subset\mathbb{R}^{d}$ is a
lattice polytope if its vertices belong to the integer lattice
$\mathbb{Z}^{d}$. In this case it is appropriate to normalize the usual
Euclidean volume by a factor of $d!$. Then the volume and the mixed volume
take integer values on lattice polytopes.

######  Problem 1.1.

Let $\operatorname{Vol}_{d}$ be the normalized volume. Describe the set of the
volume polynomials $\operatorname{Vol}_{d}(x_{1}P_{1}+\dots+x_{n}P_{n})$ over
all collections of lattice polytopes $P_{1},\dots,P_{n}$ in terms of
coefficient inequalities.

In this paper we provide a solution to Problem 1.1 in the smallest non-trivial
case $n=d=2$. We prove that every indefinite quadratic form with non-negative
integer coefficients is the volume polynomial of a pair of lattice polytopes
in $\mathbb{R}^{2}$, see Theorem 4.2. Our proof is constructive. An
implementation in Magma [BCP97] can be found at
https://github.com/isoprou/volume_polynomial.

One can also view Problem 1.1 as a discrete Blaschke-Santaló problem.
Motivated by the work of Blaschke [Bla16], Santaló studied the map which
assigns to every planar convex body a triple of its geometric invariants, such
as area, perimeter, diameter, width etc., [San61]. The image of such a map is
called a Blaschke-Santaló diagram. Now let $\phi$ be the map which sends a
pair of convex planar bodies $K,L$ to the triple
$(\operatorname{Vol}_{2}(K),\operatorname{V}(K,L),\operatorname{Vol}_{2}(L))$.
Then the diagram $\operatorname{Im}\phi$ is the closed semialgebraic set

(1.4) $\operatorname{Im}\phi=\\{(x,y,z)\in\mathbb{R}_{\geq 0}^{3}:y^{2}\geq
xz\\}$

as follows from Minkowski inequality (1.3) and Proposition 4.1. Restricting
$\phi$ to the set of pairs of lattice polytopes in the plane, we obtain the
discrete diagram which, according to our result in Theorem 4.2, is the set of
lattice points of the semialgebraic set (1.4). In a similar spirit, Scott
[Sco76] described the discrete diagram for the set of coefficients of the
Ehrhart polynomial of lattice polygons. For the actual picture of the diagram
see, for example, [BDLD+05, Thm. 2.1].

There is a strong motivation to consider mixed volumes of lattice polytopes
coming from the intersection theory in toric and tropical geometry. In the
final section (Section 5) we discuss an interpretation of our result in terms
of the intersection numbers of planar tropical curves and divisors on toric
surfaces.

### Acknowledgments

We are grateful to Gennadiy Averkov for fruitful discussions.

##  2\. Preliminaries

In this section we recall basic facts about mixed volumes of convex bodies.
For the most part we restrict ourselves to the case of lattice polytopes in
$\mathbb{R}^{2}$, as this is the focus of this paper. For the general theory
of mixed volumes we refer to [Sch14, Sec. 5.1].

A convex body is a non-empty compact convex set $K\subset\mathbb{R}^{d}$. The
support function of $K$, $h_{K}:\mathbb{R}^{d}\to\mathbb{R}$ is defined by
$h_{K}(u)=\max\\{\langle u,v\rangle:v\in K\\}$, where $\langle u,v\rangle$ is
the usual inner product in $\mathbb{R}^{d}$. A polytope
$P\subset\mathbb{R}^{d}$ is the convex hull of finitely many points in
$\mathbb{R}^{d}$. A lattice polytope $P\subset\mathbb{R}^{d}$ is the convex
hull of finitely many points in $\mathbb{Z}^{d}$. The surface area measure of
$P$ is the discrete measure supported on the set of outer unit normals to the
facets of $P$ whose values are the $(d-1)$-dimensional volumes of the
corresponding facets.

In general, the mixed volume is a polarization of the usual volume, that is,
it is the unique symmetric and multilinear function of $d$ convex bodies which
coincides with the volume when the $d$ bodies are the same [Sch14, Sec. 5.1].
In the case of dimension $d=2$, the polarization formula is given in (1.2). As
seen from (1.2), the mixed volume is invariant under linear transformations
with determinant $\pm 1$ and under independent translations of $P$ and $Q$.
There is a formula for $\operatorname{V}(P,Q)$ in terms of the support
function of $P$ and the surface area measure of $Q$, see [Sch14, Thm. 5.1.7],
which, in particular, implies the non-negativity of the mixed volume. Below we
present a lattice version of this formula which we will use in Section 4.

Assume $P,Q\subset\mathbb{R}^{2}$ are lattice polytopes. As mentioned in the
introduction, for lattice polytopes we normalize the 2-dimensional volume by a
factor of 2. For example, a lattice rectangle $P=[0,a]\times[0,b]$ for
$a,b\in\mathbb{Z}_{\geq 0}$ has normalized volume
$\operatorname{Vol}_{2}(P)=2ab$ and a right triangle $Q$ with vertices
$(0,0)$, $(a,0)$, and $(0,b)$ has $\operatorname{Vol}_{2}(Q)=ab$. Furthermore,
if $I\subset\mathbb{R}^{2}$ is a lattice segment then we define its normalized
1-dimensional volume as its lattice length, i.e.
$\operatorname{Vol}_{1}(I)=|I\cap\mathbb{Z}^{2}|-1$.

When defining the surface area measure for lattice polytopes it is more
convenient to work with primitive normals rather than unit normals. A vector
$u\in\mathbb{Z}^{2}$ is primitive if its coordinates are relatively prime. Let
$U_{P}$ be the set of outer primitive normals to the sides of $P$ and $S_{P}$
the corresponding surface area measure supported on $U_{P}$ with values
$S_{P}(u)=\operatorname{Vol}_{1}(P^{u})$. Here $P^{u}$ represents the side of
$P$ with outer primitive normal $u$. Then the normalized mixed volume can be
computed as follows (see [ABS21, Sec. 2.5] for details)

(2.1) $\operatorname{V}(P,Q)=\sum_{u\in
U_{Q}}h_{P}(u)\operatorname{Vol}_{1}(Q^{u}).$

Note that $h_{P}(u)\operatorname{Vol}_{1}(Q^{u})$ equals the inner product of
a vertex $v\in P$ and a scalar multiple of the primitive vector $u$. We give a
small example below.

######  Example 2.1.

Let $P=\operatorname{conv}\\{(0,0),\,(0,2),\,(4,1),\,(4,0)\\}$ and
$Q=\operatorname{conv}\\{(0,0),\,(0,2),\,(3,0)\\}$. Then $U_{Q}$ consists of
three primitive vectors: $(2,3)$, $(0,-1)$, and $(-1,0)$. In Figure 2.1 we
show $Q$ with the primitive vectors rescaled by the lattice lengths of the
corresponding sides, $u_{1}=(2,3)$, $u_{2}=(0,-3)$, and $u_{3}=(-2,0)$.

Figure 2.1. The vertices of $P$ and the surface area measure of $Q$

The inner product $\langle u_{i},v\rangle$ attains its maximum at the vertex
$v_{1}=(4,1)\in P$ for $i=1$ and at $v_{0}=(0,0)\in P$ for $i=2,3$. Therefore,
by (2.1),

$\operatorname{V}(P,Q)=\langle u_{1},v_{1}\rangle+\langle
u_{2},v_{0}\rangle+\langle u_{3},v_{0}\rangle=11+0+0=11.$

##  3\. Quadratic forms

In this section we review basic notions in the theory of binary quadratic
forms over the integers. We then prove a reduction algorithm for indefinite
forms similar to the classical reduction for positive definite forms (see, for
example, [Bue89, Ch. 2]).

Consider a binary quadratic form

$f(x,y)=ax^{2}+2bxy+cy^{2},$

where $a,b,c$ are integers. It is called indefinite if its discriminant
$\Delta=4(b^{2}-ac)$ is non-negative. We can express $f$ in a matrix form

$f(x,y)=\left(\begin{matrix}x&y\end{matrix}\right)M\left(\begin{matrix}x\\\
y\end{matrix}\right),$

where $M=\left(\begin{matrix}a&b\\\ b&c\end{matrix}\right)$ is an integer
symmetric matrix.

Recall that the unimodular group $\operatorname{GL}(2,\mathbb{Z})$ is the
group of invertible integer matrices $G$ with $\det G=\pm 1$. Two forms $f$
and $f^{\prime}$ are called equivalent if they are related by a unimodular
change of variables. Then $f^{\prime}$ is equivalent to $f$ whenever

$f^{\prime}(x,y)=\left(\begin{matrix}x&y\end{matrix}\right)G^{T}MG\left(\begin{matrix}x\\\
y\end{matrix}\right),$

for some $G\in\operatorname{GL}(2,\mathbb{Z})$. Here $G^{T}$ is the transpose
of $G$. In this case we say that $G$ transforms $f$ to $f^{\prime}$. More
explicitly, if $G=\left(\begin{matrix}x_{1}&x_{2}\\\
y_{1}&y_{2}\end{matrix}\right)$ then
$f^{\prime}(x,y)=a^{\prime}x^{2}+2b^{\prime}xy+c^{\prime}y^{2}$ with

$\displaystyle a^{\prime}$
$\displaystyle=ax_{1}^{2}+2bx_{1}y_{1}+cy_{1}^{2}=f(x_{1},y_{1})$
$\displaystyle b^{\prime}$
$\displaystyle=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})+cy_{1}y_{2}$
$\displaystyle c^{\prime}$
$\displaystyle=ax_{2}^{2}+2bx_{2}y_{2}+cy_{2}^{2}=f(x_{2},y_{2}).$

Clearly, equivalent forms have the same discriminant.

Our next result is a key tool for constructing lattice polytopes with a given
volume polynomial in Section 4.

######  Proposition 3.1.

Every integer indefinite binary form with positive coefficients is equivalent
to a form $f(x,y)=ax^{2}+2bxy-cy^{2}$ for some $a,b,c\in\mathbb{Z}_{\geq 0}$
such that $f(1,1)=a+2b-c>0$. Moreover, there exists a matrix
$G=\left(\begin{matrix}x_{1}&x_{2}\\\
y_{1}&y_{2}\end{matrix}\right)\in\operatorname{GL}(2,\mathbb{Z})$ satisfying
$x_{i}\geq y_{i}\geq 0$ for $i=1,2$ which transforms $f$ to the original form.

###### Proof.

Consider $F(x,y)=Ax^{2}+2Bxy+Cy^{2}$ with $A,B,C\in\mathbb{N}$ and let
$k=B^{2}-AC\geq 0$. If $A<C$ then we apply the matrix
$\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right)$ which swaps the variables.
Thus, we may assume $A\geq C$. Since $k\geq 0$ this implies that $B\geq C$.

Next, we divide $B$ by $C$ with a remainder, $B=sC+r$ where $s\geq 1$ and
$0\leq r<C$. Applying the matrix $\left(\begin{matrix}0&1\\\
1&-s\end{matrix}\right)$ we obtain a form $ax^{2}+2bxy+cy^{2}$ with $a=C$,
$b=r$, and $c=F(1,-s)$. Note that we have $a>b>c$, since $a>b$ and the
discriminant $k=b^{2}-ac\geq 0$. While $c>0$ we keep dividing $b$ by $c$ with
a remainder. Since throughout this process $c$ is getting strictly smaller, we
will eventually end up with a triple where $a>b\geq 0\geq c$. By switching the
sign of $c$, we may write the resulting form as $f(x,y)=ax^{2}+2bxy-cy^{2}$
with $a,b,c\in\mathbb{Z}_{\geq 0}$.

Now, let us look at the matrix $G$ that transforms $f$ to $F$. It is the
product of the inverses of the matrices used above, i.e.

$G=\left(\begin{matrix}s_{n}&1\\\
1&0\end{matrix}\right)\cdots\left(\begin{matrix}s_{1}&1\\\
1&0\end{matrix}\right)\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right),$

where $n\geq 1$, $s_{i}\geq 1$ for $1\leq i\leq n$, and the last matrix may or
may not be present. It is easy to see by induction that $G$ satisfies the
condition in the statement of the proposition.

Finally, to ensure that $f(1,1)>0$ we modify the last step, if needed. We
choose $s_{n}$ to be the smallest integer such that $F(1,-s_{n})\leq 0$ where
$F$ is the form in the penultimate step. Since $F$ has positive coefficients,
we must have $s_{n}\geq 1$. Also, $f(1,1)=F(1,1-s_{n})>0$, since otherwise
$s_{n}$ was not the smallest. ∎

##  4\. The volume polynomial

In this section we prove our main result (Theorem 4.2) which describes all
normalized volume polynomials of pairs of lattice polytopes in
$\mathbb{R}^{2}$. First, we look at the easier case of the usual volume
polynomial of two planar bodies.

######  Proposition 4.1.

Let $F(x,y)=Ax^{2}+2Bxy+Cy^{2}$ be an indefinite quadratic form with
$A,B,C\in\mathbb{R}_{\geq 0}$. Then there exist convex bodies
$K,L\subset\mathbb{R}^{2}$ such that $F(x,y)$ is the volume polynomial of $K$
and $L$, that is $F(x,y)=\operatorname{Vol}_{2}(xK+yL)$ for $x,y\geq 0$.

###### Proof.

We let $K$ and $L$ be rectangular boxes $K=[0,\alpha]\times[0,\beta]$ and
$L=[0,\gamma]\times[0,\delta]$ for some
$\alpha,\beta,\gamma,\delta\in\mathbb{R}_{\geq 0}$. Then
$\operatorname{Vol}_{2}(K)=\alpha\beta$,
$\operatorname{V}(K,L)=(\alpha\delta+\beta\gamma)/2$, and
$\operatorname{Vol}_{2}(L)=\gamma\delta$. We will show that the system

$A=\alpha\beta,\quad B=(\alpha\delta+\beta\gamma)/2,\quad C=\gamma\delta$

has a non-negative solution. Indeed, first assume $A>0$. Put $\alpha=1$ and so
$\beta=A$. Multiplying the middle equation by $\gamma$ and using the last
equation we obtain:

$A\gamma^{2}-2B\gamma+C=0.$

Since $k=B^{2}-AC\geq 0$ this equation has a non-negative solution
$\gamma=\frac{B+\sqrt{k}}{A}$. Note that $\gamma=0$ only if $B=0$ and $C=0$.
In this case $\delta=0$. Otherwise, $\delta=C/\gamma$.

Now if $A=0$ and $B>0$ then we put $\alpha=1$, $\beta=0$,
$\gamma=\frac{C}{2B}$, and $\delta=2B$. Finally, if $A=0$ and $B=0$ then we
put $\alpha=0$, $\beta=0$, $\gamma=1$, and $\delta=C$. ∎

######  Theorem 4.2.

Let $F(x,y)=Ax^{2}+2Bxy+Cy^{2}$ be an indefinite quadratic form with
$A,B,C\in\mathbb{Z}_{\geq 0}$. Then there exist lattice polytopes
$P,Q\subset\mathbb{R}^{2}$ such that $F(x,y)$ is the normalized volume
polynomial of $P$ and $Q$, that is $F(x,y)=\operatorname{Vol}_{2}(xP+yQ)$ for
$x,y\geq 0$.

###### Proof.

Suppose $A$ and $C$ are positive, and, hence, so is $B$. Then, according to
Proposition 3.1, there exist a form $f(x,y)=ax^{2}+2bxy-cy^{2}$ with
$a,b,c\in\mathbb{Z}_{\geq 0}$, and $f(1,1)=a+2b-c>0$, and a unimodular matrix
$G=\left(\begin{matrix}x_{1}&x_{2}\\\
y_{1}&y_{2}\end{matrix}\right)\in\operatorname{GL}(2,\mathbb{Z})$ with
$x_{i}\geq y_{i}\geq 0$, $i=1,2$ which transforms $f$ to $F$. Then

$A=f(x_{1},y_{1}),\quad
B=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})-cy_{1}y_{2},\quad C=f(x_{2},y_{2}).$

Below we show how to construct lattice polytopes $P$ and $Q$ satisfying
$\operatorname{Vol}_{2}(P)=A$, $\operatorname{V}(P,Q)=B$, and
$\operatorname{Vol}_{2}(Q)=C$. The construction differs slightly depending on
the parity of $a$ and $c$.

Case 1. Suppose first that $a$ and $c$ are even, so that $a=2d$ and $c=2e$ for
some $d,e\in\mathbb{Z}_{\geq 0}$. Then we can write $f(x,y)=ax^{2}+2bxy-
cy^{2}=2x(dx+by)-2ey^{2}$. We view this expression as the area of a rectangle
with sides $x$ and $dx+by$ with two opposite corners cut off; each corner
being a right triangle with legs $y$ and $ey$. Thus, we define

$P=\operatorname{conv}\\{\big{(}0,y_{1}\big{)},\,\big{(}0,x_{1}\big{)},\,\big{(}ey_{1},0\big{)},\,\big{(}dx_{1}+by_{1},0\big{)},\,\big{(}dx_{1}+by_{1},x_{1}-y_{1}\big{)},\,\big{(}dx_{1}+(b-e)y_{1},x_{1}\big{)}\\}.$

We define $Q$ in the same way replacing $x_{1}$ with $x_{2}$ and $y_{1}$ with
$y_{2}$ above. Since

$dx_{i}+by_{i}\geq dx_{i}+(b-e)y_{i}\geq y_{i}(d+b-e)=y_{i}(a+2b-c)/2>0,$

the relative positions of the vertices of $P$ and $Q$ are as shown in Figure
4.1. Clearly,
$\operatorname{Vol}_{2}(P)=2(dx_{1}+by_{1})-2ey_{1}^{2}=f(x_{1},y_{1})=A$ and,
similarly, $\operatorname{Vol}_{2}(Q)=f(x_{2},y_{2})=C$. In the second diagram
in Figure 4.1 we depict the surface area measure of $Q$. This allows us to
compute the mixed volume using (2.1).

Figure 4.1. Case 1: $a=2d$ and $c=2e$

$\displaystyle\operatorname{V}(P,Q)$
$\displaystyle=(x_{2}-y_{2})(dx_{1}+by_{1})+y_{2}(dx_{1}+by_{1})+ey_{2}(x_{1}-y_{1})+x_{1}(dx_{2}+(b-e)y_{2})$
$\displaystyle-
ey_{1}y_{2}=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})-cy_{1}y_{2}=B.$

Case 2. If $a=2d$ and $c=2e+1$ for $d,e\in\mathbb{Z}_{\geq 0}$, we construct
$P$ and $Q$ as above except the primitive outer normal $(1,e)$ is now replaced
with $(1,e+1)$. Then

$P=\operatorname{conv}\\{\big{(}0,y_{1}\big{)},\,\big{(}0,x_{1}\big{)},\,\big{(}ey_{1},0\big{)},\,\big{(}dx_{1}+by_{1},0\big{)},\,\big{(}dx_{1}+by_{1},x_{1}-y_{1}\big{)},\,\big{(}dx_{1}+(b-e-1)y_{1},x_{1}\big{)}\\},$

and $Q$ is obtained by replacing the indices. We have

$dx_{i}+by_{i}\geq dx_{i}+(b-e-1)y_{i}\geq(d+b-e-1)y_{i}=y_{i}(a+2b-c-1)/2\geq
0,$

and we get a diagram depicted in Figure 4.2.

Figure 4.2. Case 2: $a=2d$ and $c=2e+1$

As before, we have $\operatorname{Vol}_{2}(P)=A,\operatorname{Vol}_{2}(Q)=C$,
and

$\displaystyle\operatorname{V}(P,Q)$
$\displaystyle=(x_{2}-y_{2})(dx_{1}+by_{1})+y_{2}(dx_{1}+by_{1})+(e+1)y_{2}(x_{1}-y_{1})$
$\displaystyle+x_{1}(dx_{2}+(b-e-1)y_{2})-ey_{1}y_{2}=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})-cy_{1}y_{2}=B.$

Case 3. Suppose $a=2d+1$ and $c=2e$ for $d,e\in\mathbb{Z}_{\geq 0}$. We now
write

$f(x,y)=ax^{2}+2bxy-cy^{2}=2x(dx+by)+x^{2}-2ey^{2}.$

This represents the area of a trapezoid, which is the union of an $x$ by
$dx+by$ rectangle and the isosceles right triangle with leg $x$, with two
opposite corners cut off, where the corners have area $ey^{2}$ each. In other
words, we let $P$ have the vertices

$(0,y_{1}),\,(0,x_{1}),\,(ey_{1},0),\,((d+1)x_{1}+by_{1},0),\,(dx_{1}+(b+1)y_{1},x_{1}-y_{1}),\,(dx_{1}+(b-e)y_{1},x_{1}),$

and $Q$ is defined accordingly. Similarly to the above,

$dx_{i}+by_{i}\geq dx_{i}+(b-e)y_{i}\geq y_{i}(d+b-e)=y_{i}(a+2b-c-1)/2\geq
0,$

and we get the diagram in Figure 4.3.

Figure 4.3. Case 3: $a=2d+1$ and $c=2e$

We have $\operatorname{Vol}_{2}(P)=A$, $\operatorname{Vol}_{2}(Q)=B$, and

$\displaystyle\operatorname{V}(P,Q)$
$\displaystyle=(x_{2}-y_{2})((d+1)x_{1}+by_{1})+y_{2}(dx_{1}+(b-e)y_{1})+(e+1)x_{1}y_{2}$
$\displaystyle+x_{1}(dx_{2}+(b-e)y_{2})-ey_{1}y_{2}=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})-cy_{1}y_{2}=B.$

Case 4. Finally, suppose $a=2d+1$ and $c=2e+1$ for $d,e\in\mathbb{Z}_{\geq
0}$. Then

$dx_{i}+by_{i}\geq dx_{i}+(b-e)y_{i}\geq y_{i}(d+b-e)=y_{i}(a+2b-c)/2>0,$

and we define each of $P$ and $Q$ as the convex hull of

$\big{(}0,y_{i}\big{)},\,\big{(}0,x_{i}\big{)},\,\big{(}ey_{i},0\big{)},\,\big{(}(d+1)x_{i}+by_{i},0\big{)},\,\big{(}dx_{i}+(b+1)y_{i},x_{i}-y_{i}\big{)},\,\big{(}dx_{i}+(b-e-1)y_{i},x_{i}\big{)},$

as depicted in Figure 4.4.

Figure 4.4. Case 4: $a=2d+1$ and $c=2e+1$

Then $\operatorname{Vol}_{2}(P)=A$, $\operatorname{Vol}_{2}(Q)=C$, and

$\displaystyle\operatorname{V}(P,Q)$
$\displaystyle=(x_{2}-y_{2})((d+1)x_{1}+by_{1})+y_{2}(dx_{1}+(b-e-1)y_{1})+(e+2)x_{1}y_{2}$
$\displaystyle+x_{1}(dx_{2}+(b-e-1)y_{2})-ey_{1}y_{2}=ax_{1}x_{2}+b(x_{1}y_{2}+x_{2}y_{1})-cy_{1}y_{2}=B.$

In remains to consider the case when $A$ or $C$ (or both) is zero. Without
loss of generality, assume $A=0$. Then we take $P$ to be the unit segment
$P=\operatorname{conv}\\{(0,0),\,(1,0)\\}$. To construct $Q$ we write
$C=sB-r$, for some $s,r\in\mathbb{Z}$ with $s\geq 1$ and $0<r\leq B$, and let

$Q=\operatorname{conv}\\{(s,0),\,(1,0),\,(0,r),\,(0,B)\\}.$

Then $\operatorname{Vol}_{2}(Q)=C$ and $\operatorname{V}(P,Q)=B$, as required.
Note that when, in addition, $C=0$ we have $s=1$, $r=B$ and, hence, $Q$ is
also a segment.

∎

##  5\. Connection to intersection numbers of curves

In this section we discuss an implication of Theorem 4.2 for intersection
numbers of plane tropical curves, as well as intersection numbers of divisors
on toric surfaces. We recall basic facts below and refer to [Ful93, CLS11,
MS15] for general theory of toric and tropical varieties.

###  5.1. Planar tropical curves

A tropical bivariate polynomial $f$ is a piece-wise linear function
$f(x,y)=\min\\{c_{a,b}+ax+by:(a,b)\in S\\}$ where $S\subset\mathbb{Z}^{2}$ is
a finite set, called the support of $f$, and $c_{a,b}\in\mathbb{R}$. The
convex hull $P_{f}=\operatorname{conv}S$ is called the Newton polytope of $f$.

The set of points $(x,y)\in\mathbb{R}^{2}$ where $f$ is not differentiable
(that is where the minimum in the definition of $f$ is attained at least
twice) is called the (planar) tropical curve $V_{f}\subset\mathbb{R}^{2}$
associated to $f$. Geometrically, $V_{f}$ is a polyhedral complex of pure
dimension 1. There is a duality between the polyhedral complex $V_{f}$ and the
regular subdivision $\cal R_{f}$ of $P_{f}$ induced by lifting $(a,b)\in S$ to
height $c_{a,b}$. The vertices (0-dimensional cells) of $V_{f}$ correspond to
the polygons (2-dimensional cells) of $\cal R_{f}$ and the edges
(1-dimensional cells) of $V_{f}$ correspond to the edges of $\cal R_{f}$.
Given an edge $e\in V_{f}$, its weight $w(e)$ is the lattice length of the
corresponding edge in $\cal R_{f}$.

Two tropical curves $V_{f}$ and $V_{g}$ are said to intersect transversally if
$V_{f}\cap V_{g}$ is finite and every point $p\in V_{f}\cap V_{g}$ lies in the
relative interior of some edge $e$ of $V_{f}$ and some edge $h$ of $V_{g}$.
Define the local intersection number $(V_{f},V_{g})_{p}$ to be
$w(e)w(h)|\det(u,v)|$, where $u$ and $v$ are primitive vectors parallel to $e$
and $h$, respectively. Then the intersection number is defined as the sum of
local intersection numbers

$(V_{f},V_{g})=\sum_{p\in V_{f}\cap V_{g}}(V_{f},V_{g})_{p}.$

The following theorem is known as the tropical Bernstein-Khovanskii-
Kushnirenko theorem in dimension two, [MS15, Thm. 4.6.8].

######  Theorem 5.1.

The intersection number of two planar tropical curves $V_{f}$ and $V_{g}$
intersecting transversally equals the normalized mixed volume of their Newton
polytopes $\operatorname{V}(P_{f},P_{g})$.

Let $V_{f}$ be a tropical curve and let $V_{f^{\prime}}$ be its translate by
$(x_{0},y_{0})\in\mathbb{R}^{2}$. In other words $V_{f^{\prime}}$ is
associated to the tropical polynomial $f^{\prime}(x,y)=f(x+x_{0},y+y_{0})$
which has the same support and, hence, same Newton polytope as $f$. Note that
the lifting function $c^{\prime}_{a,b}$ differs from $c_{a,b}$ by a linear
function, $c^{\prime}_{a,b}=c_{a,b}+ax_{0}+by_{0}$, hence, the regular
subdivisions $\cal R_{f}$ and $\cal R_{f^{\prime}}$ are also the same. For
almost all values of $(x_{0},y_{0})$ the curves $V_{f}$ and $V_{f^{\prime}}$
intersect transversally, in which case we define
$(V_{f},V_{f}):=(V_{f},V_{f^{\prime}})$ and call it the self-intersection
number of $V_{f}$. By Theorem 5.1,
$(V_{f},V_{f})=\operatorname{V}(P_{f},P_{f})=\operatorname{Vol}_{2}(P_{f})$.

The following is a direct consequence of Theorems 4.2 and 5.1.

######  Theorem 5.2.

Given $(A,B,C)\in\mathbb{Z}_{\geq 0}^{3}$ satisfying $AC\leq B^{2}$, there
exist planar tropical curves $V_{f}$ and $V_{g}$ with $(V_{f},V_{f})=A$,
$(V_{f},V_{g})=B$, and $(V_{g},V_{g})=C$.

###### Proof.

Let $P$ and $Q$ be the lattice polytopes constructed in Theorem 4.2 from the
triple $(A,B,C)$. Consider tropical polynomials $f$ and $g$ with supports
$P\cap\mathbb{Z}^{2}$ and $Q\cap\mathbb{Z}^{2}$, respectively, and generic
coefficients $c_{a,b}$. Then the corresponding tropical curves $V_{f}$ and
$V_{g}$ will intersect transversally and
$(V_{f},V_{f})=\operatorname{Vol}_{2}(P)=A$,
$(V_{f},V_{g})=\operatorname{V}(P,Q)=B$, and
$(V_{g},V_{g})=\operatorname{Vol}_{2}(Q)=C$. ∎

###  5.2. Divisors on toric surfaces

A toric surface $X$ is an algebraic surface containing the torus
$(\mathbb{C}^{*})^{2}$ as a Zariski open subset whose action on itself extends
to the action on $X$. Basic examples of complete toric surfaces include the
projective plane, the product of projective lines, and the Hirzebruch surface.
One can describe the affine charts of $X$ via a rational polyhedral fan
$\Sigma$ in $\mathbb{R}^{2}$ whose 1-dimensional cones (rays) are generated by
a collection of primitive vectors $\\{u_{1},\dots,u_{r}\\}$. Each ray
corresponds to a 1-dimensional orbit in $X$ whose closure defines a torus
invariant prime divisor $D_{i}$ on $X$. Let $D=\sum_{i=1}^{r}a_{i}D_{i}$ be a
torus invariant divisor. It defines a rational polytope111Since we work with
outer normals rather than inner normals, we modify the standard definition
accordingly. $P_{D}=\\{x\in\mathbb{R}^{2}:\langle x,u_{i}\rangle\leq
a_{i},1\leq i\leq r\\}$.

As shown in [CLS11, Thm. 4.2.8], $D=\sum_{i=1}^{r}a_{i}D_{i}$ is a Cartier
divisor if and only if there is a continuous piece-wise linear function
$\phi_{D}$ on $\mathbb{R}^{2}$ which is linear on each cone of $\Sigma$ and
$\phi_{D}(u_{i})=a_{i}$ for $1\leq i\leq r$. The global sections of the
corresponding line bundle $\cal O(D)$ can be identified with the space of
Laurent polynomials supported in $P_{D}\cap\mathbb{Z}^{2}$, i.e.
$\Gamma(X,\cal
O(D))\cong\operatorname{span}_{\mathbb{C}}\\{x_{1}^{a_{1}}x_{2}^{a_{2}}:(a_{1},a_{2})\in
P_{D}\cap\mathbb{Z}^{2}\\}$, see [CLS11, Thm. 4.3.3]. Furthermore, $\cal O(D)$
is globally generated if and only if $\phi_{D}$ is convex in which case
$P_{D}$ is a lattice polytope, see [CLS11, Thm. 6.1.7].

The following theorem is a version of the Bernstein-Khovanskii-Kushnirenko
theorem for the intersection numbers of divisors on toric surfaces, see
[Ful93, Sec. 5.4].

######  Theorem 5.3.

Let $D,E$ be two globally generated divisors on a toric surface $X$. Then the
intersection number $(D,E)$ equals the normalized mixed volume of their
polytopes $\operatorname{V}(P_{D},P_{E})$.

We can now interpret the result of Theorem 4.2 as follows.

######  Theorem 5.4.

Given $(A,B,C)\in\mathbb{Z}_{\geq 0}^{3}$ satisfying $AC\leq B^{2}$, there
exist a toric surface $X$ and globally generated divisors $D$ and $E$ on $X$
such that $(D,D)=A$, $(D,E)=B$, and $(E,E)=C$.

###### Proof.

Let $P$ and $Q$ be the lattice polytopes constructed in Theorem 4.2 from the
triple $(A,B,C)$. The primitive normals of both $P$ and $Q$ belong to a set of
six vectors $\\{u_{1},\dots,u_{6}\\}$. For example, in Case 1 the vectors are
$\\{\pm(1,0),\pm(0,1),\pm(1,e)\\}$. Let $\Sigma$ be the fan generated by
$\\{u_{1},\dots,u_{6}\\}$ and $X$ the toric surface associated to $\Sigma$.
Define $a_{i}=h_{P}(u_{i})$ and $b_{i}=h_{Q}(u_{i})$ for $1\leq i\leq 6$,
where $h_{P}$ and $h_{Q}$ are the support functions of $P$ and $Q$,
respectively. Then the divisors $D=\sum_{i=1}^{6}a_{i}D_{i}$ and
$E=\sum_{i=1}^{6}b_{i}D_{i}$ are globally generated and have polytopes $P$ and
$Q$, respectively. By Theorems 5.3 and 4.2 we have
$(D,D)=\operatorname{Vol}_{2}(P)=A$, $(D,E)=\operatorname{V}(P,Q)=B$, and
$(E,E)=\operatorname{Vol}_{2}(Q)=C$. ∎

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