# Improved Limits on an Exotic Spin- and Velocity-Dependent Interaction at the
Micrometer Scale with an Ensemble-NV-Diamond Magnetometer

Diguang Wu  CAS Key Laboratory of Microscale Magnetic Resonance and School of
Physical Sciences, University of Science and Technology of China, Hefei
230026, China  CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
Hefei National Laboratory, University of Science and Technology of China,
Hefei 230088, China  Hang Liang  CAS Key Laboratory of Microscale Magnetic
Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Man Jiao<EMAIL_ADDRESS>CAS Key Laboratory of
Microscale Magnetic Resonance and School of Physical Sciences, University of
Science and Technology of China, Hefei 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei 230026, China  Yi-Fu Cai  CAS Key Laboratory
for Researches in Galaxies and Cosmology, School of Astronomy and Space
Science, University of Science and Technology of China, Hefei, Anhui 230026,
China  Department of Astronomy, School of Physical Sciences, University of
Science and Technology of China, Hefei, Anhui 230026, China  Chang-Kui Duan
CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical
Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China  Ya Wang
CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical
Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China  Hefei
National Laboratory, University of Science and Technology of China, Hefei
230088, China  Xing Rong<EMAIL_ADDRESS>CAS Key Laboratory of Microscale
Magnetic Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Hefei National Laboratory, University of Science
and Technology of China, Hefei 230088, China  Jiangfeng Du<EMAIL_ADDRESS>CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical
Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China  Hefei
National Laboratory, University of Science and Technology of China, Hefei
230088, China

###### Abstract

Searching for exotic interactions provides a path for exploring new particles
beyond the standard model. Here, we used an ensemble-NV-diamond magnetometer
to search for an exotic spin- and velocity-dependent interaction between
polarized electron spins and unpolarized nucleons at the micrometer scale. A
thin layer of nitrogen-vacancy electronic spin ensemble in diamond is utilized
as both the solid-state spin quantum sensor and the polarized electron source,
and a vibrating lead sphere serves as the moving unpolarized nucleon source.
The exotic interaction is searched by detecting the possible effective
magnetic field induced by the moving unpolarized nucleon source using the
ensemble-NV-diamond magnetometer. Our result establishes new bounds for the
coupling parameter $f_{\perp}$ within the force range from 5 to 400 $\upmu$m.
The upper limit of the coupling parameter at 100 $\upmu$m is $\lvert
f_{\perp}\rvert\leq 1.1\times 10^{-11}$, which is 3 orders of magnitude more
stringent than the previous constraint. This result shows that NV ensemble can
be a promising platform to search for hypothetical particles beyond the
standard model.

Exotic long-range spin-dependent interactions mediated by new bosons are
related to potential solutions to various mysteries in fundamental physics
[1]. Axion is such a Nambu-Goldstone boson of the spontaneously broken Peccei-
Quinn symmetry [2, 3] and is initially introduced to solve the strong _CP_
problem in QCD [4, 5, 6]. The existence of the axion or axionlike particles is
a prominent explanation of several unsolved fundamental conundrums, such as
the composition of the dark matter [7], the origin of the dark energy [8, 9],
and the hierarchy problem [10]. Exotic long-range spin-dependent interactions
mediated by axion were proposed to search for new physics [11]. Subsequently,
these exotic interactions were extended to sixteen types of potentials
involving the spin polarization and momenta of the interacting particles [12],
and revisited in [13]. These exotic interactions can be not only mediated by
pseudoscalar bosons such as axions and axionlike particles but also mediated
by axial-vector bosons such as dark photons and _Z ′_ bosons [12, 14].

Recent progress of experimental searching for exotic interactions was advanced
with a wide range of precision measurement techniques [1] utilizing atomic
magnetometer [15, 16, 17, 18, 19], spin-based amplifier [20, 21],
spectroscopic measurements [22, 23], cantilever [24], and trapped ions [25].
These advanced precision measurements in the laboratory are powerful tools for
tests of fundamental physics as complementary to astronomical observation. The
single nitrogen-vacancy (NV) center in diamonds has emerged as a solid-state
spin quantum sensor to search for exotic spin-dependent interactions at the
micrometer scale [26, 27, 28, 29], taking advantage of the ability enabling
close proximity between the sensor and the source. Recently, a nonzero result
has been reported in an experimental searching for exotic interactions [29].
It is necessary to perform experiments with improved sensitivities to
scrutinize whether the origin of the nonzero result is physical or
instrumental.

Figure 1: (a) Schematic experimental parameters. A lead sphere denoted as _M_
with the radius being _R_ vibrates parallel to the diamond surface. _d_ is the
distance between the bottom of the sphere and the surface of the diamond. _h_
stands for the thickness of the NV layer. The direction of the NV axis is
denoted as $\hat{n}$. $\hat{v}$ is the unit relative velocity vector between
the lead sphere and the NV layer. (b) The scheme of the experimental setup
based on the ensemble-NV-diamond magnetometer. NV centers are illuminated by a
532 nm laser which passes through two half-wave plates (HWP) and a polarizing
beam splitter (PBS). The CPC is the compound parabolic concentrator, LPF is
the long-pass filter, and PD is the photodiode. FM stands for the modulation
frequency of the microwave from the synthesizer (Syn.). The output magnetic
field measurement result of the magnetometer can be obtained by demodulating
PD signal with the first lock-in amplifier (LIA1). The nucleon source is
attached to a piezoelectric bender, which vibrates at the frequency $f_{M}$.
The output of the magnetometer is further demodulated by the second lock-in
amplifier (LIA2) to obtain the magnetic field component at the frequency
$f_{M}$. The operations in each commercial LIA are displayed schematically in
the figure.

In this work, we develop a technique based on an ensemble-NV-diamond
magnetometer to search for an exotic spin- and velocity-dependent interaction
between polarized electron spins and unpolarized nucleons. An ensemble of NV
centers in diamond serves as both polarized electron sources and sensitive
sensors. The extension from single NV centers to NV ensemble provides better
magnetic detection sensitivity, which makes the ensemble-NV-diamond
magnetometer a promising platform for the searches of exotic interactions [30,
31]. Our technique also takes the advantage that, by averaging the magnetic
field sensed by each NV center, the NV ensemble is insensitive to the effect
of the diamagnetism of the material of the nucleon source [31]. Here, we focus
on the exotic spin- and velocity-dependent interaction between an electron
spin and a nucleon ($V_{4,5}$ in Ref. [12]’s notation), described as

$V_{4,5}=-f_{\perp}\frac{\hbar^{2}}{8\uppi
m_{e}\mathrm{c}}{\hat{\sigma}\cdot}(\vec{v}\times\hat{r})\left(\frac{1}{\lambda
r}+\frac{1}{r^{2}}\right)e^{-r/\lambda},$ (1)

where $f_{\perp}$ is the dimensionless interaction coupling parameter, $\hbar$
is the reduced Planck’s constant, $m_{e}$ is the mass of the electron, $c$ is
the speed of light in vacuum, $\hat{\sigma}$ is the Pauli vector of the
electron spin. $\vec{v}$ is the relative velocity vector, and
$\lambda=\hbar/(m_{b}c)$ is the force range with $m_{b}$ being the mass of the
hypothetical boson. $\vec{r}$ is the displacement vector between the electron
and the nucleon, $r=|\vec{r}|$ and $\hat{r}={\vec{r}}/{r}$. The exotic spin-
and velocity-dependent interaction can be mediated by the axion or the _Z ′_
boson [14, 12]. The effective Lagrangian is
$\mathcal{L}_{a}=-a\sum_{\psi}\bar{\psi}\gamma^{\mu}(g_{S}^{\psi}+\gamma_{5}g_{P}^{\psi})\psi$
for axion, or
$\mathcal{L}_{Z^{\prime}}=Z^{\prime}_{\mu}\sum_{\psi}\bar{\psi}\gamma^{\mu}(g_{V}^{\psi}+\gamma_{5}g_{A}^{\psi})\psi$
for _Z ′_ boson, where $\psi$ stands for the fermion field, $\gamma^{\mu}$ and
$\gamma_{5}$ are Dirac matrices. For axion exchange,
$f_{\perp}=g_{s}^{e}g_{s}^{N}/2$, where $g_{s}^{e}\ (g_{s}^{N})$ denotes the
scalar electron (nucleon) coupling. In the case of _Z ′_ boson,
$f_{\perp}=g_{v}^{e}g_{v}^{N}/2$, where $g_{v}^{e}\ (g_{v}^{N})$ is the vector
electron (nucleon) coupling [14, 12]. The exotic interaction induces an
effective magnetic field sensed by the electron spin

$\displaystyle B_{\mathrm{eff}}(r)=-f_{\perp}$
$\displaystyle\frac{\hbar}{4\uppi
m_{e}c\gamma_{e}}\hat{n}\cdot(\vec{v}\times\hat{r})\left(\frac{1}{\lambda
r}+\frac{1}{r^{2}}\right)e^{-r/\lambda},$ (2)

where $\gamma_{e}=2\uppi\times 28$ GHz/T is the gyromagnetic ratio of the NV
electron spin.

Figure 1(a) shows the geometric parameters of the spin sensor and the nucleon
source. A 23-$\upmu$m-thick layer of NV ensemble at the surface of the diamond
is used as the spin sensor. The size of the layer is $660\times 661\times 23\
\upmu$m3. The substrate is a single crystal $\langle 100\rangle$ orientated
diamond with electronic grade high purity and parts-per-billion nitrogen
density. The angle between the NV axis and the surface of the diamond is
$\theta=\arcsin(1/\sqrt{3})$. A lead sphere with the diameter being
$2R=897(3)\ \upmu$m is used as the unpolarized nucleon source. The nucleon
density is $6.8\times 10^{30}\ \rm m^{-3}$ [18]. The lead sphere is attached
to the end of a piezoelectric bender, which can vibrate in the direction
parallel to the diamond surface. The NV axis is perpendicular to the velocity
vector of the lead sphere. The amplitude of the vibration is $A=538(3)$ nm and
the frequency is $f_{M}=941$ Hz. The distance between the bottom of the sphere
and the surface of the diamond is $d=5.0(5)\ \upmu$m. The experimental setup
based on an ensemble NV magnetometer is shown in Fig. 1(b). The NV ensemble is
polarized by a 532-nm laser with a diameter being 0.8 mm. The fluorescence
from the NV ensemble is collected by a compound parabolic concentrator and
converted to the photocurrent by a photodetector (PD). A long-pass filter
(LPF) is used to filter out the laser from the fluorescence. A polarizing beam
splitter (PBS) and another PD are used to monitor the power fluctuation of the
laser. The laser polarization direction and the split ratio of PBS are
adjusted by the two half-wave plates (HWP).

The NV center is featured by a ground electronic spin triplet state with
substates $|m_{s}=0\rangle$ and $|m_{s}=\pm 1\rangle$ [32]. The optical spin
initialization into $|m_{s}=0\rangle$ and spin-state-dependent fluorescence
contrast lead to the optically detected magnetic resonance [33]. An ensemble-
NV-diamond magnetometer based on the continuous-wave (CW) method [31, 34, 35]
is used to search for the exotic interaction. The optical polarization, the
microwave excitation, and the fluorescence readout occur simultaneously in
this CW magnetometry. A static bias magnetic field along the axis of NV
centers is applied to remove the degeneracy of $|m_{s}=\pm 1\rangle$. The
fluorescence of the NV ensemble varies when a change in the local magnetic
field shifts the resonance frequency. To avoid the flicker noise, the
frequency of the microwave is modulated with modulation frequency of 87.975
kHz. The magnetic field sensed by the NV ensemble is encoded in the amplitude
of the fluorescence intensity at the modulation frequency. The signals from
PDs are demodulated by the first lock-in amplifier [LIA1 in Fig.1(b)] with the
time constant being 8 $\upmu$s. Since the effects from laser power instability
are common for signals from both PDs, these effects are suppressed by
differential measurement [36]. The magnetic sensitivity of the ensemble-NV-
diamond magnetometer is shown in Fig. 2. The coefficient between the
magnetometer output voltage signal and the magnetic field is $\eta=47.2\pm
0.2\ \upmu$T/V. The magnetic sensitivity is $1.6\ \rm nT/\sqrt{Hz}$ within the
frequency range from 0.9 to 1 kHz (see Appendix B for details).

Figure 2: The magnetic sensitivity of the ensemble-NV-diamond magnetometer.
The magnetic sensitivity is $1.6\ \rm nT/\sqrt{Hz}$ within the frequency range
from 0.9 to 1 kHz. The vibration frequency $f_{M}=941$ Hz is marked with a red
arrow.

The effective magnetic field $b_{\mathrm{eff}}$ sensed by the NV ensemble can
be derived from integrating $B_{\mathrm{eff}}(r)$ over all nucleons and
electron spins [31]. Since the nucleon source vibrates at a fixed frequency
$f_{M}$, the velocity of the sphere can be represented as $v(t)=2\pi
f_{M}A\sin{(2\pi f_{M}t)}$, where _A_ is the amplitude of vibration. The
Fourier transform of the effective magnetic field is

$\displaystyle b_{\mathrm{eff}}(t)=\sum_{n=1}^{\infty}b_{n}\sin(2\pi
nf_{M}t).$ (3)

where $b_{n}$ is the amplitude of the $n$th harmonic. Here, we focus on the
measurement of the amplitude of the first-order harmonic of the velocity-
dependent effective magnetic field $b_{1}$. The output signal of LIA1 is
demodulated by a second lock-in amplifier [LIA2 in Fig.1(b)]. The
experimentally calibrated phase delay between the output of magnetometer and
the vibration of nucleon source at $f_{M}$ is $\phi=51\pm 4^{\circ}$. The
method of the phase calibration is the same as Ref. [31]. The time constant of
LIA2 is 71 ms. The quadrature component of the demodulated signal from LIA2
corresponds to $b_{1}$ (see Appendix C for details).

A typical time trace of the experimentally measured quadrature part of the
output of LIA2 is shown in Fig. 3(a) with the time duration being 600 s. The
measurement was carried out for 64 h to reduce the statistical uncertainty. A
histogram of the measured effective magnetic fields collected for 64 h is
shown in Fig. 3(b). The fit shows that the result obeys the Gaussian
distribution. The measured first-order harmonic component of the effective
magnetic fields is $b_{1}^{\mathrm{exp}}=1.0\pm 3.2$ pT and the corresponding
coupling parameter $f_{\perp}$ at $\lambda=100\ \upmu$m is $(1.5\pm 4.9)\times
10^{-12}$. The result that the mean value of the measured magnetic field is
less than the standard error shows no evidence of the exotic interaction in
this experiment. Therefore, new limits of the coupling parameter $f_{\perp}$
can be obtained.

Figure 3: (a) The black line shows the time trace of the experimentally
measured effective magnetic fields for 600 s. (b) The histogram of the
experimentally measured effective magnetic fields for 64-h collection. The red
line indicates the fit to a Gaussian distribution. The total counts are 0.2 M,
the mean of the result is $1.0$ pT, and the standard error of the mean is
$3.2$ pT.

Table 1 summarizes the systematic errors of our experiment. The uncertainties
of experimental parameters and the corrections to the coupling parameter
$f_{\perp}$ with $\lambda=100\ \upmu$m are provided. The main sources of the
systematic errors include the uncertainties of the diameter of the sphere
$2R$, the thickness of the NV layer $h$, the vibration amplitude $A$, the
angle $\alpha$ between the direction of the velocity and the $\hat{y}$
direction, the distance $d$ between the bottom of the sphere and the surface
of the diamond, the phase of the reference signal $\phi$, nucleon density, the
misalignment between the center of the sphere and the center of NV centers in
the _x-y_ plane, and the coefficient $\eta$ between the magnetometer output
voltage signal and the magnetic field (see Appendix E for details). By
quadrature summing the items of all systematic errors, the total systematic
error for the coupling parameter is $\pm 2\times 10^{-14}$ with $\lambda=100\
\upmu$m. Taking both statistical and systematic errors into account, the bound
for the coupling parameter with $\lambda=100\ \upmu$m is $\lvert
f_{\perp}\rvert\leq 1.1\times 10^{-11}$ with a 95% confidence level. The other
value of the bound with the different force range can be obtained with the
same method.

Table 1: Summary of systematic errors. The corrections to $f_{\perp}$ with $\lambda=100\ \upmu$m are listed. Parameter | Value | $\Delta f_{\perp}(10^{-13})$
---|---|---
Diameter 2R |  $897\pm 3\ \upmu$m | $\pm 0.05$
Thickness $h$ |  $23\pm 1\upmu$m | $\pm 0.08$
Amplitude $A$ |  $538\pm 3$ nm | $\pm 0.09$
Angle $\alpha$ | $0\pm 3^{\circ}$ | ${}^{+0.02}_{-0.00}$
Distance $d$ |  $5.0\pm 0.5\ \upmu$m | $\pm 0.08$
Phase $\phi$ | $51\pm 4^{\circ}$ | ${}_{-0.00}^{+0.03}$
Nucleon density | $(682\pm 2)\times 10^{28}\ \mathrm{m^{-3}}$ | $\pm 0.05$
Centroid deviation | $0\pm 10\ \upmu$m | ${}^{+0.09}_{-0.08}$
in _x-y_ plane
Coefficient $\eta$ |  $47.2\pm 0.2\ \upmu$T/V | $\pm 0.07$
Obtained $f_{\perp}$ | $15.4\times 10^{-13}$ |  $\pm 49.3$ (statistic)
$\pm 0.2$ (systematic)

Figure 4 shows the upper bounds of the coupling parameter $f_{\perp}$ of the
exotic spin- and velocity-dependent interaction established by this work
together with previous experimental constraints [15, 24]. For the force range
$\lambda<5\ \upmu$m, the limits of the coupling parameter $f_{\perp}$ were
obtained by Ding et al. with a cantilever to detect the force between a
vibrating gold sphere and a microfabricated magnetic structure [24]. The
constraints for the force range $\lambda>400\ \upmu$m were set by Kim et al.
with a spin-exchange-relaxation-free magnetometer to detect the possible
effective magnetic field induced by a rotating BGO crystal [15]. For the force
range from 5 to $400\ \upmu$m, improved experimental bounds are established by
our experiment. The upper bound for the force range $\lambda=100\ \upmu$m is
$\lvert f_{\perp}\rvert\leq 1.1\times 10^{-11}$, which is more than 3 orders
of magnitude more stringent than the previous laboratory constraint [24]. The
constraints of stellar cooling are derived by a combination of $g_{s}^{e}$
from the stellar cooling [37] and $g_{s}^{N}$ from experimental tests of the
hypothetical Yukawa interactions [38, 39, 40]. Notice that astrophysical
bounds on $g_{s}^{e}$ may suffer from the uncertainties, such as the accuracy
of stellar models [41], degree of model specificity [42], and invalidation due
to chameleon mechanism [43].

Figure 4: The upper bounds of the coupling parameter $f_{\perp}$ as a function
of the force range $\lambda$ and the boson mass $m_{b}$. Black lines are upper
bounds obtained from previous experiments [15, 24]. The blue dot denotes the
nonzero result reported in Ref.[29]. The red line is the upper limit
established by our experiment. The gray-filled area shows the excluded values.
More stringent experimental bounds are obtained by our experiment within the
force range from 5 to $400\ \upmu$m. The dashed line is set by a combination
of stellar cooling rate observation and tests of the hypothetical Yukawa
interactions [37, 38, 39, 40].

In summary, we conducted an experimental search for an exotic spin- and
velocity-dependent interaction between polarized electron spins and
unpolarized nucleons with an ensemble-NV-diamond magnetometer. Improved
constraints of the coupling parameter $f_{\perp}$ have been established within
the force range from 5 to $400\ \upmu$m. This work is also an experimental
test of the nonzero result in Ref. [29]. Different from the single NV setup in
Ref. [29], by averaging the magnetic field sensed by each NV center, our setup
is insensitive to the effect due to the diamagnetism of the nucleon source
[31]. We carefully synchronized the ensemble-NV-diamond magnetometer and the
vibration of the nucleon source using a calibration signal as in Ref. [31].
This setup is more sensitive than the single NV magnetometer in Ref. [29] with
the force range larger than 8 $\upmu$m. Our result suggests that the nonzero
result in Ref. [29] may come from instrumental artifacts (seeAppendix F for
details). In the future, more stringent constraints can be obtained by further
improving the sensitivity of the ensemble-NV-diamond magnetometer. The
dynamical decoupling technology can be utilized to optimize the dephasing time
[44, 45]. The readout fidelity can be improved by the near-infrared absorption
readout [46]. Besides, coating the diamond surface is beneficial to increase
the fluorescence collection efficiency [47]. The nitrogen delta-doping
technique can be used to fabricate the shallow NV centers with a nanoscale
thickness [48], enabling searching for the exotic interactions with a shorter
force range. Besides, the method based on an ensemble-NV-diamond magnetometer
can be utilized to explore other types of exotic interactions with additional
functional upgradation in the future. Furthermore, high-density NV ensembles
can be used not only as spin sensors but also as spin sources with well-
controlled spin polarization. This work shows that the NV ensemble is a
promising platform for searching for exotic spin-dependent interactions beyond
the standard model.

This work was supported by the Chinese Academy of Sciences (Grants No.
XDC07000000, No. GJJSTD20200001, No. QYZDY-SSW-SLH004, No. QYZDB-SSWSLH005),
the National Key R&D Program of China (Grant No. 2018YFA0306600, No.
2021YFC2203100), Anhui Initiative in Quantum Information Technologies (Grant
No. AHY050000), Innovation Program for Quantum Science and Technology
(2021ZD0302200), NSFC (11961131007, 11653002, 12150010,12205290,12261160569),
China Postdoctoral Science Foundation (2022TQ0330), and the Hefei
Comprehensive National Science Center. X. R. thanks the Youth Innovation
Promotion Association of Chinese Academy of Sciences for the support. Y. F.
C., Y. W., and M. J. thank the Fundamental Research Funds for Central
Universities. Y. F. C. is supported in part by the CSC Innovation Talent
Funds, by the USTC Fellowship for International Cooperation, and by the USTC
Research Funds of the Double First-Class Initiative. This work was partially
carried out at the USTC Center for Micro and Nanoscale Research and
Fabrication.

D.G.W. and H.L. contributed equally to this work.

##  Appendix A Devices in this experimental setup

Figure 5 shows the scheme of the experimental setup based on an Ensemble-NV-
Diamond Magnetometer, which is the same as Fig.1(b) in the main text. The
manufacturers and models of devices used in this setup are shown in Table 2.

Figure 5: The scheme of the experimental setup based on the Ensemble-NV-Diamond magnetometer. The magnetometer is composed of diamond, half-wave plates (HWP), polarizing beam splitter (PBS), compound parabolic concentrator (CPC), the long-pass filter (LPF), photodiode (PD), synthesizer(Syn.), amplifier (amp.), and the first commercial lock-in amplifier (LIA1). A piezoelectric bender drived by the second commercial lock-in amplifier (LIA2) is used to create the relative velocity term in the exotic velocity-dependent interaction. LIA2 is used to extract the magnitude of the first-order harmonic component of the effective magnetic field. The operations in each commercial LIA are displayed schematically in the figure. Table 2: List of the detailed information about the devices in our experiment Instrument | Manufacturer | Model
---|---|---
Lock-in Amplifier 1 and 2 | Zurich Instruments | HF2LI
Laser | Cobolt | 0532-05-01-1500-700
Synthesizer | National Instruments | FSW-0010
MW Amplifier | CIQTEK | GYPA2530-42
PD | Thorlabs | SM05PD1A
Piezoelectric Bender | Thorlabs | PB4VB2W

##  Appendix B The coefficient $\eta$ and The sensitivity of the ensemble-NV-
diamond magnetometer

The time-domain signal in voltage from the ensemble-NV-diamond magnetometer is
converted to the magnetic field by the coefficient $\eta$ between the
magnetometer output voltage signal and the magnetic field. The specific region
of the CW spectrum is shown in Fig. 6(a). The coefficient $\eta$ is $47.2\pm
0.2\ \upmu$T/V according to the max slope of the CW spectrum $|\frac{\partial
S}{\partial(f_{osi}-\gamma_{e}B)}|=0.7563\pm 0.0036$ V/MHz and the
gyromagnetic ratio of the NV center $\gamma_{e}=2\uppi\times 28$ GHz/T. The
sensitivity of our setup is obtained by the amplitude spectral density of the
time-domain signal [34]. Fig. 6(b) shows the magnetice detection sensitivity
of the magnetometer. The sensitivity is $1.6\ \rm nT/\sqrt{Hz}$ within the
frequency range from 0.9 to 1 kHz. Furthermore, the magnetic fields with
unstable phases don’t contribute after averaging over a long time. Because of
the above reasons, the contribution of the static external magnetic field to
the magnetic field at the vibration frequency is neglectable. Moreover, there
is no significant magnetic field with the vibration frequency in phase with
the velocity or the position.

Figure 6: (a) The solid black circle points indicate the voltage from LIA1
with different microwave frequencies. The blue line is the linear fitting. The
slope is $0.7563\pm 0.0036$ V/MHz. (b) The magnetic sensitivity of the
magnetometer. The power supply causes the peaks at 50 Hz and harmonics. The
zoomed-in view shows the vibration frequency $f_{M}=941$ Hz.

##  Appendix C Separating velocity-dependent magnetic signals from position-
dependent magnetic signals

The effective magnetic field sensed by the NV ensemble is given by:

$\displaystyle b_{\mathrm{eff}}=-f_{\perp}\frac{\hbar\rho}{4\uppi
m_{e}c\gamma_{e}V_{S}}\int_{S}\mathrm{d}V_{S}\int_{N}\hat{n}\cdot(\vec{v}\times\hat{r})(\frac{1}{\lambda
r}+\frac{1}{r^{2}})e^{-\frac{r}{\lambda}}\mathrm{d}V_{N}$ (4)

where $V_{S}$ stands for the volume of the NV ensemble, $V_{N}$ is the volume
of the nucleon source, S (N) indicates the volume integral over the space of
the spin ensemble (nucleon source), $\vec{r}$ is the displacement vector
between the electron and the nucleon, $r=|\vec{r}|$ and
$\hat{r}={\vec{r}}/{r}$. $\gamma_{e}=2\uppi\times 28$ GHz/T is the
gyromagnetic ratio of the NV center, $\rho$ is the nucleon density, and
$\hat{n}$ is the unit vector along the NV axis. The position of the center of
the lead sphere relative to the surface of diamond is $\vec{r}_{M}=A\cos(2\pi
f_{M}t)\hat{y}+(d+R)\hat{z}$. The velocity of sphere is $\vec{v}(t)=-2\pi
f_{M}A\sin(2\pi f_{M}t)\hat{y}$.

Obviously, $b_{\mathrm{eff}}(t)=b_{\mathrm{eff}}(t+1/f_{M})$ because the
sphere is at the same position with the same velocity. It is noticed that
$b_{\mathrm{eff}}(t)=-b_{\mathrm{eff}}(-t)$, because the sphere is at the same
position with the opposite velocity. The Fourier transform of
$b_{\mathrm{eff}}(t)$ is
$b_{\mathrm{eff}}(t)=\sum_{n=1}^{\infty}b_{n}\sin(2\pi nf_{M}t)$.

If a magnetic field $b_{\mathrm{other}}(t)$ arises from magnetic
nanoparticles, the magnetism of sphere, or energized coil, the magnetic field
NV sensed is dependent on the relative position and is independent on the
velocity. $b_{\mathrm{other}}(t)=b_{\mathrm{other}}(t+1/f_{M})$ and
$b_{\mathrm{other}}(t)=b_{\mathrm{other}}(-t)$ because the positions are the
same. The result of Fourier expansion of $b_{\mathrm{other}}(t)$ is
$b_{\mathrm{other}}(t)=\sum_{n=0}^{\infty}a_{n}cos(2\pi nf_{M}t)$.

Therefore, the fundamental frequency component of $b_{\mathrm{eff}}$ is in-
phase with the velocity $v(t)$ and the fundamental frequency component of
$b_{\mathrm{other}}$ is in-phase with the position $r_{M}(t)$.

The signal from LIA1 is demodulated by LIA2. By adjusting the phase of
internal reference of LIA2 to match the phase delay between the output of
magnetometer and the vibration $\phi$, the amplitude of the fundamental
frequency component of $b_{\mathrm{eff}}$ relates to output from the
quadrature channel of LIA2 and the amplitude of the fundamental frequency
component of $b_{\mathrm{other}}$ relates to output from the in-phase channel
of LIA2. It suppresses the influence of the magnetic field which is dependent
on the relative position and independent on the velocity.

The method to calibrate $\phi$ is the same as Ref.[31]. To calibrate the phase
$\phi$, one kind of $b_{\mathrm{other}}$ is temporarily introduced. A copper
wire is glued to the bender. A direct current is applied to the copper wire,
and a magnetic field that is in phase with the position of sphere $r_{M}(t)$
is generated due to the vibration. To ensure that exotic velocity-dependent
interaction does not influence the calibration, the amplitude of the magnetic
field from the copper wire is 3 nT, which is much larger than the magnetic
field when the current is not applied. As shown in Fig. 7(a), before adjusting
the phase $\phi$, the outputs of both channels are non-zero when a direct
current is applied. After calibration, the phase of the reference signal in
the second lock-in amplifier is tuned to $51^{\circ}$ where the quadrature
component is zeros when the direct current is applied. The result is shown in
Fig.7(b), and the uncertainty of the phase $\phi$ is $|\Delta\phi|\leq
4^{\circ}$. For $|\Delta\phi|=4^{\circ}$, the output of the quadrature channel
contains 99.8% of the velocity-dependent $b_{\mathrm{eff}}$ and 7% of the
position-dependent $b_{\mathrm{other}}$.

Figure 7: (a)(b) The time trace of two components when a current is applied to
generate a magnetic field that is strong enough to cover the effective
magnetic field due to the exotic interaction. The red line indicates the in-
phase component and the blue line indicates the quadrature component. (a)
Before adjusting $\phi$, the signal due to current is output through both the
in-phase channel and the quadrature channel. (b) After adjusting $\phi$, the
signal due to current is output through the in-phase channel. (c)The result of
the measured magnetic field with the time duration being 600 s. The upper one
is output through the in-phase channel and the bottom one is output through
the quadrature channel. (d) The Histogram of the measured magnetic fields
which are collected for 64 hours. The solid line is the fit to the Gaussian
distributions.

The current is turned off when performing the search for the exotic
interaction. The output from the in-phase channel and the quadrature channel
are shown in Fig. 7(c)(d), which both obey the Gaussian distribution. The
fundamental frequency component of $b_{\mathrm{other}}$ is $a_{1}^{exp}=2.3\pm
3.2$ pT and the fundamental frequency component of $b_{\mathrm{eff}}$ is
$b_{1}^{exp}=1.0\pm 3.2$ pT. The result that the mean value of the measured
magnetic field is less than the standard error of the mean shows that there is
no significant magnetic field in phase with the velocity or the position.

##  Appendix D Statistical properties of the results

The correlation coefficient $\rho(A,B)$ characterizes the correlation between
${A_{i}}$ and ${B_{i}}$. The definition of the correlation coefficient is:

$\displaystyle\rho(A,B)=\frac{1}{N-1}\sum_{i=1}^{N}(\frac{A_{i}-\mu_{A}}{\sigma_{A}})(\frac{B_{i}-\mu_{B}}{\sigma_{B}})$
(5)

where $\mu_{A}$($\mu_{B}$) and $\sigma_{A}$($\sigma_{B}$) are the mean and
standard deviation of A (B). The correlation coefficient of the in-phase
channel and the quadrature channel is $\rho(I,Q)=0.0041\pm 0.0044$. Therefore,
the output from two channels is considered to be uncorrelated.

The normalized autocorrelation of A is:

$\displaystyle
R_{A}(m)=\frac{1}{\sum_{n=0}^{N-1}A_{n}A_{n}^{*}}\times\begin{cases}\sum_{n=0}^{N-m-1}A_{n+m}A_{n}^{*}&,m\geq
0\\\ R_{A}^{*}(-m)&,m\leq 0\end{cases}$ (6)

$|R_{I}(m)|<=0.0094$ and $|R_{Q}(m)|<=0.0098$ when $m\neq 0$. There is no
correlation between the points. Therefore, the standard error of the mean can
be calculated by $\sigma/\sqrt{N}$.

The statistical properties of the output from two channels are listed in
Table3.

Table 3: Statistical properties of the results Channel | I | Q
---|---|---
Number | 203495 | 203495
Mean (pT) | 2.3 | 1.0
Standard error (nT) | 1.4 | 1.4
Standard error of mean (pT) | 3.2 | 3.2

##  Appendix E Systematicerrors

###  E.1 Uncertainty in the angle $\alpha$ between the velocity and the
$\hat{y}$ direction

The angle $\alpha$ between the velocity and the $\hat{y}$ direction is $0\pm
3^{\circ}$, which is determined by the edge of the bender and the edge of the
diamond. The correction to the coupling parameter $f_{\perp}$ is from
$0.0\times 10^{-14}$ to $0.2\times 10^{-14}$ at $\lambda=100\ \upmu$m.

###  E.2 Uncertainty in the diameter

The diameter of the lead sphere is measured to be $897\pm 3\ \upmu$m with a
micrometer. The correction to the coupling parameter $f_{\perp}$ is $\pm
0.5\times 10^{-14}$ at $\lambda=100\ \upmu$m.

###  E.3 Uncertainty in the distance

The distance between the bottom of the sphere and the surface of the diamond
is controlled by a vertically installed piezo motor carrying the piezoelectric
bender. The feedback from the piezoelectric bender is monitored when the
sphere slowly moves down with a tiny vibration amplitude. The position where
the sphere approaches the surface of the diamond is determined where the
feedback suddenly varies[31]. The distance between the bottom of the sphere
and the surface of the diamond is set to $5.0\pm 0.5\ \upmu$m. The uncertainty
is determined by measuring the position where the sphere approaches the
surface of the diamond multiple times. The correction to the coupling
parameter $f_{\perp}$ is $\pm 0.8\times 10^{-14}$ at $\lambda=100\ \upmu$m.

###  E.4 Uncertainty in the thickness

The thickness of the NV layer is determined by the thickness difference before
and after diamond growth. The NV layer is fabricated on a diamond whose
thickness is $551\pm 1\ \upmu$m. The thickness of the diamond after the growth
of the NV layer is $574\pm 1\ \upmu$m which indicates the thickness of the NV
layer is $23\pm 1\ \upmu$m. The correction to the coupling parameter
$f_{\perp}$ is $\pm 0.8\times 10^{-14}$ at $\lambda=100\ \upmu$m.

###  E.5 Uncertainty in the amplitude

The vibration amplitude is $538\pm 3$ nm which is measured by a commercial
laser vibrometer (Sunny Optical, LV-S01). The correction to the coupling
parameter $f_{\perp}$ is $\pm 0.9\times 10^{-14}$ at $\lambda=100\ \upmu$m.

###  E.6 Uncertainty in the deviation

The deviation in X-Y plane is $0\pm 10\ \upmu$m according to the CCD images.
The correction to the coupling parameter $f_{\perp}$ is from $-0.8\times
10^{-14}$ to $0.9\times 10^{-14}$ at $\lambda=100\ \upmu$m.

###  E.7 Uncertainty in the nucleon density

The density the lead sphere is measured to be $\rho=11.381\pm 0.036\
\mathrm{g/cm^{3}}$. The corresponding nucleon density is $6.82\pm 0.02\ \times
10^{30}\ \mathrm{m^{-3}}$ and the correction to the coupling parameter
$f_{\perp}$ is $\pm 0.5\times 10^{-14}$ at $\lambda=100\ \upmu$m.

###  E.8 Uncertainty in the coefficient $\eta$

The coefficient $\eta$ between the magnetometer output voltage signal and the
magnetic field is measured to be $\eta=47.2\pm 0.2\ \upmu$T/V. The correction
to the coupling parameter $f_{\perp}$ is $\pm 0.7\times 10^{-14}$ at
$\lambda=100\ \upmu$m.

The systematic error is much less than the statistical error. With the
fiducial probability of 95%, the bound of $f_{\perp}$ with $\lambda=100\
\upmu$m is $\lvert f_{\perp}\rvert\leq 1.1\times 10^{-11}$ when both
statistical and systematic errors are taken into account.

##  Appendix F Comparison with the non-zero result observed in the single-NV-
based experiment

Figure 8: The upper bounds of the coupling parameter $f_{\perp}$ as a function
of the force range $\lambda$ and the boson mass $m_{b}$. Black lines are upper
bounds obtained from previous experiments [15, 24]. The red line is the upper
limit established by our experiment. The blue dash line is the experimental
precision in Ref.[29], while the blue dot denotes the non-zero result in
Ref.[29]. The gray-filled area shows the excluded values.

Recently, a non-zero result was reported in an experimental searching for
exotic spin-denpendent interaction [29]. The upper bounds established by this
work and previous experiments are shown in Fig.8. The suggested coupling
parameter $f_{\perp}=3.93\times 10^{-8}$ with the force range $\lambda=8.07\
\upmu$m in Ref. [29] is shown as a blue dot. Our experiment sets bounds more
stringent than the experimental precision in Ref.[29] in force range larger
than 8 micrometers. Our results partially exclude the non-zero result and
suggest that the non-zero result in Ref. [29] may come from instrumental
artifacts.

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# On a generalization of two results of Happel to commutative rings

Tony J. Puthenpurakal  Department of Mathematics, IIT Bombay, Powai, Mumbai
400 076<EMAIL_ADDRESS>
###### Abstract.

In this paper we extend two results of Happel to commutative rings. Let
$(A,\mathfrak{m})$ be a commutative Noetherian local ring. Let
$D^{b}_{f}(\operatorname{mod}(A))$ be the bounded derived category of
complexes of finitely generated modules over $A$ with finite length
cohomology. We show $D^{b}_{f}(\operatorname{mod}(A))$ has Auslander-
Reiten(AR)-triangles if and only if $A$ is regular. Let
$K^{b}_{f}(\operatorname{proj}A)$ be the homotopy category of finite complexes
of finitely generated free $A$-modules with finite length cohomology. We show
that if $A$ is complete and if $A$ is Gorenstein then
$K^{b}_{f}(\operatorname{proj}A)$ has AR triangles. Conversely we show that if
$K^{b}_{f}(\operatorname{proj}A)$ has AR triangles and if $A$ is Cohen-
Macaulay or if $\dim A=1$ then $A$ is Gorenstein.

###### Key words and phrases:

bounded homotopy category, bounded derived category, Auslander-Reiten
triangles

###### 1991 Mathematics Subject Classification:

Primary 13D09, 16G70 ; Secondary 13H05, 13H10

##  1\. introduction

Let $\Gamma$ be a finite dimensional algebra over a field $k$. Happel proved
that its bounded derived category $D^{b}(\operatorname{mod}(\Gamma))$ has
Auslander-Reiten (AR) triangles if and only if $\Gamma$ has finite global
dimension, see [happel-1, 3.6], [happel-2, 1.5]. He also proved that
$K^{b}(\operatorname{proj}\Gamma)$ the homotopy category of bounded complexes
of finitely generated projective $\Gamma$-modules has right AR triangles if
and only if $\Gamma$ is a Gorenstein algebra, see [happel-3, 3.4]. As
exemplified by Auslander that concepts in the study of representation theory
of Artin algebras have natural analogues in study of maximal Cohen-Macaulay
modules over Cohen-Macaulay local rings. In this paper we study natural
analogues of Happel’s results in the context of commutative Noetherian rings.

Let $(A,\mathfrak{m})$ be a commutative Noetherian local ring of dimension
$d$. Let
$C^{b}(\operatorname{mod}(A))$ be the category of bounded complexes of
finitely generated $A$-modules. Let $K^{b}(\operatorname{mod}(A))$ be the
corresponding homotopy category and let $D^{b}(\operatorname{mod}(A)$ be the
bounded derived category of $A$. We write complexes cohomologically.

$\mathbf{X}_{\bullet}\colon\cdots\rightarrow\mathbf{X}_{\bullet}^{i}\xrightarrow{d^{i}}\mathbf{X}_{\bullet}^{i+1}\xrightarrow{d^{i+1}}\mathbf{X}_{\bullet}^{i+2}\rightarrow\cdots$

with $d^{i+1}\circ d^{i}=0$ for all $i$.

Let $C^{b}(\operatorname{proj}A)$ (and $C^{-,b}(\operatorname{proj}A)$) be
category of bounded complexes (respectively bounded above with
$H^{*}(\mathbf{X}_{\bullet})$ finitely generated for all
$\mathbf{X}_{\bullet}$) ) of finitely generated free $A$-module and let
$K^{b}(\operatorname{proj}A)$ (and $K^{-,b}(\operatorname{proj}A)$) be the
corresponding homotopy categories. Recall that in general
$D^{b}(\operatorname{mod}A)\cong K^{-,b}(\operatorname{proj}A)$ and if $A$ is
regular local then $D^{b}(\operatorname{mod}A)\cong
K^{b}(\operatorname{mod}A)$. Let $D^{b}_{f}(\operatorname{mod}(A))$ be the
bounded derived category of complexes of finitely generated modules over $A$
with finite length cohomology. Then it can be shown that
$D^{b}_{f}(\operatorname{mod}(A))$ is a Hom-finite Krull-Schmidt triangulated
category, see LABEL:tea. Our generalization of Happel’s results [happel-1,
3.6], [happel-2, 1.5] is the following:

######  Theorem 1.1.

Let $(A,\mathfrak{m})$ be a commutative Noetherian local ring. The following
conditions are equivalent:

  1. (i)

$A$ is regular.

  2. (ii)

$D^{b}_{f}(\operatorname{mod}(A))$ has AR-triangles.

Our proof is not similar to Happel’s.

To state our next result we need the following notation. Let
$K^{b}_{f}(\operatorname{proj}A)$ be the subcategory of
$K^{b}(\operatorname{proj}A)$ with finite length cohomology. By Happel’s
result if $(A,\mathfrak{m})$ is a zero-dimensional commutative Gorenstein ring
then $K^{b}(\operatorname{proj}A)$ has AR-triangles. For higher dimensional
Gorenstein rings we prove the following extension of one direction of Happel’s
result [happel-3, 3.4].

######  Theorem 1.2.

Let $(A,\mathfrak{m})$ be a complete Gorenstein local ring. Then
$K^{b}_{f}(\operatorname{proj}A)$ has AR-triangles.

######  Remark 1.3.

When $A$ is complete then Theorem 1.1 ((i) $\implies$ (ii)) follows from
Theorem 1.2. However even in the complete case it is relevant to prove
Theorem1.1 directly. The reason is that the proof of Theorem 1.2 uses a non-
trivial result due to Neeman [NK] while proof of Theorem1.1 is entirely self-
contained.

We believe the converse to Theorem 1.2 is true. More precisely

######  Conjecture 1.4.

Let $(A,\mathfrak{m})$ be a complete Noetherian local ring. If
$K^{b}_{f}(\operatorname{proj}A)$ has AR-triangles then $A$ is Gorenstein.

We prove Conjecture 1.4 under the following cases:

######  Theorem 1.5.

Let $(A,\mathfrak{m})$ be a complete Noetherian local ring. Assume
$K^{b}_{f}(\operatorname{proj}A)$ has AR-triangles. Then

  1. (1)

if $A$ is Cohen-Macaulay then $A$ is Gorenstein.

  2. (2)

if $\dim A=1$ then $A$ is Gorenstein.

then $A$ is Gorenstein.

If $A$ is also Henselian then $C^{b}(\operatorname{proj}A)$ is also a Krull-
Schmidt category. It then can be easily verified that
$K^{b}(\operatorname{proj}A)$ is also a Krull-Schmidt category. Let
$K^{b}_{f}(\operatorname{proj}A)$ denote the complexes in
$K^{b}(\operatorname{proj}A)$ with finite cohomology. _We give two results
which justify why we are concentrating on $K^{b}_{f}(\operatorname{proj}A)$_.

Our first result is the following. In a Krull-Schmidt triangulated category we
may consider the notion of right (left) AR( Auslander-Reiten) triangles. We
prove

######  Theorem 1.6.

Let $(A,\mathfrak{m})$ be a Henselian commutative Noetherian local ring. Let
$\mathbf{X}_{\bullet}\in K^{b}(\operatorname{proj}A)$. If there is a right
(left) AR-triangle ending at $\mathbf{X}_{\bullet}$ then
$H^{*}(\mathbf{X}_{\bullet})$ has finite length.

The techniques used to prove Theorem 1.6 are inspired by some techniques of
Yoshino, see [Y, Chapter 2].

To state our second result we require some preliminaries: Let
$\mathbf{X}_{\bullet}\in C^{b}(\operatorname{proj}A)$ be a minimal complex i.e
$\partial(\mathbf{X}_{\bullet})\subseteq\mathfrak{m}\mathbf{X}_{\bullet}$. Set

$0pt\mathbf{X}_{\bullet}=\sup\\{i\mid\mathbf{X}_{\bullet}^{i}\neq
0\\}-\inf\\{i\mid\mathbf{X}_{\bullet}^{i}\neq 0\\}$

and

$\operatorname{rank}\mathbf{X}_{\bullet}=\sum_{i}\operatorname{rank}\mathbf{X}_{\bullet}^{i}.$

In general if $\mathbf{X}_{\bullet}\in C^{b}(\operatorname{proj}A)$ then
$\mathbf{X}_{\bullet}=\mathbf{U}_{\bullet}\oplus\mathbf{V}_{\bullet}$ where
$\mathbf{U}_{\bullet}$ is minimal complex and $\mathbf{V}_{\bullet}$ is a
finite direct sum of shifts of the complex $0\rightarrow
A\xrightarrow{1}A\rightarrow 0$. We set

$0pt\mathbf{X}_{\bullet}=0pt\mathbf{U}_{\bullet}\quad\text{and}\quad\operatorname{rank}\mathbf{X}_{\bullet}=\operatorname{rank}\mathbf{U}_{\bullet}.$

We note that if $\mathbf{X}_{\bullet}\in K^{b}(\operatorname{proj}A)$ then we
can define width and rank of $\mathbf{X}_{\bullet}$ to be same as considered
as a complex in $C^{b}(\operatorname{proj}A)$. It can be easily seen that this
is well-defined.

Our second result justifying study of $K^{b}_{f}(\operatorname{proj}A)$ is

######  Theorem 1.7.

Let $\mathcal{T}$ be a thick subcategory of $K^{b}(\operatorname{proj}A)$ and
let $\mathbf{X}_{\bullet}\in\mathcal{T}$. Let $m=0pt\mathbf{X}_{\bullet}$ and
$r=\operatorname{rank}\mathbf{X}_{\bullet}$. Suppose $\mathcal{T}$ has only
finitely many complexes (upto shift) with width $\leq 2m$ and rank $\leq 2r$.
Then $H^{*}(\mathbf{X}_{\bullet})$ has finite length.

The proof of the above result is an abstraction of a proof due to Huneke and
Leuschke, see [HL, Theorem 1]. To get our abstraction we need an analogue of a
theorem of Miyata [Mi, Theorem 1]. Miyata proved that if $R$ is a commutative
Noetherian ring and let $\Gamma$ be an $R$\- algebra which is finitely
generated as a $R$-module. Let $M,N,L$ be finitely generated $\Gamma$ modules
such that there is an exact sequence

$s\colon 0\rightarrow M\rightarrow N\rightarrow L\rightarrow 0$

If $N\cong M\oplus L$ then $s$ is split.

Let $\mathcal{C}$ be a triangulated category. If $s\colon M\rightarrow
N\rightarrow L\xrightarrow{0}M[1]$ is a triangle then it is well-known that
$L\cong N\oplus M$; see [N, 1.2.7]. Our analogue of Miyata’s theorem is as
follows:

######  Theorem 1.8.

Let $\Gamma$ be a left Noetherian ring (not necessarily commutative). Suppose
we have a triangle in $K^{-,b}(\operatorname{proj}\Gamma)$

$s\colon\mathbf{U}_{\bullet}\xrightarrow{u}\mathbf{W}_{\bullet}\xrightarrow{w}\mathbf{V}_{\bullet}\xrightarrow{v}\mathbf{U}_{\bullet}[1],$

such that
$\mathbf{W}_{\bullet}\cong\mathbf{U}_{\bullet}\oplus\mathbf{V}_{\bullet}$ then
$v=0$ (equivalently there is
$\xi\colon\mathbf{V}_{\bullet}\rightarrow\mathbf{W}_{\bullet}$ such that
$w\circ\xi=1_{\mathbf{V}_{\bullet}}))$.

We now describe in brief the contents of this paper. In section two we discuss
some preliminary results that we need. In section three we discuss some
preliminaries on AR-sequences that we need. In section four we prove that
$D^{b}_{f}(A)$ is a Krull-Schmidt category. In the next section we give a
proof of Theorem 1.1( (i) $\implies$ (ii)). In section six we give a proof of
Theorem 1.2. In section seven we discuss a filtration of certain DG-algebras
that we need. In section eight we give a proof of Theorem 1.1( (ii) $\implies$
(i)). In section nine we give a proof of Theorem 1.5. In the next section we
give a proof of Theorem 1.6. In section eleven we prove our analogue of
Miyata’s theorem. Finally in section twelve we give a proof of Theorem 1.7.

##  2\. preliminaries

We discuss a few preliminarly notions and results that we need.

######  2.1.

_(co-chain)-Complexes:_ Let $\Gamma$ be a (not-necessarily commutative) left
Noetherian ring. We will consider co-chain complexes over $\Gamma$

$\mathbf{X}_{\bullet}\colon\cdots\rightarrow\mathbf{X}_{\bullet}^{i}\xrightarrow{\partial^{i}}\mathbf{X}_{\bullet}^{i+1}\xrightarrow{\partial^{i+1}}\mathbf{X}_{\bullet}^{i+2}\rightarrow\cdots$

with $\partial^{i+1}\circ\partial^{i}=0$ for all $i$. By
$H^{i}(\mathbf{X}_{\bullet})$ we denote the $i^{th}$ cohomology of
$\mathbf{X}_{\bullet}$. By the $m^{th}$ shift of $\mathbf{X}_{\bullet}$ we
mean the co-chain complex $\mathbf{X}_{\bullet}[m]$ with
$\mathbf{X}_{\bullet}[m]^{i}=\mathbf{X}_{\bullet}^{i+m}$ and differential
$d[m]^{i}=(-1)^{m}d^{i+m}$.


# Systematics in the SED Fitting Parameter Estimation of Composite Galaxies

Katherine Zine  Department of Astronomy, Indiana University, Bloomington, IN
47405, USA  Samir Salim Department of Astronomy, Indiana University,
Bloomington, IN 47405, USA

###### Abstract

Derivation of physical properties of galaxies using spectral energy
distribution (SED) fitting is a powerful method, but can suffer from various
systematics arising from model assumptions. Previously, such biases were
mostly studied in the context of individual galaxies. In this study, we
investigate potential biases arising from performing the SED fitting on the
combined light of two galaxies, as would be the case in post-merger systems.
We use GALEX-SDSS-WISE Legacy Catalog (GSWLC) of $z<0.3$ galaxies to identify
9,000 galaxy pairs that could eventually merge. For these we investigate if
the UV/optical SED fitting accurately determines the stellar mass and
(specific) star formation rate if the pair was unresolved (merged). The sum of
the stellar masses (and SFRs) of individual galaxies in the pair establishes
the ground truth for these quantities. For star forming galaxies no biases
($<$0.1 dex) are found in the stellar mass, SFR, or sSFRs. Moderate
systematics in SFR ($\sim$0.1 dex) are found for systems with an extreme
contrast in dust content between the two galaxies. We conclude that biases
that would arise in the determination of masses and SFRs of post-merger
systems on account of the two original galaxies having potentially very
different star formation histories and different dust properties are small and
that the approach with simple two-component star formation histories is
adequate. The approach presented in this study, using flux compositing with
empirically determined ground truth, offers new opportunities for testing the
results of SED fitting in general.

Galaxy Mergers (608) — Galaxy Properties (615) — Galaxy Masses (607) –
Spectral Energy Distribution (2129)

##  1 Introduction

Galaxy mergers play an important role in galaxy evolution. They can lead to
dramatic changes in galaxy structure and morphology. For example, when
simulating the life of a galaxy undergoing a merger, the galaxies that start
off as disk galaxies can become spheroids after a merger (e.g., Wuyts et al.,
2009). Mergers can also augment star formation efficiency and produce
starbursts (e.g., Bournaud, 2011). The study of galaxy evolution requires
unbiased determination of galaxy properties, such as the stellar mass and the
star formation rate (SFR). This is potentially more challenging for galaxies
that have recently undergone a merger for two reasons. One is that the merger
itself can produce a burst of star formation and therefore the star formation
history (SFH) of the merger, especially a recent one, is more complex than
those of the pre-merger galaxies. The second arises from the fact that mergers
can result from the combination of different types of galaxies, each with its
own distinct SFHs and dust contents prior to merging. The goal of this paper
is to shed light on the second effect, which we refer to as “compositing”.

Spectral energy distribution (SED) fitting is an important tool in the study
of galaxy evolution due to its ability to simultaneously determine many
properties of a galaxy, such as star formation history and stellar mass, based
on its light output (Conroy, 2013). In the context of SED fitting, the SEDs
are crude spectra of galaxies consisting of flux measurements at different
wavelengths, usually spanning from rest-frame UV to near IR (Walcher et al.,
2011). In SED fitting a large number of models are created by specifying
parameters relating to SFH, stellar populations, and dust attenuation (Salim
et al., 2016, hereafter S16). The observed SED is compared to model SEDs to
find the best fit, or more generally, to build the probability distribution
functions for different properties (Walcher et al., 2011). The models used in
SED fitting are based on stellar population synthesis, which uses stellar
evolution theory and libraries of observed stars of different spectral types
or model stars with different physical properties (Conroy, 2013).

Whereas the observed SEDs are the result of an underlying physics that is
being modeled, the models have limitations or simplifications that may affect
the accuracy of parameters derived from the SED fitting. Furthermore, the
observed SEDs, which have some finite precision, may not be able to accurately
distinguish between different models due to the potential degeneracies.
Various systematics involved with SED fitting have been studied previously,
but usually for the general population of galaxies, which is not likely to
have undergone major merging events in the recent past. For example, for the
stellar mass of galaxies, Sorba & Sawicki (2015, 2018), following the approach
of Zibetti et al. (2009), found that in the case of photometry that was
spatially unresolved, the mass found using SED fitting was underestimating the
true mass. van der Wel et al. (2006) found that there can be systematics at
different redshifts depending on various factors, such as the stellar
population synthesis model employed. Mitchell et al. (2013) found systematics
in stellar mass caused by using only one metallicity value and depending on
how mass recycling was treated. Systematics were also found by Simha et al.
(2014) in colors and mass-to-light ratios when using an exponentially
declining model for star formation history. S16 has discussed how two-
component star formation histories, where one component is posited to be very
old, are able to recover larger stellar masses and remove or reduce the
outshining issue (Michałowski et al., 2014). Yet, another set of systematics
emerges from the treatment of dust. For example, the use of fixed dust
attenuation curves can lead to biased SFRs and stellar masses, especially if
the assumed law differs from the average law of observed galaxies (Salim &
Narayanan, 2020; Lo Faro et al., 2017).

Additional sytematics my be present in merging or recently merged systems.
Wuyts et al. (2009) studied how systematics may arise during different phases
of a merger. They created a simulation that progressed over time and found
that during the period of enhanced star formation due to a merger the age and
mass were being most underestimated, whereas the underestimation of SFR varied
at different points during the merger (Wuyts et al., 2009). Hayward & Smith
(2015) applied MAGPHYS SED fitting code to hydrodynamical simulations of
merging galaxies and found good fits and that most of the parameters were
recovered correctly, with the exception of the dust mass, which may have had
systematics at various times, when the galaxy was not at near-coalescence.
Lanz et al. (2014) found that SEDs cannot usually be used to indicate the
interactions stage of galaxies, but this does not necessarily mean that the
SEDs of merged galaxies are not at least somewhat distinct from the galaxies
that have not experienced a merger (at least recently) and that such distinct
SEDs would still be well modeled in the course of SED fitting.

Previous studies essentially focused on the systematics arising from the
changes produced by the merger itself. However, when two galaxies merge such
that we can no longer resolve them individually and therefore cannot separate
the SEDs of the original galaxies, an additional set of systematics may arise
in the SED fitting due to the fact that we are trying to model the combined
light of the galaxies, which is a composite of two galaxies with distinct SFH
and different dust properties. The sheer number of possible combinations
precludes such composite SFHs or complex dust scenarios to be explicitly
included in the library of models, so the question we deal with in the current
work is whether the commonly used SFH parameterizations and simple dust models
can still provide unbiased results in the case of composite SEDs.

The paper is organized as follows. In Section 2 the data and sample used for
this study are described. Section 3 describes methodology used to assess
biases in the SED fitting of composite galaxies. The results are presented in
Section 4 and a summary and a discussion of the results is in Section 5.

##  2 Data and Sample

The parent sample was taken from the GSWLC-D catalogue (S16). GSWLC is a
catalog of galaxy parameters obtained from SED fitting, and consists of
galaxies contained in the SDSS spectroscopic sample (redshifts between 0.01
and 0.30) that had ultraviolet (UV) coverage from GALEX (S16). UV observations
are necessary to constrain dust-corrected SFRs. The D catalog includes
galaxies with the longest exposure times in the UV (longer than 4000 seconds)
and therefore has the highest precision of SFR estimates (S16). The catalog
contains position in the sky, SDSS identifications, and physical parameters
such as the stellar mass, dust attenuation, and star formation rates of 48,401
galaxies (S16). GSWLC is based on the photometry in the FUV, NUV, $u$, $g$,
$r$, $i$, and $z$ bands, which will be used in the SED fitting in this work as
well.

Figure 1: Projected physical and redshift separations of galaxies in candidate
pairs. The physical separation in Mpc for the two galaxies in the 33,557 pairs
plotted against their difference in redshift. Only galaxy pairs between the
two red lines were decided to be close enough to be considered potential
future mergers and were kept in the analysis.

One can in principle test biases emerging from the compositing of SEDs by
performing the SED fitting on any two galaxies from the parent sample and
combining their light. However, in order to make our assessment more pertinent
to a physical situation when this compositing may occur, we focus on physical
galaxy pairs-galaxies that are close enough to each other that they could
realistically merge. The added benefit of such an approach is that each galaxy
in the composite will lie at the same distance, so combining their light can
be accomplished by simply adding the fluxes.

We perform galaxy pair selection using a two-step process. First, we identify
33,557 candidate pairs, which are defined as a pair with the smallest angular
separation from each other not exceeding one degree. In order to establish if
the candidate pair is physically associated, we examine their redshifts. In
Figure 1 we show the physical projected separation of candidate pairs where
the galaxies have similar redshifts. We see an overdensity of pairs around
$\Delta z=0$. Informed by the appearance of this distribution, for our final
sample we require the difference in redshifts to be less than 0.003,
corresponding to a maximum radial velocity difference of 900 km $s^{-1}$. Such
pairs are all within about 2 Mpc projected distance (similar to the size of
galaxy groups), and often much closer, so we do not impose any upper limits on
the projected separation. The final sample consists of 9,032 pairs with 15,736
unique galaxies, or 32.5% of the parent sample. The number of unique galaxies
is somewhat smaller than twice the number of pairs because a given galaxy can
be a member of more than one pair.

Figure 2: Distributions of galaxy properties in our sample of physical pairs
(candidate mergers). Blue histograms show properties of individual galaxies
before combining their light, and the peach histograms are the properties of
composite galaxies (after their SEDs were combined). Properties include the
stellar mass, star formation rate, specific star formation rate, and _V_ -band
attenuation.  Figure 3: Stellar mass ratio of galaxies making the physical
pairs (candidate mergers). There is a wide range of mass ratios. The majority
of pairs have mass ratios that would correspond to a major merger. Figure 4:
Stellar mass versus specific star formation rate. The grey plot is for all
48,401 galaxies in the parent sample, whereas the composite galaxies are in
red. Composites occupy an entire parameter space, but are somewhat less
present among the low-mass galaxies.

##  3 SED Fitting

We perform the SED fitting on the galaxies in the parent sample (which
includes the individual galaxies from the 9,032 pairs) as well as on the
composite galaxies, for which we combine the fluxes of members in each pair
prior to SED fitting.

SED fitting was performed with the 2020 version of the Code Investigating
GALaxy Emission, or CIGALE (Boquien et al., 2019). We used Bruzual & Charlot
(2003) stellar population synthesis models calculated for four stellar
metallicities and for the Chabrier IMF. Nebular emission lines are included in
the models as described in Salim et al. (2018). Importantly, to model the
dust, we follow the approach from S16 based on having more than one
attenuation curve in the model library. In particular, we use the Noll et al.
(2009) modification of the Calzetti et al. (2000) attenuation curve, and make
models using two slopes each steeper than the original Calzetti curve. To this
curve we add the UV bump, in several increments up to twice the strength of
the UV bump in the MW. Note that this SED fitting does not include IR
constraints as the more recent GSWLC-2 (Salim et al., 2018), in order to stay
with the more common SED fitting methodology. The principle difference between
the SED fitting employed here and in S16 regards the parameterization of the
star formation history. Namely, prior to running the final fits, we tested the
performance of the SED fitting in order to find the parameters and variables
that would potentially improve the quality of fitting over S16, judging by the
average $\chi^{2}$. We took the first 1000 galaxies from GSWLC-D and performed
the fitting with different SFH parameter combinations (changing the
parameterizations of the of the old burst, as well as the ages and decline
times of the young burst) and we found that some improvements were possible,
as explained below. Tests were kept at 1000 galaxies in order to reduce the
computation time over many trial runs. This number is entirely sufficient to
reveal if one set of parameters performs better than another.

S16 uses a SFH parameterized as a double exponential, where the first (older)
exponential commences at a fixed age (10 Gyr before the epoch of observation)
whereas the second component is essentially flat in the SFR, but varies in
duration (age) and intensity. In contrast, for this study we used the delayed
exponential model from Appendix E of Boquien et al. (2019), but with the same
age as the old exponential component used previously (10 Gyr) and without the
extremely short decline models of 1 Myr. To this smooth delayed exponential we
add the second, exponentially declining, component, again after the Appendix
E. Unlike the second component in S16, this one includes various short decline
rates and ages. We find that the new SF parameterization has somewhat smaller
$\chi^{2}$ values for the best fitting models compared to S16, although the
improvement is not dramatic.

For the fitting, the observed errors were added in quadrature with a
calibration error ranging from 1-2% for SDSS bands and 3-5% for GALEX (S16).
These errors were used instead of the 10% calibration errors assumed by
default in CIGALE. Finally, the redshift used in the fitting of composited was
taken to be the average redshift of the two galaxies.

Figure 5: Comparison of the expected (sum of individual galaxies in a pair)
versus recovered values (from the SED fit of the composite galaxy) of stellar
mass, star formation rate, and specific star formation rate. Only pairs that
have the stellar mass ratio corresponding to major mergers are shown.

##  4 Results

###  4.1 Physical properties of parent sample and galaxy pair sample

We first explore how the properties of galaxies in pairs before combining
their light compare to those of the composite galaxies (after their SEDs were
combined). This is done for the star formation rates (SFR), the specific star
formation rates (sSFR), and the _V_ -band attenuation and is shown in Figure
2. For the stellar mass the distribution shifts by upwards some 0.5 dex as
expected because most of the massive galaxies are paired with another
similarly massive galaxy (Figure 3). For SFR values, the values obtained for
the composite galaxies have a pronounced high-SFR peak, whereas the low SFR
present in the original distribution is suppressed. We confirm that this is
not the result of there being fewer composites where both galaxies are
quiescent (an equivalent of dry merging), but rather that the low SFR galaxies
are overwhelmingly merged to higher SFR galaxies. For the sSFR, the composite
values have a peak between the two peaks for galaxies prior to combining,
which is expected given that many of the galaxy pairs are a combination of a
passive plus star forming galaxy, so the result of combining would move the
galaxy toward the green valley. Indeed it has been proposed that many of the
galaxies in the green valley are the result of such mergers (Kaviraj et al.,
2009).The two histograms for the _V_ -band attenuation look similar and don’t
have much of an offset.

The stellar mass ratios of the galaxies making the composites are shown in
Figure 3. The majority of mass ratios correspond to what would be the major
mergers, which is expected because most of the pairs are at redshifts where
only the massive galaxies ($\log M_{*}>10$) are detectable in the SDSS, so the
difference between the galaxies in a given pair cannot be very high. Having a
large fraction of composites pertain to the major mergers is a welcome aspect
of this sample because it is the major mergers in which any biases due to the
compositing of SFHs or dust properties would be accentuated.

Figure 6: Difference in the recovered and expected stellar mass (the
residuals) as a function of the contrast between the members in the pair, in
_V_ -band attenuation, stellar mass, reduced $\chi^{2}$, and star formation
rate. The white line is zero residual and the red line is the median value of
the residuals. Only the pairs with the stellar mass ratios corresponding to
major mergers are shown.

The plot of the stellar mass versus the specific star formation rate was
(Figure 4) shows all 48,401 galaxies from the parent sample. On top of that
are points for the merged galaxy pairs. The merged pairs have fewer low mass
galaxies than the parent sample (low mass galaxies live in underdense
environments and are unlikely to merge) and have a tail to lower specific star
formation rates. We will see subsequently that some of these lower values are
the result of a bias that leads to the underestimation of galaxies with
already low sSFRs.

###  4.2 Biases in parameters of composite galaxies

In order to explore potential biases in parameters recovered for the composite
galaxies, expected and recovered values were compared to see how well SED
fitting on the composite galaxies agreed with the expected values. The
expected values (“ground truth” in this comparison) were obtained by adding
the parameter values for the individual galaxies, except in the case of the
sSFR, where the sum of the SFRs divided by the sum of stellar masses is used.
Figure 5 shows comparisons of the expected versus recovered stellar masses,
star formation rates, and specific star formation rates. We include only the
composites ($M_{*,1}/M_{*,2}>0.3$). The plot of the mass comparison shows that
the data points follow the one to one line very well (to within less than 0.01
dex), demonstrating that there is no overall bias in the stellar mass. There
is also very little scatter (0.04 dex), comparable to, or even smaller than,
the formal error in individual masses as estimated from the width of the
probability distribution functions. For the star formation rate comparison,
there does not seem to be a bias for log SFR$>0$ (median difference of only
0.01 dex), but there is a tail around log SFR (expected)$=-1$, in the sense
that the recovered SFRs of these low-SFR galaxies are sometimes lower. The
specific star formation rate comparison is also mostly along the one to one
line but there is a tail around log sSFR (expected) $=-12.5$ where again the
recovered values are somewhat lower than expected. In order to find out what
may be driving the biases for some of the low (s)SFR galaxies, we compared the
estimated dust attenuation for the composite and of the original galaxy having
the larger SFR (and therefore presumable carrying greater weight in the
composite galaxy). Although there is a significant degree of scatter between
the two dust attenuations, there are no apparent biases that would suggest
that the biased dust estimate is behind the tail. We point out that the
galaxies in the -13$<$log sSFR$<$-12 range are essentially passive, and their
formal sSFRs are difficult to constrain with any degree of accuracy or
precision, with the nominal errors in sSFR being on the order of the offset
seen here between the recovered and the expected value.

Figure 7: Difference in the recovered and expected SFR for (the residuals) as
a function of the contrast between the members in the pair, in _V_ -band
attenuation, stellar mass, reduced $\chi^{2}$, and star formation rate. The
white line is zero residual and the red line is the median value of the
residuals. Only the pairs with the stellar mass ratios corresponding to major
mergers are shown.

Whereas the direct comparison does not reveal any major biases (except for a
small number of low SFR and sSFR galaxies), it is still of interest to explore
if any residuals may emerge for particular subsets of composite galaxies. For
example, one might imagine that the biases increase for pairs that have more
comparable mass ratios. In Figure 6 we show stellar mass residuals between
expected and recovered mass as a function of the contrast in: _V_ -band
attenuation, the quality of the fit (reduced $\chi^{2}$ value), the stellar
mass, and the SFR. The median value of the residuals is shown by a red line.
Residuals stay very close to zero (white line) in all four plots, showing that
there are no systematic stellar mass residuals even for cases where one galaxy
in the pair has a very different dust attenuation from the other, where
composites involve nearly equal mass galaxies, or where one fit is much worse
than the other.

Figure 7 shows the residuals as a function of the contrast between the same
parameters used in Figure 6, but now for SFR. The red line again shows the
median value of the residuals. We see that the residuals are not driven by
increasing mass contrast or the difference in the quality of the fits (upper
right and lower left panels), but they do increase somewhat when one galaxy
has much less dust than another or when a passive and active star forming
galaxy are combined. However, even in those cases the residuals are around 0.1
dex, which is comparable to or smaller than a typical SFR random error (0.1 to
0.6 dex, Figure 6 in S16).

##  5 Discussion and Conclusions

Merging can cause dramatic changes to a galaxy’s star formation (Bournaud,
2011), and this discontinuity in SFH may pose difficulties for the accurate
determination of SFR, especially if the model SFHs lack a recent burst
component. Leaving those issues aside, our study focuses on the biases that
may emerge even just from the compositing of the light of two galaxies, as in
the case of a post-merger system. While no previous study has examined this
problem, the character of our study is most similar to that of Sorba & Sawicki
(2015), who compared stellar mass estimates obtained in the UV/optical SED
fitting by treating the galaxies as unresolved and also as the sum of pixels.
Their sample consisted of 67 very nearby galaxies (mean redshift of 0.006).
They found that for galaxies with higher sSFR the mass from the integrated SED
was lower than the one from summed pixels, which was attributed to biases
arising from younger stars outshining older ones. It needs to be pointed out
that for a typical star-forming galaxy on the main sequence (log
sSFR$\sim$10), they find the bias of just 9%, or 0.04 dex. It reaches 0.11 dex
only for very strong starbursts (log sSFR=-8), which are rare in the local
universe. Our comparison of composite vs. individual masses shows no stellar
mass biases greater than 0.01 dex (Figure 6 mass panels), seemingly at odds
with Sorba & Sawicki (2015). The probable reason for that is that Sorba &
Sawicki (2015) contrast pixel-based estimates to fully unresolved ones, which
will maximize any differences, whereas we compare the sum of two already
unresolved galaxies to their composite. Indeed, they find that if their
comparison is made between the images degraded to 3 kpc (rather than pixel-
level resolution of $\sim$0.1 kpc) and the fully unresolved one, the
systematic differences between the two are no longer present. Sorba & Sawicki
(2015) investigate the effects of the resolution only for $M_{*}$, so we
cannot compare our results for the SFR.

As shown in Section 4, no major systematics were found in the determination of
the principle galaxy properties, the light of which is the composite of two
galaxy SEDs. In Figure 5, the recovered stellar masses match the expected
values well, with median differences within 0.01 dex and small standard
deviations. For SFRs, no biases are present for galaxies on the star forming
main sequence (median differences within 0.01 dex). sSFR comparisons show a
bias around log sSFR (expected) $=-12.5$, which we attribute to the general
difficulties involved in getting the accurate SFRs for practically quiescent
galaxies. The mass residuals do not depend on the contrast of the _V_ -band
attenuation, reduced $\chi^{2}$ value, stellar mass, and SFR, whereas the SFR
residuals are somewhat sensitive to extreme dust and SFR contrast.

Overall, regardless of the different dust properties and SFH, the parameters
of the composite do not suffer from significant biases in the SED fitting.
From this we conclude that despite the great complexity that the SFHs may
exhibit in reality due to the different processes that boost and suppress star
formation (e.g. Tacchella et al. (2016)), when it comes to the ability to
derive physical parameters, even relatively simple two-component prescriptions
are sufficient to obtain accurate results. This conclusion has a practical
significance for SED modelers, but also informs us that parametric models of
SFH still have their merits in the high-level understanding of galaxy
evolution.

Another important methodological contribution of this study for SED modelling
is the simplicity of the approach of combining fluxes to study potential
systematics. This approach deserves to be incorporated into a suite of
robustness diagnostics, such as the covariance analysis (corner diagrams, Leja
et al. (2017)), as well as mock SED fitting (SED fitting to best-fitting
photometry obtained from the SED fitting itself, Boquien et al. (2019), Salim
et al. (2009)).

Only the main physical parameters were explored in this study and there may be
other parameters which would be more affected by the biases in the SED fitting
in cases of combining the flux of galaxies. On the other hand, here we explore
just the more usual SED fitting involving stellar emission. The inclusion of
dust emission in the infrared, as is done in the so called “energy balance”
SED fitting, may help further reduce any systematics (da Cunha et al., 2008;
Salim et al., 2018).

The construction of GSWLC was funded through NASA awards NNX12AE06G and
80NSSc20K0440.

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