# Observation of full contrast icosahedral Bose-Einstein statistics
in laser desorbed, buffer gas cooled C60

Ya-Chu Chan  JILA, National Institute of Standards and Technology and
University of Colorado, Boulder, Boulder, CO 80309  Department of Chemistry,
University of Colorado, Boulder, CO 80309  Lee R. Liu
<EMAIL_ADDRESS>JILA, National Institute of Standards and Technology
and University of Colorado, Boulder, Boulder, CO 80309  Department of Physics,
University of Colorado, Boulder, CO 80309  Andrew Scheck  JILA, National
Institute of Standards and Technology and University of Colorado, Boulder,
Boulder, CO 80309  Department of Physics, University of Colorado, Boulder, CO
80309  David J. Nesbitt  JILA, National Institute of Standards and Technology
and University of Colorado, Boulder, Boulder, CO 80309  Department of
Chemistry, University of Colorado, Boulder, CO 80309  Department of Physics,
University of Colorado, Boulder, CO 80309  Jun Ye<EMAIL_ADDRESS>JILA,
National Institute of Standards and Technology and University of Colorado,
Boulder, Boulder, CO 80309  Department of Physics, University of Colorado,
Boulder, CO 80309  Dina Rosenberg  JILA, National Institute of Standards and
Technology and University of Colorado, Boulder, Boulder, CO 80309  Department
of Physics, University of Colorado, Boulder, CO 80309

###### Abstract

The quantum mechanical nature of spherical top molecules is particularly
evident at low angular momentum quantum number $J$. Using infrared
spectroscopy on the 8.4 $\mu$m rovibrational band of buffer gas cooled 12C60,
we observe the hitherto unseen R($J=0-29$) rotational progression, including
the complete disappearance of certain transitions due to the molecule’s
perfect icosahedral symmetry and identical bosonic nuclei. The observation of
extremely weak C60 absorption is facilitated by a laser desorption C60 vapor
source, which transfers 1000-fold less heat to the cryogenic buffer gas cell
than a traditional oven source. This technique paves the way to cooling C60
and other large gas phase molecules to much lower temperatures, providing
continued advances for spectral resolution and sensitivity.

††preprint: APS/123-QED

Due to their strong intramolecular interactions, perfect rotational
symmetries, and long-lived rotational and vibrational degrees of freedom,
large molecules are interesting platforms for probing strongly correlated many
body phenomena Liu  _et al._ (2023); Schulz  _et al._ (2015) and quantum
information processing Sawant  _et al._ (2020); Zeppenfeld (2023); Albert  _et
al._ (2024). Quantum state resolved infrared spectroscopy is the key to unlock
these capabilities. Molecules beyond $\sim 10$ atoms, however, typically
exhibit intrinsic spectral congestion in the usual infrared bands due to the
rapidly increasing vibrational density of states with increasing energy and
atom number Spaun  _et al._ (2016); Changala  _et al._ (2016). C60 is a
notable exception: due to its rigidity and symmetry, it is the largest
molecule for which quantum state resolution in the infrared was achieved Bryan
Changala  _et al._ (2019). Based on this result, a series of high resolution
spectroscopic measurements have provided the clearest understanding of
structure and dynamics of individual C60 molecules to date. By fitting high
resolution spectra to effective molecular Hamiltonians, the spectroscopic
constants for the ground and excited (T1u(3)) vibrational states were
determined Bryan Changala  _et al._ (2019). Then, detailed studies of the
collisions between C60 and a variety of atomic and molecular species explored
the intramolecular pathways of energy relaxation through rotational and
vibrational degrees of freedom Liu  _et al._ (2022). The observation of
ergodicity breaking in the rotational fine structure of C60 Liu  _et al._
(2023) offers a new perspective on emergent phenomena in mesoscopic quantum
systems made possible by free rotations, rotation-vibration coupling and
perfect rotational symmetry.

These results were obtained using a 900 K oven source coupled to a cryogenic
buffer gas cell Weinstein  _et al._ (1998); Spaun  _et al._ (2016); Patterson
and Doyle (2007). However, the significant heat load of the oven source
constrained the rotational temperature to lie $\sim$150 K and above,
preventing the concentration of significant population into states of low
angular momentum quantum number $J$ Bryan Changala  _et al._ (2019). At low
$J$, some rotational states are completely forbidden for the totally symmetric
nuclear spin species. Because of the high symmetry and 60 identical spinless
bosonic nuclei of 12C60, these forbidden states are completely absent from its
infrared spectrum, providing arguably the most striking example of symmetry
and quantum mechanical effects on spherical top spectra Bunker and Jensen
(1999).

In this work, we develop a laser desorption vapor source to produce a colder
sample of gas phase C60. This allows us to measure the R branch rotational
progression of 12C60 from $J=0-29$ and beyond using cavity-enhanced high
resolution spectroscopy on the 8.4 $\mu$m rovibrational band, demonstrating
complete absence of rotational transitions, as predicted by nuclear spin
statistics. We characterize the rotational temperature of laser desorbed C60
and find that it reaches below 100 K using both Ar and He buffer gas cooling.
Rotational and Doppler thermometry indicate that rotational and translational
degrees of freedom are essentially thermalized. We also observe large signal
from cold C60 with He buffer gas, indicating that in spite of the enormous
mismatch in masses, small He-C60 rotational quenching cross section Liu  _et
al._ (2022), and highly non thermal optical excitation by 532 nm photons, He
is able to efficiently cool the vibrational, rotational, and translational
degrees of freedom of laser desorbed C60. Our specific laser desorption
technique possesses several unique features that enable high sensitivity
molecular spectroscopy: 1) three orders of magnitude reduction in heat load on
the cryogenic buffer gas cell compared to a traditional oven source; 2) stable
pulse-to-pulse desorption yield, varying by less than 10%, due to the
uniformity of the deposited C60 film thickness; 3) the use of unusually long
duration (200 ms) and low peak power (2 W) desorption pulses to minimize
fragmentation of C60, in contrast to MW peak powers for traditional ns pulsed
lasers; 4) efficient collisional cooling by He; and 5) controllable, transient
($\sim$30 ms response time) C60 yield, enabling rapid subtraction of probe
light intensity fluctuations from absorption signal.

Figure 1: Laser desorption and buffer gas cooling of C60 A 5 $\mu$m-thick
$C_{60}$ film target is deposited on a tantalum foil via thermal evaporation.
The target is held in the center of an annular buffer gas inlet. Buffer gas
flow is directed over the face of the target. A 532 nm cw desorption laser
(green) is focused onto the target and rastered across in steps of 600 $\mu$m
using a piezo-controlled mirror. At every point, the laser is pulsed on for
200 ms and desorbs a constant amount of C60, determined by the thickness of
the target and 300 $\mu$m focal spot size. Each 1-inch diameter target yields
approximately 7,200 shots. The laser power is chosen to completely penetrate
the C60 film, thereby desorbing a nearly constant amount of C60 into the cell.
The desorbed C60 molecules cool as they diffuse through the buffer gas. The
8.4 $\mu$m mid infrared probe beam (red) resonantly excites a vibrational mode
of C60 and is absorbed. The mirrors form an optical cavity ($\mathcal{F}$ =
12,500) to enhance the absorption due to C60. Not shown, surrounding the inlet
is the buffer gas cell anchored to a liquid nitrogen cold finger, maintained
at a temperature of 82 K.

The laser desorption setup is described in Fig. 1. The heat load from laser
desorption alone is bounded from above by the total time averaged input laser
power of 1 W. The laser desorption therefore represents a 1000-fold reduction
in heat load compared to the traditional 1 kW, 900 K oven source Bryan
Changala  _et al._ (2019). Next, we characterize the shot-to-shot uniformity
of the transient C60 yield. Because of the low thermal conductivity of Ar, and
because it was shown to efficiently cool C60 in previous studies Bryan
Changala  _et al._ (2019), we primarily use Ar buffer gas cooling for
characterization. We pulse the desorption laser at 2.5 Hz with 50% duty cycle
and record the transient absorption of the 8.4 $\mu$m probe laser parked at
1184.8475 cm-1, where the C60 absorption spectrum is relatively featureless
due to spectral congestion of the Q branch. Remarkably, the peak absorption,
which is proportional to the C60 density, only fluctuates by $<$10% from shot-
to-shot. Because the desorption laser completely penetrates the C60 film, the
yield is largely dictated by the film thickness, which is uniform to within a
few percent, consistent with the shot-to-shot variation in C60 yield.

Figure 2: Characterization of transient laser desorbed C60 yields. A)(i)
Transient C60 yield converted from absorption in the Q branch region. The
desorption laser pulsing sequence is depicted by alternating green and grey
squares. (ii) C60 density profile averaged over 40 shots. Gray error band
indicates the point-wise standard deviation. Around 100 ms after the
desorption laser fires, the variation in the density is less than 10% over a
period of 50 ms (vertical rectangle) and is stable enough for acquiring
spectral features of cold C60. (iii) Mean and standard deviation of C60
density within the 50 ms window for each laser shot across three different
sections of the target. (iv) Histogram of mean density, showing uniform C60
yield. B) The 8.4 $\mu$m probe laser frequency is scanned 150 MHz around
R($J=156$) with the desorption laser on (green) and off (gray), yielding
“filled” and “empty” cavity spectra, respectively. C) Background-free
molecular absorption signal, obtained by subtraction of empty cavity
transmission from filled cavity transmission. The data is averaged over 40
shots.

Fig. 2A(ii) shows the time trace of C60 density averaged over 40 desorption
laser shots. After the desorption laser has been on for 5 ms, C60 density
increases and reaches its maximum within 30 ms, then decays over the course of
several hundred milliseconds. The variation in the density is less than 10% in
the 50 ms window around 100 ms after the desorption laser fires. At this
point, we scan the laser over 150 MHz to acquire absorption spectra in the R
branch region. Over a scan spanning $\sim$15 MHz, corresponding to the
spectral line width, the C60 density changes by less than 1% and the effects
on the line shape can be neglected. The desorption laser is shut off and the
C60 density drops to zero within 30 ms, after which another “empty cavity”
spectrum is acquired (Fig. 2B). The rapid response time of C60 density allows
for subtraction of slowly fluctuating intensity drifts such as those due to
drifting parasitic etalons. In fact, laser desorption itself is instantaneous,
but the measured 10’s of milliseconds response time is dominated by transport
through the buffer gas and our particular experimental geometry. The resulting
background-free molecular absorption spectrum for R($J=156$) is shown in Fig.
2C. From the measured absorbance, we calculate the cold 12C60 density to be
approximately $5(1)\times 10^{10}$cm-3, based on the integrated intensity of
the 8.4 $\mu$m band Iglesias-Groth  _et al._ (2011), rotational temperature of
100 K, and effective optical path length of $\sim$475 m.

Having characterized the yield of laser desorbed C60, we now turn to C60
rotational and Doppler thermometry. In particular, the 532 nm photons may
deposit energy primarily into certain (e.g., electronic) degrees of freedom,
initializing the C60 molecules in a highly non thermal state Burgess  _et al._
(1988). In contrast, an oven source produces molecules in which all internal
degrees of freedom are approximately thermalized. Therefore, a significant
concern was that laser desorbed C60 might not be efficiently cooled. On the
other hand, since vibration-to-translation energy transfer is expected to be
more favourable for smaller vibrational energy gaps, larger molecules might be
expected to vibrationally quench easily due to collisions Du  _et al._ (1991).
Furthermore, thermalization may occur rapidly in larger molecules even in the
absence of collisions due to efficient intramolecular vibrational energy
redistribution (IVR) Parmenter (1982); Smalley (1982, 1983); Nesbitt and Field
(1996). Our measurements indicate that thermalization proceeds efficiently for
laser desorbed C60. A second key question was whether C60 could be efficiently
quenched by collisions with He instead of Ar. Previous work with a 900 K oven
source of C60 found that Ar gave the largest cold C60 peak absorbance Bryan
Changala  _et al._ (2019) and that He has the smallest rotational quenching
cross section amongst all noble gases Liu  _et al._ (2022). Furthermore, He is
$10\times$ more thermally conductive than Ar, exacerbating the heat transfer
from the 1 kW oven to the nearby cryogenic buffer gas cell. Nevertheless, He
is highly desirable as a buffer gas due to its significant vapor pressure at 4
K, making it compatible with a helium cryostat. He buffer gas cooling is also
expected to mitigate possible cluster formation at lower temperatures Kłos
_et al._ (2024). Therefore, it was crucial to prove that He buffer gas
efficiently collisionally quenches laser desorbed C60 as well.

The liquid nitrogen cold finger and buffer gas cell temperatures were directly
measured with Cernox thin film resistance cryogenic temperature sensors and
silicon diodes to be 82 K. Residual heat loads, such as radiative and
conductive coupling to the room temperature dewar walls, dominate the total
heat load and prevent the temperature from reaching 77 K, even with the
desorption laser inactive. The static pressure in the cell was measured
directly with a capacitance manometer. Absorption spectra were measured for
six $J$-values scattered across the thermally populated states
R($J=66,96,126,156,186,206$) at various buffer gas pressures ($330-410$
mTorr), which we controlled by varying the buffer gas flow rate ($80-130$
sccm).

The rotational temperature was obtained from the integrated absorbance at each
of the six measured R($J$) lines. Assuming negligible population in the
vibrationally excited states, the integrated absorbances $I(J)$ are dominated
by the rotational populations in the vibrational ground state. Hence,
$I(J)=CS_{J}g_{J}e^{-E_{J}/kT_{rot}}$, where $C$ is proportional to the total
C60 population, $S_{J}$ is the intrinsic line strength factor Herzberg (1945),
$g_{J}$ is the angular momentum degeneracy factor including lab-frame and
molecular frame degeneracies for a spherical top and nuclear spin statistics
weight Bunker and Jensen (1999), $E_{J}=BJ(J+1)$ is the rotational energy in
the vibrational ground state, $k$ is the Boltzmann constant, and $T_{rot}$ is
the rotational temperature. Rearranging this equation gives
$\textrm{Ln}(I(J)/(S_{J}g_{J}))=\textrm{Ln}(C)-E_{J}/(kT_{rot})$. In this way
$T_{rot}$ can be extracted from the slope of a best-fit line to the logarithm
of $I(J)/(S_{J}g_{J})$, the relative populations per $J$ multiplet component
(Fig. 3A). For Ar buffer gas, we find that $T_{rot}$ decreases with increasing
pressure up until 400 mTorr, after which it rises again (Fig. 3C). The initial
decrease in $T_{rot}$ is expected: as pressure increases, the number of
collisions experienced by C60 increases, thereby lowering the measured
$T_{rot}$.

Figure 3: Pressure-dependent Doppler and rotational thermometry of laser
desorbed C60 in He and Ar buffer gas. A) Exemplary rotational temperature fits
for C60 in Ar and He buffer gas. Note that He buffer gas cooling yields higher
and more uniform molecular signal (as evident from the smaller error bars in
the integrated absorption). B) Exemplary fit of R($J=156$) absorption to rate
equation, with residuals plotted below. C) Rotational and translational
temperatures from Boltzmann analysis and rate equation fitting respectively,
for a range of buffer gas pressures. The rotational temperatures lie within
the error bounds of the fitted translational temperature. In addition, the
coldest rotational temperatures approach 82 K, the temperature of the liquid
nitrogen cold finger (horizontal dashed line). Note that He buffer gas cooling
achieves the lowest rotational temperature, even at much lower pressure.

The rise in Trot above 400 mTorr is also well understood. The total number of
collisions a C60 molecule experiences before reaching the 8.4 $\mu$m beam is
proportional to the product of buffer gas pressure and the time of flight from
the target. Separately, we observed a decrease in the time of flight with
increasing buffer gas flow, suggesting that buffer gas “carries” desorbed C60
to the cavity axis at a speed that depends on the flow rate. Indeed, the time
of flight ranges from a few to 10’s of milliseconds depending on the flow
rate, which is intermediate between ballistic and diffusive transport.
Therefore, while higher buffer gas flow increases pressure, it also decreases
the C60 time of flight. Pressure in the cell begins to saturate at higher flow
rates, possibly due to hydrodynamic effects Patterson and Doyle (2007); Barry
_et al._ (2011). At sufficiently high flow rates therefore, the reduction in
collision events due to time of flight overcomes the increase in collision
rate due to pressure, and the total collision number drops with increasing
flow rate. This is responsible for the measured increase in $T_{rot}$ above
400 mTorr.

Since a considerable amount of averaging was needed to obtain sufficiently
high signal to noise ratio for accurate line shape fitting, we obtained the
translational temperature $T_{trans}$ by globally fitting spectra measured for
three selected $J$ values ($J=96,156,186$) to line shapes predicted by a semi-
classical steady-state rate equation model discussed in Ref Liu  _et al._
(2022). In particular, the model accounts for optical saturation of two
counter propagating laser beams and $J$-dependent rotational quenching, which
were crucial to obtain convergence of $T_{trans}$ beyond simply fitting each
line shape to an independent Voigt profile. We fix C60-Ar collision cross
section values from Ref Liu  _et al._ (2022) and allow the buffer gas
temperature TBG and $T_{trans}$ to float, while constraining them to be equal
(TBG=Ttrans) and independent of $J$. Here, TBG affects the diffusion rate of
C60 and the collision frequency. At our buffer gas pressures, we expect
translational thermalization of C60 to occur in less than a hundred collisions
Friedrich (2009), or a few microseconds, much shorter than the 10 ms timescale
it takes for laser desorbed C60 to reach the interaction region. For each
pressure value, $T_{rot}$ was fixed by the independent rotational temperature
fit described earlier. An exemplary fit for R($J=156$) is shown in Fig. 3B.

For Ar buffer gas, we find that Ttrans falls in the range $130\pm 30$ K, with
flat dependence on buffer gas pressure. The error bars reflect the fact that
line shape fitting is generally insensitive to translational temperature,
since inhomogeneous linewidth scales approximately as $\sqrt{T_{trans}}$. The
translational and rotational temperatures agree within error bars and approach
that of the liquid nitrogen cold finger, suggesting that rotational and
translational degrees of freedom are nearly thermalized with each other and
the buffer gas. We repeated the thermometry with He at a pressure of 265 mTorr
and find that the fitted translational $T_{trans}=103(42)$ K and rotational
temperatures $T_{rot}=94(4)$ K rival the lowest temperatures achieved with Ar,
in spite of the much lower He pressure used (Fig. 3C). He buffer gas cooling
also yielded a signal of cold C60 comparable to the highest C60 signal
achieved with Ar cooling, as well as fractionally more stable C60 yield,
indicated by the smaller error bars in (Fig. 3A). We attribute this stability
to the suppression of C60-buffer gas cluster formation. This hypothesis is
supported by the shallow interaction potential and lower near-dissociation
threshold density of states of the C60-He complex compared to C60-Ar Kłos  _et
al._ (2024). The large signal achieved with He buffer gas cooling also
indicates that He collisions efficiently quench C60 vibrations, since
transitions from excited vibrational states (vibrational “hot bands”) are
expected to be red-shifted from transitions from the ground vibrational state
(fundamental band) by $\sim$60 cm-1 Wang and Daw (2018), and will not
contribute to the resolved R branch features. With a greatly reduced heat
load, He buffer gas cooling is fully compatible and in fact preferable for
laser desorbed C60.

As mentioned earlier, the apparently efficient thermalization of laser
desorbed C60 should not come as a surprise. Intramolecular redistribution,
particularly in a molecule as large as C60, might redistribute the internal
energy rapidly amongst internal degrees of freedom, even without buffer gas
cooling. Indeed, thermal C60 sublimated from a 900 K oven source possesses 7.4
eV of vibrational energy, corresponding to the energy of three desorption
photons at 532 nm. Smaller molecules, on the other hand, are expected to
exhibit quenching bottlenecks due to large energy gaps and sparse spectrum of
internal states Sinha  _et al._ (1990, 1992). Our results are promising for
laser desorption and subsequent collisional thermalization of other large
molecules Spaun  _et al._ (2016).

Figure 4: Complete R($J=0-29$) absorption spectrum in the 8.4 $\mu$m
rovibrational band of cold C60. Experimental data (black) is the average of
200-400 raw spectra, with the light gray error bands representing the point-
wise standard deviation. Simulated spectrum (gray, inverted) derived from
spectroscopic constants determined in Ref Bryan Changala  _et al._ (2019),
assuming a rotational temperature of 100 K and including nuclear spin weights.
Notably, the complete absence of many R($J$) transitions, as predicted by
bosonic exchange statistics and the molecule’s icosahedral symmetry, is
clearly visible. There is an increasing $\sim 5-15$ MHz blue-shift of
experimental line centers (gray vertical lines) from the simulated spectrum as
$J$ decreases from $29-0$, continuing the trend seen in Ref Bryan Changala
_et al._ (2019). Note, we tentatively assign R($J=6$) to the higher frequency
peak in the doublets (marked with a dash vertical line). Additional features
at $J=6$ and 22 and at very low $J$ are deemed to be real.

Having demonstrated efficient He buffer gas cooling of laser desorbed C60, as
well as the stable shot-to-shot C60 yield, we now turn to observing the
extremely weak molecular absorption in the hitherto unseen R($J=0-29$)
spectral region. For 12C60, since each carbon atom is a spinless,
indistinguishable boson, the only allowed rotational states in the vibrational
and electronic ground state are those which correlate to the totally symmetric
irreducible representation in the icosahedral group. These correspond to
$J=0,6,10,12,\cdots$, with the remaining R($J$) transitions absent Bunker and
Jensen (1999). The striking icosahedral nuclear spin weights arise from the
two-, three-, and five-fold discrete rotational symmetry axes of 12C60, and
are borne out in our observed spectra (Fig. 4). However, we also observe
perturbations to the transition intensities and frequencies, perhaps due to
high-$J$ features from the Q branch, some accidentally degenerate vibrational
combination state, or contamination from C60 isotopologues. The observation of
these subtle perturbations is facilitated by the He buffer gas cooling, which
provides stable shot-to-shot C60 yield and enables averaging of extremely weak
molecular absorption signals.

In conclusion, we have demonstrated efficient cooling of laser desorbed C60 to
below 100 K, which is sufficiently cold to observe the hitherto unseen
R($J=0-29$) transitions. This spectral region could be used for new tests of
bosonic indistinguishability involving highly symmetric molecules Angelis  _et
al._ (1996); Modugno  _et al._ (1998); Mazzotti  _et al._ (2001); Pastor  _et
al._ (2015). In addition, the three orders of magnitude reduction in heat load
of the laser desorption source compared to the oven source opens the door for
cooling using a closed cycle He cryostat, which has much less cooling power
than evaporating liquid nitrogen. At the 4 K temperature achievable by a He
cryostat, C60 population in the first 30 rotational levels in the electronic-
vibrational ground state increases by two orders of magnitude, opening the
door to understanding the origin of spectral perturbations, as well as
providing a tantalizing route toward quantum state preparation of gas phase
C60 at high density. We note that large molecules such as C60 are not amenable
to the traditional approach of laser cooling due to their large mass
(requiring correspondingly many more photon scattering events) and density of
vibrational states (increasing the number of lasers required to “repump”
vibrational populations), although a scheme involving chemical
functionalization of a fullerene has been theoretically proposed Kłos and
Kotochigova (2020). Thus, laser desorption of C60 opens up a new class of
molecules to quantum science.

We thank Adam Ellzey for technical assistance; the COSINC Fabrication facility
at CU Boulder, particularly Dylan Bartusiak; and Jutta Toscano for the
inspiration to pursue laser desorption of C60. We thank Jacob Higgins and
Baruch Margulis for providing useful comments on the manuscript. Funding
support for this work is provided by AFOSR (FA9550-19-1-0148); NSF QLCI (QLCI
OMA-2016244); NSF (Phys-1734006) and NIST. Support is also acknowledged from
the U.S. Department of Energy, Office of Science, National Quantum Information
Science Research Centers, Quantum Systems Accelerator. D.R. acknowledges
support from the Israeli CHE Quantum Science fellowship, and is an awardee of
the Weizmann Institute National Postdoctoral Award Program for Advancing Women
in Science. Y.-C.C. and D.J.N. acknowledge support from the DOE (DE-
FG02-09ER16021) and NSF (CHE 2053117).

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# First Order Topological Phase Transitions and Disorder Induced Majorana
Modes in Interacting Fermion Chains

Shruti Agarwal  Department of Physical Sciences, Indian Institute of Science
Education and Research Mohali, Sector 81, S.A.S. Nagar, Manauli PO 140306,
India  Shreekant Gawande  Department of Physical Sciences, Indian Institute of
Science Education and Research Mohali, Sector 81, S.A.S. Nagar, Manauli PO
140306, India  Satoshi Nishimoto  Department of Physics, Technical University
Dresden, 01069 Dresden  Institute for Theoretical Solid State Physics, IFW
Dresden, 01069 Dresden, Germany  Jeroen van den Brink  Department of Physics,
Technical University Dresden, 01069 Dresden  Institute for Theoretical Solid
State Physics, IFW Dresden, 01069 Dresden, Germany  Sanjeev Kumar  Department
of Physical Sciences, Indian Institute of Science Education and Research
Mohali, Sector 81, S.A.S. Nagar, Manauli PO 140306, India  Institute for
Theoretical Solid State Physics, IFW Dresden, 01069 Dresden, Germany

###### Abstract

Using a combination of the mean-field Bogoliubov deGennes (BdG) approach and
the Density Matrix Renormalization Group (DMRG) method, we discover first
order topological transitions between topological superconducting and trivial
insulating phases in a sawtooth lattice of inter-site attractive fermions.
Topological characterization of different phases is achieved in terms of
winding numbers, Majorana edge modes and entanglement spectra. By studying the
effect of disorder on the first order topological phase transitions, we
establish the disorder-induced topological phase coexistence as a mechanism
for generating a finite density of Majorana particles.

Introduction:– Discovery of topological insulators marked the beginning of a
paradigm shift in our approach to understand ordering phenomena in condensed
matter physics [1, 2, 3, 4, 5, 6]. Topological phases, since then, have become
a recurring theme of research in many fields of quantum as well as classical
physics [7, 8, 9, 10, 11, 12, 13]. Search for new topological phases is
driven, not merely by our curiosity to achieve a fundamental understanding but
also by the potential for their applications. Topological superconductors are
of particular importance as these can host Majorana particles that are
considered to be building blocks of quantum computers [14, 15, 16, 17, 18,
19]. Topological phases and phase transitions in non-interacting models can be
comprehensively understood in terms of the symmetries of the Hamiltonian [20,
21, 22, 23]. On the other hand, general understanding of topological phases in
interacting systems is a challenging theoretical problem [24, 25, 26, 27, 28].

Phase separation (PS) is a ubiquitous phenomena displayed by electronic
systems [29, 30, 31]. While the inability of an electronic system to exist in
a uniform density state appears to be an undesirable feature, PS has proven to
be of critical importance in understanding certain aspects of the physics of
strongly correlated electron systems [32, 33]. Invoking electronic PS is for
instance essential for theories of colossal magnetoresistance effect in
manganites [34]. Also some of the theories applicable in the low-doping regime
of the famous cuprate superconductors rely on electronic PS [35, 36, 37]. To
the best of our knowledge, electronic PS between a topological and a trivial
phase – henceforth referred to as topological PS (TPS) – has not been
discussed in literature. TPS has general implications that are independent of
the detailed parameter choices of the model.

In this Letter, we report the discovery of first order topological phase
transitions in a lattice model of attractively interacting spinless fermions.
Using a combination of DMRG and mean-field BdG methods, we find that the model
hosts topological superconductor (TSC) and charge-modulated insulator (CMI)
phases. Some of the topological and trivial phases are separated by first
order boundaries, leading to TPS. We characterize the topological phases with
the help of the winding numbers, edge modes and the entanglement spectra. We
explicitly demonstrate that in the presence of quenched disorder the TPS leads
to phase coexistence with Majorana modes residing at the boundaries of TSC
regions. To the best of our knowledge, this is the first mechanism for
generating a finite density of Majorana particles in a system of interacting
fermions.

Spinless fermions on sawtooth lattice:–

Figure 1: Schematic representation of the sawtooth lattice model. $t$,
$t^{\prime}$ indicate the hopping amplitudes and $V$, $V^{\prime}$ the
attractive interactions between electrons on neighboring sites. Inequivalent
sites are labeled A and B.

Let us consider spinless fermions with attractive interactions residing on a
sawtooth lattice described by the Hamiltonian,

$\displaystyle H$ $\displaystyle=$
$\displaystyle-t\sum_{i}(c^{\dagger}_{i,A}c_{i+1,A}+{\textrm{H}.c.})-t^{\prime}\sum_{i}(c^{\dagger}_{i,A}c_{i,B}+{\textrm{H}.c.})$
(1)
$\displaystyle-t^{\prime}\sum_{i}(c^{\dagger}_{i,B}c_{i+1,A}+{\textrm{H}.c.})-V\sum_{i}\hat{n}_{i,A}\hat{n}_{i+1,A}$
$\displaystyle-V^{\prime}\sum_{i}(\hat{n}_{i,A}\hat{n}_{i,B}+\hat{n}_{i,B}\hat{n}_{i+1,A})-\mu\sum_{i,s}\hat{n}_{i,s}.$

Here, $c_{i,s}(c_{i,s}^{\dagger}$) annihilates (creates) an electron at
Bravais site $i$ and sublattice $s\in\\{A,B\\}$, and $\hat{n}_{i,s}$ is the
corresponding number operator. Hopping amplitudes are denoted by $t$ and
$t^{\prime}$, and the corresponding attractive interaction strengths by $V$
and $V^{\prime}$ (see FIG. 1). We set $t^{\prime}=1$ as the basic energy
scale, and $\mu$ denotes the chemical potential. Note that this elementary
interacting model Hamiltonian reduces to a monoatomic Kitaev chain supporting
p-wave superconductivity for $t=V=0$, and the non-interacting limit displays a
flat band dispersion for $t^{\prime}/t=\sqrt{2}$.

Figure 2: Zero-temperature variations of $\Delta_{1}$, $\Delta_{2}$, $\delta
n$ and $n$ with chemical potential for $V=V^{\prime}=1$ and for, (a) $t=0.2$,
(b) $t=0.4$, (c) $t=1.0$, (d) $t=2.0$. (e) Ground state phase diagram in the
$t-\mu$ plane constructed from the data similar to that shown in panels
(a)-(d). Horizontal dashed lines indicate the scans for which the data in
panels (a)-(d) is displayed.

Mean Field Phase Diagram:– In order to obtain the mean-field ground states of
the Hamiltonian Eq. (1), we decouple the interaction term in pairing and
density channels [38, 39, 40, 41]. This leads to the effective non-interacting
Hamiltonian,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k}\Psi_{k}^{\dagger}H_{k}\Psi_{k},$ (2)

with,

$\displaystyle\Psi_{k}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}c^{\dagger}_{k,A}&c^{\dagger}_{k,B}&c_{-k,A}&c_{-k,B}\end{bmatrix}^{\dagger},H_{k}=\begin{bmatrix}\beta_{k}&\alpha_{k}\\\
\alpha^{\dagger}_{k}&-\beta^{T}_{-k}\end{bmatrix},$ $\displaystyle\beta_{k}$
$\displaystyle=$ $\displaystyle\begin{bmatrix}-\tilde{\mu}_{A}-2t\cos
k&~{}~{}~{}~{}-t^{\prime}(1+e^{-\mathrm{i}k})\\\
-t^{\prime}(1+e^{\mathrm{i}k})&~{}~{}~{}~{}-\tilde{\mu}_{B}\end{bmatrix},$
$\displaystyle\alpha_{k}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}-2\mathrm{i}V\Delta_{1}\sin
k&~{}~{}~{}~{}-V^{\prime}\Delta_{2}(1-e^{-\mathrm{i}k})\\\
V^{\prime}\Delta_{2}(1-e^{\mathrm{i}k})&~{}~{}~{}~{}0\end{bmatrix},$ (3)

$\tilde{\mu}_{A}=\mu+2n_{A}V+2n_{B}V^{\prime}$ and
$\tilde{\mu}_{B}=\mu+2n_{A}V^{\prime}$. The mean-field parameters,
$n_{A/B}=\braket{\hat{n}_{i,A/B}}$, $\Delta_{1}=\braket{c_{i+1,A}c_{i,A}}$ and
$\Delta_{2}=\braket{c_{i+1,A}c_{i,B}}=\braket{c_{i,B}c_{i,A}}$ are self-
consistently determined [38]. FIG. 2(a)-(d) displays the variation of
different mean-field parameters with chemical potential. The density per site,
$n=(n_{A}+n_{B})/2$, varies from $0$ to $1$, with the end points representing
trivial band insulator (BI) phases corresponding to empty or fully filled
bands. The pair expectation values, $\Delta_{1}$ and $\Delta_{2}$, are found
to be finite at all non-trivial densities except when there is a plateau at
$n=0.5$ (see FIG. 2(a), (c), (d)). The difference in average densities at two
sublattices, $\delta n=n_{B}-n_{A}$, remains finite in the entire region. The
finite $\delta n$ is related to the inequivalence of the two sublattices in
the non-interacting tight-binding model. Finite values of $\Delta_{1}$ and
$\Delta_{2}$ identify a superconducting state, within the mean-field
description. The $n=0.5$ plateau regions are characterized by a gap in the
single-particle density of states (DOS), together with vanishing pairing
amplitudes and finite $\delta n$. Therefore, we label these states as charge
modulated insulators (CMI). We summarize the variations of mean-field
parameters obtained for different values of $t$ in terms of a ground state
phase diagram in FIG. 2(e). The topological superconductor (TSC) phases are
further labeled as $w=\pm 1$, where $w$ denotes the winding number. Note that
the total density displays discontinuities as a function of $\mu$. This is a
direct indicator of a first-order phase transition and associated electronic
PS in the model. The discontinuities are also present in other mean-field
parameters. While most of the discontinuities seem to involve the trivial BI
as one of the states, we also find a robust jump in $n(\mu)$ between the TSC
and CMI (see FIG. 2(d)). The PS locations in $\mu$ become extended lines in
the phase diagram and are indicated with thick black lines in FIG. 2(e).

Topological characterization:– The low energy quasiparticle spectra is shown
as a function of chemical potential for both, periodic boundary conditions
(PBC) and open boundary conditions (OBC) in FIG. 3. The OBC spectra displays
states at exactly zero energy over the $\mu$-range corresponding to the
superconducting state. We have explicitly checked that these states are
Majorana zero energy modes (MZMs) localized on the edges. In the corresponding
PBC spectra, the naive expectation is that the bulk gap must close at the
transition point between topologically trivial and nontrivial phases. While
the gap seems to reduce, it does not close at the transition point. This
seemingly unusual feature is easy to understand if we note that change in
$\mu$ does not represent a continuous evolution of the effective non-
interacting Hamiltonian in parameter space. This is because the self-
consistent values of various mean-fields are also parameters of the
Hamiltonian which change discontinuously across the transitions. Note that
within the CMI phase, two states seem to be crossing the gap in the OBC
spectra. We have confirmed that these are trivial edge states that appear due
to the difference in effective on-site potentials on the boundary sites.

Figure 3: Evolution of the BdG spectrum with $\mu$ for, (a) OBC and (b) PBC.
The corresponding variations in the winding number $w$ and the charge
modulation $\delta n$ are shown in (c). The values of the parameters used for
these calculations are: $t^{\prime}=V=V^{\prime}=1$, $t=0.2$

Given that we are dealing with a one-dimensional system, winding number turns
out to be the most natural choice of the invariant that characterizes
topologically distinct phases. Using the standard approach to unitary
transform the Hamiltonian into an off-diagonal form [42], we find,

$\displaystyle UH_{k}U^{-1}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}0&\beta_{k}+\alpha_{k}\\\
(\beta_{k}+\alpha_{k})^{\dagger}&0\end{bmatrix}.$ (4)

The winding number is then defined as [42],

$\displaystyle w=\frac{1}{2\pi\mathrm{i}}\int
dk\frac{1}{\det[\beta_{k}+\alpha_{k}]}\frac{d}{dk}\det[\beta_{k}+\alpha_{k}]$
(5)

$|w|$ counts how many times $\det[\beta_{k}+\alpha_{k}]$ winds around the
origin. We find that $w$ takes values 0, $\pm$1 for the Hamiltonian, where $w$
= 0 corresponds to topologically trivial phases and a non-zero winding number
corresponds to topologically non-trivial phases. Phases with non-zero $w$ are
labeled as TSC in the phase diagrams. A representative plot displaying the
variation of $w$ with $\mu$ is shown in FIG. 3(c). We find that the sign of
$w$ is perfectly correlated with the sign of $\delta n$, and has an
interesting interpretation in terms of the SO(3) theory of intertwined charge
and superconducting orders [38].

Figure 4: (a) Inverse exponent of the density-density and pair-pair
correlation functions as well as the Tomonaga-Luttinger parameter as a
function of density per site for $t=2.0$ and $t^{\prime}=V=V^{\prime}=1.0$.
The corresponding phase diagram is also shown on the top. (b) Similar plot to
(a) for $t=t^{\prime}=V^{\prime}=1.0$ and $V=3.0$. (c) Edge component of the
single-particle correlation function for $t=2.0$,
$t^{\prime}=V=V^{\prime}=1.0$ (red) and $t=t^{\prime}=V^{\prime}=1.0$, $V=3.0$
(blue) with $N=201$ open cluster. The open and filled circles denote the
correlations between edged A sites and between edged B sites, respectively.
(d) Entanglement spectra as a function of density per site for $t=2.0$,
$t^{\prime}=V=V^{\prime}=1.0$ with $N=60$ periodic cluster.  Figure 5:
$\Delta_{i,1}$ and zero-energy LDOS as a function of site index $i$ for, (a)
$D=0.4$, $n=0.46$ and (b) $D=0.8$, $n=0.435$. The other parameters values are
$V=3$, $V^{\prime}=1$, $t=1$, $\mu=-2.93$. Note that the $\Delta_{i,1}$ values
are scaled up in order to emphasize correlations with the LDOS data.

DMRG results:– In order to check the stability of results beyond mean-field
approximation, we perform DMRG calculations for some typical parameter sets.
First, the density-density $\langle\hat{n}_{i,s}\hat{n}_{j,s^{\prime}}\rangle$
and pair-pair $\langle
P_{ii^{\prime}ss^{\prime}}^{\dagger}P_{jj^{\prime}tt^{\prime}}\rangle$, where
$P_{ijss^{\prime}}=c_{i,s}^{\dagger}c_{j,s^{\prime}}^{\dagger}$, correlation
functions are calculated to obtain the ground state phase diagram. The
electronic state is characterized by the dominant correlation function with
slower decay as a function of distance. The inverse exponent of the power-law
decayed correlation functions ($\propto 1/|i-j|^{\gamma}$) is plotted as a
function of density per site in FIG. 4(a)-(b). The resultant phase diagrams,
as a function of $\mu$, are also shown. Note that the SC correlation function
decays exponentially in the CDW phase. The phase diagrams are further
confirmed by Tomonaga-Luttinger parameter $K_{\rho}$, which can be obtained
accurately via the derivative of charge structure factor at $q=0$ as
$K_{\rho}=\frac{1}{2}\lim_{q\to 0}\langle n(q)n(-q)\rangle$ with $q=2\pi/N$
and
$n(q)=\sum_{l}(e^{-iql}c_{l,A}^{\dagger}c_{l,A}+e^{-iq(l+1/2)}c_{l,B}^{\dagger}c_{l,B}$)
[43]. The dominant SC correlation is indicated when $K_{\rho}>0.5$. Next, to
test the topological nature of the SC phases, the single-particle correlation
function
$G_{ij}=i\langle\lambda_{i,s}\bar{\lambda}_{j,s\textasciiacute}\rangle$, where
$\lambda_{i,s}=c_{i,s}+c_{i,s}^{\dagger}$ and
$\bar{\lambda}_{i,s}=(c_{i,s}-c_{i,s}^{\dagger})/i$ are Majorana fermion
operators, between two ends of open cluster is calculated [44]. In FIG. 4(c)
we find a significant enhancement of $G_{ij}$ in the SC phase around $n=0.8$
for $t=2.0$ and $t^{\prime}=V=V^{\prime}=1.0$. This indicates that the SC
phase is topologically nontrivial. The topological nature is further confirmed
by the degeneracy of entanglement spectra $\xi_{n}(X=Y)$ using the Schmidt
decomposition of the ground state $\ket{\psi}$ as
$\ket{\psi}=\sum_{\lambda}e^{-\xi_{\lambda}(X)/2}\ket{\lambda}_{X}\ket{\lambda}_{Y}$,
where $\ket{\lambda}_{X}$ and $\ket{\lambda}_{Y}$ are the orthonormal basis
for the subregions $X$ and $Y$, respectively (see FIG. 4(d)). Thus, two of the
key features of the mean-field results, the TSC phases and PS, are also
confirmed by the DMRG analysis. In contrast to the mean-field results, the
DMRG calculations also find a CDW phase and a trivial SC phase (see FIG.
4(a)-(b)). This may be because the instability to trivial PS, as a particle
condensation into a self-bound system of concentration, is missing in the
mean-field analysis. It is known that the dominance of $s$-wave conventional
SC appears near the trivial PS [45]. Accordingly, a competition between TSC
and SC could occur in the DMRG calculations [38].

Topological phase coexistence and Majorana modes:– Having discussed the phase
diagrams, the important features associated with some of the phases and their
stability beyond mean-field, we now focus on the effect of quenched disorder
on TPS. We add an on-site disorder term,
$\sum_{i}\epsilon_{i}(\hat{n}_{i,A}+\hat{n}_{i,B})$, to the Hamiltonian Eq.
(1), where $\epsilon_{i}\in(-D/2,D/2)$ are random variables drawn from a
uniform distribution. Since the translational invariance is lost in the
presence of disorder, we are forced to perform calculations in real space. The
BdG mean-field Hamiltonian in the presence of disorder can be written as,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Psi_{r}^{\dagger}H_{r}\Psi_{r}+E_{0},$ (6)

where, the Nambu spinor $\Psi_{r}$ contains $4N$ components obtained by
repeating the index $i$ from $1$ to $N$ in,

$\displaystyle\Psi_{r}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}c^{\dagger}_{i,A}&c^{\dagger}_{i,B}&c_{i,A}&c_{i,B}&...\end{bmatrix}^{\dagger}$
(7)

$H_{r}$ is a Hermitian $4N\times 4N$ matrix with the non-zero diagonal and
off-diagonal $4\times 4$ blocks given by,

$\displaystyle h_{ii}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\epsilon_{i}-\tilde{\mu}_{i,A}&-t^{\prime}&0&-V^{\prime}\Delta_{i,2}\\\
-t^{\prime}&\epsilon_{i}-\tilde{\mu}_{i,B}&V^{\prime}\Delta_{i,2}&0\\\
0&V^{\prime}\Delta^{*}_{i,2}&-(\epsilon_{i}-\tilde{\mu}_{i,A})&t^{\prime}\\\
-V^{\prime}\Delta^{*}_{i,2}&0&t^{\prime}&-(\epsilon_{i}-\tilde{\mu}_{i,B})\end{bmatrix}$
$\displaystyle h_{i,i+1}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}-t&0&-V\Delta_{i,1}&0\\\
-t^{\prime}&0&-V^{\prime}\Delta_{i,3}&0\\\ V\Delta^{*}_{i,1}&0&t&0\\\
V^{\prime}\Delta^{*}_{i,3}&0&t^{\prime}&0\end{bmatrix}=h^{*}_{i+1,i},$ (8)

where,
$\tilde{\mu}_{i,A}=\mu+V(n_{i-1,A}+n_{i+1,A})+V^{\prime}(n_{i-1,B}+n_{i,B})$
and $\tilde{\mu}_{i,B}=\mu+V^{\prime}(n_{i,A}+n_{i+1,A})$. The above
Hamiltonian is diagonalized via real-space Bogoliubov transformations [39, 40,
41], and the site-dependent quantum expectation values of density and pairing
operators are iteratively computed to obtain self-consistent solutions. We
demonstrate the key result by selecting a range of $\mu$ values that cover the
phase separation between TSC and CMI for $V=3$, $V^{\prime}=1$ and $t=1$. We
display the spatial dependence of the pairing amplitude $\Delta_{i,1}$, along
with the local DOS at zero energy, in FIG. 5. Regions with finite pairing
amplitudes coexist with regions with zero pairing amplitudes. The number of
segments corresponding to the two types of regions increase upon increasing
disorder strength (compare (a) and (b) in FIG. 5). Most importantly, we find
that at the edges of SC regions there exist sharp peaks in zero-energy LDOS.
These peaks can only arise from the Majorana zero modes that reside on the
edges of all SC segments. It is also clear from the plots that if the length
of the SC segment is small then the two edge modes can hybridize. Our
calculations explicitly show that TPS provides an interesting route for
generating a finite density of Majorana zero modes.

Conclusion:– By investigating a lattice model of attractively interacting
fermions, we have unveiled a mechanism for generating finite density of
Majorana particles. Existence of a first order phase transition and the
associated TPS are the key prerequisites for the realization of the proposed
mechanism. Within the TPS mechanism, the MZMs can be found throughout the
system in contrast to the conventional mechanism that limits the existence of
MZMs only to the edges. Furthermore, given that the on-site potential can be
controlled via suitable application of gate voltages, the MZMs can be
mobilized. Given that the implications of our study are general, it is not
necessary to have exact realization of the toy model studied here in order to
implement the mechanism in an experimental set-up. Nevertheless, for a proof-
of-principle verification, the model is realizable in optically trapped
ultracold atomic gases [46, 47, 48]. Our results also provide a simple
understanding of why the conventional idea of gap closing at topological
transitions cannot be generally applicable to interacting systems. In general,
our study establishes that the interplay of interactions and topology in many-
particle quantum systems holds many, possibly useful, surprises.

Acknowledgments:– We acknowledge the use of computing facility at IISER
Mohali. We thank Ulrike Nitzsche for technical assistance. This project is
funded by the German Research Foundation (DFG) via the projects A05 of the
Collaborative Research Center SFB 1143 (projectid 247310070) and through the
Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum
Matter-ct.qmat (Project-ID 390858490-EXC 2147).

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##  I Mean-field BdG approach

Starting with the Hamiltonian Eq. (1) in the main text, we decouple the
interaction term using mean-field approach in density and pairing channels.
The resulting Hamiltonian is given by,

$\displaystyle H=$ $\displaystyle-$
$\displaystyle\sum_{i}[\tilde{\mu}_{i,A}~{}c^{\dagger}_{i,A}c_{i,A}+\tilde{\mu}_{i,B}~{}c^{\dagger}_{i,B}c_{i,B}]$
(9) $\displaystyle-$ $\displaystyle
t\sum_{i}(c^{\dagger}_{i,A}c_{i+1,A}+{\textrm{H}.c.})$ $\displaystyle-$
$\displaystyle
t^{\prime}\sum_{i}(c^{\dagger}_{i,A}c_{i,B}+c^{\dagger}_{i,B}c_{i+1,A}+{\textrm{H}.c.})$
$\displaystyle-$ $\displaystyle
V^{\prime}\sum_{i}(\Delta_{i,3}~{}c_{i+1,A}c_{i,B}+\Delta_{i,2}~{}c_{i,B}c_{i,A}+{\textrm{H}.c.})$
$\displaystyle-$ $\displaystyle
V\sum_{i}(\Delta_{i,1}~{}c_{i+1,A}c_{i,A}+{\textrm{H}.c.})+E_{{\rm const}},$

where,

$\displaystyle E_{{\rm const}}$ $\displaystyle=$
$\displaystyle\sum_{i}[V|\Delta_{i,1}|^{2}+V^{\prime}(|\Delta_{i,2}|^{2}+|\Delta_{i,3}|^{2})]$
$\displaystyle+$
$\displaystyle\sum_{i}[Vn_{i+1,A}n_{i,A}+V^{\prime}n_{i,B}(n_{i,A}+n_{i+1,A})],$
$\displaystyle\tilde{\mu}_{i,A}$ $\displaystyle=$
$\displaystyle\mu+V(n_{i-1,A}+n_{i+1,A})+V^{\prime}(n_{i-1,B}+n_{i,B}),$
$\displaystyle\tilde{\mu}_{i,B}$ $\displaystyle=$
$\displaystyle\mu+V^{\prime}(n_{i,A}+n_{i+1,A}).$

Imposing spatial uniformity on local densities and the pairing amplitudes, we
transform the Hamiltonian to Bloch basis using the standard Fourier
transformations. The mean-field Hamiltonian in Bloch space takes the form,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{k}[\epsilon^{A}(k)c^{\dagger}_{k,A}c_{k,A}+\epsilon^{B}(k)c^{\dagger}_{k,B}c_{k,B}]$
(10) $\displaystyle-$ $\displaystyle
t^{\prime}\sum_{k}[(1+e^{-ik})c^{\dagger}_{k,A}c_{k,B}+{\textrm{H}.c.}]$
$\displaystyle-$ $\displaystyle
V\sum_{k}[\tilde{\Delta}_{1}(k)~{}c^{\dagger}_{k,A}c^{\dagger}_{-k,A}+{\textrm{H}.c.}]$
$\displaystyle-$ $\displaystyle
V^{\prime}\sum_{k}[\tilde{\Delta}_{2}(k)c^{\dagger}_{k,A}c^{\dagger}_{-k,B}-\tilde{\Delta}_{2}(-k)c^{\dagger}_{k,B}c^{\dagger}_{-k,A}+{\textrm{H}.c.}]$
$\displaystyle+$ $\displaystyle E_{{\rm const}},$

where,

$\displaystyle\tilde{\Delta}_{1}(k)$ $\displaystyle=$ $\displaystyle
2i\Delta_{1}\sin k,$ $\displaystyle\tilde{\Delta}_{2}(k)$ $\displaystyle=$
$\displaystyle\Delta_{2}-e^{ik}\Delta_{3},$ $\displaystyle\epsilon_{A}$
$\displaystyle=$ $\displaystyle-\mu-2n_{A}V-2n_{B}V^{\prime}-2t\cos{k},$
$\displaystyle\epsilon_{B}$ $\displaystyle=$
$\displaystyle-\mu-2n_{A}V^{\prime},$ $\displaystyle E_{{\rm const}}$
$\displaystyle=$ $\displaystyle
N[V(n_{A}^{2}+|\Delta_{1}|^{2})+V^{\prime}(2n_{B}n_{A}+|\Delta_{2}|^{2}+|\Delta_{3}|^{2})].$

It is convenient to represent the Hamiltonian in the following Nambu spinor
notation,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k}\Psi_{k}^{\dagger}H_{k}\Psi_{k},$ (11)

with,

$\displaystyle\Psi_{k}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}c^{\dagger}_{k,A}&c^{\dagger}_{k,B}&c_{-k,A}&c_{-k,B}\end{bmatrix}^{\dagger},H_{k}=\begin{bmatrix}\beta_{k}&\alpha_{k}\\\
\alpha^{\dagger}_{k}&-\beta^{T}_{-k}\end{bmatrix},$ $\displaystyle\beta_{k}$
$\displaystyle=$ $\displaystyle\begin{bmatrix}-\tilde{\mu}_{A}-2t\cos
k&~{}~{}~{}~{}-t^{\prime}(1+e^{-\mathrm{i}k})\\\
-t^{\prime}(1+e^{\mathrm{i}k})&-\tilde{\mu}_{B}\end{bmatrix},$
$\displaystyle\alpha_{k}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}-2\mathrm{i}V\Delta_{1}\sin
k&~{}~{}~{}~{}-V^{\prime}\Delta_{2}(1-e^{-\mathrm{i}k})\\\
V^{\prime}\Delta_{2}(1-e^{\mathrm{i}k})&0\end{bmatrix},$ (12)

$\tilde{\mu}_{A}=\mu+2n_{A}V+2n_{B}V^{\prime}$ and
$\tilde{\mu}_{B}=\mu+2n_{A}V^{\prime}$. The Hamiltonian Eq. (11) acquires a
diagonal form,
$H=E_{0}+\sum_{k,\nu}E_{k,\nu}\gamma^{\dagger}_{k,\nu}\gamma_{k,\nu}$, with
the help of Bogoliubov transformations[39],

$\displaystyle c_{k,s}$ $\displaystyle=$
$\displaystyle\sum_{\nu}u^{\nu}_{k,s}~{}\gamma_{k,\nu}+(v^{\nu}_{k,s})^{*}\gamma^{\dagger}_{-k,\nu}$
$\displaystyle c^{\dagger}_{-k,s}$ $\displaystyle=$
$\displaystyle\sum_{\nu}v^{\nu}_{-k,s}\gamma_{k,\nu}+(u^{\nu}_{-k,s})^{*}~{}\gamma^{\dagger}_{-k,\nu}$
(13)

Local states for the Hamiltonian in real space may be denoted as $E_{0}$
denotes the ground state energy and $E_{k,\nu}$ the excitation spectrum. The
constraint on the transformation to be unitary implies
$\sum_{\nu}|u^{\nu}_{k,s}|^{2}+|v^{\nu}_{k,s}|^{2}=1$ and
$u^{\nu}_{-k,s}=u^{\nu}_{k,s}$ $v^{\nu}_{-k,s}=-v^{\nu}_{k,s}$ Notice that the
basis of the Hamiltonian is doubled, making the solution redundant. Analysis
have shown that $E_{-k,\nu}=-E_{k,\nu}$[39]. Imposing this constraint, the
mean-field parameters are computed in terms of the transformation coefficients
as,

$\displaystyle n_{s}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k,\nu}|u^{\nu}_{k,s}|^{2}~{}f(E_{k,\nu})+|v^{\nu}_{k,s}|^{2}~{}(1-f(E_{k,\nu}))$
$\displaystyle{\Delta_{1}}$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{i}}{N}\sum_{k,\nu}\sin
k~{}u^{\nu}_{k,A}(v^{\nu}_{k,A})^{*}(2f(E_{k,\nu})-1)$
$\displaystyle\Delta_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k,\nu}(u^{\nu}_{k,B}(v^{\nu}_{k,A})^{*}~{}(1-f(E_{k,\nu}))-u^{\nu}_{k,A}(v^{\nu}_{k,B})^{*}~{}f(E_{k,\nu}))$
(14)

where, $f$ denotes the Fermi function and $N$ the number of sites. The mean-
field parameters are determined self-consistently by starting with a random
initial choice of the set {$\Delta_{1},\Delta_{2},n_{A},n_{B}$}. We use
$\delta_{p}=10^{-4}$ and $\delta_{n}=10^{-2}$ as the tolerance for the
convergence condition on pairing and density parameters, respectively.

##  II Additional mean-field results

###  II.1 Phase Diagram

Figure 6: Ground state phase diagram in the $t-\mu$ plane for $V^{\prime}=1$,
and (a) $V=3$, (b) $V=0.3$

In order to check that the mean-field phase diagrams do not contain any
qualitatively different phase from those already discussed in main text, we
present the ground states for other representative choices of $V$ and
$V^{\prime}$. The phase diagram discussed in the main text corresponded to
$V=V^{\prime}=1$. Here we check for strong and weak $V/V^{\prime}$ limits. We
find that there is no qualitative change in the overall structure of the phase
diagrams. Looking at the three phase diagrams together (see Figs. 6(a) and
6(b), and FIG. 2(e) in the main text), one can see the evolution of phases
with changing $V^{\prime}/V$. The $w=1$ TSC phase is missing in the
$V>V^{\prime}$ limit, and it occupies progressively larger area as
$V^{\prime}/V$ increases. Similarly, CMI phase dominates the $t-\mu$ phase
diagram in the large $V^{\prime}/V$ limit, and continues to appear in two
disjoint pockets upon reducing $V^{\prime}/V$. For $V^{\prime}/V=0.3$, one of
the CMI regions disappears and the entire density range at low $t$ hosts TSC
($w=-1$) phase.

###  II.2 Tunneling Density of States

We investigate further the nature of different ground states in terms of
sublattice-resolved tunneling density of states (TDOS), which is one of the
physical quantities that is directly accessible via tunneling spectroscopy
experiments. The sublattice-resolved TDOS is defined as,

$N_{s}(\omega)=\frac{1}{N}\sum_{k,\nu}[|u^{\nu}_{k,s}|^{2}\delta(\omega-
E_{k,\nu})+|v^{\nu}_{k,s}|^{2}\delta(\omega+E_{k,\nu})]$ (15)

where $N$ is number of sublattice sites, $u^{\nu}_{k,s}$ and $v^{\nu}_{k,s}$
are the electron-like and hole-like amplitudes of the Bogoliubov
quasiparticles residing on sublattice $s$. For the calculations, the delta
functions are approximated by the Lorentzians with broadening parameter
$0.005$. Note that, except in the CMI phase, the TDOS spectra display two
gaps. It is the gap at zero energy that is relevant as it stabilizes the order
by lowering the total energy of the system compared to a non-gapped phase. The
gap away from zero energy has its origin in the electronic bands in the non-
interacting limit. The TDOS spectra also helps in differentiating between the
SC gaps from those due to charge ordering. Note the asymmetry about zero in
the sublattice-resolved TDOS for the case of CMI.

Figure 7: (a)-(d) Sublattice-resolved and average tunneling density of states
(TDOS) for different ground states taken from the $V=1=V^{\prime}$ phase
diagram. (a) CMI: $t=0.1$, $\mu=-1.6$, (b) TSC with $w=-1$, $n>0.5$: $t=0.2$ ,
$\mu=-1.0$, (c) TSC with $w=-1$, $n<0.5$: $t=0.4$ , $\mu=-2.0$ and, (d) TSC
with $w=1$: $t=2.0$ , $\mu=-0.2$.

Another very useful quantity that possesses spatially resolved information is
the local DOS (LDOS). LDOS is defined as,

$\textrm{LDOS}~{}(i,E)=\sum_{s}[|u^{\nu}_{i,s}|^{2}\delta(E-E_{\nu})+|v^{\nu}_{i,s}|^{2}\delta(E+E_{\nu})],$
(16)

where, $u$ and $v$ are the elements of the real-space Bogoliubov
transformation matrix. LDOS at $E=0$ is especially relevant as it provides a
first signature of the presence of Majorana modes in a superconductor. In the
main text we have discussed LDOS at $E=0$ in the presence of quenched disorder
(see FIG. 5 in main text).

##  III Local Entanglement Entropy

Figure 8: Entanglement entropy $S$, charge modulation $\delta n$ and winding
number $w$ as a function of $\mu$ for, (a) $t=2.3$ and (b) $t=0.15$. (c)
Surface plot of entanglement entropy in the $t-\mu$ plane. We have used
$V=1=V^{\prime}$ and $T=0$ for all the results shown in this figure.

It has been proposed that entanglement entropy can be used efficiently to
identify phase transitions [49]. We verify this proposal for the model under
consideration by implementing the calculations of entanglement entropy. Local
states for the Hamiltonian in real space may be denoted as
$\ket{00}=\ket{0}_{A}\ket{0}_{B}$, $\ket{10}=\ket{1}_{A}\ket{0}_{B}$,
$\ket{01}=\ket{0}_{A}\ket{1}_{B}$, $\ket{11}=\ket{1}_{A}\ket{1}_{B}$. The
presence (absence) of electron at site $i,A$ or $i,B$ is denoted by $1$($0$).
The translational invariance of the Hamiltonian justifies the omission of
index $i$ in the above notation. Local density matrix for any site is then
given by tracing over all the other sites in the total density matrix. This is
given by [49],

$\rho_{i}=z\ket{00}\bra{00}+a\ket{10}\bra{10}+b\ket{01}\bra{01}+d\ket{11}\bra{11}$
(17)

where,

$\displaystyle d$ $\displaystyle=$
$\displaystyle\braket{\hat{n}_{i,A}~{}\hat{n}_{i,B}}\approx\braket{\hat{n}_{i,A}}\braket{\hat{n}_{i,B}},$
$\displaystyle a$ $\displaystyle=$
$\displaystyle\braket{\hat{n}_{i,A}}-d,b=\braket{\hat{n}_{i,B}}-d,$
$\displaystyle z$ $\displaystyle=$ $\displaystyle 1-a-b-d.$ (18)

Note that all the parameters in Eq. (18) are calculated within the mean field
approximation. Von Neumann entropy, $S$, for the density matrix is then
defined as,

$\displaystyle S=-d\ln{d}-a\ln{a}-b\ln{b}-z\ln{z}.$ (19)

In order to test the usefulness of $S$ in determining the phase boundaries, we
show the variation of $S$ with $\mu$ in the same plots that display the
variations of $w$ and $\delta n$ (see FIG. 8 (a)-(b)). We find that every
instance of change in $w$ is accompanied by an anomaly in $S$. In order to
test the usefulness of $S$ as an indicator of phase boundaries, we plot
$S$($t$, $\mu$) as a surface plot (see FIG. 8(c)). Indeed, the entire phase
diagram shown in FIG. 2(e) of the main text can be constructed based on this
surface plot of entanglement entropy.

In the main text, we have mentioned that the sign of $w$ for topologically
non-trivial phases are found to be perfectly correlated with the sign of
$\delta n$. An interesting interpretation emerges if we consider the SO(3)
theory of intertwined charge and superconducting orders [50]. Within this
theory, charge order and superconductivity are combined into a vector order
parameter with the $z$ component representing the charge order. Existence of a
finite $\delta n$ fixes the reference for orientation of the loop that defines
the winding number. Therefore the change in sign of $w$ may be interpreted as
the change in the viewpoint from positive to negative $z$ axis, as controlled
by the sign of $\delta n$.

##  IV DMRG calculations

###  IV.1 Averaged density distribution in the A and B chains

Figure 9: Averaged density distribution $n_{\rm A}$ and $n_{\rm B}$ as a
function of filling $n$ as well as chemical potential $\mu$ for (a) $t=2.0$,
$t^{\prime}=V=V^{\prime}=1.0$ and (b) $t=t^{\prime}=V^{\prime}=1.0$, $V=3.0$.
Periodic chains are used.

It is useful to see how the spinless fermions are distributed over the A and B
sites. In Fig. 9 DMRG results for the densities $n_{\rm A}$ and $n_{\rm B}$
are plotted as a function of filling $n$ as well as chemical potential $\mu$.
The chemical potential is estimated as $\mu=-\partial E_{0}/\partial N_{\rm
f}$, where $E_{0}$ is the ground-state energy and $N_{\rm f}$ is total number
of spinless fermions. We consider two parameter sets: (a) $t=2.0$,
$t^{\prime}=V=V^{\prime}=1.0$ and (b) $t=t^{\prime}=V^{\prime}=1.0$, $V=3.0$.
In the case of (a) both $n_{\rm A}$ and $n_{\rm B}$ changes continuously with
increasing $n$ although $n_{\rm A}$ is larger than $n_{\rm B}$ to gain the
transfer energy. On the other hand, in the case of (b) the most of fermions
are distributed in the A chain at $n<0.5$ as a consequence of trivial phase
separation (PS). Nevertheless, when the densities are plotted as a function of
$\mu$, we can see a jump at some $\mu$, which corresponds to half filling
$n=0.5$, for both of the parameter sets. This is a ‘charge-modulated-
insulators (CMI) plateau’ like a ‘Mott plateau’ in the Hubbard model. A
nearly-flat behavior of $n_{\rm A}$ with $\mu$ is another typical feature of
PS.

###  IV.2 Phase separation

Figure 10: Local density distribution $n_{i,{\rm A}}$ and $n_{i,{\rm B}}$ for
(a) $t=2.0$, $t^{\prime}=V=V^{\prime}=1.0$ and (b)
$t=t^{\prime}=V^{\prime}=1.0$, $V=3.0$.

For $t=t^{\prime}=V^{\prime}=1.0$, $V=3.0$ we found a trivial PS phase
signified as macroscopic condensate of fermions at low density. The PS can be
simply detected by looking at the local density distribution with open
boundary conditions in DMRG calculation. In Fig. 10 the local densities
$n_{i,{\rm A}}$ and $n_{i,{\rm B}}$ are plotted as a function of site $i$,
where a $241$-site open chain is used. We can clearly see a separation into
particle-rich and particle-poor phases at $n<0.5$ for
$t=t^{\prime}=V^{\prime}=1.0$, $V=3.0$. Note that this PS is different from
the topological PS discussed in the main text. On the other hand, a uniform
distribution of the local densities for $t=2.0$, $t^{\prime}=V=V^{\prime}=1.0$
indicate the absence of trivial PS.

###  IV.3 Density-density and Pair-Pair correlation functions

Figure 11: (a) Possible sets of sites and pairs for the density-density and
pair-pair correlation functions, respectively. Density-density correlation
functions (left panel) and SC correlation functions (right panel) for (b)
$n=0.25$, (c) $n=0.6$, and (d) $n=0.9$ with $t=2.0$,
$t^{\prime}=V=V^{\prime}=1.0$; (e) $n=0.625$ with
$t=t^{\prime}=V^{\prime}=1.0$, $V=3.0$.

In 1D system the electronic phase is generally characterized by the dominant
correlation function with slower decay as a function of distance. Depending on
parameter and filling, our system has two possible phases: charge density wave
(CDW) and superconducting (SC). Therefore, the density-density
$\langle\hat{n}_{i,s}\hat{n}_{j,s^{\prime}}\rangle$ and pair-pair $\langle
P_{ii^{\prime}ss^{\prime}}^{\dagger}P_{jj^{\prime}tt^{\prime}}\rangle$, where
$P_{ijss^{\prime}}=c_{i,s}^{\dagger}c_{j,s^{\prime}}^{\dagger}$, correlation
functions are considered. As denoted in Fig. 11(a), we calculated the density-
density correlation function between two A sites (green) or two B sites
(orange); the SC correlation function between two pairs of AB-AB (black), AA-
AB (red), or AA-AA (blue). The DMRG results with $N=2003$ open chain are shown
in Fig. 11(b-e). Both of the density-density and SC correlation functions
exhibit power-law decay with distance $r=|i-j|$ except $n\sim 0.6$ for
$t=2.0$, $t^{\prime}=V=V^{\prime}=1.0$. Since the each correlation function
shows the same decay for any set of sites or pairs, the exponent $\gamma$ of
$\langle\hat{n}_{i,s}\hat{n}_{j,s^{\prime}}\rangle\propto 1/r^{\gamma}$,
$\langle
P_{ii^{\prime}ss^{\prime}}^{\dagger}P_{jj^{\prime}tt^{\prime}}\rangle\propto
1/r^{\gamma}$ can be uniquely estimated for a given parameter and filling. The
inverse exponent $1/\gamma$ is plotted as a function of filling in Fig. 4(a,b)
of the main text.

As mentioned above, the SC correlation function exhibits an exponential decay
instead of a power-law decay in the CDW phase ($0.43\lesssim n\lesssim 0.75$)
for $t=2.0$, $t^{\prime}=V=V^{\prime}=1.0$. This means that our system
deviates from the Tomonaga-Luttinger (TL) liquid picture in this range.

###  IV.4 Tomonaga-Luttinger parameter

Figure 12: Finite-size scaling analysis of Tomonaga-Luttinger parameter for
$t=2.0$, $t^{\prime}=V=V^{\prime}=1.0$ (left panel) and
$t=t^{\prime}=V^{\prime}=1.0$, $V=3.0$ (right panel). In the SC phase, fitting
with 2nd polynomial function is performed.

To confirm the validity of estimated exponent $\gamma$, we also calculated the
TL parameter $K_{\rho}$. It can be obtained accurately via the derivative of
charge structure factor at $q=0$ as $K_{\rho}=\frac{1}{2}\lim_{q\to 0}\langle
n(q)n(-q)\rangle$ with $q=2\pi/N$ and
$n(q)=\sum_{l}(e^{-iql}c_{l,A}^{\dagger}c_{l,A}+e^{-iq(l+1/2)}c_{l,B}^{\dagger}c_{l,B}$)
using DMRG technique. Since our system is spinless fermions, the
noninteracting case corresponds to $K_{\rho}=0.5$. The CDW and SC phases are
identified by $K_{\rho}<0.5$ and $K_{\rho}>0.5$, respectively. In Fig. 12
finite-size scaling of $K_{\rho}$ is performed for several typical fillings.
In the SC phase, the scaling is simple and $K_{\rho}$ is easily extrapolated
to a value larger than $0.5$. On the other hand, in the non-TL range where the
SC correlation function exhibits an exponential decay with distance,
$K_{\rho}$ is extrapolated to $0$ in the thermodynamic limit. The extrapolated
values are plotted as a function of filling in Fig. 4(a,b) of the main text.
We can thus confirm that the relation $K_{\rho}=1/\gamma$ is perfectly
fulfilled at any filling.

###  IV.5 Binding energy

Figure 13: Binding energy as a function of filling for (a,b) $t=2.0$,
$t^{\prime}=V=V^{\prime}=1.0$ and (c,d) $t=t^{\prime}=V^{\prime}=1.0$,
$V=3.0$. Periodic and open chains are used in (a,c) and (b,d), respectively.

Finally, we calculated the binding energy between spinless fermions. The
binding energy is defined as $\Delta_{\rm B}=[E_{0}(N_{\rm f}+2)-E_{0}(N_{\rm
f})]-2[E_{0}(N_{\rm f}+1)-E_{0}(N_{\rm f})]$, where $E_{0}(N_{\rm f})$ is the
ground-state energy of our system with $N_{\rm f}$ fermions. In Fig. 13 DMRG
results for $\Delta_{\rm B}$ are plotted as a function of filling. Both
periodic and open chains are used. Although finite $\Delta_{\rm B}$ with
finite-size cluster may indicate an enhancement of short-range pairing
correlations, it is extrapolated to zero in the whole range of $n$ and
parameters except for the case of PS.


# Task-Specific Data Augmentation and Inference Processing for VIPriors
Instance Segmentation Challenge

Bo Yan, Xingran Zhao, Yadong Li, Hongbin Wang
Ant Group, China

###### Abstract

Instance segmentation is applied widely in image editing, image analysis and
autonomous driving, etc. However, insufficient data is a common problem in
practical applications. The Visual Inductive Priors(VIPriors) Instance
Segmentation Challenge has focused on this problem. VIPriors for Data-
Efficient Computer Vision Challenges ask competitors to train models from
scratch in a data-deficient setting, but there are some visual inductive
priors that can be used. In order to address the VIPriors instance
segmentation problem, we designed a Task-Specific Data Augmentation(TS-DA)
strategy and Inference Processing(TS-IP) strategy. The main purpose of task-
specific data augmentation strategy is to tackle the data-deficient problem.
And in order to make the most of visual inductive priors, we designed a task-
specific inference processing strategy. We demonstrate the applicability of
proposed method on VIPriors Instance Segmentation Challenge. The segmentation
model applied is Hybrid Task Cascade based detector on the Swin-Base based
CBNetV2 backbone. Experimental results demonstrate that proposed method can
achieve a competitive result on the test set of 2022 VIPriors Instance
Segmentation Challenge, with 0.531 AP@0.50:0.95.

##  1 Introduction

Instance segmentation applies widely in image editing, image composition,
autonomous driving, etc. Instance segmentation is a fundamental problem in
computer vision. Deep learning-based methods have achieved promising results
for image instance segmentation over the past few years, such as Mask R-CNN
[5], PANet [8], TensorMask [3], CenterMask [12]. SOLO series [10, 11]. The
main objective of the VIPriors Instance Segmentation Challenge is to tackle
the segmentation of basketball and individual humans including players,
coaches and referees on a basketball court. Different from previous studies,
VIPriors Instance Segmentation Challenge does not allow using any pre-trained
model, and training data are deficient.

In order to address the problem of data-deficient, we designed a task-specific
data augmentation(TS-DA) strategy, which mainly contains two components,
specific copy-paste and base-transform strategies. The main purpose of TS-DA
strategy is to increase the diversity of data distribution that can help with
the data-deficient problem. In order to make the most of visual inductive
priors, we designed a task-specific inference processing(TS-IP) strategy that
mainly includes two parts, inference cropping and max score filtering
strategies.

##  2 Methodology

In this section, we first illustrate the task-specific data augmentation
strategy in Sec.2.1. The task-specific inference processing strategy is
presented in Sec.2.2. And the segmentation model and training strategy is
introduced in Sec.2.3.

###  2.1 Task-Specific Data Augmentation

For VIPriors instance segmentation, although the scenario seems simple because
the task is only to segment basketball and players, coaches and referees on
images recorded of a basketball court, total images including train,
validation and test are also insufficient. In order to generate enough image
instances with wider distribution and make the model performs better and more
robust, we designed a task-specific data augmentation strategy that mainly
includes two parts, specific copy-paste and base-transform strategies.

Copy-Paste [4], which copies objects from one image to another, is
particularly useful for instance segmentation. Different from the original
copy-paste augmentation, we designed a specific copy-paste strategy including
view-specific copy-paste and ball-specific copy-paste.

We first cropped all objects including persons and basketballs on training and
validation datasets, then saved the cropped images and their corresponding
mask annotations to storage for the next specific copy-paste strategy.

We can know there are two views including the left view and right view for all
images by the visual inductive priors. And we can get the view of image by the
image name, if the prefix of image name end with “0”, the image comes from
right view, otherwise left view. We designed a view-specific copy-paste
strategy that copies and pastes the object into a specific location according
to the view of image.

For a given image, we would copy-paste several cropped persons and balls to
it. If the image is come from right view, the copy-paste location constraints
are Formula 1 and 3, otherwise are Formula 2 and 3. Our proposed view-specific
copy-paste strategy can make the copy-paste objects more evenly distributed on
the basketball court including the left and right borders, but excluding the
auditorium. Figure 1 shows examples of view-specific copy-paste strategy
method on the training set, above are original images and below are
corresponding augmented images.

$\frac{w}{5}\leq x_{min}\leq w$ (1) $0\leq x_{min}\leq w-\frac{w}{5}$ (2)
$\frac{h}{2}-\frac{h}{5}\leq y_{min}\leq\frac{h}{2}+\frac{h}{5}$ (3)

Where $w$ and $h$ are the width and height of the given image, ($x_{min}$,
$y_{min}$) denotes the top left coordinate of a person or ball that is being
pasted.

(a) image from right view (b) image from left view

Figure 1: Augmented image examples of view-specific copy-paste.

After the view-specific copy-paste phase, in order to further improve the
segmentation performance of basketball, we designed a ball-specific copy-paste
strategy including man-ball interaction copy-paste and pure-ball generation
strategies.

The purpose of the man-ball interaction copy-paste strategy is to improve the
segmentation performance of the ball when it interacts with person, For a
given image, we will select several target persons on basketball court, and
paste cropped balls around them, the copy-paste location constraints are as
follows.

$s_{x_{min}}\leq x_{min}\leq s_{x_{max}}$ (4) $s_{y_{min}}\leq y_{min}\leq
s_{y_{max}}$ (5)

Where ($s_{x_{min}}$, $s_{y_{min}}$) and ($s_{x_{max}}$, $s_{y_{max}}$) are
the top left and bottom right coordinate of selected person respectively,
($x_{min}$, $y_{min}$) denotes the top left coordinates of a ball that is
being pasted. Figure 2 shows examples of man-ball interaction copy-paste
strategy method on the training set, above are original images and below are
corresponding augmented images.

Figure 2: Augmented image examples of man-ball interaction copy-paste.

It is widely known that there are mainly two types of basketball by the visual
inductive priors, as shown in Figure 3. The basketball that likes shown in
Figure 3 (a) consists of at least two colors, we call it colorful ball,
otherwise the ball mainly consists of single color, we call it pure ball.

For a cropped ball image, we would directly change the RGB value of image but
the mask is unchanged as an approximate method of generating pure ball. For
example, if we want to get a brown pure ball, the Red value of image would be
changed into a random value from 80 to 90, and the Green and Blue value would
be changed into a random value from 50 to 60. The balls in the training and
validation set are mainly colorful balls, so we use pure-ball generation
method with probability to generate more pure balls when copy-pasting balls
including the view-specific copy-paste and man-ball interaction copy-paste
phases.

(a) colorful ball (b) pure ball

Figure 3: Image examples of common basketball types.

After the specific copy-paste pipeline, the images are input to base-
transform. The base-transform consists of two components, geometric transform,
and photometric distortion transform. The geometric transform includes three
detailed transforms, shear transform, rotate transform and translate
transform, and we randomly choose one of them as geometric transform for every
image. There are four detailed transforms for photometric distortion
transform, random brightness, random contrast, random saturation and random
hue, every image would be passed through these four transforms. After the
base-transform pipeline, we can get an augmented image. As shown in Figure 4,
above are the original images and below are examples of augmented results.

Figure 4: Augmented image examples of base-transform.

To sum up, our task-specific data augmentation mainly consists of two
components, specific copy-paste and base-transform strategies. The images are
first input to a view-specific copy-paste pipeline, we copy-paste several
persons and balls for every image with specific constraints. And then images
are input to a ball-specific copy-paste strategy, we copy-paste several balls
around persons, which could help further improve the segmentation performance
of basketball. We use the pure-ball generation method with probability to
generate more pure balls when copy-pasting balls including the view-specific
copy-paste and ball-specific copy-paste phases. Finally, the images are input
to base-transform. Followed by the specific copy-paste and base-transform
pipelines, we can get final augmented images as the input to the segmentation
network.

###  2.2 Task-Specific Inference Processing

In order to make the most of visual inductive priors, we designed a task-
specific inference processing strategy that mainly includes two parts,
inference cropping and max score filtering strategies.

Different from the common multi-scale training strategy, we resize the input
images randomly from a small scale to large scale, then randomly cropped and
padded them to a fixed scale, which is helpful to our inference cropping
strategy. We can know that the upper part of image is mainly auditorium by
visual inductive priors, so we cropped $1/5$ of image from above when
inference, then input to the trained model and get the segmentation results,
and invert the segmentation results to original image finally. Figure 5 is an
example of cropped image on the test set, left is the original image and right
is the cropped image when inference. The inference cropping strategy can
remove some invalid areas and improve model performance.

Figure 5: Image example of inference cropping on test set.

We can know that a basketball court can only have one basketball at most by
visual inductive priors, so we designed a max score filtering strategy for the
segment results of basketball. For a given test image, we can get the
segmentation results including human and basketball by the trained model, for
the segment results of basketball, we sort the segmentation results by score
and the maximum score result is preserved. And for the other segmentation
results of ball, we will calculate the Intersection over Union(IoU) of
bounding boxes and the bounding box of maximum score result, if IoU $>0$ the
result will be preserved otherwise dropped. In addition, we also filtered out
some balls with very small length and width($<10$) and very large length and
width($>40$). Figure 6 shows an example of max score filtering strategy on the
test set, left is original result of model output and right is the result of
max score filtering.

Figure 6: Result example of max score filtering on test set.

###  2.3 Segmentation Model and Training Strategy

Our segmentation model is Hybrid Task Cascade(HTC) [2] based detector on the
CBSwin-Base backbone with CBFPN [7]. The Mask R-CNN with MaskIoU head named as
Mask Scoring R-CNN [6], mask scoring can automatically learn the mask quality
instead of relying on the classification confidence of the bounding box, so we
add MaskIoU head to HTCMaskHead that can simply and effectively improve the
performance of instance segmentation due to the alignment between mask quality
and mask score.

We first train the model normally with proposed task-specific data
augmentation strategy. When the model is converged, we use SWA [13] training
strategy to finetune the model, which can make the model better and more
robust. The key components of model architecture and training pipeline are
shown in Figure 7.

Figure 7: Key components of model architecture and training pipeline for
VIPriors challenge.

##  3 Experiments

###  3.1 Training Details

We train and evaluate our model from scratch without any pre-trained model and
backbone. And we only use the training and validation datasets supplied by
VIPriors Instance Segmentation Challenge to train the model. Firstly, we
duplicated training and validation images 10 times without any change. The
total images for training are 2460 including 1840 training images and 620
validation images. Then the images are input to our proposed task-specific
data augmentation pipelines. And then the augmented images are randomly scaled
from 820 to 3080 on the short side and up to 3680 on the long side, and after
that randomly cropped and padded to (1920, 1440). Finally, the images are
input to segmentation model.

Firstly, we train the model with Adam with decoupled weight decay(AdamW) [9]
and initial learning rate(lr)=0.0001. When the model is converged, we use SWA
training strategy to finetune the model, and the optimizer is also AdamW with
initial lr=0.0001.

###  3.2 Experimental Results

As shown in Table 1, our proposed method finally achieves 0.531 AP@0.50:0.95
on the test set of VIPriors Instance Segmentation Challenge.

Models | Training Data | Data Augmentation | Methods | mAP-Test
---|---|---|---|---
HTC-CBSwinBase | Train+Val | - | - | 0.396
HTC-CBSwinBase+MaskIoU | Train+Val | - | - | 0.415
HTC-CBSwinBase+MaskIoU | Train+Val | TS-DA | - | 0.472
HTC-CBSwinBase+MaskIoU | Train+Val | TS-DA | +TTA | 0.477
HTC-CBSwinBase+MaskIoU | Train+Val | TS-DA | +Mask Threshold Binary=0.4 | 0.489
HTC-CBSwinBase+MaskIoU | Train+Val | TS-DA | +SWA | 0.512
HTC-CBSwinBase+MaskIoU | Train+Val | TS-DA | \+ Inference Cropping | 0.522
HTC-CBSwinBase+MaskIoU | Train+Val | TS-DA | \+ Max Score Filtering | 0.529
HTC-CBSwinBase+MaskIoU | Train+Val | TS-DA | +Ensemble(Mask Loss Weight=2.0) | 0.531

Table 1: Experimental results on dataset of VIPriors challenge. “+” indicates
that it is added on the basis of previous method.

###  3.3 Ablation Study

This section elaborates on how we achieve the final result by ablation study
to explain our method.

Our baseline is HTC-CBSwinBase and soft nms [1] is used at the test stage for
all experiments. The baseline achieves 0.396 mAP on test set. We add MaskIoU
head to HTC-CBSwinBase, it can improve the baseline by 0.019 mAP.

The experimental results show that TS-DA is a very effective strategy for this
task, it can bring an improvement of 0.057 mAP. Then test time
augmentation(TTA) is applied, our TTA strategy is flip horizontal and multi-
scale test with scale factors (1.0, 1.5, 2.0), and TTA can bring a little
improvement of 0.005 mAP. Then we attempt to adjust the parameters of test
configs, we adjust the threshold of rcnn mask binary from 0.5 to 0.4, which
can bring an improvement of 0.012 mAP unexpectedly. Then we use SWA training
strategy to finetune the model and it can improve 0.023 mAP. And then our
proposed inference cropping strategy can bring an improvement of 0.010 mAP.
Then we use the proposed max score filtering strategy to filter the results of
inference cropping strategy, it can improve 0.007 mAP and achieves 0.529 mAP
on the test set.

Finally, we train the model whose settings are the same as before but the loss
weight of mask head is changed from 1.0 to 2.0, and then we ensemble the
results of these two models, it achieves 0.531 mAP on the test set.

##  4 Conclusion

In this paper, we introduce the proposed method to address the VIPriors
instance segmentation problem. In order to tackle the data-deficient problem,
we designed a Task-Specific Data Augmentation strategy. And also we designed a
Task-Specific Inference Processing strategy according to the visual inductive
priors. We demonstrate the applicability of proposed method on the VIPriors
Instance Segmentation Challenge. Experimental results demonstrate that the
proposed method can achieve a competitive result on the test set without
external image or video data and pre-trained weights. Finally, our method
achieves 0.531 AP@0.50:0.95 on the test set of VIPriors Instance Segmentation
Challenge.

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