# Search for an exotic parity-odd spin- and velocity-dependent interaction
using a magnetic force microscope

Xiaofang Ren  MOE Key Laboratory of Fundamental Quantities Measurement, Hubei
Key Laboratory of Gravitation and Quantum Physics, School of Physics, Huazhong
University of Science and Technology, Wuhan 430074, China  Jianbo Wang  MOE
Key Laboratory of Fundamental Quantities Measurement, Hubei Key Laboratory of
Gravitation and Quantum Physics, School of Physics, Huazhong University of
Science and Technology, Wuhan 430074, China  Rui Luo  MOE Key Laboratory of
Fundamental Quantities Measurement, Hubei Key Laboratory of Gravitation and
Quantum Physics, School of Physics, Huazhong University of Science and
Technology, Wuhan 430074, China  Lichang Yin  Shenyang National Laboratory for
Materials Science, Institute of Metal Research, Chinese Academy of Sciences,
Shenyang 110016, China  Department of Physics and Electronic Information,
Huaibei Normal University, Anhui, Huaibei 235000, China  Jihua Ding  MOE Key
Laboratory of Fundamental Quantities Measurement, Hubei Key Laboratory of
Gravitation and Quantum Physics, School of Physics, Huazhong University of
Science and Technology, Wuhan 430074, China  Ge Zeng  School of Physics,
Huazhong University of Science and Technology, Wuhan 430074, China  Pengshun
Luo<EMAIL_ADDRESS>MOE Key Laboratory of Fundamental Quantities
Measurement, Hubei Key Laboratory of Gravitation and Quantum Physics, School
of Physics, Huazhong University of Science and Technology, Wuhan 430074, China

###### Abstract

Exotic spin-dependent interactions may be generated by exchanging hypothetical
bosons that have been proposed to solve some mysteries in physics by theories
beyond the standard model of particle physics. The search for such
interactions can be conducted by tabletop scale experiments using high
precision measurement techniques. Here we report an experiment to explore the
parity-odd interaction between moving polarized electrons and unpolarized
nucleons using a magnetic force microscope. The polarized electrons are
provided by the magnetic tip at the end of a silicon cantilever, and their
polarizations are approximately magnetized in the plane of the magnetic
coating on the tip. A periodic structure with alternative gold and silicon
dioxide stripes provides unpolarized nucleons with periodic number density
modulation. The exotic forces are expected to change the oscillation amplitude
of the cantilever which is measured by a fiber laser interferometer. Data has
been taken by scanning the tip over the nucleon source structure at constant
separation, and no exotic signal related to the density modulation has been
observed. Thus, the experiment sets a limit on the electron-nucleon coupling
constant, $g_{A}^{e}g_{V}^{N}\leq 9\times 10^{-15}$ for 15 $\mu$m
$\leq\lambda\leq$ 180 $\mu$m, using a direct force measurement method.

##  I Introduction

The experimental search for the exotic spin-dependent interactions has
received substantial attention in the last decade[1, 2, 3]. These interactions
may mediated by hypothetical bosons such as axions[4, 5, 6, 7], familons[8,
9], Majorons[10, 11], arions[12], paraphotons[13, 14, 15], and new
$Z^{\prime}$ bosons[16, 17, 18]. These new bosons are introduced in efforts to
resolve some puzzles in physics such as the strong CP problem[4, 5, 6], the
nature of dark matter and dark energy[19, 20], and the hierarchy problem [21].
Under the framework of quantum field theory, Dobrescu and Mocioiu derived
sixteen types of interaction potentials generated by any generic spin-0 or
spin-1 boson exchange assuming rotational invariance[22]. These potentials
were re-derived and classified according to the types of physical couplings
recently by Fadeev $et$ $al.$ [23]. Except for the Yukawa-type interaction,
fifteen of them are spin dependent and generated through pseudoscalar, vector
or axial-vector couplings.

Among them, extensive searches [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46] have been conducted on the
axion-mediated spin-mass or spin-spin exotic interactions that were initially
proposed by Moody and Wilczek[47]. Experimental searches have also been
performed on the spin- and velocity- dependent exotic interactions[28, 48, 49,
50, 51, 45, 52, 53, 54, 18, 55]. In this paper, we explore one of them, which
is parity-odd as given by

${V_{12+13}}=g_{A}^{e}g_{V}^{N}\frac{\hbar}{4\pi}(\hat{\sigma}\cdot\vec{v})\frac{1}{r}e^{-r/\lambda},$
(1)

where $\hbar$ is the reduced Planck’s constant, $g_{A}^{e}$ is the axial-
vector coupling constant to electrons, $g_{V}^{N}$ is the vector coupling
constant to nucleons, $\hat{\sigma}$ is the Pauli vector of electron spin,
$\vec{v}$ is the relative velocity between electron and nucleon, and
$\lambda=\hbar/m_{b}c$ is the interaction range with boson mass $m_{b}$. This
potential can be generated by the exchange of a massive spin-1 boson
$Z^{\prime}$ in a Lorentz-invariant theory described by the Lagrangian
$\mathcal{L}_{Z^{\prime}}=Z^{\prime}_{\mu}\bar{\psi}\gamma^{\mu}(g_{\psi}^{V}+\gamma_{5}g_{\psi}^{A})\psi$[22,
23], here $\psi$ denotes the fermion field. For simplicity, we assume that
$g_{V}^{p}=g_{V}^{n}=g_{V}^{N}$, where $p$ and $n$ denote proton and neutron,
respectively.

The above potential can be treated as
$V=\gamma\hbar\hat{\sigma}\cdot\vec{B}_{eff}$, and then we can see that the
spin $\hat{\sigma}$ can be used to sense the effective magnetic field
$\vec{B}_{eff}$ produced by the interaction. Atomic magnetometer operating in
the spin-exchange relaxation-free (SERF) regime has been used to detect the
$\vec{B}_{eff}$ produced by a unpolarized bismuth germanate insulator, and the
experiment sets the strongest constraints on the coupling constant between
electrons and nucleons for $\lambda>1\times 10^{-4}\ m$ [56]. The measurements
of neutron spin rotation or spin relaxation rate of polarized 3He gas have
also been used to constrain the possible new interaction between polarized
neutrons and unpolarized matter[53, 54]. The exchange of vector bosons between
electrons and nucleons induces parity-nonconserving effects (PNC) in atoms and
molecules, which was used to constrain the parity violating vector axial-
vector nucleon-electron and nucleon-proton interactions[18] at the nanometer
range. Recently, the exotic interaction between the neutron and the matter of
the Earth was sought using a spin-echo based interferometry technique [55]. On
the other hand, spin polarized torsion pendulum is one of the most sensitive
technique to search for the new interaction at centimeter scale by macroscopic
torque measurement[28, 48, 49]. Here we use a sensitive cantilever with a
magnetic tip to search for the interaction between polarized electrons and
unpolarized nucleons at the micrometer scale by direct force measurement.

Figure 1: (a) Schematic drawing of the experiment. Dimensions are not in
scale. (b) The lateral position ($x$) dependence of the exotic force amplitude
calculated at a distance of 730 nm with $\lambda$ =1 $\mu$m,
$g_{A}^{e}g_{V}^{N}=2\times 10^{-13}$.

##  II Experimental scheme

The experiment is schematically shown in Fig. 1(a). A cantilever with a
magnetic tip is used as a force sensor to measure the force between the spin-
polarized magnetic tip and a unpolarized nucleon source at the micrometer
range. The magnetic tip is coated with hard magnetic CoCr alloy to provide
spin-polarized electrons. The tip was magnetized in the perpendicular
direction to the cantilever and the electron polarizations are approximately
in the plane of the magnetic coating. As the $V_{12+13}$ potential is
proportional to $(\hat{\sigma}\cdot\vec{v})$, the relative velocity is chosen
to be along the polarization direction in order to maximize the exotic force.
This is fulfilled by driving the cantilever vibrating perpendicularly. The
source mass is made of alternative high density (gold) and low density
(silicon dioxide) materials to create a nucleon density modulation. When the
tip is oscillating over the source mass, the vibration of the cantilever is
reinforced or weakened by the exotic interaction, and the oscillation
amplitude changes depending on the density of the material underneath the tip.
Therefore, a periodic signal correlated with the density modulation would be
expected if the exotic interaction is detectable. This design helps to
separate the signal of interest from the spurious signals, such as the
electrostatic force and Casimir force.

The cantilever is driven by a piezo actuator at given amplitude and frequency,
and its displacement is measured by a home-made fiber laser interferometer.
The exotic interaction acts as an additional damping (or exciting) force
proportional to the velocity of the cantilever. The equation of motion of the
cantilever can be written as

$m\ddot{z}+m\gamma\dot{z}+kz=F_{d}\cos(\omega_{d}t)+f_{s}\dot{z},$ (2)

where $m$, $k$, and $\gamma$ are the effective mass, spring constant and
damping factor of the cantilever, respectively. $F_{d}$ is the external
driving force amplitude of the piezo, and $\omega_{d}$ is the driving angular
frequency. The $f_{s}=F_{s}/v$ is the spin-dependent exotic force ($F_{s}$)
divided by velocity. In this study, the cantilever is constantly driven at its
resonance frequency, and the cantilever’s vibrational amplitude is given by

$z_{amp}\approx z_{far}\left(1+\frac{Q\omega_{0}f_{s}}{k}\right),$ (3)

where $z_{far}=QF_{d}/k$ is the vibration amplitude when the magnetic tip is
free from any extra interaction, which can be determined when the magnetic tip
is far away from the source mass. $Q=\omega_{0}/\gamma$ is the quality factor
of the cantilever, $\omega_{0}$ is its resonance angular frequency. To avoid
the interference of constant background signals, we search for the exotic
interaction by looking for the amplitude variation when the magnetic tip scans
over different areas of nucleon density.

##  III Experimental details

###  III.1 Magnetic force probe

Figure 2: (a), (b) SEM images of the magnetic tip. (c) The hysteresis of the
CoCr coating on the silicon substrate measured with magnetic field applied in
plane and out of plane, respectively. (d) The measured first and second
resonances of the cantilever in comparison to the results from finite element
simulation. (e) MFM image ($\Delta f$) of the periodically magnetized hard-
disk plate taken with the magnetic tip.

A commercially available magnetic force probe was used in this experiment[57,
58, 59]. The cantilever is made of single crystalline silicon with dimension
of 121.90 $\mu$m $\times$ 34.77 $\mu$m $\times$ 1.62 $\mu$m as measured by
scanning electron microscopy (SEM). The eigenfrequencies of the first two
bending modes were measured to be $f_{1}=142309$ Hz and $f_{2}=915089$ Hz,
respectively. We determined the effective spring constant of the cantilever by
matching the two measured eigenfrequencies to the values calculated by finite
element analysis. The frequency match is achieved with a relative uncertainty
of 1.9% by only tuning the cantilever thickness to 1.415 $\mu$m (Fig. 2(d)).
The thickness is close to the measured value after subtracting the coating
layer. The difference in resonance frequency between the experiment and the
simulation should be due to the fact that the coating is not considered in the
simulation, because of the lack of affirmative property inputs, and its much
smaller thickness compared to the silicon cantilever. The spring constant is
derived from the simulation to be 3.6(4) N/m, and the main uncertainty is
contributed from the uncertain location of the laser spot on the cantilever.

The magnetic tip has a pyramid shape, and is coated by a layer of hard
magnetic CoCr alloy film with a Co:Cr ratio of 80:20. The nominal thickness of
the coating is 50 nm. The magnetic properties of the coating layer were
evaluated by a vibration sample magnetometer (VSM). Since the tip is too tiny
to contribute a detectable signal, we instead measured the magnetization curve
of the silicon substrate that holds the tip. As the CoCr alloy was deposited
on the silicon substrate and tip simultaneously, we assume that the coatings
have the same quality. Figure 2(c) shows the in-plane and out-of-plane
magnetization curves which is qualitatively consistent with previous
measurements[60, 61]. The results indicate that the easy axis is in the plane
and the in-plane hysteresis has a squareness ($M_{r}/M_{s}$) of 0.76, so that
we would expect that the magnetic moments of the magnetized tip would
domainantly stay parallel to the tip surface and point along the applied field
direction. The remnant magnetization per area is calculate from the
measurement to be $M_{r}^{a}=0.034$ A, which is used to calculate the exotic
forces later.

In order to qualitatively verify the magnetism of the tip, we have used the
tip to image a hard-disk plate with magnetic force microscopy (MFM). The hard-
disk plate was magnetized to periodic patterns using a home-made magnetic
writing device. The period of the pattern is $\sim$ 10 $\mu$m. In the MFM
imaging, the topography is measured in the trace scan and then the resonance
frequency shift ($\Delta f$) is measured in the retrace scan at a lifted
height. Figure 2(e) shows the frequency shift image taken at a lifted height
of 150 nm. The period of the image is consistent with the expected value,
which qualitatively verify the magnetism of the tip. The images have been
taken both before and after the experimental runs, and we observed no
significant change between those images.

###  III.2  Nucleon source structure

Figure 3: (a) The cross section of the density modulation source structure.
(b) The AFM topography image of the source structure surface.

A periodic structure with alternative high density (gold) and low density
(silicon dioxide) materials is used to provide the source of nucleons. The
density modulation structure was fabricated on a silicon wafer topped with a
200 nm thick SiO2 layer. A 190 nm thick gold film was first evaporated on the
oxide as a conductive layer, and then another 1.01 $\mu$m thick SiO2 layer was
deposited on the gold layer by plasma enhanced chemical vapor deposition
(PECVD), which serves as the low-density material. Afterwards, the SiO2 layer
was patterned to parallel trenches with a period of 20.10 $\mu$m using optical
lithography and reactive ion etching (RIE). Each trench has a width of $\sim$
10 $\mu$m and a depth of 1.01 $\mu$m. The trench was then filled with gold by
electroplating, which is used as the high nucleon density material. Finally,
the structure was glued on a glass substrate, and then the silicon wafer was
removed by alkali etching (20 % KOH with addition of hydroxylamine) followed
by HF acid etching of the oxide layer. Figure 3 presents a SEM image of the
cross section of the nucleon source structure, the dimensions of the structure
are listed in Table LABEL:parameter. The surface roughness is around 4 nm as
measured with atomic force microscopy (AFM), and the averaged periodic
corrugation is $\sim$ 2 nm.

Table 1: Mean values and uncertainties of the main experimental parameters. Parameter | Value | Error | Unit
---|---|---|---
Magnetic tip |  |  |
longer diagonal length (part 1) $l_{1}$ | 1.67 | 0.05 |  $\mu$m
longer diagonal length (part 2) $l_{2}$ | 4.55 | 0.13 |  $\mu$m
half shorter diagonal length $w_{s}$ | 2.87 | 0.13 |  $\mu$m
tip apex offset $l_{0}$ | 0.98 | 0.04 |  $\mu$m
tip height $h_{tip}$ | 8.15 | 0.27 |  $\mu$m
Cantilever |  |  |
length $l$ | 121.90 | 0.28 |  $\mu$m
width $w$ | 34.77 | 0.47 |  $\mu$m
thickness $t$ | 1.62 | 0.15 |  $\mu$m
tilt angle $\theta$ | 175.8 | 0.3 | mrad
resonance frequency $f_{1}$ | 142.31 | 0.14 | kHz
quality factor $Q$ | 6464 | 40 |
spring constant $k$ | 3.6 | 0.4 | N/m
Nucleon source structure |  |  |
width of the gold stripe | 9.87 | 0.06 |  $\mu$m
width of the SiO2 stripe | 10.23 | 0.06 |  $\mu$m
thickness | 1.01 | 0.04 |  $\mu$m

###  III.3 Displacement amplitude measurement

The experiment has been performed on a home-built scanning probe microscope as
described in ref. [62, 52]. The measurements have been conducted in frequency
modulation atomic force microscopy (FM-AFM) mode with constant drive, where
the cantilever is excited by a piezo plate with a constant drive amplitude.
The drive frequency is tracked to the resonance of the cantilever dynamically
with phase-locked loop control. The displacement of the cantilever is measured
by a fiber interferometer. Its displacement sensitivity,
$S_{int}=\mathrm{d}z/\mathrm{d}V_{int}$, is obtained through polynomial
fitting of the cavity length dependent signal $V_{int}$ near the working point
where the cavity length is adjusted to maximize the sensitivity. The
displacement amplitude is demodulated with a lock-in amplifier. The noise of
the amplitude measurement is evaluated at large tip-sample distance where the
tip is supposed to be free of sample’s action. The standard deviation is
estimated to be 14 pm for a acquisition time of 60 ms.

##  IV  Results and discussions

###  IV.1  Displacement amplitude imaging

To avoid accidentally recording data in special location, we have taken two
dimensional (2D) images at several regions of hundreds micrometer away to
check the consistency. The images were taken by scanning the magnetic tip over
the source mass at constant tip-sample separation. Both oscillation amplitude
and interferometer signal were taken simultaneously. The image size is usually
40 $\mu$m $\times$ 40 $\mu$m ($128\times 128$ pixels). The fast scan direction
is chosen to along the $x$-direction (perpendicular to the stripes). The data
acquisition time is 60 ms for every pixel. The sensitivity $S_{int}$ is
calculated for every pixel according to its interferometer signal $V_{int}$,
which can partially correct the drift of the sensitivity.

Figure 4: (a) Typical image of the oscillation amplitude taken at a tip-
surface distance of 500 nm. (b) Post-processed image after subtraction of the
linear background along the $x$-direction line by line. (c) The $x$-dependence
of the oscillation amplitude derived from the post-processed image by
averaging along the $y$-axis. A simple sine wave fit is also presented.

Figure 4(a) shows a typical image taken at a tip-surface distance of 500 nm.
The oscillation amplitude is around 2.6 nm. In this image, we observe no
obvious periodicity related to the density modulation. In order to highlight
the signal variation with respect to the density modulation other than a
constant or linear background, we subtract the data by its linear fit along
the $x$-direction line by line. The result is presented in Fig. 4(b) where
random signals are observed with a standard deviation of 14 pm, which is
identical to the value obtained at far tip-sample distance. We further average
the linearly subtracted data along the $y$-direction to look for the $x$
dependence of the signal. A simple sine wave fit to the data shows that the
amplitude is 0.13$\pm$0.17 pm. Such measurements have been performed at four
locations laterally separated more than hundreds micrometers away from each
other. All images show similar characteristics, and no obvious periodicity has
been observed with respect to the density modulation.

The distance dependence of the data can give us important information about
the exotic interaction. It is expected that the interaction should decay
exponentially with distance. We have performed such measurements at distances
of 730 nm, 980 nm, 1294 nm, 1573 nm, 1806 nm. The images taken at different
distances exhibit similar characteristics, and no obvious periodic patterns
can be observed. Except that the data points at larger distances unusually
have larger values, the signal is almost independent of distance within the
fitting error (see Fig. 5). The larger values may be due to the unstability of
the cantilever oscillation for these two data sets. The result implies that no
novel exotic interaction is observed in this experiment. We will then derive
the constraints on the coupling constant in the following sections using the
data taken at the shortest distance.

Figure 5: Distance dependence of the signal obtained by fitting the 2D images
to the sine wave surface function $z(x,y)=A\mbox{sin}[2\pi(x-x_{0})/\Lambda]$,
where $\Lambda$ is the density modulation period.

###  IV.2  Exotic force calculation

In order to extract the coupling constant of the exotic force from the
experimental data, the exotic force is calculated by numerical integration.
The interaction potential is obtained by integrating all spins over the
coating layer of the tip and all the nucleons in the source mass, which is
given by

$V_{12+13}(x,d)=g_{A}^{e}g_{V}^{N}\frac{\hbar}{4\pi}\int_{tip}\mathrm{d}S_{t}\int_{source}\mathrm{d}V_{s}n(x_{s},y_{s},z_{s})\frac{R_{so}\vec{M}_{r}^{a}\cdot\vec{v}}{\mu_{B}}\frac{1}{r}e^{-r/\lambda},$
(4)

where $n(x_{s},y_{s},z_{s})$ is the nucleon number density at
$(x_{s},y_{s},z_{s})$ in the source mass, $r$ is the distance between the spin
on the tip and the source mass element, $\mu_{B}$ is the Bohr magneton, and
$R_{so}$ is the ratio of spin magnetic moment to the total magnetic moment.
The angle ($\theta$) between the cantilever and the source surface is
considered in the calculation. The potential is a function of the relative
lateral position $x$ and distance $d$ between the tip and the source mass. By
taking derivative of Eq. (4) with respect to $d$, the $z$ component of the
exotic force is derived. The $R_{so}$ is determined to be 0.92 according to
the spin-polarized density functional theory (DFT) calculation of the CoCr
alloy (see details in Appendix. A). An example of the expected force signal is
shown in Fig. 1(b), and we can see that the force is periodic with respect to
the relative position $x$.

###  IV.3  Spurious forces

The exotic force depends on the relative velocity, so that the higher
oscillation frequency, the greater the expected signal. That is why we chose a
stiffer cantilever with a higher resonance frequency. At such high frequency,
the vibration noise of the environments is extremely small. For the spurious
forces, such as the Casimir force, electrostatic force and magnetic force,
they are all independent of the relative velocity. They affect the oscillation
amplitude of the cantilever by introducing an extra effective spring constant
equal to the force gradient ($k_{ts}=-\partial F_{ts}/\partial z$) between the
tip and source mass.

We calculated the Casimir force and electrostatic force by using the proximity
force approximation[63]. At a tip-surface distance of 500 nm, the Casimir
force gradient is estimated to be $2.5\times 10^{-9}$ N/m, and the
electrostatic force gradient is estimated to be $2.0\times 10^{-10}$ N/m with
5 mV residual potential difference between the tip and source mass. Their
impact on the oscillation amplitude is less than $10^{-8}$ nm, which is
negligible compared to the noise floor of the amplitude measurement. The
normal magnetic force between the magnetic tip and the source mass is
evaluated with finite element simulation. In order to calculate the tiny
force, the source mass is simulated as a diamagnetic material of exaggerated
magnetic susceptibility, and then the result is extrapolated to the case of
gold and silicon dioxide. The calculated force gradient is approximately 8.51
$\times 10^{-6}$ N/m on gold, and 8.54 $\times 10^{-6}$ N/m on SiO2. Their
effect on the amplitude is $\sim$ 3 fm (the difference between Au and SiO2
should be smaller), which is also negligible in this experiment. Therefore,
the influence of the above forces can be ignored thanks to the velocity
dependence of the exotic force and the resonance detection scheme.

###  IV.4  Constraints on the coupling constant

To set constraints on the coupling strength of the exotic interaction, we
analyze the images with maximum likelihood estimation. The probability density
of observing the image is calculated as a function of the position offset
$x_{0}$ and the coupling strength $f_{v}=2g_{A}^{e}g_{V}^{N}$ for every
$\lambda$ according to

$P(x_{0},f_{v},\lambda)=\frac{1}{A}\prod{\frac{1}{{\sqrt{2\pi}{\sigma_{ij}}}}{e^{-{{\left[{f_{ij}^{\exp}-f_{ij}^{T}\left({{f_{v}},\lambda}\right)}\right]}^{2}}/2\sigma_{ij}^{2}}}},$
(5)

where $A$ is the normalization coefficient, $f_{ij}^{\exp}$,
$f_{ij}^{T}\left({{f_{v}},\lambda}\right)$ and $\sigma_{ij}$ are the
experimental data, theoretical value, and total experimental uncertainty at
pixel $(i,j)$, respectively.

The exotic force per unite velocity $f_{ij}^{T}$ is calculated as described in
section IV.2. Here, we conservatively use the $2\sigma$ bound values of the
experimental parameters to calculate the theoretical values. It should be
noted that only spins in the dashed rectangle marked in Fig. 2(a) are used in
the calculation, so that the real value should be larger. The $f_{ij}^{\exp}$
is the experimental data at pixel $(i,j)$, converted from the oscillation
amplitude image by Eq. (3). The experimental uncertainty of $f_{ij}^{\exp}$ is
the quadrature sum of the errors contributed from $k$, $Q$, $\omega_{0}$,
$z_{far}$ and $z_{amp}(i,j)$. The parameters $\omega_{0}$, $Q$ and $z_{far}$
were measured at far distance where the cantilever is assumed to vibrate
freely. The mean values and uncertainties of those parameters are list in
Table LABEL:parameter. The uncertainty of amplitude measurement $\delta
z_{amp}$ is set to 14 pm as evaluated independently at far distance. An
example of the probability density function is shown in Fig. 6.

Figure 6: (a) Normalized probability density as a function of $x_{0}$ and
$f_{v}$ for $\lambda$ = 1 $\mu$m . (b) Normalized probability density as a
function of $f_{v}$ by intergrating out $x_{0}$.The curve is fitted to the
Gaussian function,which gives $f_{v}=3.3\times 10^{-16}\pm 3.6\times 10^{-13}$
at 95% confidential level.

The probability as a function of $f_{v}$ is obtained by integrating out the
$x_{0}$, and the up-bound of $f_{v}$ at 95% confidential level can be derived
for every $\lambda$. The result, based on the data taken at 730 nm distance,
is presented in Fig. 7 together with other experimental constrains reported
previously. The atomic magnetometer experiment constrains the exotic force
above 180 $\mu$m [56]. For the interaction range below 2.3 $\mu$m, the limits
are set by comparing the atomic PNC experimental result and theoretical
calculations[18]. This work sets stronger constraints than the PNC experiment
at the interaction range from 2.3 $\mu$m to 180 $\mu$m. We noticed that a
recent experiment using a single-electron spin quantum sensor reported a
stronger limit in this range[64]. Even stringent constraints may be obtained
by the combination of $g_{A}^{e}$ and $g_{V}^{N}$ from different experiments.
We derive the limit on $(g_{A}^{e})^{2}$ from the spin-spin experiments [42,
44]. By assuming $g_{s}^{N}=0$ [23, 65], the limit on $(g_{V}^{N})^{2}$ can be
obtained from the non-Newtonian gravity experiments [66, 67, 68]. Finally, we
obtain the limit on the combination $g_{A}^{e}g_{V}^{N}$, as shown by the
dashed line in Fig. 7. Nevertheless, direct experimental search with fewer
assumptions are still worthwhile.

Figure 7: Constraints on the dimensionless coupling constants
$g_{A}^{e}g_{V}^{N}$ from this work as well as previous experiments[18, 56,
64]. The dashed line shows the limit on the combination of $g_{A}^{e}$ and
$g_{V}^{N}$ as explained in the main text.

##  V  Conclusions

In conclusion, we have performed an experimental search for the exotic parity-
odd spin- and velocity-dependent interaction with a magnetic force microscope.
The magnetic probe is taken as a source of the moving spin polarized electrons
and also the force sensor to measure the exotic interaction. The magnetic
properties of the tip is evaluated quantitatively by VSM and qualitatively by
MFM measurement on a hard-disk plate. The density modulation source mass is
adopted to generate a periodic signal, so that we are able to extract the
exotic interaction from non-periodic background. We obtain the coupling
constant by using maximum likelihood estimation, and observe no new
interaction above the experimental sensitivity. The experiment sets
constraints on the coupling constant between spin polarized electrons and
unpolarized nucleons at the micrometer range using a direct force measurement
method. Future improvements could be achieved by replacing the magnetic tip
with a spherical magnet. The spherical magnet of micrometer size can be
magnetized to single domain due to its symmetry and could provide more spins
to interact with the source mass.

###### Acknowledgements.

The authors thank Fei Xue for useful discussions. This work was supported by
the National Natural Science Foundation of China under Contract Nos. 11875137,
91736312, 91436212, 51972312 and 51472249. The DFT calculations in this work
were performed on TianHe-1(A) at the National Supercomputer Center in Tianjin.

##  Appendix A Density functional theory calculation

The spin-polarized DFT calculations are performed by using the Vienna Ab
Initio Simulation Package (VASP) code[69]. Perdew-Burke-Ernzerhof (PBE)
functional is employed to describe the exchange correlation interactions
within the generalized gradient approximation. The electron-ion interactions
are represented by the projector augmented wave (PAW) method[70, 71]. The
energy cutoff of plane wave is set to be 500 eV and the convergence criterion
for the residual forces and total energies are set to be 0.01 eV/$\r{A}$ and
$10^{-6}$ eV, respectively. The primitive cell with two Co atoms, a
2$\times$2$\times$1 supercell with 6 Co and 2 Cr atoms, and a
2$\times$2$\times$2 supercell with 13 Co and 3 Cr atoms are constructed to
calculate the spin and orbital magnetic moment of $hcp$-Co, $hcp$-Co3Cr and
$hcp$-Co13Cr3 alloy, respectively. The $\Gamma$-centered 9$\times$9$\times$6,
5$\times$5$\times$6 and 5$\times$5$\times$2 Monkhorst-Pack k-point mesh are
used to sample the Brillouin zones of $hcp$-Co, $hcp$-Co3Cr and $hcp$-Co13Cr3
alloy, respectively. The $R_{so}$ values are found to be 0.96, 0.93, 0.90 for
pure Co, Co13Cr3 and Co3Cr, respectively. The value decreases with increasing
the concentration of Cr. By interpolating the data, we estimate the $R_{so}$
value for Co80Cr20 to be 0.92.

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$c_{n,\beta,\delta}=\frac{1}{\pi^{n}n!}\prod_{j=0}^{n-1}\frac{\Gamma(\frac{\beta}{2}(j+1)+1+\delta)\Gamma(\frac{\beta}{2}(j+1)+1+\bar{\delta})}{\Gamma(\frac{\beta}{2}(j+1))\Gamma(\frac{\beta}{2}(j+1)+1+\delta+\bar{\delta})}.$

This finishes the proof. ∎

Note that using the explicit description of the joint distribution of the
modified Verblunsky coefficients of the random probability measure
$\mu_{n,\beta,\delta}^{\textup{CJ}}$, and the method developed in Section 7 of
[19], one can also obtain the joint density of the perturbed circular Jacobi
beta ensemble. We omit the computation to shorten our representation.

###  7.2 Edge limit of the truncated circular Jacobi beta ensemble

In this section, we prove the edge limits of the rank-one truncation and
multiplicative perturbation of the circular Jacobi beta ensemble.

Recall the Dirac operator representation $\mathtt{CJ}_{n,\beta,\delta}$
defined in Definition 29. We denote by $\widetilde{\tau}_{n,\beta,\delta}$ the
reversed version of $\mathtt{CJ}_{n,\beta,\delta}$ via Definition 40, and by
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta}$ to be the
pulled-back version of $\widetilde{\tau}_{n,\beta,\delta}$ via Definition 41.

Our approach will be similar to the one used for the circular beta ensemble
case (which correspond to $\delta=0$). We will show that under appropriate
transformations, the operators
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta}$ and
$\mathtt{CJ}_{n,\beta,\delta}$ are orthogonally equivalent. Note however, that
in the $\delta\neq 0$ the measure $\mu_{n,\beta,\delta}^{\operatorname{CJ}}$
is no longer invariant under rotations, which requires us to develop a new
method to prove the orthogonal equivalence. The key ingredient is the
following proposition, providing equivalent descriptions of the conditioned
path parameters of $\mathtt{CJ}_{n,\beta,\delta}$.

######  Proposition 58.

Let $\mu=\mu_{n,\beta,\delta}^{\textup{CJ}}$ and let $\gamma_{k},0\leq k\leq
n-1$ be its modified Verblunsky coefficients. Let $b_{k},0\leq k\leq n$ be the
path parameters of $\mathtt{CJ}_{n,\beta,\delta}$ in ${\mathbb{D}}$. Then the
following sequences have the same joint distribution.

  1. (1)

The path parameters $b_{k},0\leq k\leq n$ conditioned on $b_{n}=1$.

  2. (2)

The path parameters corresponding to the sequence of modified Verblunsky
coefficients
$\bar{\gamma}_{0}^{\iota},\bar{\gamma}_{1}^{\iota},\dots,\bar{\gamma}_{n-2}^{\iota},1$.

  3. (3)

The ‘pulled back’ path parameters
$b_{k}^{\prime}:={\mathcal{A}}_{\hat{b}_{n-1},{\mathbb{D}}}(\hat{b}_{n-k-1}),0\leq
k\leq n$, where $\hat{b}_{-1}=1$ and $\hat{b}_{k},0\leq k\leq n-1$ are the
first $n$ elements of the path parameters produced by the sequence of modified
Verblunsky coefficients
$\bar{\gamma}_{n-2},\bar{\gamma}_{n-3},\dots,\bar{\gamma}_{0}$.

The proof relies on a special decomposition property of the
$\Theta(a+1,\delta)$ distribution and an application of Doob’s h-transform.
Before presenting the proof, we first introduce a Pearson-type distribution,
and a couple of facts about the $\Theta(a+1,\delta)$ distribution.

######  Definition 59.

For $m>1/2$ and $\mu\in{\mathbb{R}}$ we denote by $P_{IV}(m,\mu)$ the
distribution of the (unscaled) Pearson type IV distribution on ${\mathbb{R}}$
that has density function

$\displaystyle\frac{2^{2m-2}|\Gamma(m+\frac{\mu}{2}i)|^{2}}{\pi\,\Gamma(2m-1)}(1+x^{2})^{-m}e^{-\mu\arctan
x}.$ (105)

Note that the random variable $\Theta(1,\delta)$ can be connected to the
Pearson random variable $P_{IV}(\Re\delta+1,-2\Im\delta)$ via the mapping
$e^{i\theta}\mapsto-\cot(\theta/2)$.

######  Fact 60 ([24]).

Suppose that $\gamma\sim\Theta(a+1,\delta)$ with $a\geq 0$ and
$\Re\delta>-1/2$. Define $w,v\in{\mathbb{R}}$ with
$\tfrac{2\gamma}{1-\gamma}=w-iv$. Then the joint density of
$(v,1+w)\in{\mathbb{R}}\times{\mathbb{R}}_{+}$ is given by

$f_{a,\delta}(x,y)=c_{a,\delta}\,y^{\frac{a}{2}-1}(x^{2}+(1+y)^{2})^{-(\frac{a}{2}+\Re\delta+1)}e^{2\Im\delta\arctan\frac{x}{1+y}},$
(106)

with
$c_{a,\delta}=2^{a+2\Re\delta}\frac{\Gamma(a/2+1+\delta)\Gamma(a/2+1+\bar{\delta})}{\pi\Gamma(a/2)\Gamma(a/2+1+2\Re\delta)}$.

Moreover, the random variables $w$ and $\frac{v}{2+w}$ are independent. The
distribution of the random variable $\tfrac{v}{2+w}$ is given by
$P_{IV}(\tfrac{a}{2}+\Re\delta+1,-2\Im\delta)$. The distribution of $1+w$ is
the same as the distribution of $G_{1}/G_{2}$, where $G_{1},G_{2}$ are the
independent (standard) Gamma random variables with parameters $\frac{a}{2}$
and $\frac{a}{2}+2\Re\delta+1$, respectively.

Using Fact 60 and the definition of $\gamma^{\iota}$ in (94), one obtains the
following property.

######  Fact 61.

Suppose that $\gamma\sim\Theta(a+1,\delta)$ with $a\geq 0$ and
$\Re\delta>-1/2$. Then $\gamma^{\iota}\sim\Theta(\tilde{a}+1,\tilde{\delta})$
with $\tilde{a}=a+4\Re\delta+2$, $\tilde{\delta}=-(1+\delta)$.

###### Proof.

With a bit of abuse of notation, define
$w,v,w^{\iota},v^{\iota}\in{\mathbb{R}}$ with

$w-iv=\frac{2\gamma}{1-\gamma},\quad\quad
w^{\iota}-iv^{\iota}=\frac{2\gamma^{\iota}}{1-\gamma^{\iota}}.$ (107)

They satisfy the identities ${\mathcal{U}}^{-1}(\gamma)=v+i(1+w)$ and
${\mathcal{U}}^{-1}(\gamma^{\iota})=v^{\iota}+i(1+w^{\iota})$. By (107) and
(94) we have

$1+w^{\iota}=\frac{1}{1+w},\quad\quad\frac{v^{\iota}}{2+w^{\iota}}=-\frac{v}{2+w}.$

From Fact 60, we obtain that the joint density of $(v^{\iota},1+w^{\iota})$ is
proportional to

$y^{\frac{a}{2}+2\Re\delta}(x^{2}+(1+y)^{2})^{-(\frac{a}{2}+\Re\delta+1)}e^{-2\Im\delta\arctan\frac{x}{1+y}}.$

This shows that if $\gamma\sim\Theta(a+1,\delta)$, then
$\gamma^{\iota}\sim\Theta(\tilde{a}+1,\tilde{\delta})$, with
$\tilde{a}:=a+4\Re\delta+2$ and $\tilde{\delta}:=-(1+\delta)$. Note that the
random variable $\gamma^{\iota}$ is well-defined since the conditions
$\tilde{a}>0$ and $\tilde{a}/2+\Re\tilde{\delta}=a/2+\Re\delta>-1/2$ are still
satisfied. ∎

The next result shows that the right boundary point of the
$\mathtt{CJ}_{n,\beta,\delta}$ operator has the same distribution as a
$\Theta(1,\delta)$ random variable.

######  Proposition 62.

Let $b_{n}$ be the right boundary point of the Dirac operator
$\mathtt{CJ}_{n,\beta,\delta}$ in ${\mathbb{D}}$. Then
$b_{n}\stackrel{{\scriptstyle d}}{{=}}\Theta(1,\delta)$.

###### Proof.

Fix $n$, and let $b_{k}:=b_{k}^{(n)},0\leq k\leq n-1$ be the path parameters
of $\mathtt{CJ}_{n,\beta,\delta}$ in ${\mathbb{D}}$, these solve the recursion
(52) with $b_{0}=0$. We can consider the solution $b_{k,\gamma},0\leq k\leq n$
of this recursion for any initial value $b_{0,\gamma}=\gamma\in{\mathbb{D}}$.
We denote the probability density function of $b_{n,\gamma}$ by
$P_{n}(\gamma,\eta)$.

Recall from Proposition 28 that the modified Verblunsky coefficients
$\gamma_{k}=\gamma_{k}^{(n)},0\leq k\leq n-1$ of
$\mathtt{CJ}_{n,\beta,\delta}$ are independent, with
$\gamma_{k}\sim\Theta(\beta(n-k-1)+1,\delta)$.

In the case when $n=1$, we get $b_{n}=b_{1}=\gamma_{0}$ from the recursion
(52), so the statement follows. We also have

$\displaystyle
P_{1}(0,\eta)=c_{\delta}(1-\eta)^{\bar{\delta}}(1-\bar{\eta})^{\delta},\quad
c_{\delta}=\frac{\Gamma(1+\delta)\Gamma(1+\bar{\delta})}{\Gamma(1+\delta+\bar{\delta})}.$

Note that by (52) the distribution of $b_{k+1}$ given $b_{k}$ is invariant
with respect to isometries of the disk that preserve $1$. These isometries are
just the Poincaré disk version of the affine isometries of ${\mathbb{H}}$,
which can be parameterized as
${\mathcal{A}}_{a,{\mathbb{D}}}(z)=\frac{z-a}{1-\bar{a}z}\frac{1-\bar{a}}{1-a},a\in{\mathbb{D}}$
according to (49). Therefore, we have

$\displaystyle
P_{1}(\gamma,\eta)=c_{\delta}\left(\frac{(1-|\gamma|^{2})(1-\eta)}{(1-\bar{\gamma}\eta)(1-\gamma)}\right)^{\bar{\delta}}\left(\frac{(1-|\gamma|^{2})(1-\bar{\eta})}{(1-\gamma\bar{\eta})(1-\bar{\gamma})}\right)^{\delta}\frac{1-|\gamma|^{2}}{|1-\bar{\gamma}\eta|^{2}}.$

For $n\geq 2$ we will proceed by induction. Assume that
$P_{n-1}(\gamma,\eta)=P_{1}(\gamma,\eta)$ (or equivalently,
$P_{n-1}(0,\eta)=P_{1}(0,\eta)$). The proof will be completed if we can show
that $P_{n}(0,\eta)=P_{1}(0,\eta)$. Let

$\displaystyle
f_{a,\delta}=c_{a,\delta}(1-|z|^{2})^{\frac{a}{2}-1}(1-z)^{\bar{\delta}}(1-\bar{z})^{\delta},\quad
a=\beta(n-1)$

be the density function of $\gamma_{0}^{(n)}$ as in Definition 27. Note that
$\gamma_{0}^{(n)}$ is independent of $\gamma_{k}^{(n)},1\leq k\leq n-1$, which
have the same joint distribution as $\gamma_{k}^{(n-1)},0\leq k\leq n-2$.
Hence from (52) we get

$\displaystyle P_{n}(0,\eta)$
$\displaystyle=\int_{z\in{\mathbb{D}}}P_{n-1}(z,\eta)f_{a,\delta}(z)dz=\int_{z\in{\mathbb{D}}}P_{1}(z,\eta)f_{a,\delta}(z)dz$
$\displaystyle=c_{\delta}(1-\eta)^{\bar{\delta}}(1-\bar{\eta})^{\delta}\int_{z\in{\mathbb{D}}}c_{a,\delta}\frac{(1-|z|^{2})^{\frac{a}{2}+\delta+\bar{\delta}}}{(1-\bar{z}\eta)^{\delta+1}(1-z\bar{\eta})^{\bar{\delta}+1}}dz.$

Using the change of variable $z\mapsto z\bar{\eta}$ we see that the integral
does not depend on $\eta$. Hence $P_{n}(0,\eta)$ is a constant multiple of
$P_{1}(0,\eta)$ which means that it must be equal to it. This completes the
induction step and finishes the proof. ∎

Now we turn to the proof of Proposition 58.

###### Proof of Proposition 58.

We first prove the equivalence between (2) and (3). By the recursion (53) and
the fact ${\mathcal{A}}_{\gamma,{\mathbb{D}}}^{-1}(0)=\gamma$, we have

$\displaystyle b_{k}^{\prime}$
$\displaystyle={\mathcal{A}}_{\bar{\gamma}_{0},{\mathbb{D}}}\circ{\mathcal{A}}_{\bar{\gamma}_{1},{\mathbb{D}}}\circ\cdots\circ{\mathcal{A}}_{\bar{\gamma}_{n-2},{\mathbb{D}}}\big{(}{\mathcal{A}}_{\bar{\gamma}_{n-2},{\mathbb{D}}}^{-1}\circ\cdots\circ{\mathcal{A}}_{\bar{\gamma}_{k},{\mathbb{D}}}^{-1}(0)\big{)}$
$\displaystyle={\mathcal{A}}_{\bar{\gamma}_{0}^{\iota},{\mathbb{D}}}^{-1}\circ\cdots\circ{\mathcal{A}}_{\bar{\gamma}_{k-1}^{\iota},{\mathbb{D}}}^{-1}(0).$

This shows that the sequence $b_{k}^{\prime},0\leq k\leq n-1$ are the first
$n$ elements of the path produced by
$\bar{\gamma}_{0}^{\iota},\dots,\bar{\gamma}_{n-2}^{\iota}$. (This fact was
also observed in Lemma 50 of [39].) By setting $\tilde{b}_{-1}=1$, we have
$b_{n}^{\prime}={\mathcal{A}}_{b_{n-1},{\mathbb{D}}}(1)=1$. Note also, that
$b_{n}^{\prime}=1$ if and only if the corresponding last modified Verblunsky
coefficient is equal to $1$. This proves that the path parameters described in
(2) and (3) are equal in law.

We now prove that the paths described in (1) and (2) have the same
distribution. Let $z_{k}={\mathcal{U}}^{-1}(b_{k}),0\leq k\leq n$ be the path
parameters of $\mathtt{CJ}_{n,\beta,\delta}$ in ${\mathbb{H}}$. By (51) this
is a (time-inhomogeneous) Markov chain, with transition densities that are
invariant under affine transformations. By Proposition 62 it follows that
$z_{n}\stackrel{{\scriptstyle d}}{{=}}{\mathcal{U}}^{-1}(\gamma_{n-1})\sim
P_{IV}(\Re\delta+1,-2\Im\delta)$ with density

$g(q)=\frac{4^{\Re\delta}|\Gamma(1+\delta)|^{2}}{\pi\,\Gamma(2\Re\delta+1)}(1+q^{2})^{-\Re\delta-1}e^{2\Im\delta\arctan
q}.$

Moreover, the proof of the same proposition also implies that the conditional
distribution of $z_{n}$ given $z_{k}=z=x+iy$ for a fixed $0\leq k\leq n-2$ is
the same as that of $y{\mathcal{U}}^{-1}(\gamma_{n-1})+x$. This random
variable has density

$g_{x,y}(q):=y^{-1}g(\frac{q-x}{y}).$

For any fixed $z=x+iy,z^{\prime}=x^{\prime}+iy^{\prime}\in{\mathbb{H}}$ we get

$\lim_{q\to\infty}\frac{g_{x^{\prime},y^{\prime}}(q)}{g_{x,y}(q)}=\frac{h(z^{\prime})}{h(z)},\qquad
h(z):=(\Im z)^{2\Re\delta+1}.$ (108)

Let $f_{a_{k},\delta}(x,y)$ be the density function of
${\mathcal{U}}^{-1}(\gamma_{k})$ defined via (106) with parameters
$a_{k}=\beta(n-k-1)$ and $\delta$. Then the transition density of the Markov
chain $z_{k},0\leq k\leq n$ is given by

$P\Big{(}(z,k),(z^{\prime},k+1)\Big{)}=f_{a_{k},\delta}\left(\frac{\Re(z^{\prime}-z)}{\Im
z},\frac{\Im z^{\prime}}{\Im z}\right),\qquad 0\leq k\leq n-1.$

Now consider the distribution of $z_{k},0\leq k\leq n$ conditioned on
$z_{n}=q$ with $q\to\infty$. By Doob’s h-transform, the conditioned transition
density is given by

$\displaystyle Q\Big{(}(z,k),(z^{\prime},k+1)\Big{)}$
$\displaystyle=P\Big{(}(z,k),(z^{\prime},k+1)\Big{)}\frac{h(z^{\prime})}{h(z)}$
$\displaystyle=c_{a_{k},\delta}\,v^{\frac{\beta}{2}(n-k-1)+2\Re\delta}(u^{2}+(1+v^{2}))^{-\frac{\beta}{2}(n-k-1)-\Re\delta-1}e^{2\Im\delta\arctan(u/(1+v))}$

where $u=(\Re z^{\prime}-\Re z)/\Im z$ and $v=\Im z^{\prime}/\Im z$. By
Proposition 61, $Q((z,k),(z^{\prime},k+1))$ is exactly the density function of
the random variable ${\mathcal{U}}^{-1}(\bar{\gamma}_{k}^{\iota})$. This
proves the equivalence between the statements (1) and (2), and hence completes
the proof. ∎

By Proposition 58 and Fact 61 we see that the effect of conditioning the path
parameters in ${\mathbb{D}}$ to hit $1$ is equivalent to changing the
parameter $\delta\mapsto-(1+\bar{\delta})$. This coincides with a similar
factorization lemma for the generating path of the $\tau_{\beta,\delta}$
operator, see Theorem 43 of [24].

######  Corollary 63.

Consider the same setup as in Proposition 58, in particular the path
$b_{k}^{\prime},0\leq k\leq n$ defined in (3). Let $\eta\sim\Theta(1,\delta)$
be independent of $\gamma_{k},0\leq k\leq n-2$. Then the rotated path
parameters $\check{b}_{k}:=\eta b_{k}^{\prime},0\leq k\leq n$ have the same
joint distribution as the path parameters of the
$\mathtt{CJ}_{n,\beta,\delta}$ operator.

###### Proof.

By Proposition 58, the path parameters $b_{k}^{\prime},0\leq k\leq n$ can be
produced by the sequence of modified Verblunsky coefficients
$\bar{\gamma}_{0}^{\iota},\dots,\bar{\gamma}_{n-2}^{\iota},1$. Let
$(\alpha_{0}^{\prime},\dots,\alpha_{n-1}^{\prime}):={\mathcal{T}}_{n}^{-1}(\bar{\gamma}_{0}^{\iota},\dots,\bar{\gamma}_{n-2}^{\iota},1)$
be the corresponding sequence of Verblunsky coefficients. By (40) and (98) we
get

$\alpha_{k}^{\prime}=(-1)^{k}\frac{\gamma_{k}^{\iota}\cdots\gamma_{0}^{\iota}}{\gamma_{k-1}\cdots\gamma_{0}},0\leq
k\leq
n-2,\qquad\alpha_{n-1}^{\prime}=(-1)^{n-1}\frac{\gamma_{n-2}^{\iota}\cdots\gamma_{0}^{\iota}}{\gamma_{n-2}\cdots\gamma_{0}}.$

Recall also the (equivalent) description of $b_{k}^{\prime},0\leq k\leq n$
using $\alpha_{k}^{\prime},0\leq k\leq n-1$ in (38). Observe that for
$Z=\operatorname{diag}(z,1)$, we have

$\displaystyle\mathcal{P}Z^{-1}\left(\begin{array}[]{cc}1&\bar{\alpha}_{0}\\\
\alpha_{0}&1\\\
\end{array}\right)\cdots\left(\begin{array}[]{cc}1&\bar{\alpha}_{k-1}\\\
\alpha_{k-1}&1\\\ \end{array}\right)Z\left(\begin{array}[]{c}0\\\
1\end{array}\right)=\bar{z}b_{k}.$ (115)

This shows that the Verblunsky coefficients $\check{\alpha}_{k},0\leq k\leq
n-1$ corresponding to $\check{b}_{k},0\leq k\leq n$ are given by
$\check{\alpha}_{k}=\bar{\eta}\alpha_{k}^{\prime}$, in particular

$\check{\alpha}_{k}=(-1)^{k}\bar{\eta}\frac{\gamma_{k}^{\iota}\cdots\gamma_{0}^{\iota}}{\gamma_{k-1}\cdots\gamma_{0}},0\leq
k\leq
n-2,\qquad\check{\alpha}_{n-1}=(-1)^{n-1}\bar{\eta}\frac{\gamma_{n-2}^{\iota}\cdots\gamma_{0}^{\iota}}{\gamma_{n-2}\cdots\gamma_{0}}.$

(The relation between rotating the sequences of path parameters and the
corresponding Verblunsky coefficients is related to the so-called Aleksandrov
measure. We refer to [32] for more background.)

Finally, using Claim 56 for $\zeta_{0}=\eta$, $\zeta_{j}=\gamma_{j-1},1\leq
j\leq n$ we get the following distributional identity:

$\left(\gamma_{0}^{\iota}\bar{\eta},\frac{\gamma_{1}^{\iota}}{\gamma_{0}},\dots,\frac{\gamma_{n-2}^{\iota}}{\gamma_{n-3}},\frac{1}{\gamma_{n-2}}\right)\stackrel{{\scriptstyle
d}}{{=}}\left(\bar{\gamma}_{0},\frac{\bar{\gamma}_{1}}{\bar{\gamma}_{0}^{\iota}},\dots,\frac{\bar{\gamma}_{n-2}}{\bar{\gamma}_{n-3}^{\iota}},\frac{\bar{\eta}}{\bar{\gamma}_{n-2}^{\iota}}\right).$

This means that $\check{\alpha}_{k},0\leq k\leq n-1$ have the same joint
distribution as the random variables

$\dot{\alpha}_{k}=(-1)^{k}\frac{\bar{\gamma}_{k}\cdots\bar{\gamma}_{0}}{\bar{\gamma}_{k-1}^{\iota}\cdots\bar{\gamma}_{0}^{\iota}},0\leq
k\leq
n-2,\qquad\dot{\alpha}_{n-1}=(-1)^{n-1}\frac{\bar{\eta}\bar{\gamma}_{n-2}\cdots\bar{\gamma}_{0}}{\bar{\gamma}_{n-2}^{\iota}\cdots\bar{\gamma}_{0}^{\iota}}.$

Since the joint distribution of $\gamma_{0},\dots,\gamma_{n-2},\eta$ is the
same as the joint distribution of $\gamma_{0},\dots,\gamma_{n-1}$, (40) and
(98) now shows that

$(\check{\alpha}_{0},\check{\alpha}_{1},\dots,\check{\alpha}_{n-1})\stackrel{{\scriptstyle
d}}{{=}}(\dot{\alpha}_{0},\dots,\dot{\alpha}_{n-1})\stackrel{{\scriptstyle
d}}{{=}}{\mathcal{T}}_{n}^{-1}(\gamma_{0},\dots,\gamma_{n-1}),$

which implies that $\check{b}_{k},0\leq k\leq n$ have the same joint
distribution as the path parameters $b_{k},0\leq k\leq n$ for
$\mathtt{CJ}_{n,\beta,\delta}$. ∎

Recall the path $\overset{\text{\tiny${\hookleftarrow}$}}{b}_{k},0\leq k\leq
n$, the pulled back version of the reversed path $\widetilde{b}_{k},0\leq
k\leq n$ corresponding to the random measure
$\mu_{n,\beta,\delta}^{\operatorname{CJ}}$. By Proposition 55
$\widetilde{b}_{k},0\leq k\leq n-1$ has the same distribution as the path
built from the modified Verblunsky coefficients
$\gamma_{n-2},\dots,\gamma_{0}$. This path is the complex conjugate of the
path $\hat{b}_{k},0\leq k\leq n-1$ in (3) of Proposition 58. This also means
that $\overset{\text{\tiny${\hookleftarrow}$}}{b}_{k},0\leq k\leq n$ is just
the time-reversed and complex conjugated version of the path
$b_{k}^{\prime},0\leq k\leq n$ in (3) of Proposition 58. Together with
Corollary 63 this implies that applying an independent random rotation, a
complex conjugation, and a time reversal to
$\overset{\text{\tiny${\hookleftarrow}$}}{b}_{k},0\leq k\leq n$ produces a
path that has the same distribution as the driving path of the operator
$\mathtt{CJ}_{n,\beta,\delta}$. This statement allows us to show that
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta}$ and
$\mathtt{CJ}_{n,\beta,\delta}$ are orthogonally equivalent.

######  Proposition 64.

Let $\overset{\text{\tiny${\hookleftarrow}$}}{b}_{k},0\leq k\leq n$ be the
path parameters defined in Corollary 63,
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta}$ the
corresponding Dirac-type operator, and let

$Q=\frac{1}{\sqrt{1+q^{2}}}\begin{pmatrix}q&1\\\ -1&q\end{pmatrix},\qquad
q={\mathcal{U}}^{-1}(\overset{\text{\tiny${\hookleftarrow}$}}{b}_{n}).$

Then the operator
$\rho^{-1}(SQ)\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta}(SQ)^{-1}\rho$
has the same distribution as $\mathtt{CJ}_{n,\beta,\delta}$.

###### Proof.

Recall the transformation $\mathcal{Q}$ defined in Section 2.2. Observe that
in the unit-disk model, the transformation $\mathcal{Q}$ behaves exactly as a
rotation, i.e. for $z\in{\mathbb{D}}$ we have
$\mathcal{Q}(z)=\mathcal{U}\circ\mathcal{Q}\circ\mathcal{U}^{-1}(z)=\bar{\eta}z$,
where we denote $\eta:=\overset{\text{\tiny${\hookleftarrow}$}}{b}_{n}$.

Together with Propositions 55 and 58, this observation shows that the path
parameters of
$\rho^{-1}(SQ)\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta}(SQ)^{-1}\rho$
have the same joint distribution as $\check{b}_{k}=\eta b_{k}^{\prime},0\leq
k\leq n$ where $b_{k}^{\prime},0\leq k\leq n$ are distributed as in
Proposition 58. Moreover, by the same argument one can check that the left
boundary point of
$\rho^{-1}(SQ)\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta}(SQ)^{-1}\rho$
is equal to $1$ as desired. The proof is now completed by Corollary 63. ∎

Recall the limiting operator $\mathtt{HP}_{\beta,\delta}$ defined in
Definition 32, and its reversed and transformed version $\tau_{\beta,\delta}$
defined in Definition 38. By Propositions 62 and 64 the transformation
connecting $\mathtt{HP}_{\beta,\delta}$ and $\tau_{\beta,\delta}$ has the same
distribution as the one connecting $\mathtt{CJ}_{n,\beta,\delta}$ and
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta}$.

The operator level convergence of the $\mathtt{CJ}_{n,\beta,\delta}$ to the
limiting $\mathtt{HP}_{\beta,\delta}$ operator was proved in [24] and can be
summarized as follows. This is also the final ingredient to prove the
convergence of the normalized characteristic polynomial of the truncated
circular Jacobi beta ensemble.

######  Proposition 65 ([24]).

Fix $\beta>0$ and $\Re\delta>-1/2$. There exists a coupling of the operators
$\mathtt{CJ}_{n,\beta,\delta}=\mathtt{Dir}(z_{n}(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1}^{(n)}),n\geq
1$ with
$\mathfrak{u}_{0}=\binom{1}{0},\mathfrak{u}_{1}^{(n)}=\binom{-z_{n}(1)}{-1}$,
and
$\mathtt{HP}_{\beta,\delta}=\mathtt{Dir}(z(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1})$
with $\mathfrak{u}_{1}=\binom{-z(1)}{-1}$ such that as $n\to\infty$ we have
almost surely $z_{n}\to z$ pointwise on $[0,1)$, and

$\|\mathtt{r\,}\mathtt{CJ}_{n,\beta,\delta}-\mathtt{r\,}\mathtt{HP}_{\beta,\delta}\|_{\textup{HS}}\to
0,\quad\quad\mathfrak{t}_{\mathtt{CJ}_{n,\beta,\delta}}-\mathfrak{t}_{\mathtt{HP}_{\beta,\delta}}\to
0.$

Set $c_{\delta}=\frac{4}{\beta}(\Re\delta+\frac{1}{2})>0$, and for
$\varepsilon>0$ small define
$c_{1}=c_{\delta}-\varepsilon,c_{2}=c_{\delta}+\varepsilon$. Then there exists
a sequence of tight random variables $\kappa_{n},n\geq 1$ and an a.s. finite
random variable $\kappa>0$ such that for $0\leq t<1$

$\displaystyle\kappa_{n}^{-1}\left(1-\frac{\lfloor
nt\rfloor}{n}\right)^{c_{2}}\leq y_{n}(t)\leq\kappa_{n}\left(1-\frac{\lfloor
nt\rfloor}{n}\right)^{c_{1}},\quad|x_{n}(1)-x_{n}(t)|\leq\kappa_{n}\left(1-\frac{\lfloor
nt\rfloor}{n}\right)^{c_{1}}$ (116)

and similarly,

$\kappa^{-1}(1-t)^{c_{2}}\leq
y(t)\leq\kappa(1-t)^{c_{1}},\quad|x(1)-x(t)|\leq\kappa(1-t)^{c_{1}}.$ (117)

We now have all the ingredients to prove the edge scaling limit of the
truncated circular Jacobi beta ensemble.

######  Theorem 66.

For fixed $n\geq 1$, $\beta>0$ and $\Re\delta>-1/2$, let $\lambda_{i},1\leq
i\leq n-1$ be size $n$ truncated circular Jacobi beta ensemble nd set
$p_{n-1,\beta,\delta}(z)=\prod_{i=1}^{n-1}\frac{z-\lambda_{i}}{1-\lambda_{i}}$
be the normalized characteristic polynomial. Let
${\mathcal{E}}_{\beta,\delta}$ be the structure function of
$\tau_{\beta,\delta}$ defined via (20). Then there is a coupling of
$p_{n,\beta,\delta},n\geq 1$ and ${\mathcal{E}}_{\beta,\delta}$ such that

$|p_{n-1,\beta}(e^{iz/n})e^{-iz/2}-{\mathcal{E}}_{\beta,\delta}(z)|\to
0\quad\text{ almost surely, uniformly on compacts as $n\to\infty$}$

Consequently, under the edge scaling (3), the truncated circular Jacobi beta
ensembles converge weakly to the zeros of the random analytic function
${\mathcal{E}}_{\beta,\delta}(\cdot)$.

###### Proof.

Let $\widetilde{\mu}_{n}$ be the reversed version of the measure
$\mu_{n}:=\mu_{n,\beta,\delta}^{\textup{CJ}}$, and
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta}$ be the
pulled-back operator. By Propositions 13 and 55, $p_{n-1,\beta,\delta}$ has
the same distribution as the monic orthogonal polynomial of degree $n-1$
associated to $\widetilde{\mu}_{n}$, denoted as
$\widetilde{\Psi}_{n-1,\beta,\delta}$. By Proposition 42, we have

$\widetilde{\Psi}_{n-1,\beta,\delta}(e^{iz/n})=e^{iz(n-1)/(2n)}\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}((n-1)/n,z)^{{\dagger}}\binom{1}{-i},$

where $\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}$ solves the ODE (18) of
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta}$. Recall
${\mathcal{E}}_{\beta,\delta}=H_{\beta,\delta}(1,\cdot)^{{\dagger}}\cdot\binom{1}{-i}$,
where $H_{\beta,\delta}(t,z)$ solves (18) of $\tau_{\beta,\delta}$. It is
enough to show the uniform-on-compacts convergence of
$\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}(1,z)$ to
$H_{\beta,\delta}(1,z)$.

By Propositions 39 and 64, one can obtain the operators
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta,\delta},n\geq 1$ and
$\tau_{\beta,\delta}$ from $\mathtt{CJ}_{n,\beta,\delta},n\geq 1$ and
$\mathtt{HP}_{\beta,\delta}$ under the same orthogonal transformations.
Therefore, applying the coupling of the operators
$\mathtt{CJ}_{n,\beta,\delta}$ and $\mathtt{HP}_{\beta,\delta}$ in Proposition
65 gives

$\|\mathtt{r\,}\tau_{n,\beta,\delta}-\mathtt{r\,}\tau_{\beta,\delta}\|\to
0,\quad\quad|\mathfrak{t}_{\tau_{n,\beta,\delta}}-\mathfrak{t}_{\tau_{\beta,\delta}}|\to
0.$

In order to apply Proposition 43, we need to show that

$\int_{0}^{1}|\mathfrak{a}_{0,n}(s)-\mathfrak{a}_{0}(s)|^{2}ds=\int_{0}^{1}\left|\frac{1}{\sqrt{y_{n}(s)}}-\frac{1}{\sqrt{y(s)}}\right|^{2}\to
0,$ (118)

where $y_{n}=\Im z_{n}$ and $y=\Im z$ are the imaginary part of the generating
paths of $\tau_{n,\beta,\delta}$ and $\tau_{\beta,\delta}$, respectively. By a
standard subsequence argument, the path bound (116) allows one to choose a
subsequence so that $\kappa_{n}$ converges to an a.s. finite random variable.
The convergence (118) now follows from the point-wise convergence of $z_{n}\to
z$, the path bounds (116), (117) (under the time reversal), and an application
of the triangle inequality. This verifies (75) and completes the proof. ∎

Note that by Proposition 39, the structure function
${\mathcal{E}}_{\beta,\delta}$ can also be characterized as
${\mathcal{E}}_{\beta,\delta}={\mathcal{H}}_{\beta,\delta}(0,z)^{{\dagger}}\binom{1}{-i}$,
where ${\mathcal{H}}_{\beta,\delta}(u,z)$ solves the SDE (71).

Note that in the coupling of Proposition 65, the path bounds (116), (117) and
the point-wise convergence of the generating paths $z_{n}\to z$ imply that
$\mathfrak{u}_{1}^{(n)}\to\mathfrak{u}_{1}$ a.s. (see also Proposition 62).
Then using Theorem 66 and Proposition 44 one obtains the following result on
the (edge) scaling limit of the multiplicative perturbed circular Jacobi beta
ensemble.

######  Corollary 67.

For fixed $r\in[0,1]$ let
$\Lambda_{n}^{[r]}=\\{\lambda_{1},\dots,\lambda_{n}\\}$ be the set of
eigenvalues of the perturbed matrix $\mathsf{CJ}^{[r]}_{n,\beta,\delta}$. Set
$p_{n,\beta,\delta}^{[r]}(z)=\prod_{i=1}^{n}\frac{z-\lambda_{i}}{1-\lambda_{i}}$
to be the normalized characteristic polynomial, and let $H_{\beta,\delta}$ be
the solution to the ODE (18) of $\tau_{\beta,\delta}$. Let
$q\sim\Theta(1,\delta)$ be independent of $H_{\beta,\delta}$. Then under the
coupling of Proposition 65, we have almost surely as $n\to\infty$

$\displaystyle\left|p_{n,\beta,\delta}^{[r]}(e^{iz/n})e^{-iz/(2n)}-{\mathcal{E}}_{\beta,\delta}^{[r]}(z)\right|\to
0,\quad\text{ uniformly on compacts in ${\mathbb{C}}$,}$

where
${\mathcal{E}}_{\beta,\delta}^{[r]}(z)=H_{\beta,\delta}(1,z)\cdot\binom{1}{-c_{r}}$
with $c_{r}=\frac{q+i\frac{1-r}{1+r}}{1-iq\frac{1-r}{1+r}}$. In particular,
this implies the weak convergence of $\Lambda_{n}^{[r]}$ under the edge
scaling (3) to the zero set of the random analytic function
${\mathcal{E}}_{\beta,\delta}^{[r]}$ as $n\to\infty$.

Similar to the statement of Theorem 3, we have $c_{r}=q$ when $r=1$ and
$c_{r}=i$ when $r=0$, hence this result shows the connection between the
scaling limits of the truncated and the unperturbed circular Jacobi beta
ensemble.

##  8 Appendix

###  8.1 Overview of some results for $\beta=2$

Let $\Lambda$ be a locally compact Polish space (in this section it will
always be ${\mathbb{C}}$). A simple point process on $\Lambda$ is called
_determinantal_ with respect to a reference measure $\mu$ with kernel function
$K:\Lambda\times\Lambda\to{\mathbb{R}}$ if for any $k\geq 1$ the $k$th joint
intensity function of the process with respect to $\mu$ is given by

$\displaystyle\rho_{k}(x_{1},\dots,x_{k})=\det(K(x_{i},x_{j}))_{1\leq i,j\leq
k}.$ (119)

(See [16] for more on determinantal point processes.) Determinantal processes
appear naturally from joint probability densities containing the square of the
Vandermonde. In this section we will always assume that all absolute moments
of $\mu$ are finite.

######  Proposition 68 ([26, 22]).

Suppose that the $X_{1},X_{2},\dots,X_{n}$ are complex valued random variables
with joint density given by

$\displaystyle\frac{1}{Z_{n,\mu}}\prod_{1\leq i<j\leq n}|z_{i}-z_{j}|^{2}$
(120)

with respect to a product measure $\mu^{\otimes n}$ (on ${\mathbb{R}}^{n}$ of
${\mathbb{C}}^{n}$). (Here $Z_{n,\mu}$ is a finite constant.) Let
$\varphi_{k}(z)$ be the degree $k$ orthonormal polynomials with respect to
$\mu$. Then $\sum_{k=1}^{n}\delta_{X_{k}}$ is a determinantal point process
with respect to $\mu$ with kernel given by

$\displaystyle K(z,w)=\sum_{k=0}^{n-1}\varphi_{k}(z)\bar{\varphi}_{k}(w).$
(121)

Proposition 68 together with (1) and (2) immediately imply that both the
circular unitary ensemble and the truncated circular unitary ensemble are
determinantal. The size $n$ circular unitary ensemble has kernel function

$\displaystyle K_{\mathsf{Circ}_{2,n}}(z,w)=\sum_{k=0}^{n-1}z^{k}\bar{w}^{k},$
(122)

with respect to the uniform measure on the unit circle. The size $n$ truncated
circular unitary ensemble has a similar kernel function

$\displaystyle
K_{{\mathsf{Circ}}_{2,n}^{\ulcorner}}(z,w)=\sum_{k=0}^{n-1}(k+1)z^{k}\bar{w}^{k},$
(123)

with respect to the uniform measure on the unit disk. Note that [41] also
treats more general truncations of Haar unitary matrices. They showed that if
we delete the first $m$ rows and columns of a size $n+m$ Haar unitary matrix
then the eigenvalues of the resulting submatrix have joint eigenvalue density
given by

$\frac{1}{Z_{n,m}}\prod_{1\leq j<k\leq
n}|z_{j}-z_{k}|^{2}\prod_{k=1}^{n}(1-|z_{k}|^{2})^{m-1},\qquad
z_{j}\in{\mathbb{D}},$ (124)

with respect to the Lebesgue measure on the unit disk. By Proposition 68 this
is also a determinantal point process, with respect to the measure with
density $(1-|z|^{2})$ on the unit disk.

Determinantal processes have a number of nice analytic features. The following
proposition shows that if we understand the scaling limit of the kernels of a
sequence of determinantal processes then we can derive the scaling limit and
the limit is also determinantal.

######  Proposition 69 ([34, 30]).

Suppose that $\mathcal{X},\mathcal{X}_{1},\mathcal{X}_{2},\dots$ are
determinantal processes on ${\mathbb{C}}$ with respect to the common reference
measure $\mu$, with kernel functions $K,K_{1},K_{2},\dots$. Assume that

$\displaystyle K_{n}(z,w)\to K(z,w)$ (125)

uniformly on compacts in ${\mathbb{C}}^{2}$. Then $\mathcal{X}_{n}$ converges
in distribution to $\mathcal{X}$.

Using some simple transformations one can rewrite the determinantal kernel of
the circular unitary ensemble as

$\tilde{K}_{\mathsf{Circ}_{2,n}}(e^{it},e^{is})=D_{n}(t-s),\qquad
D_{n}(u)=\frac{\sin(nu/2)}{\sin(u/2)}.$

By taking the limit of this kernel as $n\to\infty$ we get the celebrated
result of Gaudin, Mehta, Dyson regarding the the scaling limit of the circular
ensemble.

######  Theorem 70 ([2, 26]).

Let $\Lambda_{n}$ be the angles in the size $n$ circular unitary ensemble
parametrized in $(-\pi,\pi]$. Then $n\Lambda_{n}\Rightarrow\Lambda$ where
$\Lambda$ is a determinantal point process on ${\mathbb{R}}$ with kernel

$K_{\textup{Sine}_{2}}(s,t)=\frac{\sin((s-t)/2)}{\pi(s-t)}$

with respect to the Lebesgue measure.

Taking the limit of the kernel (123) (without any additional scaling) we
obtain the point process studied by Peres and Virág in [28].

######  Theorem 71 ([28]).

Let $\Lambda_{n}$ be the eigenvalues of
$\mathsf{Circ}_{n+1}^{\textstyle{\ulcorner}}$. Then $\Lambda_{n}$ converges to
a determinantal point process on the unit disk with kernel

$\displaystyle K_{\textup{Bergman}}(z,w)=\frac{1}{(1-z\bar{w})^{2}}$ (126)

with respect to the uniform measure on the unit disk. The resulting point
process has the same distribution as the zero set of the Gaussian analytic
function

$\displaystyle f_{GAF}(z)=\sum_{n=0}^{\infty}\xi_{n}z^{n},$ (127)

where $\xi_{n},n\geq 0$ are i.i.d. standard complex normals.

Note that [23] provides a generalization of this result by connecting the
limit of the eigenvalues of rank $m$ truncation of Haar unitary matrices with
the singular points of a matrix valued Gaussian analytic function that
generalizes (127).

The circular Jacobi beta ensemble (9) for $\beta=2$ is also determinantal,
this is also called the Hua-Pickrell distribution. The kernel function can be
expressed in terms of the orthogonal polynomials with respect to the
probability measure $\mu_{\delta}$ that has probability density function
proportional to

$(1-\bar{z})^{\delta}(1-z)^{\bar{\delta}}$

on the unit circle. The Hua-Pickrell distribution can be realized as the joint
eigenvalue distribution of the random unitary matrices that have density
proportional to $|\det(1-U)^{\delta}|^{2}$ with respect to the Haar measure.

In [25] the authors studied the truncations of these random matrices. They
showed that the eigenvalues of the rank-$m$ truncated matrices form a
determinantal point process on the unit disk, and derived their kernel
function. Moreover, they showed that the scaling limit of the joint eigenvalue
distribution (without any additional scaling) leads to the same limits as in
the Haar unitary case (as described in [28] and [23]).

If we are interested in the (hard) edge scaling limit of the eigenvalues of
the truncated matrix $\mathsf{Circ}_{2,n+1}^{\textstyle{\ulcorner}}$ then one
needs to transform the kernel (123) according to (3), and take the limit. This
was considered in [1] where the following result was shown.

######  Theorem 72 ([1]).

Let $\Lambda_{n}$ be the eigenvalues of
$\mathsf{Circ}_{2,n+1}^{\textstyle{\ulcorner}}$. Then the sequence
$-ni\log\Lambda_{n}$ converges to a determinantal point process
$\mathcal{X}_{2}$ supported on the upper half plane ${\mathbb{H}}$ with kernel
function

$\displaystyle K_{\textup{edge}}(z,w)=f(z-\bar{w}),\quad
f(u)=\frac{1}{\pi}\int_{0}^{1}te^{itu}dt,\quad z,w\in{\mathbb{H}},$ (128)

with respect to the Lebesgue measure on ${\mathbb{H}}$.

In fact, this is a special case of a more general problem that [1] considers:
the product of independent copies of rank-$m$ truncations of Haar unitary
matrices. The point process $\mathcal{X}_{2}$ also appears in [11] as the
scaling limit of the rank-one additive anti-Hermitian perturbation for the
Gaussian unitary ensemble.

Note that our Theorem 2 for $\beta=2$ provides a new characterization for the
determinantal point process $\mathcal{X}_{2}$. It would be interesting to see
whether the determinantal structure could be proved directly from that result.

###  8.2 Open problems

We end with a couple of open problems.

Problem 1. Our results for general $\beta>0$ consider the models with the edge
scaling (3). It would be interesting to explore the limiting behavior under
the bulk scaling (i.e. when we do not scale at all). In other words, it would
be interesting to see whether one could extend the results of [28] (see
Theorem 71) and [25] for general $\beta$.

Problem 2. In the $\beta=2$ case Theorem 71 provides a description of the bulk
scaling limit of the truncated circular unitary ensemble as the zero set of a
Gaussian analytic function. It would be interesting to see if one could
connect this result to some sort of a scaling limit of the characteristic
polynomial of the truncated circular ensemble under the bulk scaling.

Problem 3. A simple calculation shows that under the scaling $z\mapsto
c^{-1}z,c\to\infty$ the kernel $\eqref{eq:edge_det_kernel}$ converges to a
transformed version of the kernel (126), where the transformation is the
Cayley transform mapping the unit disk ${\mathbb{D}}$ to the upper half plane
${\mathbb{H}}$. This implies that as $c\to\infty$ the scaled edge limit
process $c^{-1}\mathcal{X}_{2}$ converges to the image of the bulk limit
process of Peres-Virág under the Cayley transform. Note that similar ‘edge-to-
bulk’ limits are known for other random matrix ensembles, see e.g. [35] for a
similar transition involving the point process limits of the Gaussian beta
ensemble. It would be interesting to see if a similar limit exists for
$\mathcal{X}_{\beta}$ for general $\beta>0$. Presumably, this could also
provide a way to prove the ‘edge-to-bulk’ transition in the $\beta=2$ case
without using the determinantal process framework.

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Yun Li, Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084,
China<EMAIL_ADDRESS>
Benedek Valkó, Department of Mathematics, University of Wisconsin – Madison,
Madison, WI 53706, USA<EMAIL_ADDRESS>