# Dielectric relaxation in the quantum multiferroics Rb2Cu2Mo3O12 and
Cs2Cu2Mo3O12

D. Flavián  Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich,
Switzerland  P. A. Volkov  Department of Physics, University of Connecticut,
Storrs, Connecticut 06269, USA  S. Hayashida  Laboratory for Solid State
Physics, ETH Zürich, 8093 Zürich, Switzerland  Neutron Science and Technology
Center, Comprehensive Research Organization for Science and Society (CROSS),
Tokai, Ibaraki 319-1106, Japan  K. Yu. Povarov  Dresden High Magnetic Field
Laboratory (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat,
Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  S. Gvasaliya
Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland  P.
Chandra  Department of Physics and Astronomy, Center for Materials Theory,
Rutgers University, Piscataway, NJ 08854, USA  A. Zheludev  Laboratory for
Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland

###### Abstract

Motivated by the recent discovery of dielectric relaxation by quantum critical
magnons in Cs2Cu2Mo3O12, we conduct a detailed analysis of its dielectric
response and compare it to that in the isostructural compound Rb2Cu2Mo3O12.
Measurements in the vicinity of the field-induced magnon softening show that
its description in terms of 3D Bose-Einstein condensation of magnons quantum
critical point is unaltered by the presence of dielectric relaxation. We also
demonstrate the existence of dielectric relaxation anomalies at 19 K in
Rb2Cu2Mo3O12 and discuss the implications for the microscopic origin of
dielectric activity in two compounds.

In our previous work we reported unusual dielectric properties at magnetic
quantum phase transitions in two isostructural frustrated quantum
antiferromagnets, namely Rb2Cu2Mo3O12 [1] and Cs2Cu2Mo3O12 [2]. Thanks to
magneto-electric coupling, the magnetically ordered phases of Rb2Cu2Mo3O12 are
also ferroelectric under a finite magnetic field. In applied fields, the
electric polarization becomes directly coupled to the antiferromagnetic order
parameter. As a result, dielectric permittivity shows a critical divergence at
the magnetic phase transitions. This includes the magnetic field-induced
saturation transition and the corresponding quantum critical point (QCP).
Since the latter can be described in terms of Bose-Einstein condensation (BEC)
of magnons [3, 4, 5, 6, 7], dielectric experiments enabled a direct
excitation-probe measurement of the critical susceptibility and the critical
exponent $\gamma$ [1]. This is significant, as in conventional BEC those
quantities are not physically accessible.

An apparently similar behavior of the dielectric constant at magnetic phase
boundaries was also detected in Cs2Cu2Mo3O12. However, the focus of previous
work was on an additional new dielectric anomaly, one that is not seen in the
Rb-based system [2]. This broad anomaly does not explicitly follow the
magnetic phase boundary, but nevertheless shows critical characteristics at
the QCP. Its origin is entirely different. It represents Debye-like relaxation
of some intrinsic dipolar degrees of freedom mediated by quantum-critical soft
magnons. This demonstrates that magnetic excitations in this system show
electric activity of their own, regardless the existence of long-ranged
ordering.

Several important questions remain unanswered. Is the sharp dielectric anomaly
at the phase boundaries in Cs2Cu2Mo3O12 also described in terms of critical
BEC susceptibility, as in Rb2Cu2Mo3O12? How is the relaxation anomaly in
Cs2Cu2Mo3O12 affected by the spin-flop transition, which is present in that
material but absent in the Rb-counterpart? What is the origin of the dipolar
degrees of freedom in Cs2Cu2Mo3O12 and are they also present in Rb2Cu2Mo3O12?
And if they are ineed also present there, why does the relaxation anomaly seem
to be absent? In the present work we address these issues via new dielectric
measurements on the Cs-compound, highlighting similarities and differences
between the materials (Fig.1).

Figure 1: Schematic phase diagrams of (a) Cs2Cu2Mo3O12 and (b) Rb2Cu2Mo3O12
showing their antiferromagnetic (AFM) and ferroelectric (FE) long range order
domes (shaded) and the presence of relaxation anomalies (dashed line) as
discussed in detail in the text.

##  I Results

Figure 2: False color plots of complex capacitance $\Delta C=\Delta
C^{\prime}+i\Delta C^{\prime\prime}$ in Cs2Cu2Mo3O12: (a) real and (b)
imaginary parts. The electric field is applied parallel to the c∗ axis, and
magnetic field along the b axis. A dashed line shows the phase boundary as
extracted from specific heat measurements in Ref. [8], while a solid line
represents the spin-flop field. Arrows indicate the relaxing feature at
$T^{\ast}$ in both real and imaginary plots.

###  I.1 Capacitance for $\mathbf{H}\|\mathbf{b}$

In the present study we used the same 2.4$\times$0.25$\times$0.18 mm3, and
0.25 mg single crystal Cs2Cu2Mo3O12 sample and experimental setup as in Ref.
[2]. Sample capacitance is measured with an alternating current (AC) probing
field at 1 kHz, as are all subsequent measurements in this text. Data for
$\mathbf{H}\|\mathbf{b}$ and $\mathbf{E}\|\mathbf{c}^{\ast}$ is presented in
Fig. 2 as a function of magnetic field and temperature. False color plots show
the real ($\Delta C^{\prime}$) and imaginary ($\Delta C^{\prime\prime}$)
components of the complex signal. These plots are to be compared against Fig.
1 in Ref. [2], which shows similar data for other magnetic field directions.
The same anomalies are also found here: a sharp feature at the phase boundary
as a result of magnetic/ferroelectric order, and a relaxing feature with
significant real and imaginary parts. As previously, the temperature at which
the imaginatry part of the latter is a maximum is denoted as $T^{\ast}$.

The presence of a spin-flop transition at 2 T for this field configuration
notably affects both kinds of anomalies. On the one hand, the feature at the
phase boundary retains a significant weight as the magnetic field is lowered
almost all the way down to zero field. It is particularly prominent in the
vicinity of the bicritical termination point found at 2 T and 1.4 K. On the
other hand, the value of $T^{*}$ is visibly reduced around the spin-flop
field, echoing the reduction of the ordering temperature, $T_{c}$.

###  I.2 Effect of a bias electric field

Figure 3: Effect of a DC bias electric field along the crystallographic c∗
axis on the complex dielectric response of Cs2Cu2Mo3O12 close to the QCP for:
(a,b) H$\parallel$c∗ and (c,d) H$\parallel$a. Note that data in (c,d) have
been multiplied by 10 in order to match the scale in (a,b). Colors show field
scans at a constant temperature of and different bias fields from 0 to
$\sim$250 kV/m. Only the diverging feature at the phase boundary is modified
by a uniform electric field, as expected for a critical susceptibility. The
broad relaxing features, shown by asterisks in (a-d), are unchanged.

For Rb2Cu2Mo3O12 the fact that the dielectric polarization is a primary order
parameter of the magnetic phase transitions was established by studying the
effect of a static electric field. Under such an applied field, the divergence
of dielectric permittivity at the phase boundary was shown to be suppressed,
and the corresponding peak broadened. We now conducted the same measurement in
Cs2Cu2Mo3O12 to understand how a static field affects both the phase boundary
and relaxation anomalies. In all cases the direct current (DC) electric field
was applied parallel to the probing AC field. Figure 3 shows data obtained for
two orthogonal field directions, where the boundary signal is strongest
($\mathbf{H}\|\mathbf{c}^{*}$) and weakest ($\mathbf{H}\|\mathbf{a}$).

Figure 4: Analysis of the phase boundaries of antiferro- magnetic-
ferroelectric (AFM-FE) long-range order (LRO) dome in Cs2Cu2Mo3O12. Magnetic
field is applied along the crystallographic directions (a) a, (b) b, and (c)
c∗. Circles show the phase boundary extracted from typical constant-
temperature field scans. The shaded area shows the long-range-ordered region.
A dashed line is the best fit to a power-law dependence. The best fits (shown
in Table 1) are obtained in a windowing analysis, as exemplified in (d) and
(e) for H$\parallel$a. $T_{\mathrm{max}}$ in the abscissa represents the
maximum temperature of the window used in each individual fit. The boundary
exponent is in perfect agreement with BEC criticality ($\phi=2/3$).

The suppression of the phase boundary anomaly with an increasing electric
field is fully analogous to what was seen in Rb2Cu2Mo3O12. Note that the
effect on the imaginary part is particularly strong. This component of
dielectric permittivity is associated with ferroelectric domain-wall dynamics
[9]. A static field suppresses domain formation and thereby the imaginary part
of the response.

In contrast, the behavior of the relaxing features is completely different.
The external field has virtually no effect on the signal at fields close to
the QCP for either magnetic field orientation. The relaxing features in the
real part of capacitance, that appear as a broad plateau of about 0.5 T in
width below the divergent peak [as shown by asterisks in Figs. 3(a) and 3(c)],
are not altered by fields as high as 250 kV/m (50 V). Similarly, the imaginary
part [asterisks Figs. 3(b) and 3(d)] remains unchanged over the whole range of
electric fields. This observation confirms that the relaxation anomaly is
entirely unrelated to ferroelectric long-range order in the system.

###  I.3 Shift exponents

The very sharp and prominent dielectric anomaly at the phase boundary allows a
very precise measurement of the field dependence of the ordering temperature,
and with it the shift exponent $\phi$ at the magnetic saturation QCP. The
positions of the dielectric peaks are plotted versus field and temperature in
Fig. 4 for three different applied field directions. In order to determine
both the critical field and the shift exponents for the three studied
geometries, we employ a shrinking-window fit analysis [10], as was previously
done for Rb2Cu2Mo3O12. This procedure makes the best of the experimental data
while trying to restrict the analysis on a region as narrow as possible around
the critical point. The critical field $H_{c}$, the power law exponent $\phi$
and a scale factor are used as parameters in a least-squares fitting, as a
function of temperature window given by $T_{\mathrm{max}}$. The optimal
fitting window is narrowed until a further reduction stops producing a
statistically significant change in the fit results. An example of such
analysis is provided in Figs. 4(d) and 4(e). The best fits from the windowing
analysis are shown in Figs. 4(a)-4(c) and summarized in Table 1.

The fitted shift exponents are in good agreement with the expected $\phi=2/3$
for a 3D BEC QCP [3, 4, 5, 6, 7]. The present case reveals a much better
agreement with the BEC framework than Rb2Cu2Mo3O12, where the presence of a
narrow presaturation phase for certain field orientations is associated with a
distorted phase boundary. Even though the presence of presaturation phases in
Cs2Cu2Mo3O12 has been found for all field orientations [8], these seem to have
a minimal effect on the shape of the long-range-ordered dome.

Table 1: Critical fields and boundary exponents at the QCP of Cs2Cu2Mo3O12. Values are extracted from best fits in the range given by $T_{\mathrm{max}}$ around the QCP for the investigated field geometries.  |  $\mu_{0}H_{c}$ (T) |  $\phi$ |  $T_{\mathrm{max}}$ (K)
---|---|---|---
$\textbf{H}\parallel\textbf{a}$ | 7.683(4) | 0.67(2) | 1.00
$\textbf{H}\parallel\textbf{b}$ | 7.384(4) | 0.68(3) | 0.90
$\textbf{H}\parallel\textbf{c}^{\ast}$ | 8.726(5) | 0.712(5) | 0.90

###  I.4 Critical susceptibility

Figure 5: (a) Scaling of the critical dielectric susceptibility ($\chi$) of
Cs2Cu2Mo3O12 along the critical trajectory for H$\parallel$ c∗ (fixed
$\mu_{0}H_{c}=8.726$ T), after subtraction of the non-critical contribution
from Fig. 7(a). A double logarithmic plot shows indeed power-law behavior
below 2 K and down to the lowest measured temperatures. Best fit to a power-
law trend is shown as a solid thick line, along with the obtained critical
exponent. Solid thin lines are guides to the eye representing other typical
critical exponents. (b) Results of a windowing analysis performed to analyze
the critical susceptibility. The optimal fitting range is highlighted.

Having determined the critical fields we turn to study the BEC critical
susceptibility, $\chi$, at the QCP. It was already shown in [2] that a non-
critical contribution to dielectric permittivity exists at low temperature, as
a result of the existence of intrinsic electric dipoles in Cs2Cu2Mo3O12. This
field-independent contribution ($\Delta C_{0}$) is analyzed below in detail
(see Fig. 7(a)) and is subtracted from the measurement of $\Delta C$ to obtain
the critical dielectric susceptibility, $\chi$. Figure 5(a) shows precisely
this quantity for H$\parallel$ c∗ along the critical trajectory with
$\mu_{0}H_{c}=8.726$ T. The double logarithmic scale in Figure 5(a) shows a
robust power-law trend for $\chi$ over a decade in temperature and down to the
lowest accessible values. A shrinking window analysis is used to extract a
consistent power-law exponent, just as in Fig. 4 and is illustrated in Fig.
5(b). Given the high density of data points, these are binned together every
40 points for the analysis. Best fit to a $T^{-\gamma}$ dependence is obtained
with $T_{\mathrm{max}}=1.31$ K and provides a critical exponent
$\gamma=1.49(2)$, in fantastic agreement with $\gamma=3/2$ as expected for a
three-dimensional BEC QCP [11].

Just as in the case of Rb2Cu2Mo3O12, the saturation transition in Cs2Cu2Mo3O12
is an excellent example of magnon-BEC criticality. The results in Fig. 5 show
that the BEC description of the transition is unaltered by the presence of
dipolar relaxation in Cs2Cu2Mo3O12, and further establishes the independence
of ferroelectric ordering and relaxation processes.

###  I.5 Dielectric relaxation by magnons

Figure 6: Typical constant-field measurements of complex capacitance of
Cs2Cu2Mo3O12 for $\textbf{H}\parallel\textbf{b}$ and
$\textbf{E}\parallel\textbf{c}^{\ast}$. (a) Measurements at different fields
are shown in colors, showing a perfect collapse at high temperature, from
which the static response $\Delta C_{0}$ is obtained (thick shaded line).
(b,c) Capacitance measurements normalized by $\Delta C_{0}$: (b) real, and (c)
imaginary components. Black solid lines show a fit to (1). Triangles show the
value of $T^{\ast}$ as defined in the text. Each data set is offset vertical
by 0.5 units for visibility.  Figure 7: Typical constant-field measurements of
complex capacitance of Cs2Cu2Mo3O12 for $\textbf{H}\parallel\textbf{c}^{\ast}$
and $\textbf{E}\parallel\textbf{c}^{\ast}$. (a) Measurements at different
fields are shown in colors, showing a perfect collapse at high temperature,
from which the static response $\Delta C_{0}$ is obtained (thick shaded line).
(b,c) Capacitance measurements normalized by $\Delta C_{0}$: (b) real, and (c)
imaginary components. Black solid lines show a fit to (1). Triangles show the
value of $T^{\ast}$ as defined in the text. Each data set is offset vertical
by 0.5 units for visibility. The prominence of the boundary dielectric anomaly
close to the QCP is apparent in this geometry.

We extend the analysis presented in Ref. [2] to a wider set of dielectric
data, including two additional orientations of the external magnetic field.
The magnon-mediated relaxation of electric dipoles manifests itself in a Cole-
Cole relaxation process [12, 13], modeled after the following dependence:

$\Delta C=\frac{\Delta C_{0}(T)}{1+[i\omega\tau]^{1-\alpha}}$ (1)

where best agreement is found for activated behavior of the relaxation time
$\tau(T,H)=\tau_{0}\exp[E_{b}(H)/k_{B}T]$. Fig. 6 and Fig. 7 show capacitance
data for the geometries with $\textbf{H}\parallel\textbf{b}$ and
$\textbf{H}\parallel\textbf{c}^{\ast}$, respectively. All the data are
analyzed following the prescriptions in [2]. A field-independent static
response, $\Delta C_{0}$, is obtained for every geometry, with
$T_{0,\textbf{b}}=0.44(7)$ K and $T_{0,\textbf{c}^{\ast}}=0.39(2)$ K, as shown
in Fig. 6(a) and Fig. 7(a), and in good agreement with the results reported in
[2]. Further, the entirety of the data set is fitted to (1) using a single
parameter $\alpha=0.2$ and a single attempt time $\tau_{0}=10^{-6}$ s as
global variables, same values as in in the previous report, emphasizing the
field independence of these parameters. Optimization of the activation energy,
$E_{b}(H)$, for each magnetic field value results in a good agreement with the
experimental data as shown in Fig. 6(b,c) for $\textbf{H}\parallel\textbf{b}$,
and in Fig. 7(b,c) for $\textbf{H}\parallel\textbf{c}^{\ast}$.

Figure 8: Characteristic energy barriers ($E_{b}$) in Cs2Cu2Mo3O12 extracted
from fits of complex capacitance to a Cole-Cole model (1). Different symbols
represent magnetic fields along different crystallographic directions; solid
lines are a guide to the eye and reach zero at the corresponding critical
fields, as given in Table 1.

The resulting energy barriers for all three studied geometries are displayed
in Fig.8. As was shown in [2] the energy barrier approaches zero at the QCP.
Above $H_{c}$, the energy barrier follows the Zeeman-induced single-magnon
energy gap $E_{b,i}=g_{i}\mu_{B}\mu_{0}(H-H_{c,i})$, with $g_{i}$ the
$g$-factor and $i=a,b,c^{*}$. Linear fits close to the transition result in
$g$-factors $g_{\textbf{a}}=2.13$, $g_{\textbf{b}}=2.11$ and
$g_{\textbf{c}^{*}}=2.10$, in fantastic agreement with the isotropic
$g$-tensor found in bulk measurements [8].

In addition, results for $\textbf{H}\parallel\textbf{b}$ evidence the
influence of the spin-flop transition in the dipolar relaxation rate. The
height of the energy barrier shows a dip at the spin-flop field, the same way
$T_{c}$ shows a dip. This indicates that the relaxation rate is linked to the
underlying magnetic structure and the associated magnetic excitations.
However, we reiterate that long range magnetic order is not a requirement for
magnon-mediated relaxation to take place.

###  I.6 Dielectric relaxation in Rb2Cu2Mo3O12

Figure 9: Zero-field complex capacitance on single crystals of Cs2Cu2Mo3O12
and Rb2Cu2Mo3O12. A black arrow in (a) real and (b) imaginary panels indicate
the presence of a dielectric anomaly in Rb2Cu2Mo3O12, which is completely
absent in Cs2Cu2Mo3O12. Figure 10: (a) Capacitance measured on coaligned
single crystals of Rb2Cu2Mo3O12 at several fields as reported in [1]. A
phenomenological background ($\Delta C_{0}$) is determined from the high-
temperature capacitance signal. Data for different fields are perfectly
overlapping. (b,c) The reduced real (b) and imaginary (c) components of the
signal show good agreement with (1) as given by the black line. Measurements
for different fields have a vertical offset of 0.05 fF for visibility. (d)
Activation energy as a function of field, result of the Cole-Cole fits. The
extracted energy gap and attempt time are consistent with optical phonon
activity. The values show a negligible field dependence around the average
value of 27.2 meV, shown with dashed line.

We finally turn to Rb2Cu2Mo3O12 and revisit the single crystal capacitance
data at high temperature reported in [1]. There we have observed a curious
dielectric anomaly at much higher temperatures, around 19 K [Fig.2(c) of Ref.
[1] and Fig. 9]. It involves both a peak in the imaginary part of capacitance
and a step in the real component, reminiscent of Debye relaxation. This is in
contrast to the dielectric response of Cs2Cu2Mo3O12, where no dielectric
anomalies were found in this temperature range (Fig. 9) or up to room
temperature (not shown). We note, however, the striking similarity of this
plot with the relaxation anomaly seen in the Cs-based compound at much lower
temperatures.

Assuming that in Rb2Cu2Mo3O12 we are dealing with a similar relaxation process
of intrinsic dipolar degrees of freedom, we analyze these data using the Cole-
Cole relaxation model in (1). At such high temperatures the lattice expansion
is no longer negligible and contributes to a significant variation on $C$. To
account for this effect, a field-independent linear background is subtracted
from the data as shown in Fig. 10(a). The resulting complex capacitance
components ($\Delta C^{\prime}$ and $\Delta C^{\prime\prime}$) are displayed
in Fig. 10(b), 10(c), and 10(e).

Cole-Cole plots in Fig. 10(e) clearly show the characteristic semicircle
response of a relaxation process for all values of magnetic field. Note that
data are collected at a constant frequency, and the plots show the temperature
variation of complex capacitance. A value of $\alpha=0.14$ is extracted from
fitting the data and is used in the subsequent analysis. A good fit to (1)
with activated behavior is obtained with a single $\alpha$, a flat $\Delta
C_{0}$ = const., and a single attempt time $\tau_{0}\approx 10^{-11}$ s (or a
frequency of 0.1 THz), shown as black lines in Fig.10(b) and 10(c). The
characteristic energy barrier is optimized for each value of magnetic field,
showing a negligible field dependence around an average value of
$E_{b}/k_{B}=27.2$ meV, as illustrated in Fig.10(d). As opposed to
Cs2Cu2Mo3O12, relaxation here is occurring over a much higher energy barrier,
and with a much higher attempt rate.

##  II Discussion

From the above it becomes clear that the phase-boundary anomalies in
Cs2Cu2Mo3O12 and Rb2Cu2Mo3O12 are, essentially, identical. The same applies to
quantum criticality found at the saturation field, constituting both fantastic
instances of 3D QCP BEC of magnons.

That the relaxation anomaly in Cs2Cu2Mo3O12 is something entirely distinct and
independent of ferroelectric order is apparent from its lack of response to a
DC electric field. Such behavior may be consistent with defects being the
origin of the low-energy electric dipolar degrees of freedom. A crude estimate
of the energy level shift induced by the DC electric field can be given as
$\Delta\sim Zea_{0}E\approx 25$ mK, assuming the maximum field used in Fig. 3
of $E=$ 250 kV/m, $Z\sim$1 and a displacement $a_{0}$ of 0.1 Å as typically
found in ferroelectric materials [14, 15, 16]. This shift is well over an
order of magnitude smaller than the energy barrier mediating relaxation
responsible for the anomalies in Fig. 3, of the order of $E_{b}\sim$ 600-800
mK.

Let us now discuss the potential implications of the qualitative difference
between dielectric relaxation in the Cs- and Rb-based compounds. In the latter
compound, dipolar degrees of freedom appear to also be present as evidenced in
Fig. 10. We may speculate that they are the result of crystallographic defects
or oxygen vacancies. However, in Rb2Cu2Mo3O12 these electric dipoles relax
over a much larger barrier via a different mechanism. As we have previously
shown, the 19 K feature in Rb2Cu2Mo3O12 is not affected by the magnetic field.
The relaxation is thus unlikely to be of a magnetic origin. The effects of
trapped charge carriers or polarons [17, 18, 19] can also be ruled out in such
a crystalline stoichiometric insulator. A more likely mechanism is one
involving an optical phonon. Low temperature dielectric relaxation features
like those in Fig. 10 have been observed in SrTiO3 as a collection of
dielectric and ultrasound anomalies between 8 K and 67 K [20, 21]. Relaxation
with an attempt time of 10-12 s was found and associated with an optical
phonon at 33 meV in energy. Identification of that phonon as a ”silent” mode
that can not directly couple to charge indicated that the coupling must be
through a defect, which locally breaks symmetry [21]. These numbers are fully
consistent with our observations for Rb2Cu2Mo3O12, suggesting that the
relaxation mechanism in this case may likely be due to an optical phonon with
a similar coupling.

##  III Conclusions

The dielectric anomalies associated with the ferroelectric-antiferromagnetic
ordering in Cs2Cu2Mo3O12 and Rb2Cu2Mo3O12 have been shown to be extremely
similar, as is the corresponding magnon-BEC behavior at the magnetic field
induced quantum critical point. While both materials feature some intrinsic
dipolar degrees of freedom, the mechanisms leading to relaxation are very
different. In the Cs-compound this happens is via magnons that soften at the
magnetic quantum critical point and show electrical activity regardless the
ground state. In the Rb-system relaxation occurs likely via optical phonon
modes.

##  IV Acknowledgments

This work was partially supported by the Swiss National Science Foundation,
Division II. This work was supported by the Deutsche Forschungsgemeinschaft
through ZV 6/2-2, the Würzburg-Dresden Cluster of Excellence on Complexity and
Topology in Quantum Matter - ct.qmat (EXC 2147, Project No. 390858490) and the
SFB 1143 (Project No. 247310070), as well as by HLD at HZDR, member of the
European Magnetic Field Laboratory (EMFL). PAV acknowledges support by the
University of Connecticut OVPR Quantum CT seed grant.

## References

  * Hayashida  _et al._ [2021] S. Hayashida, L. Huberich, D. Flavián, Z. Yan, K. Y. Povarov, S. Gvasaliya, and A. Zheludev, Critical dielectric susceptibility at a magnetic bec quantum critical point, Phys. Rev. Research 3, 033053 (2021).
  * Flavián  _et al._ [2023] D. Flavián, P. A. Volkov, S. Hayashida, K. Y. Povarov, S. Gvasaliya, P. Chandra, and A. Zheludev, Dielectric relaxation by quantum critical magnons, Phys. Rev. Lett. 130, 216501 (2023).
  * Batyev and Braginski [1984] E. G. Batyev and L. S. Braginski, Antiferromagnet in a strong magnetic field: analogy with Bose gas, Sov. Phys. JETP 60, 781 (1984).
  * Giamarchi and Tsvelik [1999] T. Giamarchi and A. M. Tsvelik, Coupled ladders in a magnetic field, Phys. Rev. B 59, 11398 (1999).
  * Nikuni  _et al._ [2000] T. Nikuni, M. Oshikawa, A. Oosawa, and H. Tanaka, Bose-einstein condensation of dilute magnons in ${\mathrm{tlcucl}}_{3}$, Phys. Rev. Lett. 84, 5868 (2000).
  * Giamarchi  _et al._ [2008] T. Giamarchi, C. Rüegg, and O. Tchernyshyov, Bose–Einstein condensation in magnetic insulators, Nat. Phys. 4, 198 (2008).
  * Zapf  _et al._ [2014] V. Zapf, M. Jaime, and C. D. Batista, Bose-Einstein condensation in quantum magnets, Rev. Mod. Phys. 86, 563 (2014).
  * Flavián  _et al._ [2020] D. Flavián, S. Hayashida, L. Huberich, D. Blosser, K. Y. Povarov, Z. Yan, S. Gvasaliya, and A. Zheludev, Magnetic phase diagram of the linear quantum ferro-antiferromagnet Cs2Cu2Mo3O12, Phys. Rev. B 101, 224408 (2020).
  * Liu  _et al._ [2015] G. Liu, S. Zhang, W. Jiang, and W. Cao, Losses in ferroelectric materials, Materials Science and Engineering: R: Reports 89, 1 (2015).
  * Hüvonen  _et al._ [2012] D. Hüvonen, S. Zhao, M. Månsson, T. Yankova, E. Ressouche, C. Niedermayer, M. Laver, S. N. Gvasaliya, and A. Zheludev, Field-induced criticality in a gapped quantum magnet with bond disorder, Phys. Rev. B 85, 100410(R) (2012).
  * Continentino [2011] M. A. Continentino, Interplay of quantum and classical fluctuations near quantum critical points, Braz. J. Phys. 41, 201 (2011).
  * Debye [1929] P. J. W. Debye,  _Polar molecules_ (Dover publications, 1929).
  * Cole and Cole [1941] K. S. Cole and R. H. Cole, Dispersion and absorption in dielectrics i. alternating current characteristics, The Journal of Chemical Physics 9, 341 (1941).
  * Kwei  _et al._ [1993] G. H. Kwei, A. C. Lawson, S. J. L. Billinge, and S. W. Cheong, Structures of the ferroelectric phases of barium titanate, The Journal of Physical Chemistry 97, 2368 (1993).
  * B. Ravel and Kraizman [1998] R. I. V. B. Ravel, E. A. Stern and V. Kraizman, Local structure and the phase transitions of batio3, Ferroelectrics 206, 407 (1998).
  * Dawber  _et al._ [2005] M. Dawber, K. M. Rabe, and J. F. Scott, Physics of thin-film ferroelectric oxides, Rev. Mod. Phys. 77, 1083 (2005).
  * Emin [1993] D. Emin, Optical properties of large and small polarons and bipolarons, Phys. Rev. B 48, 13691 (1993).
  * Zhang and Tang [2004] L. Zhang and Z.-J. Tang, Polaron relaxation and variable-range-hopping conductivity in the giant-dielectric-constant material $\mathrm{Ca}{\mathrm{Cu}}_{3}{\mathrm{Ti}}_{4}{\mathrm{O}}_{12}$, Phys. Rev. B 70, 174306 (2004).
  * Bidault  _et al._ [1995] O. Bidault, M. Maglione, M. Actis, M. Kchikech, and B. Salce, Polaronic relaxation in perovskites, Phys. Rev. B 52, 4191 (1995).
  * Ang  _et al._ [1999] C. Ang, J. F. Scott, Z. Yu, H. Ledbetter, and J. L. Baptista, Dielectric and ultrasonic anomalies at 16, 37, and 65 K in ${\mathrm{SrTiO}}_{3}$, Phys. Rev. B 59, 6661 (1999).
  * Scott [1999] J. F. Scott, Comment on the physical mechanisms of the 37 K and 65 K anomalies in strontium titanate, Journal of Physics: Condensed Matter 11, 8149 (1999).


An antimagic labeling of a graph $G(V,E)$ is a bijection $f: E \to \{1,2, \dots, |E|\}$ so that $\sum_{e \in E(u)} f(e)  \neq \sum_{e \in E(v)} f(e)$ holds for all $u, v \in V(G)$ with $u \neq v$, where $E(v)$ is the set of edges incident to $v$.  We call $G$ antimagic if it admits an antimagic labeling. A forest is a graph without cycles; equivalently, every component of a forest is a tree.
It was proved by Kaplan, Lev, and Roditty [2009], and by Liang, Wong, and Zhu [2014] that every tree with at most one vertex of degree-2 is antimagic. A major tool used in the proof is the zero-sum partition introduced  by Kaplan, Lev, and Roditty [2009]. In this article, we provide an algorithmic representation for the zero-sum partition method and apply this method to show that every forest with at most one vertex of degree-2 is also antimagic.

Antimagic labeling; antimagic graphs; trees; rooted trees; forests

05C78; 05C05

§ INTRODUCTION

The notion of magic graphs was motivated by magic squares. A magic square is an $n\times n$ array of integers $\{1,2,\dots, n^2\}$ so that each row, column, and the main and back-main diagonals of the square sum to the same value (see <Ref> for an example). In 1963, Sedláček [13] extended this concept to graphs by labeling the edges of a graph with numbers and defining the vertex-sum of each vertex to be the total of the labels  assigned to the edges incident to that vertex. <Ref> shows how a magic square is used to create a magic  labeling on a complete bipartite graph, where all the vertex-sums are identical.

An edge labeling of a graph $G$ is a function $f$ that assigns to each edge of $G$ a positive integer. For a vertex $u \in V(G)$, denote $E(u)$ the set of edges incident to $u$.  The vertex-sum of $u$ is defined as

\begin{equation}
\label{defn}
\phi_f(u)=\sum_{e\in E(u)} f(e).
\end{equation}

A magic square along with a magic $K_{3,3}$ graph. Each row and column of the magic square sum to 15. Each vertex in $K_{3,3}$ has vertex-sum 15.

An edge labeling $f$ is called magic if all vertices have the same vertex-sum (see <Ref>). Many variations of magic labelings have been studied (cf. [3]). Among them, the antimagic labeling has been studied widely in recent decades. The definition is given below. For positive integers $a \leq b$, denote $[a,b]=\{a, a+1, \dots, b\}$.

Let $G=(V,E)$ be a graph with $m$ edges. A bijective function  $f:  E \to [1,m]$  is an antimagic labeling for $G$ if $\phi_f(u)\neq \phi_f(v)$ for any two vertices $u\neq v$, where the vertex-sum $\phi_f(u)$ is defined in <Ref>.  If $G$ admits such an antimagic labeling, then $G$ is said to be antimagic.

By <Ref>, it is clear that $K_2$, the simple graph with only one edge and two vertices, is not antimagic. The notion of antimagic labelings was introduced by Hartsfield and Ringel in [10] in 1990, who conjectured that every connected graph other than $K_2$  is antimagic. Since then, this conjecture has been studied extensively. Many families of graphs are known to be antimagic, yet the conjecture remains open (cf. [2, 3, 4, 5, 7, 8, 9]).

A tree is a connected graph without cycles. For a graph $G$ and a vertex $v \in V(G)$, the degree of $v$, denoted by $\deg(v)$, is the number of edges incident to $v$. The following result was due to Kaplan, Lev, and Roditty [11],  and Liang, Wong, and  Zhu [12]:

A tree $T \neq K_2$ with at most one degree-2 vertex  is antimagic.

A major tool used in proving <Ref> is called the zero-sum partition. It was first devised by Kaplan, Lev, and Roditty in [11]. In Section 2, we provide a step-by-step process of this  method.

A forest is a graph without cycles. Thus,  every component of a forest is a tree, called a component tree. We aim to investigate antimagic labelings of forests by utilizing the zero-sum partition method of antimagic labelings for trees. The main result of this article is:

A forest $F$ with at most one degree-2 vertex and without  $K_2$ component trees is antimagic.

The proof of <Ref> is presented in Section 3. In Section 2, we introduce the zero-sum partition method that will be used in the proof of <Ref>. In Section 4, we discuss possible directions and open problems for future study.

§ THE ZERO-SUM PARTITION METHOD

The zero-sum partition method is based on the following two results. Here we present proofs that provide a step-by-step partition algorithm.

Let $s,l$ be non-negative integers and let $k=2s+6l$. Then there is a partition of $[1, k]$ into subsets $Q_1, Q_2, \dots, Q_{s+2l}$ such that the following hold:

\[
\begin{array}{llll}
\mbox{For $i \in [1, l]$:}    &|Q_i|=3 \  \mbox{and} \ \sum\limits_{a\in Q_i}a=k+1. \\
\mbox{For $i \in [1+l, s+l]$:} &|Q_i|=2 \ \mbox{and} \  \sum\limits_{a\in Q_i}a=k+1. \\
\mbox{For $i \in [s+l+1, s+2l]$:} &|Q_i|=3 \ \mbox{and} \ \sum\limits_{a\in Q_i}a=2(k+1).
\end{array}
\hspace{2in}
\]

The proof in [11] provides formulas of sets $Q_i$ directly. To assist in understanding, we provide a step-by-step method to obtain these formulas.
In general, <Ref> partitions the set of numbers in $[1, k]$ ($k=2s+6l$)  into three types of sets, called  $A$-, $B$-, and $C$-sets, where each $A$-set has three elements with sum $k+1$, each $B$-set has two elements with sum $k+1$, and each $C$-set has three elements with sum $2(k+1)$.
Precisely, there are $l$ $A$-sets (denoted as  $A_1, A_2, \dots, A_l$), $s$ $B$-sets (denoted as $B_{1}, B_{2}, \dots, B_{s}$), and $l$ $C$-sets (denoted as $C_{1}, C_{2}, \dots, C_{l}$).

The strategy of getting these $A$-, $B$-, and $C$-sets is expressed  in the following four steps. Along the process, we use the following <Ref> to illustrate each step.

Suppose $s=5$ and $l=2$. Then $s+3l=11$ and $k=2(s+3l) =22$.
By <Ref>, we can partition the numbers in $[1, 22]$ into subsets $A_1,A_2,B_1, \dots ,B_5, C_1, C_2$.

Step 1: Arranging all the labels into a two-row matrix.  List the numbers in $[1,k]$ as a $2 \times (\frac{k}{2})$ matrix $M$ where the first row has numbers $1, 2, \dots, \frac{k}{2}$ in increasing order, and the second row has numbers $k, k-1, \dots, \frac{k}{2} + 1$ in decreasing order.  Precisely, $a_{1,i}=i$ and $a_{2,i} = k-i+1$, for $i \in[1,\frac{k}{2}]$. See <Ref> as an illustration for <Ref>. Consequently, the two numbers in each column sum up to $k+1$. Note that there are  $\frac{k}{2}=s+3l$ columns.

Arrange $[1,22]$ into a two-row matrix $M$. Group the columns as shown. In total, there are $s+3l$ columns.

Step 2: Determining the B-sets.
Fix the columns of $[l+1, l+s]$ in $M$ as the $B$-sets. Precisely,

\mbox{{\bf B-sets}}: B_i = \{a_{1, l+i}, \ a_{2,l+i}\} = \{l+i, \ k-l-i+1\},  \ i \in [1, s].

Note that the sum of each $B$-set is $k+1$.
See <Ref> for an illustration of Step 2 for <Ref>.

Use these $s$ columns to create the $B$-sets. The $B$-sets are $B_1 = \{3,20\}, B_2=\{4,19\},  B_3=\{5,18\}, B_4=\{6,17\}$, and $B_5=\{7,16\}$.

Step 3: Determining the $A$-sets.   We use the first $l$ elements on the first row of $M$ to create A-sets. Recall that each column $i \in [1,l]$ in $M$, the sum of the two elements is $k+1$, since $a_{1,i}+a_{2,i}=(i)+(k-i+1)=k+1$.
We replace $a_{2,i}$ by two elements, $a_{1, l+s+i}$ and $a_{2, l+s+2i}$, since:

a_{1, l+s+i} +  a_{2, l+s+2i} = l+s+i + [ k-(l+s+2i)+1]= k-i+1=a_{2,i}.

Hence, we obtain the $l$ $A$-sets as follows:

\[
\mbox{{\bf $A$-sets}}: A_i =\set{a_{1,i}, \ a_{1,s+l+i}, \   a_{2,s+l+2i}}=\{i, \ l+s+i, \  k-(l+s+2i)+1\} ,\  i \in [1,l].
\]

See <Ref> for an illustration of this step for <Ref>.

The orange boxes are used to create $A$-sets. The process is to exchange the bottom numbers from the first $l$ columns with numbers in the last $2l$ columns. Notice that, 22=8+14 and 21 = 9+12. The $A$-sets are $A_1=\{1,8,14\}$ and $A_2 = \{2, 9, 12\}$.

After Steps 1, 2, and 3, the remaining unused labels are

\[
\begin{array}{lll}
a_{1, i}, &i \in [s+2l+1, s+3l] \\
a_{2, i}, &i \in [1,l] \cup \{s+l+1, s+l+3, \dots, s+3l-1\}.
\end{array}
\]

Step 4: Determining the $C$-sets.
For $i \in [1, l]$, first fix $a_{2, i} = k+1-i \in C_{i}$.
Next, we combine the unused labels
$a_{2, s+3l-2i}$ and $a_{1, s+3l-i}$ to form $C_i$:

\mbox{{\bf C-sets}}:
C_{i} = \{a_{2,i}, a_{1, s+3l+1-i}, \ a_{2, s+3l+1-2i}\} = \{k-i+1, \ s+3l+1-i, \ k-(s+3l+1-2i)+1\}.

One can easily check that $C_i$ is a $C$-set since the sum of the three elements is $2(k+1)$. See <Ref> for an illustration of Step 4 and the final partition for <Ref>.

The remaining numbers are used to create $C$-sets. The $C$-sets are $C_1=\{22,11,13\}$ and $C_2=\{21,10,15\}$.

After the four steps, each number in $[1,k]$ belongs to exactly one of the $A$-, $B$-, $C$-sets.
It is clear that the numbers in the first row and the numbers in the second row up to $a_{2, l+s}$ are used exactly once. For the remaining numbers, $a_{2, i}$, $i \in [s+l+1, s+3l]$, exactly half are in the $A$-sets while the other half are in the  $C$-sets, according to the parity of $i$.  See <Ref> for an illustration.

This figure helps visualize how $[1,k]$ is partitioned. $A$-sets cover the blocks in orange, $B$-sets cover the blocks in blue, and $C$-sets cover the blocks in red.

The next corollary is an immediate consequence of <Ref>, which allows us to partition the set of  integers $[1,k]$ into subsets of any sizes greater than one.

Let $k=r_1+r_2+\dotsb+r_t$ be a partition of the positive integer $k$, where $r_i\geq 2$ for $i \in [1,t]$.
Then the set $[1,k]$ can be partitioned into pairwise disjoint subsets, $D_1, D_2, \dots, D_t$, such that for every $1\leq i \leq t$, $|D_i|=r_i$ and $\sum_{a\in D_i} a \equiv 0 \pmod{k'}$, where $k'=k+1$ if $k$ is even, and $k'=k$ if $k$ is odd.

Let $k=r_1+r_2+\dotsb+r_t$, where each $r_i\geq 2$ for $i=1,2,\dots,t$. First, assume $k$ is even. Write each $r_i$ as the sum of multiples of 2 and multiples of 3, $r_i=2m_i+3n_i$, $m_i, n_i \geq 0$. Then $k = \sum_{i=1}^t (2m_i + 3n_i)$. Because $k$ is even, $n_1+n_2 +\dots+n_t$ must be even. Denote

$$s=m_1+m_2+\dotsb+m_t \ \ \mbox{and} \ \    l=\frac{n_1+n_2+\dotsb +n_t}{2}.

Then, $k=2s+6l$. By <Ref>,
$[1,k]$ can be partitioned into sets $Q_1$, $Q_2,\dots, Q_{s+2l}$.

We denote this partition of $[1,k]$ as ${\cal Q}$. To create the set $D_i$ with $|D_i|=r_i=2m_i+3n_i$, take $m_i$ sets in ${\cal Q}$ that are 2-element sets,  and take $n_i$ sets in ${\cal Q}$ that are 3-element sets. With the selected sets in ${\cal Q}$, let $D_i$ be the union of those sets. Since each 2- or 3-element set in ${\cal Q}$ has the elements sum to $0  \pmod{k+1}$, it implies that the elements in $D_i$ also sum to $0  \pmod{k+1}$.

Now assume $k=r_1+r_2 + \dots + r_t$ is odd. Then there exists some odd $r_j$, $r_j \geq 3$. Then $k-1=r_1+r_2+\dotsb (r_j-1)+\dotsb +r_t$. Note that $r_j -1 \geq 2$. As $k-1$ is even, following the procedure described above, we can partition the set $[1,k-1]$ into the union of sets $D_i$, $i \in [1, t]$, where $|D_i|=r_i$ for $i \neq j$ and $|D_j|=r_j-1$, and the sum of each set $D_i$ is a multiple of $k$, $i \in [1,t]$. Once the sets are created,
replace $D_j$ by   $\{k\}\cup D_{j}$ and keep the other $D_i$, $i \neq j$, to create a partition for $[1,k]$ which satisfies the statement.

The following definitions show how we apply  <Ref> to obtain an antimagic labeling for some trees. A leaf of a graph is a degree-1 vertex; a non-leaf vertex is called an internal vertex.

An (oriented) rooted tree is a tree where one vertex is designated to be the root. We draw the root at the top of the tree and orient edges from top to down.  Define the root to be at level 0. A vertex $v$ is at level-$i$ if the distance from $v$ to the root is $i$.  All the edges of the tree are oriented from vertices in level-$i$ to vertices in level-$(i+1)$. For a vertex $v$, the children of $v$ are the vertices that are adjacent to $v$ and are one level higher than $v$. We call $v$ the parent of its children. We say the tree is oriented from parent to children.

Let $T$ be a rooted tree. For every non-root vertex $v$, we denote $e^v$ as the incoming edge (the edge from the parent) of $v$. This is well-defined since each vertex (other than the root) has exactly one incoming edge.

Let $T$ be a rooted tree with $m$ edges. For $v \in V(T)$, denote  $E^+(v)$ the set of outgoing edges  from $v$ and denote $|E^+(v)|=n_v$.

Let  $v_1, v_2, \dots,  v_t$ be the vertices of $T$ with $n_{v_i} \geq 1$. It readily follows that

\begin{equation}
m=|E(T)|=n_{v_1} + n_{v_2} + \dots + n_{v_t}.
\label{equ1}
\end{equation}

For a rooted tree $T$ with $m$ edges, we call a bijective mapping  $f:E(T)\to [1,m]$ a zero-sum labeling of $T$ if for each $i \in [1,t]$,

\sum_{e \in E^+(v_i)} f(e) \equiv 0 \  \mbox{(mod $m'$)},

where $m'=m$ if $m$ is odd; and $m'=m+1$ if $m$ is even.

[11, 12]

Let $T$ be a tree with an even number of edges and at most one vertex of degree-2.
Then there exists a zero-sum labeling for $T$ that is antimagic.

Let $T$ be a tree with $m$ edges, where $m$ is even. Root $T$ at the degree-2 vertex if $T$ has a degree-2 vertex; otherwise, root $T$ at any internal vertex. Denote this root as $w$. Let $v_1, v_2, \dots, v_t$ be the internal vertices of $T$ and $n_{i}$ the number of outgoing edges of $v_i$. By our assumption, for $i\in[1,t]$, $n_i \geq 2$. By <Ref> and <Ref>, there exists a zero-sum labeling $f$ for $T$ by assigning numbers in $D_i$ to the outgoing edges of $v_i,\  i \in[1,t]$.  Further, for every non-root vertex $v \in V(T)$, $\phi_f(v) = f(e^{v}) + \sum_{e \in E^+(v)} f(e) \equiv f(e^{v}) \pmod{m+1}$ (if $v$ is a leaf then $E^+(v)=\emptyset$). For the root vertex $w$, $\phi_f(w) \equiv 0 \not\equiv f(e^{v}) \pmod{m+1}$ for any $v$. Since each $v$ has a distinct incoming edge, no two internal vertices will have congruent sums$\pmod{m+1}$. Thus, all vertices have distinct vertex sums, implying $f$ is an antimagic labeling for $T$.

A zero-sum antimagic labeling of a tree using the partition of <Ref> in <Ref>.

See <Ref> for an illustration of a zero-sum labeling of a tree by using <Ref>. In the proof of <Ref> in [11, 12] the tree is carefully rooted so that a zero-sum labeling is also an antimagic labeling. Following this idea, in the proof of <Ref> (next section), we carefully root each component tree of a given forest $F$ at a vertex and then identify all these roots as a  single vertex to form a single tree $T$. We find a zero-sum labeling for $T$ using the zero-sum partition method. Then we use this labeling to produce an antimagic labeling for $F$.

§ PROOF OF THEOREM 2

Let $F$ be a forest with $s$ component trees and $m=|E(F)|$. Denote the component trees of $F$ by $T_i$ with root at $w_i$ for  $i \in[1,s]$. If $s=1$, then $F$ is a tree, and the result follows by <Ref>. Henceforth we assume $s\geq 2$. We proceed with the proof by considering cases.

Case 1: $m$ is odd. Consider two sub-cases.

Sub-case 1.1: $F$ has no degree-2 vertex. Let $w_i\in V(T_i)$ be a leaf of $T_i$ for $i \in [1, s]$. Root each $T_i$ at $w_i$ and orient from parents to children as defined in <Ref>. Denote $w_i'$ to be the child of $w_i$ and denote the edge between them as $e_i = w_iw_i'$.

Let $T$ be the tree obtained by identifying the vertices $w_1, w_2, \dots, w_s$ into a single vertex $w$, where $w$ is the root of $T$. Let $v_1, v_2, \dots, v_t$ be the non-leaf vertices of $T$. Assume each vertex $v_i$ has $n_i$ outgoing edges, $n_i \geq 2$. Recall from <Ref>, the number of edges of $T$ is
$m = n_1 + n_2 + \dots + n_t$. Since $m$ is odd, there exists some odd $n_j$ so that $n_j \geq 3$.
By  <Ref> one can partition $[1,m]$ into sets $D_1,  D_2,  \dots, D_t$, where $|D_i|=n_i \geq 2$, and the elements in each  $D_i$ sum up to a multiple of $m$. Further, the label $m$ is assigned to an outgoing edge $v_jv_{j'}$ of $v_j$ with $n_j \geq 3$.   This gives a zero-sum labeling $f$ for $T$, where $\phi_f(v) \equiv  f(e^{v})$ (mod $m$) holds for all vertex, except the root $w$.  Thus all values of $\phi_f(v_i)$, $i \in [1,t]$, are distinct, except that $\phi_f(w) \equiv \phi_f(v_{j'}) \equiv 0$ (mod $m$), as $f(v_jv_{j'})=m$.

Let $g$ be the labeling for $F$ defined by

\[
g(e) = \begin{cases}
f(ww_i') & \textrm{if } e=w_iw_i',\\
f(e) & \textrm{otherwise.}
\end{cases}
\]

Note that $v_{j'}$  is the only vertex with $\phi_g(v_{j'})=\phi_f(v_{j'}) \equiv 0$ (mod $m$).  For every non-root vertex $u$ of $F$ (i.e., $u \neq w_i$ for all $i \in [1,s]$), we have $\phi_g(u) = \phi_f(u) \equiv f(e^u) \pmod{m}$. Thus $\phi_g(u)\pmod{m}$ are pair-wisely distinct. If $w_i$ and $w_j$ are two root vertices of the component trees, $i \neq j$, then $\phi_g(w_i)=f(e_i) \neq f(e_j) = \phi_g(w_j)$. Thus the roots have different vertex-sums. Further, $\phi_g(w_i)\equiv \phi_g(w_i') \equiv f(e_i) \pmod{m}$. Since $w_i'$ has at least two children, we have $\phi_g(w_i') \geq f(e_i)+ m > f(e_i) = \phi_g(w_i)$, implying $\phi_g(w_i)\neq \phi_g(w_i')$.
Hence, $g$ is an antimagic labeling for $F$. See <Ref> as an example.

An example for Sub-case 1.1. The forest $F$ to the left has two component trees, $T_1$ and $T_2$, and $m=11$ edges. Root $T_1$ and $T_2$ at a leaf, $w_1$ and $w_2$,  and identify $w_1$ and $w_2$ into a single root $w$ to form a tree $T$ shown on the right. A zero-sum labeling on $T$ gives an antimagic labeling on $F$ when $T$ is split back into the components of $F$.

Sub-case 1.2: $F$ contains a degree-2 vertex. Suppose $F$ has a degree-2 vertex, $v'$. Without loss of generality, assume $v' \in V(T_1)$. Root each $T_i$ at a leaf $w_i\in V(T_i)$ for $i \in [1,s]$.
Let $w_i'$ be the child of $w_i$ and let $v''$ be the (only) child of $v'$. Let $T$ be the tree obtained by identifying $w_1, w_2, \dots, w_s$ into a vertex $w$, which is the  root of $T$. Note that it is possible that $v'$ is the child of $w_1$ in $T_1$.  Let $w=v_1, v_2, \dots, v_t$ be the internal vertices of $T \setminus \{v'\}$.
By <Ref>, we can partition the numbers in the set $[1,m-1]$ into sets that sum up to multiples of $m$ and assign these sets to the outgoing edges of $v_i$, $i \in [1,t]$. Finally,  we assign $m$ to the edge $v'v''$.  This labeling, denoted by $f$, is a zero-sum labeling for $T$, where $\phi_f(u) \equiv f(e^u) \pmod{m}$ holds for every non-root vertex $u$ and $f(v'v'')=m$. Note that $\phi_f(v') \equiv f(e^{v'}) \pmod{m}$.
Let $g$ be the labeling for $F$ defined as follows:

\[
\begin{cases}
f(ww_i') & \textrm{ if } e=w_iw_i',\\
f(e) & \textrm{ otherwise.}
\end{cases}
\]

Then $\phi_g(u)\equiv g(e^u) \pmod{m}$ if $u \neq w_i$. Note that $v''$ is the only vertex with $\phi_g(v'')\equiv 0 \pmod{m}$. If $u\neq v$, then $\phi_g(u)\not\equiv \phi_g(v) \pmod{m}$ since $u$ and $v$ have different incoming edge labels. However, we have $\phi_g(w_i)\equiv \phi_g(w_i') \pmod{m}$.  Each $w_i'$ has at  least two children or one child if $w_i'=v'$ (the degree-2 vertex), so $\phi_g(w_i')\geq g(w_iw_i')+m > g(w_iw_i') = \phi_g(w_i)$. Thus $g$ is an antimagic labeling for $F$. See <Ref> as an example.

An example for Sub-case 1.2. The forest $F$ on the left consists of two trees $T_1$ and $T_2$, and $m=9$ edges, where $T_1$ has a degree-2 vertex $v'$ and roots of $T_1$ and $T_2$  are leaves. We identify these two roots to create a tree $T$ (on the right) and find a  zero-sum labeling on $F$ using <Ref> and labeling $v'v''$ with $m$. Afterwards, split $T$ back into $T_1$ and $T_2$ to get an antimagic labeling for $F$ as shown.

Case 2: $m$ is even.

Sub-case 2.1: $F$ has no degree-2 vertex.  For $i \in [1, s]$, root $T_i$ at a leaf $w_i$.
Let $w_i'$ be the child of $w_i$. Let $T$ be the tree obtained by identifying the vertices $w_1, w_2, \dots, w_s$ into a single vertex $w$. Denote $v_1, v_2, \dots, v_t$ the non-leaf vertices of $T$. Assume each vertex $v_i$ has $r_i$ children. Then $|E(T)|=m = r_1 + r_2 + \dots + r_t$, where $r_i \geq 2$ for $i \in [1,t]$.
By <Ref> there is a zero-sum labeling $f$ for $T$, which is antimagic.

Let $g$ be the labeling for $F$ defined by

\[
g(e) = \begin{cases}
f(ww_i') & \textrm{for } e=w_iw_i', \ i \in[1,s];\\
f(e) & \textrm{otherwise.}
\end{cases}
\]

Similar to Case 1, one can easily show that $g$ is an antimagic labeling of $F$.

Sub-case 2.2. $F$ contains a degree-2 vertex.
Assume $F$ has a degree-2 vertex $u$. Without loss of generality, assume $u$ is located in $T_1$. Consider two possibilities.

$\diamond$ Sub-case 2.2.1.  $s=2$.  Root $T_1$ at $w_1=u$. Root $T_2$ at  $w_2$ where $\deg_{T_2}({w_2}) \geq 3$. By <Ref> there is a zero-sum labeling $f$ for $F$. In this labeling, we assign a $B$-set to edges of $E^+(w_1)$. The vertex-sums are distinct in modulo $m+1$, excluding the two roots $w_1$ and $w_2$.
Since $w_1$ is degree-2,
$\phi_f(w_1)=m+1$. Further, as deg$(w_2) \geq 3$, we can choose the labels for edges incident to $w_2$ to sum up to at least $2(m+1)$ (if deg$(w_1)=3$, we use a $C$-set from <Ref>).  Thus the vertex-sums are all distinct. Therefore $f$ is an antimagic labeling for $F$. See <Ref>.

$\diamond$ Sub-case 2.2.2. $s \geq 3$.  Let $w_1=u$ be the root of $T_1$, and $E^+(u)=\{e', e''\}$.  For the remaining component trees $T_i$, $i \in [2, s]$, root $T_i$ at a leaf $w_i \in V(T_i)$. Let $T$ be the tree obtained by identifying $w_1,w_2,\dots, w_s$ into a single vertex $w$, and root $T$ at $w$.
By <Ref>, there exists a zero-sum labeling for $T$ such that $f(e')$ and $f(e'')$ belong to the same $B$-set in the partition, $f(e')+ f(e'')=m+1$. This is possible as $s \geq 3$, and so deg$(w) \geq 4$.

Let $g$ be the labeling for $F$ defined by

\[
g(e) = \begin{cases}
f(wv) & e=w_iv, \ i \in [1,s];\\
f(e) & \textrm{otherwise.}
\end{cases}
\]

Then $\phi_g(w_1) = m+1 \equiv 0 \pmod{m+1}$.
Similarly to the above cases, one can show that $g$ is an antimagic labeling for $F$.
See <Ref> as an example.
This completes the proof of <Ref>.

An example of Sub-case 2.2.1. This forest has two components, $T_1$ and $T_2$, $m=12$ edges, and a degree-2 vertex $w_1$. Root $T_1$ at $w_1$, and root $T_2$ at a vertex $w_2$ of degree-3 or higher. By <Ref> there exists a zero-sum antimagic labeling so that $\phi(w_1)=m+1$ and $\phi(w_2) \geq 2(m+1)$.

An example of Sub-case 2.2.2. The forest $F$ has 3 components, $T_1,T_2,$ and $T_3$. Root $T_1$ at the degree-2 vertex $w_1$, and root the other trees at a leaf. When labeling the combined tree $T$, the edges incident to $w_1$  get a $B$-set, so that it is the only vertex with sum 0 (mod 15).

§ FUTURE STUDY

In this article, we proved that every forest without $K_2$ as a component tree and having at most one vertex of degree-2 is antimagic. It would be interesting to investigate the forests where each component tree contains at most one degree-2 vertex.

Suppose $G$ is a graph and $e=uv$ is an edge of $G$. A subdivision of $e$ is the operation of replacing $e=uv$ with a path $(u,w_e,v)$, where $w_e$ is a new vertex.
For a tree $T$,
denote by $T^*$ the tree obtained by subdividing each edge in $T$.
The following result was proved in [12].

If $T$ is a tree without degree-2 vertices, then $T^*$ is antimagic.

Define $F^*$ in a similar way to $T^*$.
Suppose $F$ is a forest.
Denote by $F^*$ the forest obtained by subdividing each edge in $F$.

Question. For any forest $F$ without a degree-2 vertex and without $K_2$ as a component, is it true that $F^*$ is antimagic?

§ ACKNOWLEDGMENTS

We are grateful for the support of NSF PUMP (Preparing Undergraduates through Mentoring towards Ph.D.'s) grant. We would like to thank the anonymous referee for careful reading of the  manuscript with helpful comments.

[2]
N. Alon, G. Kaplan, A. Lev, Y. Roditty, and R. Yuster,
Dense graphs are antimagic,
J. Graph Theory,
47 (2004),

[3]  M. Bača, M. Miller,
J. Ryan and A. Semaničová-Feňovčíková,  Magic and Antimagic Graphs, Springer International Publishing, 2019.

[4]
K. Bérezi, A. Bernáth, and M. Vizer,
Regular graphs are antimagic,
Electric J. Combinatorics, 22 (2015), paper P3.34.

[5] F.-H. Chang, Y.-C. Liang, Z. Pan, X. Zhu, Antimagic labeling of regular graphs, J. Graph Theory, 82 (2016), 339–349.

[6] A. Chavez, P. Le, D. Lin, D. D.-F. Liu, and M. Shurman, Antimagic labeling for unions of graphs with many three-paths, Discrete Math., 346 (2023), 113356, on-line available.

[7] D. W. Craston, Regular bipartite graphs are antimagic, J. Graph Theory, 60 (2009), 179–182.

[8]
K. Deng, Y. Li,
Caterpillars with maximum degree 3 are antimagic,
Discrete Math..,
342 (2019), 1799–1801.

[9] K. D. E. Dhanajaya, and W.-T. Li, Antimagic labeling of forests with sets of consecutive integers, Discrete Applied Math., 309 (2022), 75–84.

[10] N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, INC, Boston, 1990, pp. 108–109, Revised version 1994.

[11] G. Kaplan, A. Lev, and Y. Roditty, On zero-sum partitions and anti-magic
trees, Discrete Math., 309 (2009), 2010–2014.

[12] Y.-C Liang, T.-L. Wong, and X. Zhu, Anti-magic labeling of trees, Discrete Math., 331 (2014), 9–14.

[13] J. Sedláček, Problem 27, In: Theory and Its Appl., Proc. Symp. Smolenice (1963), 163–164.

Johnny Sierra

California State University, Los Angeles

5151 State University Dr.

Los Angeles, CA 90032

E-mail<EMAIL_ADDRESS>
Daphne Liu

California State University, Los Angeles

5151 State University Dr.

Los Angeles, CA 90032

E-mail<EMAIL_ADDRESS>
Jessica Toy

California State University, Los Angeles

5151 State University Dr.

Los Angeles, CA 90032

E-mail<EMAIL_ADDRESS>


# Quantum Phase Transitions of the Distorted Diamond Spin Chain

Tomosuke Zenda1, Yuta Tachibana1, Yuki Ueno1,
Kiyomi Okamoto1 and Tôru Sakai1,2 1Graduate School of Material Science,
University of Hyogo, Hyogo 678-1297, Japan
2National Institutes for Quantum and Radiological Science and Technology
(QST), SPring-8, Hyogo 679-5148, Japan

(Received September 8, 2019)

###### Abstract

The frustrated quantum spin system on the distorted diamond chain lattice
suitable for the alumoklyuchevskite is investigated using the numerical
diagonalization of finite-size clusters and the level spectroscopy analysis.
It is found that this model exhibits three quantum phases; the ferrimagnetic
phase, the spin gap one, and the gapless Tomonaga-Luttinger liquid depending
on the exchange coupling parameters. The ground state phase diagram is
presented.

##  I Introduction

Frustrated quantum spin systems have attracted a lot of interest in the field
of strongly correlated electron systems. The $S=1/2$ distorted diamond spin
chain is one of strongly frustrated quantum spin systems. It was proposed as a
good theoretical model of the compound Cu3(CO3)2(OH)2, called azuritekikuchi .
The previous theoretical workokamoto1  using the perturbation analysis, the
numerical exact diagonalization of finite clusters, and the level spectroscopy
method, indicated that the system exhibits various quantum phases; the spin
gap phase, the ferromagnetic one, and gapless Tomonaga-Luttinger liquid (TLL)
one in the ground state, depending on the exchange coupling parameters.
Recently another candidate material of the distorted diamond spin chain was
discovered. It is the compound K3Cu3AlO2(SO4)4, called
alumoklyuchevskitefujihara ; morita ; fujihala2 . This material has a
different structure of the distortion from azurite. Thus it would be useful to
investigate the suitable theoretical model for alumoklyuchevskite. In this
paper, the $S=1/2$ distorted diamond spin chain model suitable for
alumoklyuchevskite is studied by the numerical exact diagonalization of
finite-size clusters and the level spectroscopy analysis.

##  II Model

We investigate the model described by the Hamiltonian

$\displaystyle{\mathcal{H}}={\mathcal{H}}_{0}+{\mathcal{H}}_{1}$ (1)
$\displaystyle{\mathcal{H}}_{0}=\sum_{j}^{L}\left\\{J_{2}\mbox{\boldmath$S$}_{j,1}\cdot\mbox{\boldmath$S$}_{j,2}+J_{5}\mbox{\boldmath$S$}_{j,2}\cdot\mbox{\boldmath$S$}_{j,3}+J_{1}\mbox{\boldmath$S$}_{j,3}\cdot\mbox{\boldmath$S$}_{j,1}\right\\}$
(2)
$\displaystyle{\mathcal{H}}_{1}=J_{1}\sum_{j}^{L}\left\\{\mbox{\boldmath$S$}_{j,3}\cdot\mbox{\boldmath$S$}_{j+1,2}+\mbox{\boldmath$S$}_{j,3}\cdot\mbox{\boldmath$S$}_{j+1,1}\right\\}$
(3)

where $\mbox{\boldmath$S$}_{j,i}$ is the spin-1/2 operator, $J_{1}$, $J_{2}$,
and $J_{5}$ are the coupling constants of the exchange interactions. The
schematic picture of the model is shown in Fig. 1.

For alumoklyuchevskite, it is thought that the interactions corresponding to
four sides of diamond differ from one another. Since such a model, however,
has many parameters, we use a simplified model sketched in Fig.1. When $J_{5}$
is much larger than other couplings, the spins coupled by $J_{5}$ are going to
form a singlet pairs, which make $\mbox{\boldmath$S$}_{j,1}$ spins nearly
free. If the direct or effective interactions between
$\mbox{\boldmath$S$}_{j,1}$ spins are antiferromagnetic, the ground state will
be the TLL state. This is the essential mechanism for the TLL ground state
observed in almoklyuchevskite. On the other hand, when $J_{2}$ is much larger
than other couplings, singlet pairs locate at the $J_{2}$ bonds, which yields
nearly free $\mbox{\boldmath$S$}_{j,3}$ spins. If the direct or effective
interactions between $\mbox{\boldmath$S$}_{j,3}$ spins are antiferromagnetic,
the ground state will be the TLL state, which is nothing but the essential
mechanism for the TLL ground state of azurite. Thus our model is a minimal
model describing both TLL ground states of alumoklyuchevskite and azurite. We
note that the direct interactions between nearly free spins are very important
to explain experimental results both of almoklyuchevskite and azurite jeschke
.

Figure 1: The model of the $S=1/2$ distorted diamond spin chain.

For $L$-unit systems, the lowest energy of ${\cal H}_{0}$ in the subspace
where $\sum_{j}S_{j}^{z}=M$, is denoted as $E(L,M)$. The reduced magnetization
$m$ is defined as $m=M/M_{\rm s}$, where $M_{\rm s}$ denotes the saturation of
the magnetization, namely $M_{\rm s}=2L/3$ for this system. $E(L,M)$ is
calculated by the Lanczos algorithm under the periodic boundary condition
($\mbox{\boldmath$S$}_{L+1,i}=\mbox{\boldmath$S$}_{1,i}$) for $L=$4, 6 and 8.

##  III Ground state phase diagram

We consider the ground state phase diagram of the model (1). Since the three
different exchange interactions $J_{1}$, $J_{2}$ and $J_{5}$, we fix $J_{1}=1$
and vary $J_{2}$ and $J_{5}$ in this paper. On the analogy of the azurite-type
model, the ferrimagnetic, the spin gap and the gapless TLL phases are expected
to appear.

###  III.1 Ferrimagnetic phase

The ferrimagnetic phase is easily distinguished from other phases. In this
phase the finite magnetization $m=1/3$ appears in the ground state. When
$J_{5}=0.5$ is fixed, the $J_{2}$ dependence of the lowest energies with $M=0$
and $M=4$ for $L=8$ are shown in Fig. 2. The phase boundary between the
ferrimagnetic and singlet phases can be detected as the intersection of two
energy levels. Since the phase boundary is almost independent of the system
size, the phase boundary is estimated from the result for $L=8$.

Figure 2:  $J_{2}$ dependence of $E(L=8,M=0)$ (black line) and $E(L=8,M=4)$
(red line) for $J_{5}=0.5$. The phase boundary between the ferrimagnetic and
singlet phases can be detected as the intersection of two energy levels.

###  III.2 Spin gap and TLL phases

In order to determine the phase boundary between the spin gap and the TLL
phases, the level spectroscopy analysis okamoto2 ; nomura  is one of the best
methods. According to this method, we should compare the excitation energies
of the lowest singlet excitation and the lowest triplet one. Namely, we define
two excitation energies

$\displaystyle\Delta(L,M=0)=E_{1}(L,M=0)-E_{0}(L,M=0),$ (4)
$\displaystyle\Delta(L,M=1)=E_{0}(L,M=1)-E_{0}(L,M=0),$ (5)

where $E_{0}(L,M)$ and $E_{1}(L,M)$ are, respectively, the lowest energy and
first excited energy within the subspace of $M$ for the $L$-unit system, The
ground state is in the spin gap phase or the TLL phase according as
$\Delta(L,M=0)>\Delta(L,M=1)$ or $\Delta(L,M=0)<\Delta(L,M=1)$. The $J_{2}$
dependences of $\Delta$’s with fixed $J_{5}=0.5$ for $L=4$, 6 and 8 are shown
in Fig. 3. Assuming the finite-size correction of the cross points between
$\Delta(L,M=0)$ and $\Delta(L,M=1)$ is proportional to $1/L^{2}$, we estimate
the phase boundary in the thermodynamic limit. This analysis indicates that
the spin gap phase is adjacent to the ferrimagnetic phase.

Figure 3:  $J_{2}$ dependences of $\Delta(L,M=0)$ and $\Delta(L,M=1)$ with
$J_{5}=0.5$ for $L=4$ (black lines), 6 (blue lines) and 8 (red lines). Solid
and dashed lines correspond to the $\Delta(L,M=0)$ and $\Delta(L,M=1)$,
respectively.

###  III.3 Phase diagram

According to the above analyses, the ground state phase diagram is obtained as
shown in Fig. 4. As expected, it includes the ferrimagnetic, the spin gap and
the TLL phases. Takano et al. takano  indicated that the dimer-monomer state
with high degeneracy is the exact ground state on the line of $J_{5}=1$ and
$J_{2}>2$. They also found that the doubly degenerate tetramer-dimer state is
the ground state on the line of $J_{5}=1$ and $0.909<J_{2}<2$. Reflecting this
fact, our spin gap state is also doubly degenerate which is consistent with
the level spectroscopy method to determine the boundary between the spin-gap
phase and the TLL phase.

Figure 4:  Ground state phase diagram of the present model. It includes the
ferrimagnetic, the spin gap and the TLL phases. On the line of $J_{5}=1$, the
dimer-monomer state is the exact ground state for $J_{2}>2$ and the tetramer-
dimer state is the exact ground state for $0.909<J_{2}<2$ takano .

##  IV Summary

Using the numerical exact diagonalization and the level spectroscopy analysis,
the $S=1/2$ distorted diamond spin chain suitable for the alumoklyuchevskite
is investigated. The obtained ground state phase diagram includes the
ferrimagnetic, the spin gap and the TLL phases. We believe that the upper TLL
state is attributed to nearly free $\mbox{\boldmath$S$}_{j,1}$ spins
(alumoklyuchevskite type), while the lower TLL state to nearly free
$\mbox{\boldmath$S$}_{j,3}$ spins (azurite type). More detailed analysis will
be a future problem.

## Acknowledgment

This work was partly supported by JSPS KAKENHI, Grant Numbers 16K05419,
16H01080 (J-Physics) and 18H04330 (J-Physics). A part of the computations was
performed using facilities of the Supercomputer Center, Institute for Solid
State Physics, University of Tokyo, and the Computer Room, Yukawa Institute
for Theoretical Physics, Kyoto University.

## References

  * (1) H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T. Tonegawa, K. Okamoto, T. Sakai, T. Juwai, and H. Ohta, Phys. Rev. Lett. 94, 227201 (2005).
  * (2) K. Okamoto T. Tonegawa, Y. Takahashi, and M. Kaburagi, J. Phys.: Condens. Matter 11, 10485 (1999).
  * (3) M. Fujihala, H. Koorikawa, S. Mitsuda, M. Hagihala, H. Morodomi, T. Kawae, A. Mitsudo, and K. Kindo, J. Phys. Soc. Jpn. 84, 073702 (2015).
  * (4) K. Morita, M. Fujihala, H. Koorikawa, T. Sugimoto, S. Sota, S. Mitsuda and T. Tohyama, Phys. Rev. B 95, 184412 (2017).
  * (5) M. Fujihala, H. Koorikawa1, S. Mitsuda, K. Morita, T. Tohyama, K. Tomiyasu, A. Koda, H. Okabe, S. Itoh, T. Yokoo, S. Ibuka, M. Tadokoro, M. Itoh, H. Sagayama, R. Kumai and Y. Murakami, Sci. Rep. 7, 16785 (2017)
  * (6) H. Jeschke, I. Opahle, H. Kandpal, Roser Valentí, H. Das, T. Saha-Dasgupta, O. Janson, H. Rosner, A. Brühl, B. Wolf, M.l Lang, J. Richter, S. Hu, X. Wang, R. Peters, T. Pruschke and A. Honecker, Phys. Rev. Lett. 106, 217201 (2011)
  * (7) K. Okamoto and K. Nomura, Phys. Lett. A 169, 433 (1992).
  * (8) K. Nomura and K. Okamoto, J. Phys. A: Math. Gen. 27, 5773 (1994).
  * (9) K. Kubo, K. Takano and H. Sakamoto, J. Phys: Cond. Matter 8, 6405 (1996)


# Fast Benchmarking of Accuracy vs. Training Time with Cyclic Learning Rates

Jacob Portes Davis Blalock Columbia University, MosaicML MosaicML
<EMAIL_ADDRESS><EMAIL_ADDRESS>Cory Stephenson Jonathan Frankle
MosaicML MosaicML<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding
author. Work done during a research internship at MosaicML. Code available at
https://github.com/jacobfulano/cyclic-learning-rate-schedules

###### Abstract

Benchmarking the tradeoff between neural network accuracy and training time is
computationally expensive. Here we show how a multiplicative cyclic learning
rate schedule can be used to construct a tradeoff curve in a single training
run. We generate cyclic tradeoff curves for combinations of training methods
such as Blurpool, Channels Last, Label Smoothing and MixUp, and highlight how
these cyclic tradeoff curves can be used to efficiently evaluate the effects
of algorithmic choices on network training.

In order to make meaningful improvements in neural network training
efficiency, ML practitioners must be able to compare between different choices
of network architectures, hyperparameters, and training algorithms. One
straightforward way to do this is to characterize the tradeoff between
accuracy and training time with a “tradeoff curve.” Tradeoff curves can be
generated by varying the length of training for each model configuration;
longer training runs take more time but tend to reach higher quality (Figure
1C). For a fixed model and task configuration, this method of generating
tradeoff curves is an estimate of the theoretical Pareto frontier, i.e. the
set of all of the best possible tradeoffs between training time and accuracy,
where any further attempt to improve one of these metrics worsens the
other.111We choose to use the language of “tradeoff curve” instead of Pareto
frontier with the understanding that the tradeoff curve is likely a lower
bound of the true Pareto frontier.

Accuracy-time tradeoff curves also allow us to make quantitative statements
about different hyperparameter choices and training algorithms (Leavitt, 2022;
Blakeney et al., 2022). For example, training the same network with Blurpool
(Zhang, 2019), Channels Last,222We refer here to the Channels Last memory
format in PyTorch https://pytorch.org/blog/tensor-memory-format-matters/ Label
Smoothing (Szegedy et al., 2016; Müller et al., 2019), and MixUp (Zhang et
al., 2017) shifts the tradeoff curve left along the time axis and up along the
accuracy axis, leading to a Pareto improvement in both accuracy and speedup
relative to the baseline (Figure 1C). Our main research question in this work
is: Given a fixed training configuration (e.g. architecture, objective
function, and regularization methods), how can we efficiently generate an
accuracy vs. training time tradeoff curve?

Figure 1: Accuracy-time tradeoff curves can be efficiently constructed with
cyclic learning rate schedules. (A) Multiplicative cyclic learning rate
schedules increase and decrease the learning rate with increasing periods
(top), while cosine decay learning rate schedules decrease monotonically after
linear warm up. Cyclic learning rate schedules lead to cycles in accuracy
during training (bottom). (B) Total wall clock time for generating a tradeoff
curve with 6 sample points at 16, 32, 64, 128, 256 and 512 epochs. Training 6
separate networks by scaling the cosine decay schedule (Standard, dark blue
and crimson) takes twice as long as training a single network for 512 epochs
with a cyclic learning rate schedule (Cyclic, light blue and crimson). Total
wall clock time is shown for baseline training (blue) and training with
Blurpool (BP), Channels Last (CL), Label Smoothing (LS), and MixUp (MX). (C)
Cyclic peaks in accuracy for cyclic learning rate schedules can be used to
generate accuracy vs. time tradeoff curves. This is shown for baseline
training of ResNet-50 on ImageNet (blue, dashed line) and for training
ResNet-50 with BP, CL, LS, and MX, (crimson, dashed line).

One approach might be to do multiple training runs with different training
durations and construct a tradeoff curve from the final accuracies. For
example, we could train a network for three different durations of 64, 128,
and 256 epochs, and fit a parameterized function to interpolate the rest of
the tradeoff curve. In this scenario, training would take 448 epochs total.
For the remainder of this paper, we refer to any tradeoff curve generated this
way as a standard tradeoff curve. Another approach might be to do one long
training run (e.g. for 256 epochs) and sample points on the tradeoff curve
during training. However, modern training best practices involve gradually
decaying the learning rate over the course of training. Many of the points
sampled in this procedure will therefore be at intermediate states where
training is not complete and the learning rate schedule is atypically high,
leading to lower-than-normal model quality (this can be seen in Figure 1A).
Neither of these approaches is satisfactory. The first approach is too
computationally expensive, and the second produces low-fidelity results that
are not indicative of standard performance.

Cyclic learning rate schedules were first proposed in Smith (2017), and
essentially increase and decrease the learning rate periodically during
training. One version of a cyclic schedule consists of repeated cosine decay
with periods that increase multiplicatively, which we call multiplicative
cyclic learning rate schedules (Figure 1A).333In the original paper, the
schedule is called SGDR, i.e. Stochastic Gradient Descent with Restarts. In
PyTorch, the scheduler is called CosineAnnealingWarmRestarts We suspected that
cyclic learning rate schedules could be used to efficiently construct
accuracy-time tradeoff curves, by storing validation accuracy peaks that track
cyclic learning rate troughs (Figure 1C).

In this paper, we show that multiplicative cyclic learning rate schedules can
be used to efficiently construct accuracy-time tradeoff curves for
benchmarking on ImageNet. In particular, we show that multiplicative cyclic
learning rate schedules can be used to construct an accuracy-time tradeoff
curve in a single training run, saving time and money. We then explore how
these curves change for regularization and speedup methods such as Blurpool,
Channels Last, Label Smoothing, and MixUp, both individually and in
combination. Although using this method necessarily changes the outcome of
training, we still find that it provides similar-fidelity information about
the relative efficacy of different training methods. This is particularly
useful for characterizing the accuracy-time tradeoff more efficiently, and for
making meaningful comparisons between choices of architectures,
hyperparameters and network training methods.

Related Work While Smith (2017) proposed various sawtooth-like schedules,
Loshchilov and Hutter (2016) showed that a schedule designed with cycles of a
cosine decay followed by a restart led to competitive accuracy on CIFAR-100.
In practice, cyclic learning rate schedules have been used for ensembling
methods (Huang et al., 2017) and Stochastic Weight Averaging (Izmailov et al.,
2018); they have also been the subject of theoretical work (Goujaud et al.,
2022). The use of Pareto frontiers in neural network research is surprisingly
limited, and is mostly restricted to the subfield of multi-objective
optimization (MOO) for multitask learning; recent examples include Lin et al.
(2020) and Navon et al. (2020). We extend this framework to evaluate the
effects of algorithmic choices on network training efficiency.

Results We trained ResNet-50 on ImageNet with (noncyclic) cosine decay
learning rate schedules for various training durations, and cyclic learning
rate schedules for 512 epochs. All runs began with a linear warm-up over 8
epochs to the maximum learning rate, and all learning rate schedules were
stepwise and not epoch-wise. For standard (noncyclic) training, runs were done
across 3 seeds for each data point at 16, 32, 64, 128, 256, and 512 epochs.
Cyclic runs were done across 3 seeds. All experiments report top-1 validation
accuracy.

Multiplicative Cyclic Learning Rate Schedules Efficiently Estimate Tradeoff
Curves How efficient might a cyclic learning rate schedule be? The cyclic
learning rate schedule in Loshchilov and Hutter (2016) is defined as
$\eta_{t}=\eta_{min}+\frac{1}{2}(\eta_{max}-\eta_{min})(1+\cos(\frac{T_{cur}}{T_{i}}\pi))$,
where $\eta_{t}$ is the learning rate at epoch $t$, $\eta_{max}$ and
$\eta_{min}$ are the maximum and minimum learning rates, period $T_{i}$ is the
number of epochs between two cycles, and $T_{cur}$ is the number of epochs
since the end of the previous cycle. For a multiplicative cyclic learning rate
schedule, the period for each cycle increases such that $T_{i}=rT_{i-1}$,
where $r$ is the multiplicative factor and $i$ denotes the cycle.

The ratio of the time to construct a standard tradeoff curve versus the time
to construct the cyclic tradeoff curve, where both tradeoff curves sample the
same exact points, can be expressed in the following formula:

$\textrm{speedup
ratio}=\frac{\sum_{i=0}^{n-1}T_{0}r^{i}}{T_{0}r^{n-1}}=\frac{1-r^{n}}{(1-r)r^{n-1}}$
(1)

where $T_{0}$ is the starting number of epochs, $r$ is the multiplicative
factor, and $n-1$ is the total number of sampled points (and number of
cycles). When $r=2$, this ratio is approximately 2, meaning that using a
cyclic learning rate schedule should in principle lead to $\sim$$2\times$
speedup (Figure 1B, Figure 3D).

We used multiplicative cyclic learning rate schedules with doubling periods,
and found that they approximated the baseline standard tradeoff curve quite
well. For the rest of this work, we refer to these multiplicative curves as
cyclic tradeoff curves. We designed the learning rate schedule to linearly
warm up for 8 epochs to a maximum value of 2.048, and then cycle with a cosine
decay for increasing periods of 8, 16, 32, 64, 128 and 256 epochs, for a total
of 512 epochs of training. We then stored maximum accuracy values at the end
of each cycle, and plotted an accuracy-time tradeoff curve. This cyclic
tradeoff curve matched our ResNet-50 ImageNet baseline standard tradeoff curve
(with a difference of $-0.084\%$ on average across three seeds for all points,
Figure 1C, Figure 2). This result held across choices of starting periods
(e.g. starting with $T_{0}=5$ epochs, then doubling to $T_{1}=10$, $T_{2}=20$
etc. epochs; data not shown). We found that cyclic learning rate schedules
with constant periods were not able to approximate tradeoff curves (Appendix
B, Figure S1).

Figure 2: Cyclic learning rate tradeoff curves are similar to standard
tradeoff curves across training methods (A) Blurpool (BP) tradeoff curve
constructed with the standard method (solid gray) and cyclic approach (dashed
gray) are similar. (B) The simple Channels Last (CL) modification on 3080 GPUs
leads to a leftward shift along the time axis. (C) Label Smoothing (LS) leads
to improvements later in training (green). (D) MixUp (MX) leads to accuracy
improvements later in training, and prevents overfitting (pink). (E) Tradeoff
curve for networks trained with both Blurpool and Channels Last (BP+CL)
(gold). (F) Tradeoff curves for Label Smoothing and MixUp (LS+MX) (magenta).

Cyclic Learning Rate Tradeoff Curves are Predictive Across Training Methods
There are a handful of demonstrated methods that modify the training procedure
to achieve better tradeoffs between the final model accuracy and the time to
train the model. Some of these methods include Blurpool (Zhang, 2019),
Channels Last, Label Smoothing (Szegedy et al., 2016; Müller et al., 2019),
and MixUp (Zhang et al., 2017). In our benchmarking work, we wanted to find
methods in the literature that led to clear Pareto improvements. We then asked
whether multiplicative cyclic learning rate schedules could be used to
construct competitive accuracy-time tradeoff curves for separate methods such
as Blurpool (BP), Channels Last (CL), Label Smoothing (LS), and MixUp (MX). We
found that using multiplicative cyclic learning rate schedules allowed us to
generate tradeoff curves of comparable quality to standard tradeoff curves,
but at substantial time savings ($\sim$$2\times$).

Cyclic learning rate tradeoff curves were able to match baseline standard
tradeoff curves for BP and CL almost exactly (BP exceeded by $0.07\%$, and CL
by $0.09\%$, on average across all points, Figures 2A-B). Note that the simple
CL modification on RTX 3080 GPUs led to a leftward shift of the tradeoff curve
along the time axis (solid cyan line); thus training a network for 512 epochs
took approximately 30 hours instead of 40 hours. BP also led to strong
improvements for shorter training times (e.g. 16 epochs). Cyclic learning rate
tradeoff curves were slightly below baseline standard tradeoff curves for LS
and MX, on average differing roughly by $-0.44\%$ for LS and $-0.39\%$ for MX
across all points (Figures 2C-D). Importantly, cyclic tradeoff curves for
these two cases still trended with the standard tradeoff curves.

Constructing a separate accuracy-time tradeoff curve for every training method
and/or combination of methods (for example LS + MX) is also combinatorially
expensive. We therefore asked whether cyclic learning rate schedules could
accurately construct tradeoff curves for combinations of training methods. We
trained ResNet-50 with both BP and CL (BP + CL, Figure 2E), and found that the
cyclic learning rate schedule tradeoff curve matched the standard tradeoff
curve (exceeded by $0.08\%$, on average across all points).

We then trained ResNet-50 with the combined methods of LS and MX (LS + MX,
Figure 2F), and found that cyclic tradeoff curves trended with standard
tradeoff curves, but were slightly lower (by $-0.46\%$ on average). Finally,
we trained ResNet-50 with the combined methods of BP, CL, LS and MX
(BL+CL+LS+MX, Figure 1C), and found that cyclic tradeoff curves trended with
the standard tradeoff curves (crimson dashed line, lower by $-0.24\%$ on
average across all points).

When comparing training methods such as LS and MX, can we really trust that
improvements in the Pareto frontier over the baseline will be matched by a
tradeoff curve constructed with a cyclic learning rate? One way to assess this
is simply to compare the relative improvements over standard or cyclic
baseline for each method. We found that relative improvements held between
standard and cyclic tradeoff curves (Figure 3A, Figure S2).

For example, when the baseline standard tradeoff curve is subtracted from the
Blurpool standard tradeoff curve, it is clear that Blurpool provides a $2\%$
accuracy boost early on in training, but provides no accuracy boost after 256
epochs (Figure 3A, gray solid line). Subtracting the baseline cyclic tradeoff
curve from the Blurpool cyclic tradeoff curve shows the same trend (Figure 3A,
gray dashed line).

We then plotted the total wall clock time for standard tradeoff curves and
cyclic tradeoff curves with six sampled points at 16, 32, 64, 128, 256 and 512
epochs (Figure 3D). As expected, standard tradeoff total wall clock time was
almost twice as long as cyclic tradeoff curve total wall clock time across all
training methods and combinations of methods (see Equation 1).

Figure 3: Relative improvements hold between standard and cyclic tradeoff
curves (A) Relative improvements over baseline for each method can be plotted
by subtracting the baseline standard/cyclic tradeoff curve from the
standard/cyclic tradeoff curve of a particular method. Relative improvements
over baseline for standard tradeoff curves (solid lines) are captured by
cyclic tradeoff curves (dashed lines). (A) Blurpool and Channels Last (B)
Label Smoothing and MixUp (C) Combinations of methods, including Blurpool +
Channels Last (gold), Label Smoothing + Mixup (magenta), and all four methods
combined (crimson) (D) Total wall clock time (WCT) for standard tradeoff
curves with six sampled points at 16, 32, 64, 128, 256 and 512 epochs is
almost twice as large as cyclic tradeoff curves constructed with the same six
sampled points.

Discussion We have shown here that multiplicative cyclic learning rate
schedules can be used to construct an accuracy-time tradeoff curve in a single
training run, saving time and money. This is a practical means of assessing
changes in tradeoff curves caused by different architectures, hyperparameters,
and training methods such as Label Smoothing and MixUp. We hope that this is a
small step towards framing training choices in terms of tradeoff curves and
Pareto improvements.

It is somewhat surprising that multiplicative cyclic learning rates are able
to approximate Pareto frontiers when constant period cyclic learning rates do
not (Figure S1). Why does this work, particularly for increasing periods? One
possibility might have to do with different phases of learning (Golatkar et
al., 2019; Frankle et al., 2020). For example, rapid changes in learning rate
might help find better minima early in training (10-20 epochs), while very
slow changes in learning rate might help find minima in already “good” basins
late in training ($>$90 epochs). As training continues, it might become more
difficult to find better minima, implying that it might be beneficial to take
longer for each new cycle.

Cyclic learning rate schedules were popularized by Fast.ai (Howard and Gugger,
2020), but are not widely used in the research community. This is likely due
to the fact that, with current best practices on benchmarks like ImageNet,
cyclic learning rate schedules do not necessarily lead to improved accuracy
when compared to cosine decay schedules.

In many instances, the cyclic tradeoff curve underestimated the standard
tradeoff curve by a margin of 0.5% validation accuracy. There are many
possible variations to our approach that might improve this gap. One idea is
to allow the entire cyclic learning rate schedule to decay over the course of
learning; this is one of many schedule variations explored by the literature
(Schmidt et al., 2021). Another idea would be to apply a cyclic learning rate
schedule to other hyperparameters such as weight decay, batch size, and
gradient clipping (in line with Smith (2022)).

Our results primarily hold for convolutional networks such as ResNet-50 and
ResNet-101 (data not shown) applied to computer vision tasks such as CIFAR-10
(data not shown) and ImageNet. Can a similar approach be applied to other
architectures such as transformers (that take significantly longer to train
than ResNets) and to other domains such as NLP? We hope to explore this in
future work.

Acknowledgements An earlier version of this work appeared as a blog post
https://www.mosaicml.com/blog/efficiently-estimating-pareto-frontiers. Thanks
to Abhinav Venigala, Matthew Leavitt, and Kobie Crawford for fruitful
discussions.

## References

  * Blakeney et al. (2022) C. Blakeney, J. Z. Forde, J. Frankle, Z. Zong, and M. L. Leavitt.  Reduce, reuse, recycle: Improving training efficiency with distillation.  _arXiv preprint arXiv:2211.00683_ , 2022.
  * Frankle et al. (2020) J. Frankle, D. J. Schwab, and A. S. Morcos.  The early phase of neural network training.  _arXiv preprint arXiv:2002.10365_ , 2020.
  * Golatkar et al. (2019) A. S. Golatkar, A. Achille, and S. Soatto.  Time matters in regularizing deep networks: Weight decay and data augmentation affect early learning dynamics, matter little near convergence.  _Advances in Neural Information Processing Systems_ , 32, 2019.
  * Goujaud et al. (2022) B. Goujaud, D. Scieur, A. Dieuleveut, A. B. Taylor, and F. Pedregosa.  Super-acceleration with cyclical step-sizes.  In _International Conference on Artificial Intelligence and Statistics_ , pages 3028–3065. PMLR, 2022.
  * Howard and Gugger (2020) J. Howard and S. Gugger.  Fastai: a layered api for deep learning.  _Information_ , 11(2):108, 2020.
  * Huang et al. (2017) G. Huang, Y. Li, G. Pleiss, Z. Liu, J. E. Hopcroft, and K. Q. Weinberger.  Snapshot ensembles: Train 1, get m for free.  _arXiv preprint arXiv:1704.00109_ , 2017.
  * Izmailov et al. (2018) P. Izmailov, D. Podoprikhin, T. Garipov, D. Vetrov, and A. G. Wilson.  Averaging weights leads to wider optima and better generalization.  _arXiv preprint arXiv:1803.05407_ , 2018.
  * Leavitt (2022) M. Leavitt.  MosaicML ResNet Deep Dive, 2022.  URL https://www.mosaicml.com/blog/mosaic-resnet-deep-dive.
  * Lin et al. (2020) X. Lin, Z. Yang, Q. Zhang, and S. Kwong.  Controllable pareto multi-task learning.  _arXiv preprint arXiv:2010.06313_ , 2020.
  * Loshchilov and Hutter (2016) I. Loshchilov and F. Hutter.  Sgdr: Stochastic gradient descent with warm restarts.  _arXiv preprint arXiv:1608.03983_ , 2016.
  * Loshchilov and Hutter (2017) I. Loshchilov and F. Hutter.  Decoupled weight decay regularization.  _arXiv preprint arXiv:1711.05101_ , 2017.
  * Müller et al. (2019) R. Müller, S. Kornblith, and G. E. Hinton.  When does label smoothing help?  _Advances in neural information processing systems_ , 32, 2019.
  * Navon et al. (2020) A. Navon, A. Shamsian, G. Chechik, and E. Fetaya.  Learning the pareto front with hypernetworks.  _arXiv preprint arXiv:2010.04104_ , 2020.
  * Schmidt et al. (2021) R. M. Schmidt, F. Schneider, and P. Hennig.  Descending through a crowded valley-benchmarking deep learning optimizers.  In _International Conference on Machine Learning_ , pages 9367–9376. PMLR, 2021.
  * Smith (2017) L. N. Smith.  Cyclical learning rates for training neural networks.  In _2017 IEEE winter conference on applications of computer vision (WACV)_ , pages 464–472. IEEE, 2017.
  * Smith (2022) L. N. Smith.  General cyclical training of neural networks.  _arXiv preprint arXiv:2202.08835_ , 2022.
  * Szegedy et al. (2016) C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna.  Rethinking the inception architecture for computer vision.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 2818–2826, 2016.
  * Wu et al. (2019) Y. Wu, L. Liu, J. Bae, K.-H. Chow, A. Iyengar, C. Pu, W. Wei, L. Yu, and Q. Zhang.  Demystifying learning rate policies for high accuracy training of deep neural networks.  In _2019 IEEE International conference on big data (Big Data)_ , pages 1971–1980. IEEE, 2019.
  * Zhang et al. (2017) H. Zhang, M. Cisse, Y. N. Dauphin, and D. Lopez-Paz.  mixup: Beyond empirical risk minimization.  _arXiv preprint arXiv:1710.09412_ , 2017.
  * Zhang (2019) R. Zhang.  Making convolutional networks shift-invariant again.  In _International conference on machine learning_ , pages 7324–7334. PMLR, 2019.

##  Appendix A Experimental Details

We trained ResNet-50 on ImageNet with noncyclic and cyclic learning rate
schedules for various training durations. All ImageNet experiments used 8 RTX
3080s with mixed precision training, 8 dataloader workers per GPU, a batch
size of 2048, and a gradient accumulation of 4. We used SGDW [Loshchilov and
Hutter, 2017] with a maximum learning rate of 2.048, momentum of 0.875, and
weight decay 5e-4 (these values were found to be optimal in a hyperparameter
sweep; the learning rate of 2.048 was chosen to be proportional to the batch
size). All runs began with a linear warm-up over 8 epochs to the maximum
learning rate, and all learning rate schedules were stepwise and not epoch-
wise. For standard (noncyclic) training, runs were done across 3 seeds for
each data point at 16, 32, 64, 128, 256, and 512 epochs, and for 1 seed for
other data points. Cyclic runs were done across 3 seeds. Experiments using
cosine decay learning rate schedules were run using the CosineAnnealingLR
scheduler in PyTorch, while cyclic learning rate schedules were implemented
using the PyTorch CosineAnnealingWarmRestarts scheduler. Training methods such
as Blurpool Zhang [2019], Channels Last, Label Smoothing Szegedy et al.
[2016], Müller et al. [2019] and MixUp Zhang et al. [2017] were implemented
using the MosaicML Composer library for PyTorch
https://github.com/mosaicml/composer.

##  Appendix B Cyclic Learning Rate Schedules with Constant Periods

We used cyclic learning rate schedules with constant periods to construct
accuracy-time tradeoff curves, and examined the effect of varying the periods
($T=10$, 20, 30, 40, and 50 epochs). We found that cyclic learning rate
schedules with shorter periods led to suboptimal tradeoff curves, while cyclic
learning rates with longer (constant) periods led to curves that were on the
standard tradeoff curve but were not well sampled (Figure S1A-B).

Historically, there was a flurry of work exploring alternative learning rate
schedules from exponential decays to trapezoids; cyclic learning rates fall on
the more exotic end of the spectrum. We found mixed results for other cyclic
schedules such as sawtooth/triangle schedules as in Smith [2017]. For a review
of different schedule shapes, see Table 3 of Appendix A in Schmidt et al.
[2021] as well as Wu et al. [2019].

Figure S1: Constructing accuracy-training time tradeoff curves with constant
and multiplicative cyclic learning rate schedules (A) Cyclic learning rate
schedule with a constant period of 25 epochs (top) leads to a validation
accuracy that is also cyclic (bottom). (B) Accuracy-training time tradeoff
curves can be constructed with cyclic learning rate schedules of different
constant periods, for example 10 (gray), 20 (cyan), 25 (crimson), 40 (gold)
and 50 epochs (magenta). (C) Cyclic learning rate schedule with multiplicative
periods of 16, 32, 64 and 128 epochs (top) also lead to a validation accuracy
that is cyclic (bottom). (D) An accuracy-time tradeoff curve can be
constructed with a multiplicative cyclic learning rate schedule (dashed line).
This tradeoff curve closely resembles the standard tradeoff curve constructed
from individual training runs with scaled cosine decay schedules (solid line).
Figure S2: Relative ordering of tradeoff curves between training methods is
preserved. (A) Standard tradeoff curves across training techniques. (B)
Accuracy vs. training time tradeoff curves using multiplicative cyclic
learning rate schedules across training methods. Same data as in Figures 1 and
2.