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@@ -0,0 +1,757 @@
+Draft version January 12, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Gravitational wave source populations: Disentangling an AGN component
+V. Gayathri,1, 2 Daniel Wysocki,2 Y. Yang,1 R. O’Shaughnessy,3 Z. Haiman,4 H. Tagawa,4 and I. Bartos1
+1Department of Physics, University of Florida, PO Box 118440, Gainesville, FL 32611-8440, USA
+2Leonard E. Parker Center for Gravitation, Cosmology, and Astrophysics, University of Wisconsin–Milwaukee, Milwaukee, WI 53201,
+USA∗
+3Center for Computational Relativity and Gravitation, Rochester Institute of Technology, Rochester, NY 14623, USA
+4Department of Astronomy, Columbia University, 550 W. 120th St., New York, NY, 10027, USA
+ABSTRACT
+The astrophysical origin of the over 90 compact binary mergers discovered by the LIGO and Virgo
+gravitational wave observatories is an open question. While the unusual mass and spin of some of the
+discovered objects constrain progenitor scenarios, the observed mergers are consistent with multiple
+interpretations. A promising approach to solve this question is to consider the observed distributions of
+binary properties and compare them to expectations from different origin scenarios. Here we describe a
+new hierarchical population analysis framework to assess the relative contribution of different formation
+channels simultaneously.
+For this study we considered binary formation in AGN disks along with
+phenomenological models, but the same framework can be extended to other models. We find that
+high-mass and high-mass-ratio binaries appear more likely to have an AGN origin compared to the
+same origin as lower-mass events. Future observations of high-mass black hole mergers could further
+disentangle the AGN component from other channels.
+1. INTRODUCTION
+Understanding the origin of binary black hole merg-
+ers is the first step in utilizing black hole mergers to
+probe a range of astrophysical processes.
+The LIGO
+(Aasi et al. 2015) and Virgo (Acernese et al. 2015) grav-
+itational wave observatories have discovered about 90
+binary mergers so far (Abbott et al. 2021a), provid-
+ing important information on the astrophysical popu-
+lation of mergers. Binary black hole systems can form
+through various channels, including isolated stellar bi-
+naries (Portegies Zwart & Yungelson 1998; Belczynski
+et al. 2002; Marchant et al. 2016; de Mink & Mandel
+2016) or triples (Antonini et al. 2014; Kimpson et al.
+2016; Veske et al. 2020), dynamical interactions in star
+clusters (Sigurdsson & Hernquist 1993; Portegies Zwart
+& McMillan 2000), primordial black holes formed in the
+early universe (Carr & Hawking 1974), and in the ac-
+cretion disks of active galactic nuclei (AGNs; McKernan
+et al. 2012; Bartos et al. 2017; Stone et al. 2017; Tagawa
+et al. 2020b; McKernan et al. 2020, 2022).
+The increased number of binary black hole observa-
+tions allows for a more detailed investigation of the pop-
+ulation’s mass and spin distributions.
+While individ-
+ual events may provide anecdotal suggestions hinting at
+∗ gayathri.v@ligo.org
+one formation channel or another, only an interpreta-
+tion of the full census can enable one to disentangle the
+potential contributions from multiple formation scenar-
+ios. Several studies have previously explored how to dis-
+entangle multiple channels, largely relying on compar-
+ison to phenomenologically-motivated estimates of the
+detailed outcomes of full formation scenarios (Doctor
+et al. 2020; Gerosa & Fishbach 2021; Gayathri et al.
+2021, 2020; Yang et al. 2020b; Tagawa et al. 2021; Kim-
+ball et al. 2021).
+Some studies also shown a mixture
+of channels is strongly preferred over any single channel
+dominating the detected population (Zevin et al. 2021).
+The latest population analysis carried out by LIGO-
+Virgo-KAGRA, GWTC3 (Abbott et al. 2021a,b), iden-
+tified several population features that may be indica-
+tive of the binaries’ origin. First, there appears to be
+a peak in the binary black hole mass spectrum around
+30-40 M⊙ compared to a more simple power-law type
+population (Tiwari & Fairhurst 2021; Talbot & Thrane
+2018; Edelman et al. 2022; Sadiq et al. 2022; Fishbach &
+Holz 2017). At the same time a few observed black holes
+had unusual properties, such as masses in the so-called
+upper mass gap (≳ 50 M⊙), highly unequal masses in
+the binary, high spin and precessing mergers, which are
+rare in stellar evolution and might be indicative of al-
+ternative formation scenarios.
+arXiv:2301.04187v1  [gr-qc]  10 Jan 2023
+
+2
+Here we introduce a flexible approach to compare
+the predictions of detailed formation models with
+observations while simultaneously accounting for the
+potentially-confounding contributions from a flexible
+phenomenologically-parameterized model for compact
+binary formation.
+Our paper is organised as follows. In section 2 we in-
+troduce binary formation in AGN disks, phenomenolog-
+ical descriptions for formation in non-AGN sources, and
+our flexible parametric population inference method. In
+section 3, we talk about the analysis’s key findings. In
+section 4, we summarize our findings and comment on
+future directions.
+2. METHODS
+2.1. Binary mergers in AGN disks
+We construct a one-parameter model for BBH forma-
+tion and merger within an AGN disk, parameterized
+by the maximum mass mmax of the natal BH distri-
+bution. Specifically, we adopt a seed BH mass distri-
+bution which follows the Salpeter mass function with
+index 2.35, dN/dm ∝ m−2.35 with given mmax.
+For
+neutron stars (NS) we assume a normal distribution
+m/M⊙ ∼ N(1.49, 0.19).
+The BHs and NSs are as-
+sumed to orbit a supermassive black hole in an AGN,
+migrating into the disk and inward from their natal lo-
+cations. Close to the AGN, these objects undergo multi-
+ple encounters, facilitating binary formation and merger.
+Other AGN parameters are fiducial values that are ex-
+pected to be typical, while there are large uncertainties.
+The range of possible values in the AGN models param-
+eter space is discussed in (McKernan et al. 2018).
+Following (Bartos et al. 2017), we adopted a geometri-
+cally thin, optically thick, radioactively efficient, steady-
+state accretion disk expected in AGNs. We used a vis-
+cosity parameter α = 0.1, radioactive efficiency ϵ = 0.1,
+fiducial supermassive BH mass M• = 106 M⊙ and ac-
+cretrion rate 0.1 ˙MEdd, where
+˙MEdd is the Eddington
+accretion rate.
+Using Yang et al. (2020) and Tagawa
+et al. (2020b,a), we have computed the expected mass
+and spin distributions of binary mergers in AGNs.
+Figure
+1
+shows
+the
+binary
+black
+hole
+merger
+intrinsic
+parameter
+distribution
+for
+the
+AGN
+model with different initial mass limits (mmax
+=
+[15M⊙, 35M⊙, 50M⊙, 75M⊙]).
+As the natal BH mass
+upper limit mmax increases, more massive binary com-
+ponents and total masses are allowed. At the same time,
+for high mmax, asymmetric systems are more frequent.
+By contrast, we have observed that the spin distribu-
+tion properties are largely independent of our choice for
+mmax limit.
+Figure 1.
+Parameter distributions for binary black holes
+formed in our AGN disk formation models; line color indi-
+cates the maximum natal BH mass. As expected, as the max-
+imum mass increases, total (M) mass upper limits increase.
+Additionally, a higher BH maximum natal mass mmax in-
+creases the relative frequency of asymmetric mergers, partic-
+ularly highly asymmetric mergers with q < 1/10.
+2.2. Phenomenology of AGN and non-AGN sources
+To allow for binary black holes which have a non-
+AGN origin, we follow previous work and introduce a
+few-parameter mixture model family. As illustrated in
+Figure 2, for the non-AGN component, we allow binary
+black holes to arise from a mixture of a power law and
+Gaussian components, as detailed in Table 11 of Abbott
+et al. (2020).
+In the power law component, the pri-
+mary is drawn from a pure power law mass distribution
+(with some unknown mmax,pl < 50M⊙ and unknown
+primary power law index); the mass ratio is drawn from
+another power law; and the spins are drawn from an
+
+10-
+mmax = 15
+mmax = 35
+10-2,
+mmax = 50
+mmax = 75
+10-3
+PDF
+10-4
+10-5
+10-6
+50
+100
+150
+200
+250
+300
+Taotal Mass (Mo)100
+PDF
+10-1
+mmax= 15
+mmax=35
+mmax= 50
+mmax = 75
+0.2
+0.4
+0.6
+0.8
+1.0
+Mass Ratio3
+unknown Beta distribution.
+In the Gaussian compo-
+nent, the primary and secondary are drawn from two
+independent Gaussian distributions with unknown mean
+and variance, with both means confined a priori to be
+near 30 − 40M⊙ to be consistent with expectations for
+PISN supernovae.
+By including a second component
+this phenomenological model can allow for non-power-
+law features and also allow for spin distributions that
+vary with mass. Each component k has some undeter-
+mined overall rate Rk. The top panel of Figure 2 shows
+the general non-AGN model.
+Our overall model is therefore a mixture model, pa-
+rameterized by the unknown (continuous) AGN merger
+rate and its (discrete) maximum mass mmax, along with
+all parameters of the non-AGN mixture model.
+The
+overall merger rate density dN/dV dXdt can therefore
+be expressed as a sum
+dN
+dV dXdt = Ragnpagn(X|mmax)+Rgpg(X|Λg)+Rplppl(X|Λpl)
+where X are binary parameters and where pq, Λq are the
+model distributions and parameters for the qth compo-
+nent (AGN, Gaussian, and power-law respectively).
+To systematically assess how well the distinctive fea-
+tures in AGN formation scenarios can be disentangled
+from this large model family, we will perform a se-
+quence of calculations with increasing model complexity,
+as shown in the panels of Figure 2. Specifically, we con-
+sider without any non-AGN component; the power-law
+and Gaussian (PL+G) model only; the power-law and
+AGN model (PL+AGN), without any Gaussian compo-
+nent; and finally the most general model with all three
+components.
+2.3. Population inference
+We describe and demonstrate a flexible parametric
+method to infer the event rate as a function of compact
+binary parameters, accounting for Poisson error and se-
+lection biases. In (Abbott et al. 2021c), analysed the
+Multi-Spin model which is the joint mass-spin model for
+binary black holes. Independent analyses have shown
+that there is a feature in the BBH mass spectrum around
+30-40 M⊙, which is modeled as a Gaussian peak on top
+of a power law continuum. It is an empirical way of mod-
+eling extra features, but here we tried to understand it
+feature with AGN models.
+In this section we review the population inference
+Wysocki et al. (2019) used for multi-formation chan-
+nel contributions. Binaries with intrinsic parameters x
+would merge at a rate dN/dVc dtdx = R p(x), where N
+is the number of detections, Vc is the comoving volume,
+R is the space-time-independent rate of binary coales-
+cence per unit comoving volume and p(x) is the probabil-
+ity of x from detected binaries. The binaries intrinsic pa-
+rameters includes mass mi and spins Si, where i = 1, 2.
+The likelihood of the astrophysical BBH population at a
+Figure 2. Graphical representations of the BBH population
+analysis in m1 − m2−spin parameter.
+Top panel for the
+PL+G, the middle panel for the PL+AGN and the bottom
+panel for the PL+G+AGN model.
+given merger rate R (Loredo 2004; Mandel et al. 2019;
+Thrane & Talbot 2019) and given binary intrinsic pa-
+rameters X ≡ (m1, m2, χ1, χ2), where χi = Si/m2
+i ,
+given the data for N detections D = (d1, ..., dN). This
+the likelihood is given by
+L(R, X) ≡ p(D|R, X),
+(1)
+L(R, X) ∝ e−µ(R,X)
+N
+�
+n=1
+�
+dx ℓn(x) R p(x, X),
+(2)
+where µ(R, X) is the expected number of detections un-
+der a given population parametrization X. Using Bayes’
+
+m2(Mo)
+spin
+BBH
+spin
+PL
+mi(Mo)m2(Mo)
+BBH
+spin
+PL+AGN
+m1
+1(Mo)m2(Mo)
+spin
+G
+BBH
+spin
+PL +AGN
+mi(Mo)4
+theorem one may obtain a posterior distribution on R
+and X after assuming some prior p(R, X). For compu-
+tational efficiency and to enable direct comparison with
+discrete formation models, we use the Gaussian likeli-
+hood approximation technique introduced in (Delfavero
+et al. 2022) to characterize each BBH observation’s like-
+lihood ℓn(x).
+Using this formalism, we estimate what fraction of
+binary black holes are generated via the AGN channel
+using detected gravitational wave merger information.
+To do this study we have upgraded current parametric
+methods with a mixture model feature. Here we have a
+freedom to do the analysis with number of models which
+has different astrophysical binary distributions.
+3. ANALYSIS
+As we discussed before, we perform population infer-
+ence analysis with mixture model feature and detected
+confident detection binary black hole detection. For this
+study we have considered different the astrophysical bi-
+nary distributions from AGN ( with different mmax),
+power-law, Gaussian peak and its combinations (see Fig-
+ure 2).
+3.1. Astrophysical merger rate
+We have estimated the astrophysical black hole merger
+rate for a given astrophysical model with a given
+number of GW detection.
+For this study, we have
+used the parameter estimation samples from obtained
+by LIGO-Virgo-KAGRA Collaboration (Abbott et al.
+2021d,e, 2019, 2021f) and it available in the Gravita-
+tional Wave Open Science Center (https: //www.gw-
+openscience.org) with the mixed model. Here we follow
+the same selection criteria as (Abbott et al. (2021b)),
+we have considered events with a false alarm rate of
+< 0.25yr−1.
+Table
+1
+shows
+the
+merger
+rates
+inferred
+for
+joint PL only,
+G only,
+AGN only (with different
+mmax), PL+AGN model (with different mmax) and
+PL+G+AGN models (with different mmax). We have
+estimated the merger rate results derived using each in-
+dividual model component as well as combined. We have
+observed that the inferred merger rate from the AGN-
+only models with different choices for mmax produces
+largely consistent results peaking near 50 Gpc−3yr−1.
+For the smallest maximum BH natal mass mmax =
+15M⊙, the inferred single-component AGN merger rate
+peaks around 80 Gpc−3yr−1. For all other models, they
+peak around the same R. Similarly, we have inferred
+the merger rate distribution for the single-component
+Gaussian, and power-law components. As expected, the
+merger rate from the power-law component dominates
+overall, as it incorporates and describes many frequent
+mergers of the lowest-mass binary black holes. In the
+case of PL+AGN, the inferred AGN and PL compo-
+nents have a well-determined merger rate, with median
+AGN merger rate ≃ 8/, Gpc−3yr−1 and PL merger rate
+≃ 30/, Gpc−3yr−1. Changing the maximum natal BH
+mass mmax has a very mild impact on the inferred AGN
+merger rate, and almost no effect on the inferred PL
+merger rate.
+In
+the
+case
+of
+PL+G+AGN
+analyses,
+the
+in-
+ferred AGN, G and PL components have a well-
+determined merger rate, with median AGN merger rate
+≃ 7/, Gpc−3yr−1, G merger rate ≃ 4/, Gpc−3yr−1 and
+PL merger rate ≃ 30/, Gpc−3yr−1.
+As we have seen
+in the PL+AGN study, we have not seen any major
+effect on PL or G merger rate when we change the
+AGN model. Our inferred merger rates deduced from
+the multi-component model are consistent with infer-
+ences performed using single-component models alone,
+suggesting that inference isolates only the contribution
+from each component.
+3.2. Inferred merger rate versus mass
+To better appreciate how well our inference directly
+projects out the relative contribution from each com-
+ponent, Figure 3.2 and 3.2 shows our inferred merger
+rate versus mass for PL+AGN and PL+G+AGN mod-
+els analyses respectively. In each plot we have shown
+each model component in mass space.
+As we expected the low mass region is highly con-
+tributed by the PL model compared to other models.
+We have observed a peak in PL model distribution
+around 7M⊙ for both PL+AGN as well as PL+G+AGN
+analyses. The peak is prominent for the PL+G+AGN
+study compared to PL+AGN. The high mass region is
+represented by only AGN, as we expected to see. Note
+that, the AGN model not only contributes in high mass
+region it also contributes full mass space as shown in 3.2
+and 3.2.
+3.3. Inferred merger rate versus mass ratio
+Similarly here we show the inferred merger rate ver-
+sus mass ratio for PL+AGN and PL+G+AGN models
+analyses.
+Figure 3.2 and 3.2 shows our inferred merger rate ver-
+sus mass ratio for PL+AGN and PL+G+AGN mod-
+els analyses respectively. In each plot we have shown
+each model component in mass ratio space. As we ex-
+pected the low mass ratio region contributed by AGN
+model for PL+AGN analysis and AGN & G models for
+PL+G+AGN analysis. For high q, the dominate contri-
+bution from PL model, that is consistence with detected
+events.
+
+5
+Analysis
+models
+mmax = 15
+mmax = 35
+mmax = 50
+mmax = 75
+PL only
+71.9+19.9
+−20.4
+-
+-
+-
+-
+G only
+19.1+2.5
+−2.3
+-
+-
+-
+-
+AGN only
+84.7+19.5
+−18.5
+49.8+6.1
+−5.5
+52.9+7.8
+−6.1
+53.2+7.7
+−6.2
+PL+AGN
+PL
+29.3+9.9
+−7.2
+28.8+11.3
+−6.8
+25.7+7.6
+−6.3
+26.8+9.5
+−6.2
+AGN
+8.7+6.3
+−4.5
+8.3+8.3
+−3.7
+12.7+6.5
+−4.1
+11.9+4.6
+−3.2
+PL+AGN+G
+PL
+21.3+7.3
+−5.2
+23.5+6.6
+−5.5
+21.6+6.7
+−5.0
+23.3+7.1
+−5.0
+AGN
+6.7+5.6
+−4.0
+6.2+5.0
+−3.5
+12.7+5.0
+−4.9
+12.9+4.3
+−4.3
+G
+10.2+2.2
+−2.3
+9.3+2.6
+−3.7
+5.9+2.9
+−1.9
+5.6+2.2
+−1.5
+Table 1. The astrophysical rates from PL+AGN and PL+AGN+G models. Each row corresponds to each analysis and each
+column corresponds to different AGN models with different initial mass limit.
+Figure 3.
+The inferred merger rate versus mass for
+PL+G+AGN and PL+G+ AGN model analyses. The solid,
+dashed and dotted lines for AGN, G and PL models compo-
+nents.
+3.4. Power-law model parameters
+As we discussed before, the contribution of a power-
+law model to the overall merger rate does not change
+substantially if we include or omit other model com-
+Figure 4. The inferred merger rate versus mass ratio for
+PL+AGN and PL+G+ AGN model analyses.
+The solid,
+dashed and dotted lines for AGN, G and PL models compo-
+nents.
+ponents like AGN or G. Among the models we con-
+sider, this quasi-universality is expected: the PL model
+most effectively reproduces the merger rate versus mass
+for the lowest-mass and most frequently merging binary
+
+AGN mmax = 15
+100
+AGN mmax = 35
+AGN mmax = 50
+AGN mmax = 70
+10-1
+R*p(m1)
+10-2
+10-3
+10-4
+101
+102
+mi(Mo)101
+AGNmm=15
+AGN mm = 35
+AGN mm = 50
+100
+AGN mm = 70
+10-1
+(Tw)d
+*
+10-2
+R
+10-3
+10-4
+101
+102
+mi(Mo)AGN mmax = 15
+AGN mmax = 35
+102
+AGN mmax = 50
+AGN mmax = 70
+101
+R* p(q)
+100
+10-1
+10-1
+100
+bAGNmm=15
+AGN mm = 35
+102
+AGN mm = 50
+AGN mm = 70
+101
+R*p(q)
+100
+10-1
+10-1
+100
+b6
+black holes.
+While the overall merger rate from this
+component is stable to our choice of the mixture, the
+model parameters recovered for PL depend strongly on
+which other confounding contributions are also present,
+as suggested by Figure 3.2 and 3.2.
+While the PL
+mass ratio distribution does not depend strongly on in-
+cluding or omitting AGN or G, the power law slope
+α and minimum mass mmin do change substantially.
+The estimated α median value with 65% credible inter-
+vals from different analysis as 1.6+0.2
+−0.2, 6.7+3.1
+−2.4,8.4+2.3
+−3.4,
+and 1.8+0.5
+−0.5 for PL-only, PL+G, PL+AGN (mmin=50)
+and PL+G+AGN (mmin=50) respectively. Similarly,
+mmin estimation are 2.5+0.3
+−0.3, 8.4+0.2
+−0.3, 8.5+0.2
+−0.4, and
+6.7+1.6
+−1.3 for PL-only, PL+G, PL+AGN (mmin=50) and
+PL+G+AGN (mmin=50) respectively.
+For example,
+the α estimation suggests that while a pure power-law
+model favours mmin close to the lower limit our priors al-
+low, incorporating other components causes the power-
+law component’s minimum mass to favour larger masses.
+With the pertinent mass range for the power-law chang-
+ing substantially via different mmin, unsurprisingly. The
+α estimation has a wide range of inferred power law ex-
+ponents, as the PL may dominate only an extremely
+narrow range of masses; see Figure 3.2.
+4. CONCLUSION
+In this paper, we have directly compared a one-
+parameter model for AGN binary black hole formation
+with the reconstructed sample of binary black holes
+identified via gravitational wave observations. To decon-
+volve the AGN component from binaries with different
+origin, we allow for BBH formation in both AGN and
+phenomenological channels. We consistently find a sig-
+nificant contribution to the merger rate from the AGN
+component (≃ O(5/Gpc−3yr−1).
+Our inferred AGN
+contribution follows by our prior belief on the maximum
+mass of BBH formed from other channels, which we pre-
+sume is less than 50M⊙ due to pair-instability impacts
+on stellar evolution and death.
+As in previous studies (Yang et al. 2020a,b; Gayathri
+et al. 2020, 2021; Vajpeyi et al. 2022), our models for
+AGN BBH formation predict a wide range of BBH mass
+ratios and frequent significant spins. At present, because
+the distinctive signatures of AGN formation are prefer-
+entially imparted only to the most massive BBH, the
+extant BBH sample does not yet contain enough events
+to provide overwhelming evidence in favour of an AGN
+component, consistent with prior work (Vajpeyi et al.
+2022) subsequent observations could support or rule out
+this channel.
+Acknowledgements We gratefully acknowledge the
+support of LIGO and Virgo for the provision of com-
+putational resources. G.V. and D.W. acknowledge the
+support of the National Science Foundation under grant
+PHY-2207728.
+I.B. acknowledges the support of the
+National Science Foundation under grants #1911796,
+#2110060 and #2207661 and of the Alfred P. Sloan
+Foundation.
+This research has made use of data,
+software and/or web tools obtained from the Gravita-
+tional Wave Open Science Center (https: //www.gw-
+openscience.org), a service of LIGO Laboratory, the
+LIGO Scientific Collaboration and the Virgo Collabora-
+tion. LIGO is funded by the U.S. National Science Foun-
+dation. Virgo is funded by the French Centre National
+de Recherche Scientifique (CNRS), the Italian Istituto
+Nazionale della Fisica Nucleare (INFN) and the Dutch
+Nikhef, with contributions by Polish and Hungarian in-
+stitutes. This material is based upon work supported by
+NSF’s LIGO Laboratory, which is a major facility fully
+funded by the National Science Foundation.
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+

2dFRT4oBgHgl3EQfnDfl/content/tmp_files/2301.13604v1.pdf.txt

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+Nonlinearities in Macroeconomic Tail Risk through
+the Lens of Big Data Quantile Regressions
+Jan Pr¨user
+TU Dortmund
+Department of Statistics
+Florian Huber1
+University of Salzburg
+Department of Economics
+Abstract.
+Modeling and predicting extreme movements in GDP is notoriously difficult
+and the selection of appropriate covariates and/or possible forms of nonlinearities are key in
+obtaining precise forecasts.
+In this paper, our focus is on using large datasets in quantile
+regression models to forecast the conditional distribution of US GDP growth.
+To capture
+possible non-linearities we include several nonlinear specifications. The resulting models will
+be huge dimensional and we thus rely on a set of shrinkage priors.
+Since Markov Chain
+Monte Carlo estimation becomes slow in these dimensions, we rely on fast variational Bayes
+approximations to the posterior distribution of the coefficients and the latent states.
+We
+find that our proposed set of models produces precise forecasts. These gains are especially
+pronounced in the tails. Using Gaussian processes to approximate the nonlinear component
+of the model further improves the good performance in the tails.
+JEL: C11, C32, C53
+KEYWORDS: Growth at risk, quantile regression, global-local priors, non-linear models,
+large datasets.
+1We would like to thank the editor, Mike McCracken, three anonymous referees, and participants at the
+International Symposium on Forecasting 2022 at the University of Oxford and at the Statistische Woche in M¨unster
+for helpful comments. Jan Pr¨user gratefully acknowledges the support of the German Research Foundation (DFG,
+468814087). Please address correspondence to: Jan Pr¨user. Department of Statistics, TU Dortmund. Address:
+CDI Building, Room 122, 44221 Dortmund, Germany. Email: prueser@statistik.tu-dortmund.de.
+arXiv:2301.13604v1  [econ.EM]  31 Jan 2023
+
+1
+Introduction
+Modeling and predicting the conditional distribution of output growth has attracted considerable
+academic attention in recent years. Starting at least with the influential paper by Adrian et al.
+(2019), focus has shifted towards analyzing whether there exist asymmetries between a predictor
+(in their case financial conditions) and output growth across different quantiles of the empirical
+distribution. Several other papers (Adrian et al., 2018; Ferrara et al., 2019; Gonz´alez-Rivera
+et al., 2019; Delle Monache et al., 2020; Plagborg-Møller et al., 2020; Reichlin et al., 2020;
+Figueres and Jaroci´nski, 2020; Adams et al., 2021; Mitchell et al., 2022) have started to focus on
+modeling full predictive distributions using different approaches and information sets. However,
+most of these contributions have been confined to models which exploit small datasets and, at
+least conditional on the quantile analyzed, assume linear relations between GDP growth and the
+predictors.2
+Times of economic stress such as the global financial crisis (GFC) or the Covid-19 pandemic
+have highlighted that exploiting information contained in many time series and allowing for
+nonlinearities improves predictive performance in turbulent periods (see, e.g., Huber et al.,
+2023). Since economic dynamics change in volatile economic regimes, models that control for
+structural breaks allow for different effects of economic shocks over time or imply nonlinear
+relations between GDP growth and its predictors often excel in forecasting applications (see
+D’Agostino et al., 2013; Carriero et al., 2016; Adrian et al., 2021; Clark et al., 2022b; Pfarrhofer,
+2022; Huber et al., 2023). Moreover, another important empirical regularity is that the set of
+predictors might change over time. This is because variables which are seemingly unimportant
+in normal periods (such as financial conditions) play an important role in recessions and yield
+important information on future behavior of output growth.
+This discussion highlights that the effect of predictors on output growth depends on the
+quantile under consideration and thus appears to be state dependent and modeling the tran-
+sition might call for nonlinear econometric models. The key challenge, however, is to identify
+the different determinants of GDP growth across quantiles while taking possible nonlinearities
+into account. In this paper, we aim to solve these issues by proposing a Bayesian quantile re-
+gression (QR) which can be applied to huge information sets, and which is capable of capturing
+nonlinearities of unknown form. Our model is a standard QR model that consists of two parts.
+2A recent exception is Kohns and Szendrei (2021) who estimate large-scale quantile regressions and then apply
+ex-post sparsification to sharpen predictive inference.
+2
+
+The first assumes a linear relationship between the covariates and quantile-specific GDP growth
+whereas the second component assumes an unknown and possibly highly nonlinear relationship
+between the two. The precise form of nonlinearities is captured through three specifications.
+One is parametric and based on including polynomials up to a certain order, whereas the re-
+maining two are nonparametric. Among these nonparametric specifications we include B-splines
+(see Shin et al., 2020) and Gaussian processes (see Williams and Rasmussen, 2006). Both have
+been shown to work well when it comes to function estimation and forecasting.
+The combination of a linear and nonlinear term implies that the dimension of the parame-
+ter space increases substantially. Since all these models can be cast in terms of a linear regression
+conditional on appropriately transformed covariates, we can use regularization techniques to de-
+cide on whether more flexibility is necessary and which variables should enter the model. We
+achieve this through several popular shrinkage priors that have excellent empirical properties in
+large dimensions and are relatively easy to implement. These shrinkage priors enable us to select
+promising subsets of predictors and the degree of nonlinearities for each quantile separately.
+Posterior inference using Markov Chain Monte Carlo (MCMC) techniques in these dimen-
+sions proves to be an issue because we have to estimate a large-scale regression model for all
+quantiles of interest. This procedure needs to be repeated a large number of times if we wish to
+carry out an out-of-sample forecasting exercise. To reduce the computational burden enormously
+we estimate the QRs using Variational Bayes (VB).3 This estimation strategy approximates the
+exact full conditional posterior distributions with simpler approximating distributions. These
+approximating densities are obtained by minimizing the Kullback-Leibler (KL) distance be-
+tween some known density q and the exact posterior distribution p. Hence, integration in huge
+dimensions is replaced by a simpler optimization routine. Our approach is fast and allows for
+computing all results of our forecasting exercise without the use of high performance computing
+environments.
+We apply our techniques to the large dimensional FRED-QD dataset (McCracken and Ng,
+2016) and focus on single and multi-step-ahead forecasting of US GDP growth over a hold-out
+period ranging from 1991Q2 to 2021Q3. The different nonlinear models we consider are high
+dimensional and feature up to around 1,000 coefficients per equation.
+The empirical results can be summarized as follows. Using huge information sets and
+nonlinear models in combination with priors that introduce substantial shrinkage pays off for tail
+3For an introduction, see Blei et al. (2017) and an algorithm for QRs is provided in Bufrei (2019).
+3
+
+forecasts. In both tails, forecast improvements relative to the small-scale QR model developed in
+Adrian et al. (2019) are sizable. When we focus on the center of the distribution the differences
+become smaller. Once we allow for nonlinearities we find modest improvements in predictive
+accuracy. Comparing the different nonlinear specifications reveals that Gaussian processes offer
+the largest improvements vis-´a-vis the linear QR. This indicates that a successful tail forecasting
+model should be able to extract important information from huge datasets, while controlling for
+possibly nonlinear relations. When we focus on the key properties of the proposed priors we
+observe that priors that imply a dense model (characterized by many small coefficients) yield
+good tail forecasts.
+The paper is structured as follows.
+The next section introduces the general QR and
+the scale-location mixture representation to cast the model in terms of a standard generalized
+additive regression with auxiliary latent variables. We then focus on the different priors used,
+provide additional details on the nonlinear components of the models, briefly discuss VB, outline
+how we estimate the posterior distributions of the parameters and latent quantities, and illustrate
+the computational properties of our approach. Section 3 discusses our empirical findings. The
+final section summarizes and concludes the paper.
+An Online Appendix includes additional
+technical details, empirical results and more precise information on the used dataset.
+2
+Bayesian analysis of general QRs
+2.1
+The likelihood function
+In this paper, our goal is to model the dependence between the qth quantile of GDP growth
+yt and a panel of K predictors in {xt}T
+t=1 with K being huge. The covariates include a wide
+range of macroeconomic and financial indicators. Possible nonlinearites between yt and xt are
+captured through a function gq(xt), with gq : RK → R.
+The fact that K is large and the
+inclusion of nonlinear functions of xt implies that the number of parameters is large relative to
+the number of observations T.
+Our workhorse model is the QR developed in Koenker and Bassett (1978). As opposed
+to the standard QR, our model decomposes the qth quantile function Qq(yt) in a linear and
+nonlinear part and a non-standard error distribution:
+(1)
+yt = x′
+tβq + gq(xt) + εt,
+4
+
+where βq is a K−dimensional vector of quantile-specific regression coefficients and εt is a shock
+term with density fq such that the qth quantile equals zero:
+� 0
+−∞
+fq(εt)dεt = q.
+Conditional on the quantile, this model resembles a generalized additive model (GAM), see
+Hastie and Tibshirani (1987).
+We approximate gq(xt) using nonlinear transformations of xt:
+(2)
+gq(xt) ≈
+M
+�
+m=1
+γqmzm(xt) = z′
+tγq
+with γq = (γq1, . . . , γqM)′, zt = (z1(xt), . . . , zM(xt))′ and zm(xt) denotes a basis function that
+depends on xt with γqm denoting the corresponding basis coefficient. This basis function de-
+pends on the specific approximation model used to infer the nonlinear effects and our additive
+representation nests models commonly used in the machine learning literature (such as Gaussian
+processes, splines, neural networks but also more traditional specifications such as time-varying
+parameter models). We will discuss the precise specification of zm (and thus zt) in more detail
+in Sub-section 2.3. Here it suffices to note that depending on the specification, M could be very
+large. For instance, in the Gaussian process case, M = T and thus the number of regression
+coefficients would be K + T.
+If fq remains unspecified, estimation of βq and γq is achieved by solving the following
+optimization problem:
+arg max
+{βq,γq}
+=
+T
+�
+t=1
+ρq(yt − x′
+tβq − z′
+tγq),
+with ρq(l) = l[q − I(l < 0)] denoting the loss function. This optimization problem is straight-
+forward to solve but, if K + M is large, regularization is necessary. This motivates a Bayesian
+approach to estimation and inference.
+From a Bayesian perspective, carrying out posterior inference requires the specification of
+a likelihood and suitable priors. Following Yu and Moyeed (2001) we assume that the shocks εt
+follow an asymmetric Laplace distribution (ALD) with density:
+fq(εt) = q(1 − q) exp (−ρq(εt)).
+The key thing to notice is that the qth quantile equals zero and the parameter q controls the
+5
+
+skewness of the distribution. Kozumi and Kobayashi (2011) show that one can introduce auxil-
+iary latent quantities to render the model with ALD distributed shocks conditionally Gaussian.
+This is achieved by exploiting a scale-location mixture representation (West, 1987):
+εqt = θqνqt + τq√σqνqtut,
+θq = 1 − 2q
+q(1 − q),
+τ 2
+q =
+2
+q(1 − q),
+νqt ∼ E
+� 1
+σq
+�
+,
+ut ∼ N(0, 1),
+where E
+�
+1
+σq
+�
+denotes the exponential distribution and σq is a scaling parameter. Hence, con-
+ditional on knowing νq = (νq1, . . . , νqT )′, θq, τq, σq and appropriately selecting gq, the model is a
+linear regression model with response ˆyt = yt −θqνqt and Gaussian shocks that are conditionally
+heteroskedastic. This conditional likelihood will form the basis of our estimation strategy.
+To complete the model specification we assume that
+1
+σq ∼ G(c0, d0), where c0 is the shape
+and d0 the rate parameter of the Gamma distribution which we set both to zero in order to
+obtain a flat prior. The choice of the prior distribution on βq and γq is essential for our high
+dimensional QRs. We discuss different suitable choices in the next section.
+2.2
+Priors for the quantile regression coefficients
+For the large datasets we consider in this paper, M + K ≫ T and thus suitable shrinkage priors
+are necessary to obtain precise inference. Kohns and Szendrei (2021) and Mitchell et al. (ming)
+use flexible shrinkage priors in large-scale QRs and show that these work well for tail forecasting.
+We build on their findings by considering a range of different priors on βq and γq. All these
+priors belong to the class of so-called global-local shrinkage priors (Polson and Scott, 2010) and
+have the following general form:
+βq|ψβ
+q1, . . . , ψβ
+qK, λβ
+q ∼
+K
+�
+j=1
+N(0, ψβ
+qjλβ
+q ),
+ψβ
+qj ∼ u,
+λβ
+q ∼ π,
+γq|ψγ
+q1, . . . , ψγ
+qM, λγ
+q ∼
+M
+�
+j=1
+N(0, ψγ
+qjλγ
+q),
+ψγ
+qj ∼ u,
+λγ
+q ∼ π,
+with λs
+q (s ∈ {β, γ}) denoting a quantile-specific global shrinkage parameter and ψs
+qj are local
+scaling parameters that allow for non-zero coefficients in the presence of strong global shrinkage
+(i.e., with λs
+q close to zero). The functions u and π refer to mixing densities which, if suitably
+chosen, translate into different shrinkage priors. In this paper, all the priors we consider can be
+cast into this form but differ in the way the mixing densities u and π are chosen. Since these
+6
+
+priors are well known, we briefly discuss them in the main text and relegate additional technical
+details to the Online Appendix.
+We focus on five shrinkage priors that have been shown to work well in a wide variety of
+forecasting applications (see, e.g., Huber and Feldkircher, 2019; Cross et al., 2020; Chan, 2021;
+Pr¨user, 2022). The first prior we consider is the Ridge prior. The Ridge prior is a special case
+of a global-local prior with local parameters set equal to 1 and a global shrinkage parameter
+which follows an inverse Gamma distribution. Formally, this implies setting ψs
+qj = 1 for all q, j
+and ��s
+q ∼ G−1(e0, e1). The hyperparameters e0 and e1 control the tightness of the prior. We
+set these equal to e0 = e1 = 0. This prior shrinks all coefficients uniformly towards zero and
+provides little flexibility to allow for idiosyncratic (i.e., variable-specific) deviations from the
+overall shrinkage pattern.
+This issue is solved by estimating the local shrinkage parameters. The Horseshoe (HS,
+see, Carvalho et al., 2010), our second prior, does this. This prior sets u and π to a half-Cauchy
+distribution:
+�
+ψs
+qj ∼ C+(0, 1) and �λsq ∼ C+(0, 1).
+The HS possesses excellent posterior
+contraction properties (see, e.g., Ghosh et al., 2016; Armagan et al., 2013; van der Pas et al.,
+2014). Moreover, it does not rely on any additional tuning parameters.
+Another popular global-local shrinkage prior is the Normal-Gamma (NG) prior of Griffin
+and Brown (2010). This prior assumes that u and π are Gamma densities. More formally,
+ψs
+qj ∼ G(ϑ, λs
+qϑ/2) and λs
+q ∼ G(c0, d0), with ϑ being a hyperparameter that controls the tail
+behavior of the prior, and c0 and d0 are hyperparameters that determine the overall degree of
+shrinkage. We set c0 = d0 = 0 and ϑ = 0.1. This choice implies heavy global shrinkage on the
+coefficients but also implies fat tails of the marginal prior of the coefficients after integrating
+out the local scaling parameters. The Bayesian LASSO is obtained as a special case of the NG
+prior with ϑ = 1.
+Finally, the Dirichlet-Laplace prior (Bhattacharya et al., 2015) assumes that the local scal-
+ing parameter ψs
+qj is a product of a Dirichlet-distributed random variate φs
+qj ∼ Dir(α, . . . , α) and
+a parameter ˜
+ψs
+qj ∼ E(1/2) that follows an exponential distribution. Hence, the Dirichlet-Laplace
+prior sets ψs
+qj = (φs
+qj)2 ˜
+ψs
+qj. On the global scaling parameters we use a Gamma distribution
+�
+λβ
+q ∼ G(Kα, 1/2) and
+�
+λγ
+q ∼ G(Mα, 1/2). We set α = 1
+K for the linear part and α =
+1
+M for
+the non-linear part.
+7
+
+2.3
+Capturing nonlinearities in high dimensional QRs
+In extreme periods such as the GFC or the Covid-19 pandemic, nonlinearities in macroeconomic
+data become prevalent. We control for this by having a nonlinear part in our QR. As stated in
+(2), we capture possible nonlinearities in xt through nonlinear transformations zm(xt).
+The first and simplest nonlinear specification maps xt into the space of polynomials. Bai
+and Ng (2008) capture nonlinearities in macro data through polynomials and by relying on
+factor-based predictive regressions. We follow this approach and define the corresponding basis
+function as follows:
+zt = ((x2
+t )′, (x3
+t )′, . . . , (xN
+t )′)′.
+Deciding on the order of the polynomial N is a model selection issue and suitable shrinkage
+priors can be adopted. In our empirical work, we focus on the cubic case. This specification
+will overweight large movements in xt and should thus be suitable for quickly capturing sharp
+downturns in the business cycle. In this case, the number of coefficients triples since M = 3K.
+The resulting nonlinear model is called Polynomial-QR.
+Adding cubic terms allows us to capture nonlinearities in a relatively restricted manner.
+Since the precise form of nonlinearities is typically unknown, the remaining two specifications
+we consider are nonparametric and only require relatively mild prior assumptions on the form
+of nonlinear interactions. The first of these two is the B-Spline (see, e.g., De Boor, 2001, for a
+review). B-Splines have a proven track record in machine learning and computer science (Shin
+et al., 2020).
+For the B-spline, we assume that each element in xt exerts a (possibly) nonlinear effect
+on yt that might differ across covariates. This implies that gq(xt) equals:
+gq(xt) ≈
+K
+�
+k=1
+Φk(x•,j)γq,k.
+Here, we let Φk denote a T ×r matrix of B-spline basis functions that depend on the jth covariate
+in X = (x′
+1, . . . , x′
+T )′, x•,j and r is the number of knots. In this case, the number of nonlinear
+coefficients is M = rK. In our empirical work we place the knots at the following quantiles of
+x•,j: {0, 0.05, 0.1, 0.25, 0.50, 0.75, 0.90, 0.95, 1}, implying that r = 9 and thus M = 9K. We
+will henceforth call this model Spline-QR.
+The last specification we consider is the Gaussian process (GP) regression. GP regression
+8
+
+is a nonparametric estimation method that places a GP prior on the function gq(xt):
+gq(xt) ∼ GP(µq(xt), K(xt, xt)).
+The mean function µq(xt) is, without loss of generality, set equal to zero and K(xt, xt) is a
+kernel function that encodes the relationship between xt and xt for t, t = 1, . . . , T. It is worth
+noting that our additive specification implies that if the mean function is set equal to zero, the
+model is centered on a standard QR.
+Since xt is observed in discrete time steps, the GP prior implies a Gaussian prior on
+gq = (gq(x1), . . . , gq(xT ))′:
+gq ∼ N(0T , K(w)),
+where K(w) is a T × T-dimensional matrix with (t, t)th element K(xt, xt). w = (w1, w2)′ is
+a set of hyperparameters that determine the properties of the kernel (and thus the estimated
+function).
+The GP regression is fully specified if we determine the kernel function K. In this paper,
+we use the Gaussian (or squared exponential) kernel:
+K(xt, xt) = w1 × exp
+�
+−w2
+2 ||xt − xt||2�
+.
+The hyperparameters w are set according to the median heuristic proposed in Arin et al. (2017).
+What we discuss above is the function-space view of the GP regression. An alternative
+way of expressing the GP is the so-called weight-space view. The weight-space view is obtained
+by integrating out gq, yielding the following regression representation:
+y = Xβq + Zγq + ε,
+with y denoting the stacked dependent variables, Z is the lower Cholesky factor of K and
+γq ∼ N(0, IT ). Notice that gq = Zγq. Hence, the Cholesky factor of the kernel matrix provides
+the basis functions, and the parameters can be readily estimated. In this case, the number
+of nonlinear coefficients is M = T. Since we use a shrinkage prior on γq, the corresponding
+implied kernel is given by ZBγ
+q Z′. The M × M matrix Bγ
+q is a prior covariance matrix with
+Bγ
+q = λγ
+q × diag(ψγ
+q1, . . . , ψγ
+qM). Approximating gq using GPs leads to the GP-QR specification.
+This completes our choice of nonlinear techniques used in the big data QR. Alternative
+9
+
+choices (such as allowing for time-varying parameters, neural networks or Bayesian additive
+regression trees) can be straightforwardly introduced in this general framework.
+2.4
+A brief introduction to variational Bayes
+The high dimensionality of the state space calls for alternative techniques to carry out posterior
+inference. We opt for using variational approximations to the joint posterior density. In this
+section, we provide a discussion on how VB works in general. For an excellent in-depth introduc-
+tion, see Blei et al. (2017). In machine learning, variational techniques have been commonly used
+to estimate complex models such as deep neural networks (see, e.g., Polson and Sokolov, 2017).
+In econometrics, recent papers use VB in huge dimensional multivariate time series models such
+as VARs (Gefang et al., 2022; Chan and Yu, 2020) or state space models to speed up estimation
+(Koop and Korobilis, 2023). In a recent paper, Korobilis and Schr¨oder (2022), propose a QR
+factor model and estimate it using VB techniques.
+To simplify the exposition, we fix the prior variances. The appendix provides information
+on how we estimate the prior variances (and associated hyperparameters) using VB. Let ξq =
+(βq, γq, σq, νq) denote a generic vector which stores all unknowns of the model, with νq =
+(νq1, . . . , νqT ) denoting the latent components.
+Our aim is to approximate the joint posterior distribution p(ξq|y) using an analytically
+tractable approximating distribution q(ξq). This variational approximation is found by mini-
+mizing the Kullback-Leibler (KL) distance between p and q. One can show that minimization
+of the KL distance is equivalent to maximizing the evidence lower bound (ELBO) defined as:
+(3)
+ELBO = Eq(ξq) (log p(ξq, y)) − Eq(ξq) (log q(ξq)) ,
+with Eq(ξq) denoting the expectation with respect to q(ξq). This implies that finding the approx-
+imating density q replaces the integration problem (which is typically solved through MCMC
+sampling) with an optimization problem (which is fast and thus scales well into high dimensions).
+A common and analytically tractable choice of approximating densities assumes that q(ξq)
+is factorized as follows:
+q(ξq) =
+S
+�
+s=1
+qs(ξqs),
+where ξqs denotes a partition of ξq. A particular example (which we use in this paper) would
+specify ξq1 = (β′
+q, γ′
+q)′, ξq2 = σq and ξq3 = νq.
+10
+
+This class is called the mean field variational approximation and assumes that the different
+blocks ξqs are uncorrelated.4 Notice that all our priors on ξq can be written as:
+p(ξq) =
+S
+�
+s=1
+p(ξqs),
+and using the fact that:
+Eq(ξq)(log p(ξq, y)) = Eq(ξq)(log p(y|ξq)) +
+S
+�
+s=1
+Eq(ξq)(log p(ξqs)),
+the ELBO can be stated as:
+ELBO = Eq(ξq)(log p(y|ξq)) +
+S
+�
+s=1
+Eq(ξq)(log p(ξqs)) −
+S
+�
+s=1
+Eq(ξq)(log q(ξqs)).
+Wand et al. (2011) prove that under the variational family the optimal approximating densities
+are closely related to the full conditional posterior distributions:
+q∗
+s(ξq) = exp
+�
+Eq(ξq)(log p(ξqs|y, ξq,−s)
+�
+,
+where ξq,−s is the vector ξq with the sth component excluded. Hence, if p(ξqs|y, ξq,−s) is known
+(which is the case for the QR regression based on the auxiliary representation discussed in the
+previous subsection), the elements in ξqs can be updated iteratively (by conditioning on the
+expected values of ξq,−s) until the squared difference of the ELBO or of all elements of ξqs is
+smaller than some small ϵ between two subsequent iterations.
+2.5
+Approximate Bayesian inference in general QRs
+In this section we briefly state the three approximating densities (q∗
+s(ξ)) used to estimate the
+parameters and latent quantities in the QR regression. We provide derivations for the three
+approximating densities of the three parameter groups: ˜βq = (β′
+q, γ′
+q)′, σq and νq in the Online
+Appendix.
+We start by discussing the approximating densities for the regression and basis coefficients.
+4Frazier et al. (2022) state that mean field VB approximations might perform poorly in models with a large
+number of latent variables. However, they also note that the resulting model forecasts could still perform well in
+practice.
+11
+
+A Gaussian distribution approximates the posterior of ˜βq:
+p( ˜βq|•) ≈ N
+�
+E( ˜βq), ˆΣκq
+�
+,
+with variance and mean given by, respectively:
+ˆΣ ˜βq =
+� T
+�
+t=1
+ftf ′
+t
+τ 2q
+E
+� 1
+νqt
+�
+E
+� 1
+σq
+�
++ B−1
+0q
+�−1
+,
+E( ˜βq) = ˆΣ ˜βq
+�
+���E
+� 1
+σq
+�
+T
+�
+t=1
+E
+� 1
+νqt
+� ft
+�
+yt − θq
+�
+E
+�
+1
+νqt
+��−1�
+τ 2q
+�
+��� .
+ft = (x′
+t, z′
+t)′ and B−1
+0q = diag(Bβ
+q , Bγ
+q )−1 is a prior precision matrix with Bβ
+q = λβ
+q ×diag(ψβ
+q1, . . . , ψβ
+qK)
+and Bγ
+q = λγ
+q × diag(ψγ
+q1, . . . , ψγ
+qK). The approximating densities used to estimate the prior hy-
+perparameters are provided in Section 1 of the Online Appendix.
+The latent variable νqt follows a generalized inverse Gaussian (GIG) distribution: GIG(r, A, B)5
+with
+p(νqt|•) ≈ GIG
+�
+�
+�
+�
+�
+1
+2, 2E
+� 1
+σq
+�
++ θ2
+q
+τ 2q
+E
+� 1
+σq
+�
+�
+��
+�
+Aq
+,
+E
+�
+1
+σq
+�
+τ 2q
+��
+yt − f ′
+tE( ˜βq)
+�2
++ f ′
+t ˆΣ ˜βqft
+�
+�
+��
+�
+Bq
+�
+�
+�
+�
+� .
+The moments of νqt are given by
+E
+�
+νj
+qt
+�
+=
+��
+Bq
+�
+Aq
+�j K1/2+j
+��
+AqBq
+�
+K1/2
+��
+AqBq
+� ,
+where Kx denotes the modified Bessel function of the second kind.
+Finally, we approximate
+p
+� 1
+σq
+|•
+�
+≈ G(cq1, dq1)
+5We use the following parametrization of the GIG distribution: log (GIG(x)) ∝ (0.5 − 1) log(x) −
+�
+Ax + 1
+2
+B
+x
+�
+.
+12
+
+with
+cq1 = c0 + 1.5T,
+dq1 = d0 +
+T
+�
+t=1
+E(νqt) +
+1
+2τ 2q
+T
+�
+t=1
+(E
+� 1
+νqt
+� �
+yt − f ′
+tE( ˜βq)
+�2
++ 2θq(f ′
+tE( ˜βq) − yt)
++ E(νqt)θ2
+q + E
+� 1
+νqt
+�
+f ′
+t ˆΣ ˜βqft),
+and E
+�
+1
+σq
+�
+= cq1
+dq1 .
+2.6
+Comparing computation times between VB and MCMC
+These steps, in combination with the updating steps for the priors detailed in the Online Ap-
+pendix, form the basis of our VB algorithm. As stated in the introduction, the key advantage of
+using VB instead of more precise MCMC-based techniques is computational efficiency. Before
+we turn to our empirical work, we illustrate this point using synthetic data.
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+1800
+2000
+Number of Variables
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+Time in Minutes
+Runtime: VB vs MCMC
+VB
+MCMC
+Figure 1: Comparison of computation times against the number of covariates M + K
+To illustrate the computational merits of employing VB-based approximations, Fig. 1
+shows the estimation times for different values of M + K using our VB-based QR (for a specific
+quantile) and the QR estimated through the Gibbs sampler. The MCMC algorithm is repeated
+10, 000 times. The figure shows that the computational burden increases lightly in the number
+of covariates for VB. When we focus on MCMC estimation, the computational requirements
+increase sharply in the number of covariates. Especially in our empirical work, where K + M
+is often above 1, 000, VB proves to be a fast alternative to MCMC-based quantile regressions.
+It is also worth stressing that if the number of quantiles to estimate is large (and no parallel
+computing facilities are available), MCMC-based estimation becomes excessively slow.
+13
+
+3
+Forecasting output growth using huge dimensional QRs
+In this section, we present our forecasting results. The next sub-section provides information on
+the dataset and the forecasting setup. We then proceed by discussing the results from QRs that
+exclude the nonlinear part in Sub-section 3.2. The question whether nonlinearities are important
+is investigated in Sub-section 3.3, and Sub-section 3.4 deals with how forecast accuracy changes
+over time. Sub-section 3.5 discusses the determinants of the different tail forecasts and differences
+in the shrinkage properties across priors.
+3.1
+Data overview and forecasting setup
+We use the quarterly version of the McCracken and Ng (2016) dataset. The data set covers in-
+formation about the real economy (output, labor, consumption, orders and inventories), money,
+prices and financial markets (interest rates, exchange rates, stock market indexes). All series
+are seasonally adjusted and transformed to be approximately stationary. The set of variables
+included in xt and their transformation codes are described in Table 1 of the Online Appendix.
+All models we consider also include the first lag of GDP growth.6 Forecasts are carried out using
+direct forecasting by appropriately lagging the elements in xt.
+Our sample runs from 1971Q1 to 2021Q3 and we use the period 1991Q2 to 2021Q3 as
+our hold-out period.
+The forecasting design is recursive.
+This implies that we estimate all
+our models on an initial training sample with data until 1991Q1 and produce one-quarter-
+and four-quarters-ahead predictive distributions for 1991Q2 and 1992Q1, respectively. After
+obtaining these, we add the next observation (1991Q2) and recompute the models to obtain the
+corresponding predictive densities for 1991Q3 and 1992Q2. This procedure is repeated until we
+reach the end of the hold-out period.
+As a measure of overall forecasting accuracy we focus on the continuous ranked probability
+score (CRPS). The CRPS is a measure of density forecasting accuracy and generalizes the mean
+absolute error (MAE) to take into account how well a given model predicts higher order moments
+of a target variable.
+The CRPS measures overall density fit. Considering overall CRPSs possibly masks rele-
+vant idiosyncrasies of model performance across quantiles. If a decision maker is interested in
+downside risks to GDP growth, she might value a model more that does well at the critical 5 or
+10 percentiles as opposed to the remaining regions of the predictive distribution. To shed light
+6We find that including more lags of GDP growth only has small effects on the empirical results.
+14
+
+on asymmetries across different predictive quantiles, we focus on the quantile score (QS):
+QSqt = (yt − Qqt)(q − 1{yt≤Qqt}),
+where Qqt is the forecast of the qth quantile of yt and 1{yt≤Qqt} denotes the indicator function
+that equals one if yt is below the forecast for the qth quantile.
+The QS can also be used to construct quantile-weighted (qw) CRPS scores (Gneiting and
+Ranjan, 2011). These qw-CRPSs can be specified to put more weight on certain regions of the
+predictive distribution. In general, the qw-CRPS is computed as:
+qw-CRPS =
+2
+J − 1
+J−1
+�
+j=1
+ω(ζj)QSsjt,
+with ζj = j/J, J − 1 = 19 denoting the number of quantiles we use to set up the qw-CRPS and
+sj selects the jth element from the set of quantiles we consider. This set ranges from 0.05 to
+0.95 with a step size of 0.05 and thus, s1 = 0.05, s2 = 0.10, . . . , s19 = 0.95.
+We use two weighting functions ω(ζj) that focus on different regions of the predictive
+density. These schemes are motivated in Gneiting and Ranjan (2011). The first (CRPS-left)
+puts more weight on the left tail (i.e. downside risks) and is specified as ω(ζj) = (1 − ζj)2,
+while the second (CRPS-tails) puts more weight on both tails as opposed to the center of the
+distribution: ω(ζj) = (2ζj − 1)2.
+Notice that if we use equal weights we obtain a discrete
+approximation to the CRPS.
+3.2
+Results based on linear QRs
+We start discussing the QRs that set g(xt) = 0 for all t. Here, our goal is to show that including
+more information pays off relative to the model proposed in Adrian et al. (2019). Hence, we
+benchmark the QR models to the model which only includes lagged GDP growth and the NFCI.
+This model is henceforth called the ABG model and estimated in the same way as in the original
+paper.
+Table 1 shows average (over time) qw-CRPSs relative to the ABG model. Numbers smaller
+than one suggest that a given model outperforms the ABG benchmark whereas numbers exceed-
+ing unity indicate that the model produces less precise density forecasts.
+The table reveals a great deal of heterogeneity with respect to different priors. Popular GL
+priors such as the HS, the NG or the DL lead to forecasts that are often slightly worse than the
+15
+
+Table 1: CRPS for linear models
+One-quarter-ahead
+Four-quarters-ahead
+Model
+CRPS
+CRPS-tails
+CRPS-left
+CRPS
+CRPS-tails
+CRPS-left
+HS
+1.06
+1.06
+1.05
+0.99
+0.95
+1.02
+RIDGE
+0.88
+0.84
+0.83
+0.87
+0.86
+0.87
+NG
+1.01
+0.98
+0.98
+1.01
+0.96
+1.03
+LASSO
+0.91
+0.89
+0.91
+0.87
+0.85
+0.88
+DL
+1.10
+1.09
+1.08
+1.09
+1.06
+1.14
+Notes: We highlight in light gray (dark gray) rejection of equal forecasting accuracy against the
+benchmark model at significance level 10% (5%) using the test in Diebold and Mariano (1995) with
+adjustments proposed by Harvey et al. (1997). Results are shown relative to the AGB model and
+are based on the full sample.
+ones obtained from the benchmark. However, priors such as the Ridge or the LASSO (which is
+particular known for over-shrinking significant signals (see, e.g., Griffin and Brown, 2010)) yield
+forecasts that are better than the benchmark forecasts for both forecast horizons and across the
+different variants of the CRPS. Our findings corroborate recent results in Carriero et al. (2022)
+who show that large QRs with shrinkage improve upon the ABG benchmark.
+This is especially pronounced in the case of the Ridge prior. In this case, the accuracy
+gains vis-´a-vis the ABG benchmark reach 17 percent and, in most cases, accuracy differences
+are statistically significant according to the Diebold and Mariano (1995) test.
+Turning to the different forecast horizons reveals that specifications that do well in terms
+of short-term forecasting also produce precise longer-term predictions. For the LASSO-based
+model, four-quarters-ahead accuracy gains are slightly more pronounced whereas for the Ridge
+we do not find discernible differences across both forecast horizons.
+Next, we drill deeper into the quantile-specific forecasting performance by considering QSs
+for q ranging from q ∈ {0.05, 0.1, 0.25, 0.5, 0.75, 0.95, 0.99}. These, for one-step-ahead forecasts,
+are shown in Fig.
+2 and Fig.
+3 provides the four-steps-ahead results.
+Before starting our
+discussion it is worth stressing that many of these differences are statistically significant with
+respect to the DM test. The corresponding results are provided in the Online Appendix (see
+Fig. 15 and 16).
+Similar to the findings based on the CRPSs, there is a great deal of heterogeneity across
+priors. Both the LASSO and the Ridge prior improve upon the ABG benchmark for all quantiles
+by relatively large margins. These gains appear to be more pronounced in the tails, reaching
+over 20 percent in terms of the QSs. When focusing on the center of the distribution (i.e., the
+median forecast), the gains are much smaller. In general, the other priors perform considerably
+16
+
+5%
+10%
+25%
+50%
+75%
+90% 
+95%
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+HS
+RIDGE
+NG
+LASSO
+DL
+One-quarter-ahead quantile scores
+Figure 2: One-quarter-ahead quantile scores for different values of q, averaged over the hold-out
+period.
+5%
+10%
+25%
+50%
+75%
+90% 
+95%
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+HS
+RIDGE
+NG
+LASSO
+DL
+One-year-ahead quantile scores
+Figure 3: Four-quarters-ahead quantile scores for different values of q, averaged over the hold-
+out period.
+worse. The only exception turns out to be the NG prior which, displays an excellent performance
+in the left tail, while being still outperformed by the LASSO and the Ridge prior.
+Considering four-quarters-ahead tail forecasts yield a similar but less pronounced picture.
+17
+
+For higher-order forecasts, priors that did well at the one-quarter-ahead horizon (LASSO and
+Ridge) also yield precise tail forecasts. One remarkable difference from short-term forecasts is
+that higher order median forecasts appear to be much more precise than the ones obtained from
+the ABG benchmark specification.
+This brief discussion gives rise to a simple recommendation for practitioners. If interest is
+on producing precise tail forecasts (irrespective of the forecast horizon) it pays off to use large
+QRs coupled with either a LASSO or Ridge-type prior. Since the Ridge prior is much simpler
+(i.e., it only features a single hyperparameter) and the empirical performance is very similar to
+the LASSO, our focus from now on will be on comparing the Ridge-based QR with a range of
+non-linear specifications.
+3.3
+Allowing for nonlinearities in large scale QRs
+In the previous sub-section we have shown that using big QRs leads to tail forecasts that are
+superior to the ones of the benchmark ABG specification. Conditional on the quantile, these
+models are linear in the parameters. However, recent literature (see, e.g., Clark et al., 2022b)
+suggests that nonlinearities become more important in the tails. Hence, we now address this
+question within our approximate framework.
+Table 2 shows relative CRPSs for the different nonlinear models. As opposed to Table 1,
+all results are now benchmarked against the QR with the Ridge prior. This allows us to directly
+measure the performance gains from introducing nonlinearities relative to setting gq(xt) = 0.
+Notice that the absence of gray shaded cells in the table indicates that the DM test does not
+point towards significant differences in forecast accuracy between the linear and the different
+nonlinear QRs.
+Despite this, a few interesting insights emerge from the table. First, many numbers in the
+table are close to unity and differences are not statistically significant from the best performing
+linear QR.7 This indicates that once we include many predictors, additionally controlling for
+nonlinearities of different forms only yields small positive (and sometimes negative) gains in
+terms of tail forecasting accuracy. Second, the first finding strongly depends on the approxi-
+mation techniques chosen. Among all three specifications, using GPs is superior to using either
+polynomials or B-Splines to approximate the unknown function gq. Second, and focusing on
+7For the GP-QR specifications with ridge prior we obtain p-values between 0.1 and 0.2 using the test in
+Diebold and Mariano (1995) with adjustments proposed by Harvey et al. (1997).
+18
+
+Table 2: CRPSs for nonlinear models
+One-quarter-ahead
+Four-quarters-ahead
+Model
+CRPS
+CRPS-tails
+CRPS-left
+CRPS
+CRPS-tails
+CRPS-left
+Polynomials
+HS
+1.02
+1.00
+1.08
+0.96
+0.97
+1.00
+RIDGE
+0.98
+0.94
+1.01
+0.96
+0.97
+1.00
+NG
+1.03
+0.99
+1.07
+0.95
+0.96
+1.00
+LASSO
+1.05
+1.04
+1.08
+1.02
+1.00
+0.99
+DL
+1.07
+1.04
+1.15
+1.22
+1.20
+1.29
+B-Splines
+HS
+1.13
+1.16
+1.18
+1.10
+1.14
+1.05
+RIDGE
+1.08
+1.08
+1.08
+1.09
+1.13
+1.04
+NG
+1.15
+1.17
+1.20
+1.13
+1.17
+1.07
+LASSO
+1.07
+1.06
+1.09
+1.02
+1.01
+0.99
+DL
+0.98
+1.00
+1.01
+1.04
+1.08
+1.02
+Gaussian Processes
+HS
+0.96
+0.94
+0.97
+1.05
+1.02
+1.10
+RIDGE
+0.97
+0.95
+0.98
+0.96
+0.95
+0.97
+NG
+0.98
+0.95
+0.98
+1.06
+1.02
+1.08
+LASSO
+1.04
+1.04
+1.07
+1.01
+0.99
+1.00
+DL
+1.02
+0.97
+1.00
+1.22
+1.20
+1.29
+Results are shown relative to the linear QR with a Ridge prior and are based on the
+full sample.
+GP-QR specifications, the specific prior chosen matters appreciably. Whereas the results for the
+conditionally linear models clearly suggest that the LASSO and Ridge priors are producing the
+most precise density forecasts. The results for the nonlinear models tell a slightly different story.
+We observe that the Ridge does well again but, for one-quarter-ahead tail forecasts, is outper-
+formed by the HS. The LASSO, by contrast, is the weakest specification. Since the LASSO is
+known to overshrink significant signals (see, e.g., Griffin and Brown, 2010), it could be that it
+misses out important information arising from the GP-based basis functions. Third, and finally,
+if we consider four-quarters-ahead predictions the QR coupled with a GP and a Ridge prior
+becomes the single best performing model again.
+To again gain a better understanding on which quantiles of the predictive distribution
+drive the CRPSs, Figs. 4 and 5 are similar to Figs. 2 and 3 and show the QSs for different
+quantiles. These are normalized to the linear QR with ridge prior so that numbers smaller than
+one indicate that nonlinearities improve predictive accuracy for a given quantile and numbers
+exceeding one imply that nonlinearities decrease forecasting accuracy.
+In general, both figures tell a consistent story: nonlinearities help in the right tail across
+19
+
+both forecast horizons, for all three nonlinear specifications, and for most priors considered. The
+only exception to this pattern are four-quarters-ahead right tail forecasts of GDP growth when
+B-Splines are used. When there are gains, they are often sizable. For instance, in the case of
+the QR-GP model we observe accuracy improvements up to 25 percent relative to the linear QR
+model.
+5%
+10%
+25%
+50%
+75%
+90% 
+95%
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+Polynomials     
+5%
+10%
+25%
+50%
+75%
+90% 
+95%
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+B-Splines       
+5%
+10%
+25%
+50%
+75%
+90% 
+95%
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+Gaussian process
+HS
+RIDGE
+NG
+LASSO
+DL
+One-quarter-ahead quantile scores
+Figure 4: One-quarter-ahead quantile scores for different values of q, averaged over the hold-out
+period and normalized to the QR with a Ridge prior.
+When we focus on the left tail, accuracy premia often turn negative. In some cases (such as
+for GP models with Ridge, NG and HS priors) there are accuracy gains for predicting downside
+risks but these gains are only rather small (reaching five percent in the case of the QR-GP
+regression with a Ridge prior).
+3.4
+Heterogeneity of forecast accuracy over time
+Up to this point, our analysis focused on averages over time. In the next step we will focus on
+how forecasting performance changes over the hold-out period. To shed light on the importance
+of nonlinearities over time, we again compare the different nonlinear specifications to the linear
+QR regression with a Ridge prior. Figs. 5 and 6 show the cumulative CRPSs relative to the
+linear benchmark QR for one-quarter and four-quarters-ahead forecasts.
+20
+
+5%
+10%
+25%
+50%
+75%
+90% 
+95%
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+Polynomials     
+5%
+10%
+25%
+50%
+75%
+90% 
+95%
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+B-Splines       
+5%
+10%
+25%
+50%
+75%
+90% 
+95%
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+Gaussian process
+HS
+RIDGE
+NG
+LASSO
+DL
+One-year-ahead quantile scores
+Figure 5: Four-quarters-ahead quantile scores for different values of q, averaged over the hold-
+out period and normalized to the QR with a Ridge prior.
+We start by focusing on the one-quarter-ahead forecasts first. For this specification, the
+density accuracy performance is heterogenous over time. In the first part of the sample, models
+using either polynomials or Gaussian processes coupled with a DL prior yield CRPSs that are
+superior to the linear benchmark. However, these accuracy gains vanish during the GFC. When
+we put more weight on tail forecasting accuracy (and consider GP-QRs), the gains disappear as
+early as during the 2001 recession that followed the 9/11 terrorist attacks and the burst of the
+dot-com bubble.
+In the pandemic, we observe a sharp increase in predictive accuracy for several priors (most
+notably the Ridge and NG priors). This pattern is more pronounced for the weighted variants of
+the CRPSs. Considering the other nonlinear model specifications gives rise to similar insights.
+Spline-based approximations to gq generally perform poorly up until the pandemic. During the
+pandemic, even this specification improves sharply against the linear benchmark specification.
+This pattern is particularly pronounced for the GP-QRs.
+Considering the performance of the models and priors that did well on average (GP-
+QRs with Ridge and the HS) reveals that most of these gains are actually driven by superior
+performance during the pandemic.
+21
+
+1992
+1994
+1996
+1998
+2000
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+CRPS      
+Polynomials     
+1992
+1994
+1996
+1998
+2000
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+B-Splines       
+1992
+1994
+1996
+1998
+2000
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+1.05
+1.1
+Gaussian process
+1992
+1994
+1996
+1998
+2000
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8
+CRPS-tails
+1992
+1994
+1996
+1998
+2000
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+1992
+1994
+1996
+1998
+2000
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+0.8
+0.85
+0.9
+0.95
+1
+1.05
+1.1
+1.15
+1992
+1994
+1996
+1998
+2000
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+CRPS-left 
+1992
+1994
+1996
+1998
+2000
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+1
+1.5
+2
+2.5
+3
+3.5
+1992
+1994
+1996
+1998
+2000
+2002
+2004
+2006
+2008
+2010
+2012
+2014
+2016
+2018
+2020
+0.85
+0.9
+0.95
+1
+1.05
+1.1
+1.15
+1.2
+HS
+RIDGE
+NG
+LASSO
+DL
+CRPS over time One-Quarter-ahead
+Figure 6: Cumulative one-quarter-ahead CRPS relative to the linear QR with the Ridge prior
+over the hold-out period.
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+CRPS      
+Polynomials     
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+2
+B-Splines       
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+0.5
+1
+1.5
+2
+2.5
+3
+Gaussian process
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+CRPS-tails
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+CRPS-left 
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+5.5
+HS
+RIDGE
+NG
+LASSO
+DL
+CRPS over time One-Year-ahead
+Figure 7: Cumulative four-quarter-ahead CRPS relative to linear QR with a Ridge prior over
+the hold-out period.
+22
+
+Turning to four-quarters-ahead forecasts provides little new insights. Models using the
+DL prior do not excel in the first part of the hold-out period and are generally outperformed by
+the linear QR. However, accuracy improvements during the GFC and the pandemic are quite
+pronounced for splines and the GP-QRs.
+To sum up this discussion, our results indicate that forecast performance is heterogenous
+over time. Different models such as the Polynomial-QR and the GP-QR with a DL prior out-
+perform in the early part of the hold-out period. This performance premium vanishes during
+the first two recessions observed in the sample. By contrast, other models such as QR-GP with
+either the NG or the LASSO do not gain much in tranquil periods but excel during recessions.
+3.5
+Properties and determinants of the quantile forecasts
+The previous sub-sections have outlined that QRs and QRs with nonlinear components perform
+well in terms of tail forecasting. In this sub-section, our goal is to investigate which variables
+determine the quantile forecasts and in what respect successful shrinkage priors differ from their
+less successful counterparts.
+The presence of nonlinearities complicates our investigation since it is not clear on how
+to measure the effect of xt on a given quantile of yt in the presence of nonlinearities. As a
+simple solution, we follow Clark et al. (2022a) and approximate the nonlinear, quantile-specific
+model using a linear posterior summary (see Woody et al., 2021). Specifically, we estimate the
+following regression model:
+Qq,t = x′
+t ˆαq + ˆεt,
+ˆεt ∼ N(0, σ2
+ˆεq).
+On the linearized coefficients we use a Horseshoe prior and on the error variances an inverse
+Gamma prior. To achieve interpretability and decouple shrinkage and selection (see Hahn and
+Carvalho, 2015), we then apply the SAVS estimator proposed in Ray and Bhattacharya (2018)
+to the posterior mean of ˆαq.8 This will yield a sparse variant of ˆαq, that is easy to interpret
+and can be understood as the best linear approximation to the corresponding quantile forecast
+arising from the nonlinear model. For brevity, we focus on one-step-ahead forecasts (i.e. xt
+includes a single lag of all variables). Results for four-quarters-ahead are included in the Online
+Appendix.
+8Huber et al. (2021) and Hauzenberger et al. (2021) apply SAVS to multivariate time series models and show
+that it works well for forecasting.
+23
+
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.3
+-0.2
+-0.1
+0
+Linear          
+10%
+M1REAL
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.4
+-0.2
+0
+0.2
+0.4
+50%
+PCESVx
+PAYEMS
+M1REAL
+PCESVx
+M1REAL
+PCESVx
+M1REAL
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.5
+0
+0.5
+90%
+PCESVx
+M1REAL
+M1REAL
+PCESVx
+USSERV
+M1REAL
+USSERV
+M1REAL
+CLAIMS
+PCESVx
+USSERV
+M1REAL
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.4
+-0.2
+0
+0.2
+0.4
+Polynomials     
+TCU
+CUMFNS
+PAYEMS
+CUMFNS
+M1REAL
+CUMFNS
+PAYEMS
+PRFIx
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+UNRATE
+M1REAL
+PCESVx
+PRFIx
+UEMP5T
+CLAIMS
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+PCESVx
+M1REAL
+TLBCBB
+TLBBBD
+PCESVx
+HWIx
+M1REAL
+TLBCBB
+TLBBBD
+UNRATE
+M1REAL
+TLBBBD
+PCECC9
+UNRATE
+CLAIMS
+TNWMVB
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+B-Splines       
+CUMFNS
+USPRIV
+CUMFNS
+USCONS
+M1REAL
+CUMFNS
+USPRIV
+SRVPRD
+FPIx
+M1REAL
+UMCSEN
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.6
+-0.4
+-0.2
+0
+UEMP15
+UEMP15
+UEMP15
+M1REAL
+M1REAL
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+UEMP15
+HWIx
+HWIURA
+TLBCBB
+TLBBBD
+UEMP15
+HWIx
+HWIURA
+TLBCBB
+TLBBBD
+UEMP15
+HWIx
+HWIURA
+TLBCBB
+TLBBBD
+USSERV
+M1REAL
+PCECC9
+UEMP15
+M1REAL
+TLBCBB
+TLBBBD
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+Gaussian process
+PRFIx
+PRFIx
+M1REAL
+SRVPRD
+PRFIx
+LNS120
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.5
+0
+0.5
+M1REAL
+M1REAL
+M1REAL
+M1REAL
+PCESVx
+M1REAL
+HS
+RIDGE
+NG
+LASSO
+DL
+-0.5
+0
+0.5
+1
+PCESVx
+USSERV
+UEMP15
+UEMP15
+M1REAL
+PCESVx
+UEMP15
+USSERV
+M1REAL
+CLAIMS
+PCESVx
+M1REAL
+Figure 8: One-quarter-ahead linearized posterior summaries across quantiles
+Figure 8 shows the results of this exercise across nonlinear specifications and priors. Start-
+ing with the left tail forecasts and linear models suggests that most quantile forecasts are not
+related to elements in xt in a robust manner. There are only two exceptions. The first relates
+to the NG prior. In this case, real money growth (M1real) survives the sparsification step and
+the relationship indicates that declines in money growth imply an increase in tail risks (i.e. a
+decline in GDP growth in the ten percent quantile). The other exception relates to the DL
+prior. In this case, employment growth in education and health services (USEHS) remains. If
+we focus on nonlinear models other variables appear to be correlated with forecasts of tail risks.
+Among the different priors, we find some variables which show up repeatedly. Among these
+are all nonfarm employees (PAYEMS), money growth, capacity utilization in manufacturing
+(CUMFNS) and private fixed investment (both residential and non-residential). Most of these
+variables are forward looking in nature and thus consistent with our intuition that economic
+agents form expectations about the state of the economy in the future and thus change their
+investment decisions accordingly. Notice that the relationship between private fixed investment
+is particularly pronounced for GP-QRs under the HS, NG and the DL prior. Another pattern
+worth mentioning is that the LASSO-based forecasts are generated from sparse models across
+both linear and nonlinear specifications.
+Once we focus on the center of the distribution we find that forecasts from linear models
+24
+
+are driven by one or two variables. Most prominently, specifications that do well in terms of point
+forecasts (such as the Ridge and Lasso) yield point forecasts that display a strong relationship
+with (lagged) money growth. In case we adopt a nonlinear specification, some differences arise
+across specifications.
+For polynomials, median forecasts under all priors except the DL are
+related to very few predictors, with money growth and short-run unemployment showing up
+for the NG and LASSO models. The DL prior implies a more dense model. This could be a
+possible reason for the rather weak performance of this specification. When we turn to spline-
+based models we again find a similar pattern. Money growth shows up in the case of the NG and
+LASSO and short-run unemployment predicts median output growth if we adopt a HS, Ridge
+or NG prior. Models that capture nonlinearities through GPs, our best performing nonlinear
+specifications, give rise to a very consistent pattern across priors. In all cases, lagged money
+growth appears to be a robust predictor of GDP growth. And it impacts GDP growth forecasts
+negatively.
+Finally, when our focus is on right-tail forecasts, all models become much more dense.
+Variables that have been showing up in the case of left-tail and point forecasts again show
+up (most notably money growth and short-term unemployment). Additional variables such as
+initial unemployment claims or prices remain in the sparse model as well. But there is no clear
+pattern across models except for the fact that money growth also remains in the set of robust
+predictors even if much shrinkage is introduced.
+The analysis based on linearized coefficients provides information on which variables are
+predictive for output growth forecasts across quantiles. However, the analysis in Sub-sections 3.2
+and 2.3 suggests that differences in forecast performance are driven by the prior. To understand
+which properties of a given prior exert a positive effect on predictive accuracy, we now focus
+on the shrinkage hyperparameters of the different priors. Comparing the amount of shrinkage
+introduced through the different priors is not straightforward. Here, our measure of choice is
+based on using the re-scaled log determinant of the prior covariance matrices as a measure of
+overall shrinkage for each respective prior. Since all prior covariance matrices are diagonal this
+simply amounts to summing over the log of the diagonal elements of B0q and then normalizing
+through by the number of diagonal elements.
+This constitutes a rough measure of overall
+shrinkage and we can compute it for each quarter in the hold-out period. Again, we will focus
+on shrinkage introduced in one-quarter-ahead predictive regressions. The four-quarters-ahead
+results are qualitatively similar and included in the Online Appendix.
+25
+
+Log-determinants of the prior covariance matrices over the hold-out period are depicted
+in Fig. 9. The figure includes (if applicable) solid lines which refer to the amount of shrinkage
+introduced on the linear coefficients and dashed lines which refer to the log-determinants of the
+prior covariances that relate to the shrinkage factors on the basis coefficients of the different
+nonlinear models.
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-8
+-7.8
+-7.6
+-7.4
+-7.2
+-7
+-6.8
+-6.6
+-6.4
+-6.2
+10%
+Linear          
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-9.5
+-9
+-8.5
+-8
+-7.5
+-7
+-6.5
+-6
+-5.5
+-5
+Polynomials     
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-10
+-9
+-8
+-7
+-6
+-5
+-4
+-3
+B-Splines       
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-9.5
+-9
+-8.5
+-8
+-7.5
+-7
+-6.5
+-6
+-5.5
+-5
+Gaussian process
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-8
+-7.5
+-7
+-6.5
+-6
+-5.5
+50%
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-9
+-8
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-9
+-8
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-10
+-9
+-8
+-7
+-6
+-5
+-4
+-3
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-8.2
+-8
+-7.8
+-7.6
+-7.4
+-7.2
+-7
+-6.8
+-6.6
+-6.4
+-6.2
+90%
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-9.5
+-9
+-8.5
+-8
+-7.5
+-7
+-6.5
+-6
+-5.5
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-10
+-9
+-8
+-7
+-6
+-5
+-4
+-3
+1992
+1995
+1997
+2000
+2002
+2005
+2007
+2010
+2012
+2015
+2017
+2020
+-9.5
+-9
+-8.5
+-8
+-7.5
+-7
+-6.5
+-6
+HS
+RIDGE
+NG
+LASSO
+DL
+HS-nonlinear
+RIDGE-nonlinear
+NG-nonlinear
+LASSO-nonlinear
+DL-nonlinear
+Figure 9: Overall shrinkage in one-step-ahead predictive QRs
+From this figure, a few interesting insights emerge. First, the different priors introduce
+different degrees of shrinkage. Overall, two priors stand out in terms of the amount of shrinkage
+they introduce.
+The first one is the DL. This is rather surprising given the fact that this
+prior performs worst in the forecasting horse race but also leads to posterior summaries which
+feature several non-zero coefficients. Our conjecture is that this prior forces the vast majority of
+coefficients to effectively zero but several coefficients remain sizable and the corresponding set
+of variables is still too large and overfitting issues arise. The prior that introduces the largest
+amount of shrinkage is the LASSO. In this case, almost all coefficients are very small. These
+observations are corroborated by boxplots, included in the Online Appendix (see Figs. 3 to 6 in
+the Online Appendix), which show the scaling parameters over three sub-samples. Our results
+imply that models which feature a large number of shrunk coefficients provide better forecasts
+than models which feature many coefficients that are effectively zero and some coefficients that
+are non-zero and sizable. This is consistent with findings in Giannone et al. (2021) who provide
+26
+
+empirical evidence that macroeconomic data is rather dense as opposed to sparse. Notice that
+the fact that dense models produce accurate tail forecasts is not inconsistent with our analysis
+based on linearized posterior summaries. This is because the linearized model under a shrinkage
+and sparsification approach strikes a balance between achieving a good model fit while keeping
+the model as simple as possible. Hence, if the covariates in the panel co-move, shrinkage and
+sparsification techniques will select one of these variables.
+Second, in almost all cases the amount of shrinkage introduced on the nonlinear part of
+the different models is much larger than the degree of shrinkage on linear coefficients. This holds
+for most priors, nonlinear methods and over all time periods. One exception is the Spline-QR
+specification with a DL prior and when the right tail is considered. Interestingly, this specific
+combination of much stronger shrinkage on the linear part of the model and less shrinkage on
+the nonlinear part leads to good forecasts in the right tail (see Fig. 4).
+Third, and finally, there is (with some notable exceptions) relatively little time-variation
+in the amount of shrinkage over the hold-out period. The only exception are the GP-QRs. In
+this case, the amount of shrinkage decreases appreciably from 2013 onward.
+4
+Concluding remarks
+In this paper, we have shown that combining QRs with nonlinear specifications and large datasets
+leads to precise quantile forecasts of GDP growth.
+Since the resulting models are high di-
+mensional, we consider several popular shrinkage priors to regularize estimates. MCMC-based
+estimation of these huge dimensional models is slow. Hence, we speed up computation by us-
+ing VB approximation methods that approximate the joint posterior distribution using simpler
+approximating densities.
+The empirical results indicate that our methods work remarkably well when the CRPS is
+taken under consideration. When we put more weight on the tail forecasting performance, we
+find that most of the overall gains are driven by a strong performance in both the left and right
+tail while the performance in the center of the distribution is close to the predictive accuracy of
+the simple quantile regression proposed in Adrian et al. (2019). These results, however, differ
+across priors and nonlinear specifications. In principle, it can be said that models featuring
+simple shrinkage priors, such as the LASSO or Ridge, in combination with GPs to capture
+nonlinearities of arbitrary form yield the most precise forecasts.
+27
+
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+31
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+A Fast and Scalable Method for
+Inferring Phylogenetic Networks from Trees
+by Aligning Lineage Taxon Strings
+Louxin Zhang1 ∗, Niloufar Abhari2, Caroline Colijn2, Yufeng Wu3
+1 Dept. of Mathematics and Centre for Data Science and Machine Learning
+National University of Singapore, Singapore 119076
+* Corresponding author: matzlx@nus.edu.sg; +65-65166579
+2 Dept. of Mathematics
+Simon Fraser University, Burnaby, B.C. Canada V5A 1S6
+3 Dept. of Computer Science and Engineering
+University of Connecticut, Storrs, CT 06269, USA
+Abstract
+The reconstruction of phylogenetic networks is an important but challenging problem in phy-
+logenetics and genome evolution, as the space of phylogenetic networks is vast and cannot be
+sampled well. One approach to the problem is to solve the minimum phylogenetic network prob-
+lem, in which phylogenetic trees are first inferred, then the smallest phylogenetic network that
+displays all the trees is computed. The approach takes advantage of the fact that the theory of
+phylogenetic trees is mature and there are excellent tools available for inferring phylogenetic trees
+from a large number of biomolecular sequences. A tree-child network is a phylogenetic network
+satisfying the condition that every non-leaf node has at least one child that is of indegree one.
+Here, we develop a new method that infers the minimum tree-child network by aligning lineage
+taxon strings in the phylogenetic trees. This algorithmic innovation enables us to get around the
+limitations of the existing programs for phylogenetic network inference. Our new program, named
+ALTS, is fast enough to infer a tree-child network with a large number of reticulations for a set of
+up to 50 phylogenetic trees with 50 taxa that have only trivial common clusters in about a quarter
+of an hour on average.
+arXiv:2301.00992v1  [q-bio.PE]  3 Jan 2023
+
+1
+Introduction
+In this study, phylogenetic networks are rooted,
+directed acyclic graphs in which the leaves are
+labeled with taxa, the non-leaf indegree-1 nodes
+represent speciation events and the nodes with
+multiple incoming edges represent reticulation
+events. The non-leaf indegree-1 nodes are called
+tree nodes and the other non-leaf nodes are called
+reticulate nodes. Phylogenetic trees are phyloge-
+netic networks with no reticulate nodes.
+Now that a variety of genomic projects have
+been completed, reticulate evolutionary events
+(e.g. horizontal gene transfer, introgression and
+hybridization) have been demonstrated to play
+important roles in genome evolution [9, 12, 19,
+21, 26]. Although phylogenetic networks are ap-
+pealing for modeling reticulate events [18], it is
+extremely challenging to apply phylogenetic net-
+works in the study of genome evolution.
+One
+reason for this is that a computer program has
+yet to be made available for analyzing data as
+large as what current research is interested in
+[23, 31], although recently, Bayesian methods
+have been used to reconstruct reassortment net-
+works, which describe patterns of ancestry in
+which lineages may have different parts of their
+genomes inherited from distinct parents [24, 25].
+Here, we focus on reconstructing phylogenetic
+networks from (phylogenetic) trees by comput-
+ing the smallest phylogenetic network displaying
+a given set of multiple trees [2, 8, 30, 28, 29].
+In this approach, trees are first inferred from
+biomolecular sequences and then used to recon-
+struct a phylogenetic network with the smallest
+hybridization number (HN) that displays all the
+trees (see [8]), where the HN is defined as the
+sum over all the reticulate nodes of the differ-
+ence between the indegree and outdegree of each
+reticulate node. This approach takes advantage
+of the fact that the theory of phylogenetic trees
+is mature and there are excellent tools available
+for inferring trees from a large number of se-
+quences. Here, we focused on the parsimonious
+inference of phylogenetic networks from multiple
+trees, which computes a phylogenetic network
+with the minimum HN that displays all the trees.
+This problem is NP-hard even for the special case
+when there are only two input trees [4].
+For
+the two-tree case, the fastest programs include
+MCTS-CHN [32] and HYBRIDIZATION NUM-
+BER [29]. For the general case where there are
+multiple input trees, HYBROSCALE [1] and its
+predcessor [2], PRIN [30] and PRINs [22], have
+been developed.
+All these methods are based
+on the process of searching through inserting
+reticulate edges or other editing operations in
+the space of phylogenetic networks, by reducing
+the problem to the maximum acyclic agreement
+forests of the input trees or both. Unfortunately,
+none of them can be used for inferring a network
+from a so-called irreducible set of 30 trees with
+30 taxa in which the trees do not contain any
+non-trivial common clusters.
+Since the whole network space is vast and
+cannot be fully sampled, attention has been
+switched to the inference of the tree-child net-
+works, in which every non-leaf node has at least
+one child that is not reticulate [28], or, recently,
+a member of a subclass of the tree-child network
+[26]. Tree-child network [6] is a superclass of phy-
+logenetic trees with a completeness property that
+for any set of phylogenetic trees, there is always
+a tree-child network (whose reticulate nodes can
+be of indegree 2 or more) that displays all the
+trees [20]. Other desired properties of tree-child
+networks include the fact that all the tree-child
+networks are efficiently enumerated [33]. Most
+importantly, the validation results in [28] and our
+results (reported in Section 4) suggest that the
+HN of a tree-child network solution is close to
+the optimal HN of a phylogenetic network that
+displays the trees.
+The program for inferring tree-child networks
+that appears in [28] is based on a fixed-parameter
+algorithm. The time-complexity of the algorithm
+is O((8r)rpoly(k, n)), where k and n are, respec-
+tively, the number of taxa and the input trees; r
+is the HN of the network solution.
+The new program we introduce here, ALTS,
+1
+
+takes a different approach that reduces the in-
+ference problem to aligning the lineage taxon
+strings of all the input trees.
+Algorithmic in-
+novations in ALTS enable us to get around some
+of the limitations associated with parsimonious
+inference by efficiently sampling the orderings of
+the taxa and progressively computing the short-
+est common supersequence (SCS) of the lineage
+taxon strings derived for each taxon in all the
+input trees. ALTS is fast enough to infer a par-
+simonious tree-child network for a set of 50 trees
+on 50 taxa in a quarter of an hour on average.
+We also added a feature of inferring a weighted
+tree-child network if the input trees are weighted.
+2
+Concepts and notation
+A directed graph G consists of a set V of nodes
+and a set E of directed edges that are ordered
+pairs of distinct nodes. Let e = (u, v) ∈ E. We
+call e an outgoing edge of u and an incoming
+edge of v. For a node v ∈ V , its outdegree and
+indegree are defined as the number of outgoing
+and incoming edges of v, respectively.
+For a graph, subdividing an edge (u, v) involves
+replacing it with a directed path from u to v
+that passes one or more new nodes. Conversely,
+an edge contraction at a node v of indegree one
+and outdegree one is to remove v and replace
+the path u → v → w with an edge (u, w), where
+(u, v) and (v, w) are the unique incoming and
+outgoing edge of v, respectively.
+2.1
+Phylogenetic networks
+A phylogenetic network on a set X of taxa is
+a rooted, directed acyclic graph in which (i) all
+the edges are oriented away from the root, which
+is of indegree 0 and outdegree 1; (ii) the nodes
+of indegree 1 and outdegree 0, called leaves, are
+uniquely labeled with the taxa; and (iii) all the
+non-root and non-leaf nodes are either tree nodes
+that are of indegree 1 and outdegree 2 or reticu-
+late nodes that are of indegree more than 1 and
+outdegree 1. Reticulate nodes represent evolu-
+tionary reticulation events. A phylogenetic net-
+work is said to be binary if the indegree of every
+reticulate node is exactly 2 (Figure 1).
+Let N be a phylogenetic network.
+We use
+V(N) and E(N) to denote the node and edge set
+of N, respectively. We also use R(N) to denote
+the set of reticulate nodes, and use T (N) to de-
+note the set of all non-reticulate nodes, including
+the root, tree nodes and leaves. Let u, v ∈ V(N).
+The node v is a child of u if (u, v) is an edge; v is
+a descendant of u if there is a directed path from
+u to v. If v is a descendant of u, v is said to be
+below u.
+A phylogenetic network N is a tree-child net-
+work if every non-leaf node has a child that is
+not reticulate.
+Equivalently, N is a tree-child
+network if and only if for every non-leaf node,
+there is a path from that node to some leaf that
+passes only tree nodes. Figure 1 presents a bi-
+nary tree-child network (left) and two non-tree-
+child networks.
+Consider a tree-child network N with k retic-
+ulate nodes. Let the root be r0 and let the retic-
+ulate nodes be r1, r2, · · · , rk. After the removal
+of the incoming edges of every ri, N becomes
+the union of k + 1 subtrees, which are rooted
+at r0, r1, · · · , rk, respectively, and have network
+leaves as their leaves (see Figure 1). These sub-
+trees are called the tree-node components of N.
+Tree-node decomposition is a useful technique in
+the study of phylogenetic networks [11, 13, 14].
+2.2
+Phylogenetic trees
+A phylogenetic tree on X is a phylogenetic net-
+work with no reticulate nodes. In fact, a tree is
+a tree-child network. Let T be a phylogenetic
+tree on X and u ∈ V (T). The node cluster of
+u, denoted as C(u), is the subset of taxa that
+are represented by the leaves below u. Clearly,
+C(u) ∩ C(v) ∈ {C(u), C(v), ∅} for any two nodes
+u and v. The node u and its descendants induce
+a unique subtree on C(u). We use Tu or T(C(u))
+to denote the subtree.
+Let S be a set of binary phylogenetic trees on
+2
+
+d
+b
+a
+c
+c
+b
+a
+c
+b
+a
+x
+4
+2
+1
+3
+y
+x
+4
+2
+1
+3
+y
+Edge insertion
+Figure 1: A binary tree-child network (left) in which
+there are four tree-node components (shaded grey)
+and two non-tree-child networks (middle) and (right).
+In the middle network, the child of the top reticulate
+node is also reticulate. In the right network, the chil-
+dren of a tree node in the middle are both reticulate.
+X. A common cluster of S is a subset of X that is
+a node cluster in every tree of S. Obviously, each
+single taxon is common cluster of S, and so is X.
+Any other common clusters of S are called non-
+trivial common clusters. S is a reducible tree set
+if there is a non-trivial common cluster for S, and
+it is irreducible otherwise. A non-trivial common
+cluster C of S is maximal if any subset C′ such
+that C ⊂ C′ ⊂ X is not a common cluster of S.
+Clearly, for any two maximal common cluster C1
+and C2 of S, C1 ∩ C2 = ∅; and any non-trivial
+common cluster X′ of S must be contained in a
+unique maximal cluster of S if X′ is not maximal.
+2.3
+Tree display and network infer-
+ence problems
+Let T be a binary phylogenetic tree on X and let
+N be a tree-child network with k reticulate nodes
+on X. T is displayed by N if T can be obtained
+from N by applying edge contraction from N
+after the removal of all but one incoming edge
+for each reticulation node (Figure 2). For any
+set of binary phylogenetic trees over X, there is
+always a tree-child network that displays all the
+trees [20]. However, such a solution network may
+not be binary.
+Let P by a phylogenetic network. Its
+c
+d
+a
+b
+c
+d
+a
+b
+c
+d
+b
+a
+A.)                      B.)                     C.)
+Figure 2: (A.) A tree-child network with two retic-
+ulate nodes on the taxa (a to d). (B.) A subtree of
+the network in (A) that can be obtained by the re-
+moval of the dashed incoming edges of the reticulate
+nodes. (C.) A tree displayed in the network in (A),
+which was obtained from the subtree in (B) by edge
+contraction.
+reticulate number is defined as the number of
+reticulate nodes. Its HN, denoted as H(P), is
+defined as the sum over all the reticulate nodes
+of the difference between the indegree and the
+outdegree of that reticulate node. If P is binary,
+H(P) is equal to the reticulate number. Here,
+we studied the following minimum tree-child
+network inference problem:
+Input: A set of phylogenetic trees on X.
+Output: A parsimonious tree-child network P
+on X (with the smallest H(P)) that displays
+all input trees.
+2.4
+The SCS problem
+Let s and t be two sequences in an alphabet. The
+sequence s is said to be a supersequence of t if t
+can be obtained from s by the deletion of one or
+more letters. The SCS problem is, given a set of
+sequences, to find the shortest sequence that is
+a supersequence of every given sequence.
+The SCS problem can be solved in a quadratic
+time for two sequences. However, it is NP-hard
+in general.
+3
+
+3
+The methods
+In this section, we assume that the input trees
+are binary phylogenetic trees.
+3.1
+The Inference Algorithm
+Let X be a taxon set and let π = π1π2 · · · πn,
+representing a (total) ordering of X by which
+πi is ‘less than’ πi+1 for each i < n.
+For any
+non-empty subset X′ of X, we use minπ(X′)
+and maxπ(X′) to denote the minimum and
+maximum taxon of X′ with respect to (w.r.t.)
+π, respectively. Consider a tree T on X. Since
+the root of T is of outdegree 1, T has n non-leaf
+nodes, called internal nodes.
+We label the n
+internal nodes one-to-one with X w.r.t. π using
+the following algorithm:
+Labeling
+Input A tree T on X and an ordering π of X
+1. Label the degree-1 root of T by minπ(X).
+2. Label each internal node u with two
+children v and w with
+maxπ{minπ(C(v)), minπ(C(w))}, where
+C(v) consists of all taxa below v in T.
+For instance, let X = {a, b, c, d, e} and π be an
+ordering of X such that b < c < a < d < e,
+Figure 3B gives two trees in which their internal
+nodes are labeled w.r.t. π by using Labeling.
+For each taxon τ, there is a unique internal
+node w that is labeled with τ, which is an an-
+cestor of the leaf τ. The sequence of the taxon
+labels appearing in the path from w to the leaf
+τ exclusively is called the lineage taxon string
+(LTS) of τ.
+The LTSs computed in the trees
+in Figure 3B are listed in Figure 3C. It is not
+hard to see that a tree can be recovered by using
+the LTSs derived from the given ordering of X
+in the tree. In addition, we have the following
+proposition, the proof of which appears in the
+Supplementary Document.
+Proposition 1 Let π be an ordering of X,
+|X| = n. For a phylogenetic tree T on X, the
+LTS sπ(i) of each taxon πi obtained by applying
+the Labeling algorithm has the following prop-
+erties:
+(i) sπ(1) is always not empty;
+(ii) sπ(n) is always empty;
+(iii) every taxon πk (k > 1) appears in the LTS
+2
+1
+4
+3
+5
+2
+5
+1
+3
+4
+1
+2
+3
+4
+5
+×
+×
+×
+×
+Ordered Taxa: b < c < a < d < e 
+b
+a
+d
+c
+e
+b
+c
+a
+e
+d
+b
+e
+a
+c
+d
+b
+c
+e
+d
+a
+The lineage taxon strings
+b
+c, a
+c, e
+Taxon     Left tree  Right tree 
+c
+e, d
+d, a
+a
+empty   empty
+d
+empty   empty
+e
+empty   empty
+A.) 
+B.) 
+D.) 
+ E.) 
+b
+c
+a
+d
+e
+c
+e
+a
+d
+e
+a
+b
+c
+a
+d
+e
+C.) 
+Figure 3: The construction of a tree-child network
+that displays two phylogenetic trees. (A) An order-
+ing on {a, b, c, d, e}. (B) Two trees, where the inter-
+nal nodes are labeled using the Labeling algorithm.
+(C) The LTSs of the taxa obtained from the label-
+ing in Panel B. (D) The rooted directed graph con-
+structed from the shortest common supersequences
+(SCS) of the LTSs of the taxa (in Panel C) using
+the Tree-child Network Reconstruction algo-
+rithm. Here, the SCS is [c, e, a] for [c, a] and [c, e], and
+is [e, d, a] for [e, d] and [d, a]. (E) The tree-child net-
+work obtained after contraction of the degree-2 nodes.
+4
+
+of πj for a unique j < k;
+(iv) π1 does not appear in any LTS.
+Let Ti (1 ≤ i ≤ k) be k trees on X and
+let π = π1π2 · · · πn, an ordering of X, where
+n = |X|. Let αij be the LTS of πj in Ti for each
+j from 1 to n − 1. Assume that, for each j, βj
+is a common supersequence of α1j, α2j, · · · , αkj
+such that βj does not contain any symbol not
+in X.
+We can construct a tree-child network
+Nπ(β1, β2, · · · , βn−1) on X using the following
+algorithm.
+Tree-Child Network Construction
+1. (Vertical edges) For each βi, define a
+path Pi with |βi| + 2 nodes:
+hi, vi1, vi2, · · · , vi|βi|, ℓπi,
+where βn is the empty sequence.
+2. (Left–right edges) Arrange the n paths
+from left to right as P1, P2, · · · , Pn. If
+the m-th symbol of βi is πj, we add an
+edge (vim, hj) for each i and each m.
+3. Contract each hi if it is of indegree 1.
+Tree-Child
+Network
+Construction
+is
+illustrated in Figure 3D, where the SCSs are
+[c, e, a] and [e, d, a] for π1 = b and π2 = c, and
+the empty sequence for π3 = a and π4 = d.
+Clearly, the network output from the algorithm
+is a tree-child network.
+Proposition 2 Let Ti (1 ≤ i ≤ k) be k trees
+on X such that |X| = n and let π be an order-
+ing of X. Let αij be the LTS of πj in Ti with
+respect to π, 1 ≤ j ≤ n − 1. If βj is a com-
+mon supersequence of α1j, α2j, · · · , αkj on X for
+each j from 1 to n − 1, the Tree-Child Net-
+work Construction algorithm outputs a tree-
+child network that displays all k trees.
+Conversely, we assume that P is a tree-child
+network with the smallest HN, H(P), compared
+with those that displays all k trees Ti. The prop-
+erty that P has the smallest HN implies that, for
+each i, any display of Ti in P must use one in-
+coming edge for each reticulate node of P.
+A.) 
+B.) 
+ C.) 
+D.)
+b
+a
+d
+c
+b
+c
+a
+d
+b
+d
+a
+c
+d
+a
+b
+d
+a
+a
+b
+c
+d
+a
+b
+c
+a
+b
+b
+c
+The lineage taxon strings w.r.t. 
+the ordering: c < d < b < a
+Taxon  Left tree  Middle tree  Right tree 
+c
+b, d
+d, a
+d,  b
+d
+empty 
+ b 
+ a
+b
+a
+empty 
+empty
+a
+empty  empty
+empty
+Figure 4: Mapping a tree-child network on n taxa
+that displays multiple trees to (n−1) common super-
+sequences of the LTSs of the taxa in the trees with
+respect to a selected ordering.
+(A) Three trees on
+taxa {a, b, c, d}. (B) A tree-child network with the
+smallest HN (4) that displays all three trees. (C) De-
+termine an ordering: c < d < b < a, label all internal
+tree nodes, and derive the LTS for each taxon: [b, d,
+a, b] (Taxon c), [a, b] (Taxon d), [a] (Taxon b), empty
+(Taxon a). (D). The LTS of the taxa in the trees. For
+each taxon, the LTS obtained in the network is the
+SCS of the LTSs obtained in the trees.
+Let P contain t reticulate nodes ri (1 ≤
+i ≤ t).
+N has t + 1 tree-node components
+C0, C1, C2, ..., Ct such that C0 is rooted at the
+root r0 of P, and Ci is rooted at ri for i ≥ 1.
+Since P is acyclic, its nodes can be topologically
+5
+
+sorted into a list such that u appears before v for
+every edge (u, v) of P. By using such a topolog-
+ical ordering of P, we can order all the taxa into
+π1, π2, · · · , πn such that (i) all the taxa in each
+tree-node component appear consecutively, and
+(ii) if the reticulate node ri has a parent in Cj,
+the taxa of Cj appear before the taxa of Ci in
+the list. This is because there is a directed path
+from rj to every node of Ci. For instance, for
+the tree-child network in Figure 4B, C0 contains
+Taxa c and d; the tree component rooted below
+the left reticulate node contains Taxon b; the
+tree component below the right reticulate node
+contains Taxon a. Therefore, we can order the
+taxa as either c < d < b < a or d < c < b < a,
+where b must appear before a.
+For an ordering π = π1π2 · · · πn satisfying the
+property given in the last paragraph, we label the
+tree nodes of P using the following algorithm.
+• Label the network root with the smallest
+taxon in C0 (i.e. π1).
+Label each parent
+of the reticulate node ri with the smallest
+taxon in Ci for every i > 1.
+• Let u be a tree node that is not a parent of
+any reticulate node. In this case, u has two
+children x and y in the same tree-node com-
+ponent C.
+We label u with maxπ(ax, ay),
+where ax and ay are the smallest taxon be-
+low x and y in C, respectively. (For exam-
+ple, the tree-node component C0 contains
+only one such tree node and this node is la-
+beled with d in Figure 4C.)
+As in the case of trees, we can obtain a LTS for
+each taxon.
+For the smallest taxon τ of each
+tree-node component Ci, its LTS is composed of
+the taxon labels of the tree nodes in the unique
+path from ri to Leaf τ. For the other taxa τ of
+Ci, there is a unique tree node w that is labeled
+with τ. The LTS of τ is composed of the taxon
+labels of the tree nodes (excluding w) in the path
+from w to Leaf τ. (For example, in Figure 4C,
+C0 contains Taxa c and d. The LTS for c and d
+are [b, d, a, b] and [a, b], respectively.
+Proposition 3 Let Ti (1 ≤ i ≤ k) be k trees on
+X and let P be a tree-child network on X with
+the smallest HN, compared with those that dis-
+play all Ti. For any ordering π of X obtained
+above and for each taxon τ, if we label the tree
+nodes of P as described above, the LTS Sτ ob-
+tained for τ is the SCS of the LTS obtained for
+τ in the trees Ti. Moreover, applying the Tree-
+child Construction algorithm to the obtained
+supersequences Sτ gives the same network as P.
+The proof of Proposition 3 appears in Sup-
+plementary document.
+By Propositions 2 and
+3, we obtain the following exact algorithm for
+inferring the minimum tree-child network that
+displays the trees.
+Algorithm A
+Input: Trees T1, T2, · · · , Tk on X, |X| = n.
+0. Define M = ∞ and n string variables
+S1, S2, · · · , Sn−1;
+1. For each ordering π1π2 · · · πn of X:
+1.1. Call the Labeling algorithm to
+label the internal nodes in each Ti;
+1.2. For each taxon πj, compute its
+LTS sij in each Ti;
+1.3. Compute the SCS sj of
+s1j, s2j, · · · , skj for each j < n;
+1.4. If M > �n−1
+j=1 |sj|, update M to
+the length sum; update Sj to sj
+for each j;
+2. Call the
+Tree-Child Network Construction
+algorithm to compute a tree-child network
+P from the strings S1, S2, · · · , Sn−1.
+Step 1.1 and Step 1.2 of Algorithm A take a
+linear time of O(n). Note that the SCS problem
+is a special case of the multiple sequence align-
+ment problem. Since the the total length of the
+(n − 1) LTSs computed in Step 1.2 is n − 1 for
+each Ti, Step 1.3 takes a time of O((n − 1)k) at
+most. Step 2 takes a quadratic time of O(n2).
+Therefore, the worst-case time complexity of
+6
+
+Algorithm A is O
+�
+n!(n − 1)k�
+.
+3.2
+A Scalable Version
+Since there are n! possible orderings of n taxa,
+Algorithm A is not fast enough for a set of
+multiple trees with 15 taxa if all the trees do not
+have any common clusters other than the single-
+ton cluster and the whole taxa. Another obstacle
+to scalability is computing the SCS for the LTS
+of each taxon. We achieved high scalability by
+using an ordering sampling and a progressive ap-
+proach for the SCS problem.
+First, the ordering sampling starts with an ar-
+bitrary ordering of the taxa and finishes in ⌊n/2⌋
+iterative steps. Assume that Πm is the set of or-
+derings obtained in the m-th step (m ≥ 1), which
+contains at most K orderings (K is a predefined
+parameter to bound the running time). In the
+(m+ 1) step, for each ordering π = π1π2 · · · πn ∈
+Πm, we generate (n−2m+1)(n−2m) orderings
+by interchanging π2m−1 with πi and interchang-
+ing π2m with πj for every possible i and j such
+that i ̸= j, i > 2m and j > 2m. For each new or-
+dering π′ = π′
+1π′
+2 · · · π′
+n, we compute a SCS si of
+the LTSs of Taxon π′
+i in the input trees for each
+i ≤ 2m. We compute Πm+1 by sampling at most
+K orderings that have the smallest length sum
+�
+1≤i≤2m |si| from all the generated orderings of
+the taxa.
+Second, different progressive approaches can
+be used to compute a short common superse-
+quence for LTSs in each sampling step [10]. We
+use the following approach, which had good per-
+formance for our purposes according to our sim-
+ulation test.
+A common supersequence of n se-
+quences is computed in n − 1 iterative
+steps. In each step, a pair of sequences
+si, sj for which the SCS of si and sj,
+SCS(si, sj), has the minimum length,
+over all possible pairs of sequences, is
+selected and replaced with SCS(si, sj).
+After the sampling process finishes, we obtain
+a set Π⌊n/2⌋ of good ordering; for each ordering,
+we obtain a short common supersequence of the
+LTS of a taxon, which might not the shortest
+one for each taxon. To further improve the tree-
+child network solution, we also use the dynamic
+programming algorithm to recalculate the SCS
+for the LTS of each taxon w.r.t. each obtained
+ordering, subject to the 1G memory usage limit.
+We then use whichever is shorter to compute a
+network.
+3.3
+A program for network inference
+Another technique for improving the scalability
+is to decompose the input tree set into irreducible
+sets of trees if the input trees are reducible [2,
+30]. Let S be a reducible set of k trees on X,
+which are ordered as: ⟨T1, T2, · · · Tk⟩. We assume
+that C1, C2, · · · , Ct are all the maximal common
+clusters of S. We introduce t new taxa yi and
+let Y = {y1, y2, · · · , yt}.
+By replacing Ti(Cj)
+with yj in Ti for each i and j, we obtain a set
+S′ of k trees T ′
+i on Y ∪
+�
+X \
+�
+∪t
+i=1Ci
+��
+. In this
+way, we decompose S into an irreducible tree set
+S′ = ⟨T ′
+1, T ′
+2, · · · , T ′
+k⟩ and t ordered sets of trees
+S′
+i = ⟨T1(Ci), T2(Ci), · · · , Tk(Ci)⟩, 1 ≤ i ≤ t.
+Combining the tree-child networks constructed
+from S′ and all of S′
+i gives tree-child networks
+that display all the trees of S.
+Our program is named ALTS, an acronym
+for “Aligning Lineage Taxon Strings”.
+It
+can
+be
+downloaded
+from
+the
+Github
+site
+https://github.com/LX-Zhang/AAST. We also
+developed a program that assigns a weight to
+each edge of the obtained tree-child network if
+the input trees are weighted. The least squares
+method for estimating edge weights is presented
+in Section B of the Supplementary Document.
+In summary, the process of reconstructing a
+parsimonious tree-child network involves the fol-
+lowing steps. (i) Decompose the input tree set
+S into irreducible tree sets, say S1, S2, · · · , St.
+(ii) Infer a set Ni of tree-child networks for each
+Si.
+(iii) Assemble the tree-child networks in
+N1, N2, · · · , Nt to obtain the networks that dis-
+play all the trees in S. (iv) If the input trees are
+7
+
+weighted, the branch weights are estimated for
+the output tree-child networks.
+4
+Validation Experiments
+We assessed the accuracy and scalability of
+ALTS on a collection of simulated datasets that
+were generated using an approach reported in
+[30]. For each k ∈ {20, 30, 40, 50}, a phylogenetic
+network on k taxa was first generated by simulat-
+ing speciation and reticulation events backwards
+in time with the weight ratio of reticulation to
+speciation ratio being set to 3:1. Fifty trees dis-
+played in the networks were then randomly sam-
+pled. This process was repeated to generate 2500
+trees for each k. The test tree datasets are avail-
+able together with the code for ALTS on Zhang’s
+Github site mentioned in Section 3.3.
+We compared ALTS with two heuristic net-
+work inference programs: PRINs [22], which in-
+fers an arbitrary phylogenetic network, and van
+Iersel et al.’s method [28], which infers a tree-
+child network. We tested the three methods on
+50 sets of trees on 20 and 30 taxa, each con-
+taining 10 trees. Van Iersel et al.’s program is a
+parallel program. It could run successfully only
+on 44 (of 50) tree sets in the 20-taxon case and
+27 (of 50) tree sets in the 30-taxon case. It was
+aborted for the remaining datasets after 24 hours
+of clock time (or about 1000 CPU hours) had
+elapsed.
+To assess the scalability of ALTS, we further
+ran it on 100 datasets, each containing 50 trees
+on 40 or 50 taxa. PRINs finished on five 50-taxon
+50-tree datasets. Van Iersel et al.’s method did
+not run successfully on these datasets.
+4.1
+The optimality evaluation
+ALTS computed the same tree-child HN as van
+Iersel et al.’s method on all but three datasets
+where the latter ran successfully.
+The HN of
+the tree-child networks inferred with ALTS was
+one more than that inferred with the latter on
+two 20-taxon 10-tree datasets and three more
+than that with the latter on one 30-taxon 10-tree
+dataset.
+Moreover, Van Iersel et al.’s method
+only outputted a tree-child network, whereas
+ALTS computed multiple tree-child networks
+with the same HN.
+PRINs ran successfully on all but one dataset
+in the 20-taxon case. In theory, the HN is in-
+herently equal to or less than the HN of the op-
+timal tree-child networks for every tree set. In
+the 20-taxon 10-tree case, the tree-child HN in-
+ferred with ALTS was equal to that inferred with
+PRINs on 20 datasets. The 29 discrepancy cases
+are summarised in the row one of Table 1
+Table 1: Summary of the HN discrepancy be-
+tween ALTS and PRINs in 20-taxon and 30-
+taxon datasets each containing 10 trees.
+HNALTS − HNPRINs
+Date type
+-1
+0
+1
+2
+3
+4
+20 taxa
+20
+11
+9
+6
+3
+30 taxa
+1
+5
+13
+14
+16
+1
+The HN discrepancies between the two pro-
+grams in the 30-taxon case are summarised in
+the row two of Table 1. Like the 20-taxon case,
+the difference in HN was also at most four. The
+tree-child HN inferred by ALTS was even one less
+than the HN inferred by PRINs on one dataset.
+We also noted that the difference in HN of-
+ten occurred when the HNs inferred by the two
+methods were greater than 15, when van Iersel’s
+method could not run successfully.
+In summary, ALTS is almost as accurate as
+van Iersel et al.’s method in terms of minimizing
+network HN. The comparison between ALTS and
+PRINs indicated that the tree-child HN is rather
+close to the HN for multiple trees.
+4.2
+The scalability evaluation
+The wall-clock time of the three methods on 100
+datasets, each having 10 trees on 20 or 30 taxa,
+are summarized in Figure 5. In the 20-taxa 10-
+tree case, the HN inferred by PRINs ranged from
+8
+
+1
+10
+100
+1000
+10000
+100000
+PRINs
+van Iersel et al.
+ALTS
+0.01
+0.1
+1
+10
+100
+1000
+10000
+Prin
+van Iersel et al.
+ALTS
+Run Time (log scale)
+Fifty 20-taxon 10-tree Datasets
+Run Time (log scale)
+Fifty 30-taxon 10-tree Sets
+1
+10
+100
+1000
+10000
+PRINs
+AAST
+Fifty 40-taxon 10-tree Sets
+Run Time (log scale)
+Figure 5: Run time (in seconds) of the three
+methods on 100 datasets, each containing 10
+trees on 20 or 30 taxa. Here, the datasets are
+sorted in the increasing order according to the
+HN output from PRINs. Missing data points for
+van Iersel et al.’s method are explained in the
+main text.
+5 to 17. The run time of ALTS ranged from 0.09
+s to 25 m 14 s (with the mean being 2 m 21 s).
+On the 49 (out of 50) 20-taxa 10-tree datasets on
+which PRINs finished, its run time ranges from
+2.94 s to 17 m 19 s (with the mean being 2 m 58
+s). ALTS was faster than PRINs on 35 tree sets.
+On average, PRINs and ALTS were comparable
+in time.
+On the 44 20-taxa 10-tree datasets on which
+van Iersel et al.’s method finished, its run time
+ranged from 0.07 s to 82 m 22 s (with the mean
+being 13 m 3 s). Van Iersel et al.’s method ran
+faster than ALTS on 26 datasets where the HN
+inferred by PRINs was less than 11. One reason
+for this is probably that the former is a parallel
+program. However, ALTS was faster than van
+Iersel et al.’s method on the remaining 18 tree
+1
+10
+100
+1000
+10000
+100000
+PRINs
+ALTS
+1
+10
+100
+1000
+10000
+100000
+PRINs
+ALTS
+Fifty 50-taxon 50-Tree Sets
+Run Time (log scale)
+Fifty 40-taxon 50-Tree Sets
+Run Time (log scale)
+Figure 6: The run time (in seconds) of ALTS on
+100 datasets, each containing 50 trees on 40 or
+50 taxa. The datasets are sorted in the increas-
+ing order according to the HN of the tree-child
+networks inferred by ALTS.
+sets where the HN inferred by PRINs was 12 or
+more.
+In the 30-taxon 10-tree case, the HN of the
+solution from PRINs ranged from 8 to 21. As
+shown in Figure 5, ALTS was faster than PRINs
+on each of the 50 datasets. Van Iersel et al.’s
+method finished on 31 (out of 50) datasets, for
+which the HN of the solution obtained with
+PRINs was 15 or more.
+ALTS was faster on
+23 datasets where the HN of the solution was
+larger than 10.
+Van Iersel et al.’s method
+was faster than ALTS on the remaining eight
+datasets where the HN ranged from 8 to 10. On
+average, ALTS was 24 and 53 times faster than
+PRINs and the van Iersel et al.’s method, respec-
+tively, in the 30-taxon 10-tree case.
+Lastly, ALTS was also able to infer tree-child
+networks on 100 datasets, each containing 50
+trees with 40 or 50 taxa. In the 40-taxon 50-tree
+case, the tree-child HN inferred by ALTS ranged
+from 9 to 64. The run time of ALTS ranged from
+9
+
+Simplified network 1
+Network 1
+Network 2
+Simplified network 2
+20 trees
+Figure 7:
+The box and whisker plots for the
+dissimilarity scores for the original network and
+that inferred by ALTS in four cases.
+In each
+plot, the four bars from left to right summarize
+the dissimilarity scores for the original network
+and 10 networks inferred from 20-, 30-, 40-, and
+50-tree sets, respectively. The four networks are
+presented in Figure S3–S6.
+3 s to 31 m 52 s (with the mean being 7 m 14 s).
+On contrast, PRINs finished on 28 tree sets. Its
+run time ranged from 3 m 19 s to 15 h 34 m 52
+s (with the mean being 3 h 49 m 46 s) (Fig. 6).
+In the 50-taxon 50-tree case, the tree-child HN
+inferred by ALTS ranged from 10 to 61. The run
+time of ALTS ranged from 2 s to 45 m 12 s (with
+the mean being 9 m 24 s) (Figure 6). In contrast,
+van Lersel et al’s method could not finish on any
+irreducible set of 50 trees on 50 taxa.
+PRINs
+finished on five tree sets in 2 h 25 m on average
+(Fig. 6).
+Taken together, these results suggest that
+ALTS has high scalability and is fast enough to
+infer a tree-child network on an irreducible tree
+set with a size comparable with those of the cur-
+rent focus of biological research.
+4.3
+The accuracy evaluation
+Evaluating the accuracy of ALTS (and the other
+two methods) is not straightforward. The ran-
+dom networks that were used to generate the tree
+sets used in the last two subsections are not tree-
+child networks and have frequently a large num-
+ber of deep reticulation events.
+On the other
+hand, by the principle of parsimony, the net-
+works inferred by the three programs contain far
+fewer reticulation events. As such, we assessed
+the accuracy of ALTS by considering the sym-
+metric difference of the set of taxa clusters in
+the original networks and the set of cluster in
+the network inferred by ALTS [15]. Here, a clus-
+ter in a network consists of all taxa below a tree
+node in that network, as the cluster of a retic-
+ulate node x is always equal to the cluster of
+its child if x has only one child. Precisely, for
+two phylogenetic networks N1 and N2 over X,
+we use C(Ni) to denote the multiset of clusters
+appearing in Ni for i = 1, 2, and define their
+dissimilarity score s(N1, N2) as the Jaccard dis-
+tance of C(N1) and C(N2), i.e.
+s(N1, N2) =
+1 − |C(N1) ∩ C(N2)|/|C(N1) ∪ C(N2)|.
+We considered two simulated networks con-
+taining 16 binary reticulations (network 1, Fig-
+ure S3) and 19 binary reticulations (network
+2, Figure S5) and their simplified version (Fig-
+ure S4 and S6). The two networks were produced
+using the same simulation method just with a
+low rate of reticulation events; the two simpli-
+fied networks were obtained by merging a retic-
+ulate node and its child if the reticulate node
+has a unique child and the child is also a reticu-
+lation node, which have 9 and 10 multiple reticu-
+lations, respectively. For each network and each
+k = 20, 30, 40, 50, we generated 10 k-tree sets.
+In total, we used 160 tree sets.
+For each tree
+set, we inferred a network using ALTS and com-
+puted the dissimilarity score for it and the origi-
+nal network. The dissimilarity score analyses are
+summarised in Figure 7.
+Network 1 (and its simplified version) contains
+less reticulation events than Network 2. We had
+slight better reconstruction accuracy for Net-
+work 1 than Network 2 (mean dissimilarity score
+range [0.3 to 0.45] vs.
+[0.55, 0.65], Figure 7).
+Also, the reconstruction from the trees sampled
+from each network was not significantly better
+than that from its simplified version. Given that
+10
+
+0.60
++
+0.55
+0.50
+0.45
+0.40
+0.35
+0.300.70
+0.65
+0.60
+0.55
+0.50
+0.45
+0.400.70
+0.65
+X
+0.60
+0.55
+0.50
+0.45
+0.400.55
+0.50
+0.45
+0.40
+X
+X
+0.35
+0.30all four networks can contain as many as 217
+trees, the results suggest that 50 trees are far
+fewer than enough for accurate reconstruction of
+both networks.
+Since we could not run Iersel et al’s program
+on the most of tree sets, we were unable to assess
+its accuracy for comparison.
+5
+A Phylogenetic Network for
+Hominin Relationships
+Hominins’ phylogenetic relationships are not
+fully established.
+As an application of ALTS,
+we reconstructed a network model for hominin
+species using 10 phylogenetic trees derived from
+the Bayesian analysis of the morphological data
+of hominin evolution presented in [7] (Fig. 8).
+To choose 10 phylogenetic trees, we grouped the
+posterior trees into five clusters using the dis-
+tance metric and approach described in previ-
+ous work [17, 16], using Ward clustering.
+We
+chose two trees from each of the five clusters.
+Due to the nature of the morphological data, the
+trees were discordant, and no single tree captures
+a highly-supported pattern of ancestry among
+the taxa. This motivates using a network to il-
+lustrate the ancestral relationships among these
+data.
+The resulting network model contains 12 retic-
+ulation events with the HN being 24.
+The
+top tree-node component contains the two out-
+group species G. gorilla and P. troglodytes, as
+well as the oldest hominin species, S. tchaden-
+sis.
+The three earliest members of the genus
+Homo ( African H. erectus, H. rudolfensis and
+H. habilis), together with Au.
+africanus, ap-
+pear in a tree-node component, whereas four re-
+cent members of the genus Homo (H. heidelber-
+gensis, H. neanderthalensis, H. sapiens and H.
+naledi ) compose another tree-node component.
+The three members of the genus Paranthropus,
+together with Au. garhi, compose a tree-node
+component. The model also reflects the high un-
+certainty about the phylogenetic position of H.
+floresiensis, who lived in the island of Flores, In-
+donesia [3, 5, 27].
+This network provides an illustration of the
+performance of ALTS on hominin morphological
+data. We find that its HN is unexpectedly high.
+Since the evolutionary time of hominin species
+is relatively short, some discrepancies in the 10
+trees are perhaps a result of incomplete lineage
+sorting (ILS) [31], (with impacts on morphology,
+in order that they are implicitly detected in these
+data), or of convergent evolution, ambiguity in
+the morphological data, or other factors. With-
+out genetic data, we cannot assess the extent to
+which ILS or other factors affects the phyloge-
+netic trees and consequently this network model.
+6
+Conclusions
+We have presented ALTS, a fast and scalable
+method for inferring tree-child networks from
+multiple trees. It is based on a novel algorith-
+mic innovation that reduces the minimum tree-
+child network problem to computing the SCS of
+the LTSs obtained w.r.t. a predefined ordering
+on the taxa in the input trees. Another contri-
+bution is an algorithm for assigning weights to
+the tree edges of the reconstructed tree-child net-
+work if the input trees are weighted. Our work
+makes network reconstruction more feasible in
+the study of evolution.
+The accuracy analyses in Section 4.3 suggest
+that 50 trees are likely not enough for accurately
+inferring a phylogenetic network model that has
+10 or more reticulation events.
+Therefore, a
+program that can process over hundred trees is
+definitely wanted.
+We remark that ALTS can
+be made even more scalable by distributing the
+computing tasks for taxon orderings into a large
+number of processors using the distributed com-
+puting programming. This is because the com-
+puting tasks for different orderings are indepen-
+dent from each other.
+We will further investigate how to improve the
+accuracy of ALTS by incorporating the genomic
+sequences of the taxa or/and ILS into network
+11
+
+4
+5
+7
+6
+20
+21
+22
+23
+3
+9
+17
+11
+16
+13
+15
+14
+12
+24
+1
+2
+18
+10
+19
+8
+Figure 8: A network model of hominin relationships. 1: G. gorilla; 2: P. troglodytes; 3: H. floresiensis;
+4: Ar. ramidus; 5: Au. anamensis; 6: Au. afarensis; 7: K. platyops; 8: Au. africanus; 9: Au. sediba; 10:
+African H. erectus; 11: Asian H. erectus; 12: H. heidelbergensis; 13: H. neanderthalensis; 14: H. sapiens; 15:
+H. naledi; 16: H. antecessor; 17: Georgian H. erectus; 18: H. rudolfensis; 19: H. habilis; 20: Au. garhi; 21:
+P. robustus; 22: P. boisei; 23: P. aethiopicus; 24: S. tchadensis.
+inference.
+Acknowledgements
+We thank Cedric Chauve and Aniket Mane
+for discussion in the beginning of this project.
+We also thank anonymous reviewers for con-
+structive comments on an earlier version of our
+manuscript.
+L. Zhang was partly supported
+by Singapore MOE Tier 1 grant R-146-000-318-
+114. Y. Wu was partly supported by U.S. Na-
+tional Science Foundation grants CCF-1718093
+and IIS-1909425.
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+Supplementary Document
+A. Propositions and their proof
+A1. Total ordering, trees and child-tree networks
+Let X be a set of taxa. A (total) ordering R on X is a binary relation on X such that (i) R is
+anti-symmetric, i.e. if x1Rx2, then x2 ̸R x1. (ii) R is transitive, i.e., if x1Rx2 and x2Rx3, then
+x1Rx3. (iii) For any x1, x2, x1Rx2 or x2Rx1. For convention, we write x <R y if x is related y
+under R or even x < y if R is clear.
+Any non-empty subset X′ of X whose elements are ordered according to R has a unique minimum
+(resp.
+maximum) element.
+We use minR X′ (resp.
+maxP X′) to denote the minimum (resp.
+maximum) element of X′.
+Let X = {x1, x2, · · · , xn}. We use π = π1π2 · · · πn on {1, 2, .., n} to denote the following ordering:
+xπ1 < xπ2 < · · · < xπn.
+A tree-child network is a phylogenetic network in which every non-leaf node has at least one child
+that is a tree node or a leaf (i.e., a node of indegree 1).
+Let P be a tree-child network on X. If P has k reticulations, the removal of all incoming edges
+for every reticulate node results in a union of k + 1 subtrees, the root of which are each either
+the network root or a reticulate node. Each of these subtrees contains at least one taxon, and is
+called a tree-node component (see Figure 1 in the main text). For each node x of P, the tree-node
+component containing x is denoted as Cx.
+A phylogenetic tree is a tree-child network without reticulate nodes. Recall that the root of a
+phylogenetic tree is of indegree 0 and outdegree 1; every non-leaf, non-root node is of indegree-1
+and outdegree-2.
+A2. Proof of Propositions
+We use the following algorithm to derive another representation of a phylogenetic tree on |X|
+given an ordering on X.
+Labeling
+Input A tree T on X and an ordering π of X
+1. Label the degree-1 root of T by minπ(X).
+2. Label each internal node u with two children v and w with
+maxπ{minπ(C(v)), minπ(C(w))}, where C(v) consists of all taxa below v in T.
+Proposition 1.
+Let π be an ordering of X, |X| > 1.
+For a phylogenetic tree T on X,
+the ancestor sequence sπ(t) of each taxon t obtained by applying the Labeling algorithm to T
+and π has the following properties:
+(i) sπ(π1) is always not empty;
+(ii) sπ(πn) is always empty;
+(iii) for each 1 < i ≤ n, πi appears in the ancestor sequence of πj for a unique j such that j < i;
+15
+
+(iv) the smallest taxon π1 does not appear in any ancestor sequence.
+Proof. Let the degree-1 root of T be ρ. Let the ancestors of Leaf π1 be:
+ρ = u0, u1, u2, · · · , uk
+and uk+1 = π1, where ui is the parent of ui+1 for 0 ≤ i ≤ k. Recall that each non-leaf, non-root
+node has two children. We let u′
+i+1 be another child of ui for 0 ≤ i ≤ k.
+(i) Since |X| > 1, k ≥ 1. Clearly, minπ C(ui) = π1 for each i ≤ k. Since π1 is the smallest taxon, in
+Step 2 of the Labeling algorithm, ui is labeled with maxπ{minπ(ui+1), minπ(u′
+i+1)} = minπ(u′
+i+1)
+for i = 1, 2, · · · , k. Therefore, that k ≥ 1 implies that sπ(π1) contains at least one taxon.
+(ii) Let the parent and sibling of Leaf πn be v and v′. In Step 2 of the Labeling algorithm, v
+is labeled with maxπ{minπ(v′), πn} = πn. Since there is no node between v and Leaf πn, sπ(πn) is
+empty.
+(iii) and (iv) We prove the statement by mathematical induction. If |X| = 2, clearly, the root ρT
+is labeled with π1 and the other internal node is labeled with π2. In this case, sπ(1) contains only
+π2 and sπ(2) is empty. Thus, the fact is true.
+For |X| > 2, from the proof of Part (i), we have that ui is labeled with the minimum taxon
+appearing in C(u′
+i+1) for i = 1, 2, · · · , k. Moreover, the internal nodes in each subtree T ′
+i rooted at
+u′
+i are labeled with the taxa of C(u′
+i) \ { minπ C(u′
+i) } according to the algorithm. Since each T ′
+i is
+a proper subtree of Ti, by induction, the fact holds. □
+Remark.
+The ancestor sequences of the taxa obtained according to an ordering on X give
+a unique phylogenetic tree T. This can be generalized to an algorithm to reconstruct a tree-child
+network using ancestor sequences of taxa.
+Tree-Child Network Construction
+1. (Vertical edges) For each βi, define a path Pi with |βi| + 2 nodes:
+hi, vi1, vi2, · · · , vi|βi|, ℓπi, where βn is the empty sequence.
+2. (Left–right edges) Arrange the n paths from left to right as P1, P2, · · · , Pn. If the
+m-th letter of βi is πj, we add an edge (vim, hj) for each m and each i.
+3. Contract each hi (i > 1) if it is of indegree 1 and outdegree 1.
+Proposition 2.
+Let Ti (1 ≤ i ≤ k) be k trees on X such that |X| = n and π be an
+ordering on X.
+Let αij
+= βTi,π(πj), the ancestor sequences of πj in Ti with respect to
+π, 1 ≤ j ≤ n − 1. If βj is a common supersequence of α1j, α2j, · · · , αkj for each j, the Tree-
+Child Network Construction algorithm outputs a tree-child network that displays the k trees.
+Proof. Let N be the directed network constructed by applying the algorithm to β1, β2, · · · , βk.
+First, N is acyclic due to the two facts: (i) the edges of each path Pi are oriented downwards, and
+(ii) the so-called left–right edges (u, v) are oriented from a node u in a path defined for πi to a
+node v in a path defined for πj such that i < j.
+Second, N is tree-child. This is because all the nodes of each Pi are tree nodes except hi for each
+i > 1 (see Figure 3 in main text). The node h1 is the network root. For i > 1, hi may or may not
+be a reticulation node. Therefore, every non-leaf node has a child that is not reticulate.
+16
+
+Lastly, we prove that Ti is displayed by N as follows. By assumption, βj is a supersequence of
+{αij | i = 1, 2, · · · , k} for each j = 1, 2, · · · , n − 1. Following the notation used in the Tree-Child
+Network Construction algorithm, we let:
+βj = βj1βj2 · · · βjtj, tj ≥ 1,
+where tj is the length of βj. Since αij is a subsequence of βj, there is an increasing subsequence
+1 ≤ m1 < m2 < · · · < mℓj ≤ tj such that
+αij = βim1βim2 · · · βimℓj
+and ℓj = |αij| ≥ 1.
+According to Step 1 of the algorithm, in N, each taxon βjx of βj corresponds one-to-one a node
+vjx in the path Pj; and there is a (left-right) edge from vjx to the first node hy(x) of the path Py(x)
+that ends with the taxon πy(x) = βjx, where y(x) ≥ j.
+Conversely, after removing the edge (vjx, hy(x)) for each x ̸= m1, m2, · · · , mℓj, we obtain a
+subtree T ′
+i of N. This is because each taxon πt appears exactly once in αi1, αi2, · · · , αi(n−1) and
+thus the node ht is of indegree 1 in the resulting subgraph, where t = 2, 3, · · · , n. It is not hard
+to see that after contracting degree-2 nodes of T ′
+i, the resulting subtree T ′′
+i has the same ancestor
+sequence as Ti for each πj. Thus T ′′
+i is equal to Ti. □
+The proof of Proposition 3 is divided into several lemmas.
+Lemma 1.
+Let π be an ordering on X and let T1, T2, · · · , Tk be k phylogenetic trees on
+X.
+For each x ∈ X and each Ti, we use βx(Ti, π) to denote the ancestor sequence of x ob-
+tained from π using the Labeling algorithm on Ti. Assume βx is a common supersequence of
+{βx(T1, π), βx(T2, π), · · · , βx(Tk, π)} for each x ∈ X.
+For the tree-child network P constructed
+from {βx | x ∈ X} and π using the Tree-Child Network Construction algorithm,
+H(P) = �
+x∈X |βx| − |X| + 1.
+Proof. Since only the first node hi of each path can be a reticulate node and that each node in
+the middle of each path is a parent of some hi, H(P) = �|X|
+i=2(din(hi) − 1) = �
+x∈X |βx| − |X| + 1,
+where din(hi) is the indegree of hi. □
+Definition 1.
+Let P be a phylogenetic network on X, where |X| > 1 and π be an order-
+ing on X. P is said to be compatible with π if for each reticulate edge (s, r) of P, the minimum
+taxon below s in the tree-node component Cs is less than the minimum taxon in the tree-node
+component Cr.
+Remark.
+For a tree-child network P, we can construct a compatible ordering π as follows.
+We first compute a topological sorting on the vertices of P. Assume the reticulate nodes and the
+network root ρ appear in the sorted list as: r0 = ρ, r1, r2, · · · , rk. We construct a desired ordering
+by listing the taxa in the tree-node component Cri before the taxa in the tree-node component
+Cri+1 for every i ≤ k − 1.
+Let π be an ordering on X and P be a tree-child network on X that is compatible with π. The
+compatibility property implies that the smallest taxon is in the tree-node component Cρ that is
+17
+
+rooted at the network root ρ. We use the following generalized Labelling algorithm to label all
+the tree nodes of P, which is identical to Labelling when P is a phylogenetic tree.
+Generalized Labelling
+S1: For every reticulate node r, label all parents of r with the smallest taxon in
+the tree-node component Cr. Similarly, the network root ρ is labeled with
+the smallest taxon in Cρ.
+S2: For each tree node z that is not a parent of any reticulate node, label x with
+maxπ(minπ(C(x)), minπ(C(y)), where x and y are the two children of z, and
+C(x) and C(y) are the set of taxa below x and y in the tree-node component
+where they belong to.
+Lemma 2. Let C be a tree-node component of P and let it contain t taxa x1, x2, · · · , xt in P.
+All t − 1 tree nodes that are not a parent of any reticulate node are uniquely labeled with some
+xj ̸= minπ{xi | 1 ≤ i ≤ t} (blue labels in Figure S1B).
+Proof. This can be proved using the same mathematical induction as in Prop. 1.iii. □
+Definition 2.
+Let π be an ordering on X and N be a tree-child network on X that is
+compatible with π.
+Assume the tree nodes of N are labeled by using the Generalized La-
+belling algorithm. The ancestor sequence of a taxon x obtained according to π is defined to
+be the sequence of the labels of the x’s ancestors that are in Cx, if x is the smallest taxon in
+C; it is the sequence of the labels of the x’s ancestors that are below the unique tree node la-
+beled with x in Cx otherwise. The ancestor sequence of x obtained in this way is denoted by βN,π(x).
+Definition 3.
+Let P be a tree-child network on X and let (s, r) be a reticulate edge.
+P − (r, s) is defined to be the tree-child network obtained through the removal of (s, r) and
+contraction of s (and also r if r is of indegree 2 in N).
+Lemma 3.
+Let π be an ordering on X and P be a tree-child network on X such that
+H(P) ≥ 1 and P is compatible with π. For any reticulate node r and each parent s of r, the
+tree-child network P − (s, r) has the following properties:
+1. P − (s, r) is also compatible with π;
+2. For each taxon x, βP,π(x) is a supersequence of βP−(s,r),π(x).
+Proof. These properties are illustrated in Figure S1. Let (s, r) be a reticulate edge. We have that
+s is a tree node, and r is a reticulate node. Recall that CN(z) denotes the tree-node component
+containing z for each node z and for N = P, or P − (s, r). We consider the two cases.
+Case 1. The r is of indegree 3 or more.
+In this case, after (s, r) is removed, s will be contracted and all the other nodes remains the same
+in P − (s, r). Moreover, P − (s, r) has the same tree-nodes components as P and also has the same
+18
+
+labelling as P. For any reticulate edge (s′, r′), CP−(s,r)(s′) = CP (s′) and CP−(s,r)(r′) = CP (r′). As
+such, the constraint is also satisfied for (s′, r′) in P − (s, r). Therefore, the first fact holds.
+Let x be a taxon.
+If βP,π(x) contains the label y of s, say βP,π(x)
+=
+β1yβ2, then,
+βP−(s,r),π(x) = β1β2.
+If βP,π(x) does not contain the label of s, βP−(s,r),π(x) = βP,π(x).
+This concludes that βP,π(x) is a supersequence of βP−(s,r),π(x). Therefore the second fact is true.
+Case 2. The r is of indegree 2.
+This case is illustrated in Figure S1b. Let s′ be another parent of r. After (s, r) is removed, the
+r becomes a node of indegree 1 and outdegree 1 and thus is contracted, together with s. All the
+other nodes remains in P − (s, r). Therefore, s′ becomes a tree node in P − (s, r). The tree-node
+component CP−(s,r)(s′) is the merge of CP (s′) and CP (r). Assume (s′′, r′) be a reticulate edge of
+P − (s, r).
+If CP−(s,r)(s′′) ̸= CP−(s,r)(s′) and CP−(s,r)(r′) ̸= CP−(s,r)(s′), then, CP−(s,r)(s′′) = CP (s′��) and
+1
+3
+2
+4
+5
+6
+7
+8
+9
+r
+1
+3
+2
+4
+5
+6
+7
+8
+9
+9
+8
+8
+7
+7
+6
+6
+5
+5
+4
+3
+2
+1
+1
+3
+2
+4
+5
+6
+7
+8
+9
+9
+8
+8
+7
+7
+6
+6
+5
+5
+4
+3
+2
+1
+(a) 
+   (b) 
+(c)
+Figure S1: Illustration of the Generalized Labelling algorithm and the proof of Lemma 3. (a)
+A tree-child network on the taxa from 1 to 9, which has two tree-node components each containing
+at least two taxa. (b) Labelling all the tree nodes in a tree-child network using the increasing order
+of taxa: i < i + 1, i = 1, 2, ..., 8, which is compatible. The labels of the parents of a reticulation
+node are in blue; while the labels of other tree-nodes are in red. (c) the resulting network after the
+removal of the left incoming edge of the reticulation node r, in which the tree-nodes are labeled
+identically if the same ordering is used.
+19
+
+CP−(s,r)(r′) = CP (r′). The constraint is satisfied for (s′′, r′).
+If CP−(s,r)(s′′) ̸= CP−(s,r)(s′) and CP−(s,r)(r′) = CP−(s,r)(s′), the constraint is satisfied for s′′, r′
+because of the fact that minπ CP−(s,r)(r′) = minπ CP (r′).
+If CP−(s,r)(s′′) = CP−(s,r)(s′) and CP−(s,r)(r′) ̸= CP−(s,r)(s′), then the minimum taxon below s′′
+in CP−(s,r)(s′′) is equal to that in CP (s′′), the constraint is satisfied for (s′′, r′).
+We have proved the first statement. We prove the second statement as follows. To this end, we
+use cP (r) to denote the unique child of r in P.
+Recall that after (s, r) was removed, s and r were contracted to obtain P − (r, s). Note that in
+P − (r, s), s′ becomes the parent of cP (r). Since P is compatible with π, the minimum taxon y
+below cP (r) is larger than the minimum taxon below s′ in π. This implies that s′ is labeled with
+y, as s′ is not a parent of any reticulate node in P − (s, r). Therefore, for any taxon x ∈ X, if
+βP,π(x) contains the label y of s, say βP,π(x) = β1yβ2, then, βP−(s,r),π(x) = β1β2. If βP,π(x) does
+not contain the label of s, βP−(s,r),π(x) = βP,π(x). This concludes that βP,π(x) is a supersequence
+of βP−(s,r),π(x) for each x ∈ X. □
+Proposition 3.
+Let T1, T2, · · · , Tk be k trees on X and P be a tree-child network on X
+with the smallest H(P), compared with those displaying all Ti. For any ordering Π of X such
+that P is compatible with it, if we label the tree nodes of P using the Generalized Labelling
+algorithm, the ancestor sequence βP,Π(x) of each taxon x is a shortest common supersequence of
+{βTi,Π(x) | i = 1, 2, · · · , k}. Moreover, applying the Tree-child Construction algorithm to the
+obtained supersequences βP,Π(x) produces the same network as P.
+Proof.
+Let P be a tree-child network on X with the smallest H(P), compared with those
+displaying all Ti. For each i, Ti can be obtained from P by deleting all but one incoming edge
+for each reticulate node. For convention, we assume that all removed reticulate edges are (sj, rj),
+1 ≤ j ≤ H(P). Let x be a taxon. By Lemma 3, βP,Π(x) is a supersequence of βP−(s1,r1),Π(x) and
+βP−�j
+t=1(st,rt),Π(x) is a supersequence of βP−�j+1
+t=1(st,rt),Π(x) for each j = 1, .., H(P) − 1. Therefore,
+for any x, βP,Π(x) is a supersequence of βTi,π(x) for each Ti, as Ti = P − �H(P)
+j=1 (sj, rj).
+Let P contain m reticulate nodes. P has m+1 tree-node components. In a tree-node component
+C, there are |X(C)| − 1 tree nodes that are not the parents of any reticulation nodes, where X(C)
+is the set of taxa in C. Hence
+�
+x∈X
+|βP,Π(x)|
+=
+�
+C
+(|X(C)| − 1) +
+�
+r∈R(P)
+din(r)
+=
+|X| − (m + 1) + H(P) + m
+=
+|X| − 1 + H(P).
+This implies that H(P) = �
+x∈X |βP,Π(x)| − |X| + 1.
+Assume βP,Π(x) is not a shortest supersequence of βTi,Π(x) (i = 1, 2, · · · , k) for some x. Let βx
+be a shortest supersequence of βTi,Π(x) (i = 1, 2, · · · , k). Then, |βx| < |βP,Π(x)|. By Lemma 1,
+we can use the Tree-Child Network Construction algorithm to obtain a tree-child network
+with the HN smaller than H(P), a contradiction.
+It is obvious that the Tree-Child Network Construction algorithm to obtain P. □
+20
+
+B. Computing the branch weights of inferred tree-child network
+A phylogenetic network is weighted if every branch has a non-negative value, which represents
+time or other evolutionary measures. A weighted phylogenetic tree T is said to be displayed in a
+weighted network N if the tree is displayed in the network when the branch weights are ignored. For
+a display T ′ of T in N, its fitness score ||T −T ′||2 is defined as
+��
+e∈E(T) |wT (e) − wT ′(P(u′, v′))|2,
+where wT (e) is the weight of e = (u, v) in T and wT ′(P(u′, v′)) is the weight of the unique path
+between the images u′ and v′ of u and v under the display mapping, respectively.
+Recall that a tree can be displayed multiple times in a network. The score of the display of T in
+N is the smallest fitness score which a display of T in N can have, denoted d(T, N). If d(T, N) = 0,
+we say that N perfectly displays T.
+If the input trees are weighted, we will first compute tree-child networks that each display all the
+trees. We then use branch weights of trees and the information on how the trees are displayed in
+a tree-child network to compute the weights of the network branches.
+We model the branch weight assignment problem as an optimization problem with the following
+assumption on the inferred tree-child network N that displays all the trees:
+For any reticulate edge e, the tree-child network P − e obtained after removal of e fails to
+display one input tree at least.
+By ordering the edges of N on X, we may assume
+E(N) = {e1, e2, · · · , em}.
+Let S = {T1, T2, · · · , Ts}, where |S| = s. We further assume that T ′
+k is a display of Tk in N. Then,
+each edge e′
+i of Tk is mapped to a path P ′
+i of T ′
+k, where 1 ≤ i ≤ 2|X| − 2. Since N displays Ti, we
+derive the following linear equation system from the display of Tk:
+�
+1≤j≤m
+aijw(ej) = w(e′
+i), i = 1, 2, · · · , 2|X| − 2,
+(1)
+where
+aij =
+� 1
+ej ∈ E(P ′
+i);
+0
+ej ̸∈ E(P ′
+i).
+Let the coefficient matrix of Eqn. (1) be Ak = (aij), which is a (2|X| − 2) × m matrix, and let:
+Wk =
+�
+�
+�
+�
+�
+�
+w(e′
+1)
+w(e′
+2)
+...
+w
+�
+e′
+2|X|−2
+�
+�
+�
+�
+�
+�
+�
+.
+Since N displays every tree of S, we then determine the edge weights of N by solving the following
+linear equation system:
+�
+�
+�
+�
+�
+A1
+A2
+...
+As
+�
+�
+�
+�
+� ×
+�
+�
+�
+�
+�
+x1
+x2
+...
+xm
+�
+�
+�
+�
+� =
+�
+�
+�
+�
+�
+W1
+W2
+...
+Ws
+�
+�
+�
+�
+�
+(2)
+21
+
+e1: (  0, 13)
+e2: (10, 12)
+e3: (10, 16)
+e4: (11, 12)
+e5: (11,   1)
+e6: (12, 18)
+e7: (13, 10)
+e8: (14,   3)
+e9: (13, 15)
+e10: (14, 15)
+e11: (15, 17)
+e12: (16, 11)
+e13: (16,   4)
+e14: (22,   2)
+e15: (17, 19)
+e16: (18, 14)
+e17: (19,   6)
+e18: (18, 20)
+e19: (19, 20)
+e20: (20, 21)
+e21: (21,   5)
+e22: (21, 22)
+e23: (17, 22)
+e2
+e6
+e16
+e10
+e11
+e15
+e19
+e20
+e22
+e14
+e21
+e17
+e8
+e3
+e12
+e13
+e5
+e’1: (11, 3)
+e’2: (10, 11)
+e’3: (10, 12)
+e’4: (11, 13)
+e’5: (13,   6)
+e’6: (12,   1)
+e’7: (13, 14)
+e’8: (14,   2)
+e’9: (12,   4)
+e’10: (14,   5)
+A                                        B                                                 C
+D
+Figure S2: An illustration of how to derive linear equations from a tree display. (A) The list of the
+edges of a tree-child network. (B) A display of the tree in C. (C) a phylogenetic tree on six taxa (1
+to 6). (D) the list of the edges of the tree in C.
+Note that Eqn. (2) is a linear equation system that contains 2s(|X| − 1) equations and at most
+5|X| − 4 variable, as each Ti contains 2|X| − 2 edges and N contains 3r + 2|X| − 1, where r is the
+number of reticulations, which is at most |X| − 1.
+Example 1. The edge list of a tree-child network is given in Figure S2A, where the full network is
+not given here. Figure S2B presents a particular display of the tree in Figure S2C, whose edges are
+listed in Figure S2D. In the display of the tree, the edge e′
+2 is mapped to the path from the node
+10 to the node 14, which consists of three edges e2, e6, e16 (Figure S2B). From e′
+2 and its image,
+we obtain the following equation in the linear equation system Eqn. (2):
+x2 + x4 + x16 = w(e′
+2).
+In general, N may not perfectly display every T when branch weights are considered. Therefore,
+let us set:
+A =
+�
+�
+�
+�
+�
+A1
+A2
+...
+As
+�
+�
+�
+�
+�
+(3)
+W =
+�
+�
+�
+�
+�
+W1
+W2
+...
+Ws
+�
+�
+�
+�
+� .
+(4)
+22
+
+10
+1416
+18
+1
+3
+15
+20Noticing that
+s
+�
+i=1
+||T ′
+i − Ti||2
+2 = ||AX − W||2
+2,
+we determine the branch weights of N by solving the following quadratic optimization problem:
+min ||AX − W||2
+2
+(5)
+subject to:
+xj ≥ 0, 1 ≤ j ≤ m.
+(6)
+Remark. Let r be a reticulation node that has incoming e1, e2, · · · , ed and the outgoing ed+1.
+For each input tree Ti, one of edge pairs (e1, ed+1), (e2, ed+1), ..., (ed, ed+1) appears in the display
+of Ti exclusively. Thus, solving the above optimization problem can only determine the value of
+w(ei) + w(ed+1) for i ≤ d.
+C. Tree distance and clustering in the hominin analysis
+We analysed the morphological data in [7] by sampling 500 phylogenetic trees from a posterior
+collection of trees estimated from the morphological data. We computed the distance between each
+pair of trees using the rooted tree metric described in [17]. Briefly, this metric is the Euclidean
+distance between two vectors (one for each tree). The vector captures the amount of shared ancestry
+between each pair of tips, as well as each tip’s distance from its parent. We used the tree topology
+only (λ = 0 in the tree metric in the ‘treespace’ function in the ‘treespace‘ package in R [16]). The
+amount of shared ancestry is the length of the path (in a phylogeny) between the root and the most
+recent common ancestor of a pair of tips. Having found pairwise distances between all pairs of trees
+in our sample of 500, we clustered the trees into five clusters using Ward clustering. We chose two
+trees uniformly at random from each of the five clusters, as input for the analysis presented here.
+23
+
+Figure S1.  Network 1 used in the accuracy assessment in Section 4.3. 
+It has 16 binary reticulation events.
+Figure S3: Network 1 used in the accuracy assessment in Section 4.3. It has 16 binary reticulation
+events.
+Figure S2. Simplified network 1  used in the accuracy assessment in 
+Section 4.3. It has 9 reticulation events.
+Figure S4: Simplified network 1 used in the accuracy assessment in Section 4.3. It has 9 reticulation
+events.
+24
+
+Figure S3.  Network 2 used in the accuracy assessment in Section 4.3. 
+It has 19 binary reticulation events.
+Figure S5: Network 2 used in the accuracy assessment in Section 4.3. It has 19 binary reticulation
+events.
+Figure S4. Simplified network 2 used in the accuracy assessment in 
+Section 4.3. It has 10 reticulation events.
+Figure S6: Simplified network 2 used in the accuracy assessment in Section 4.3. It has 10 reticulation
+events.
+25
+

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+Quantum-to-classical transition enabled by quadrature-PT symmetry 
+  Wencong Wang,1 Yanhua Zhai,2 Dongmei Liu,1* Xiaoshun  
+Jiang, 3† Saeid Vashahri Ghamsari,4 and Jianming Wen4‡ 
+1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,  
+School of Physics and Telecommunication Engineering,  
+South China Normal University, Guangzhou 510006, China 
+2 Physics Department, Spelman College, Atlanta, Georgia 30314, USA 
+3National Laboratory of Solid State Microstructures,  
+College of Engineering and Applied Sciences, Nanjing University, Nanjing 210093, China 
+4Department of Physics, Kennesaw State University, Marietta, Georgia 30060, USA 
+Quantum Langevin noise makes experimental realization of genuine quantum-optical 
+parity-time (PT) symmetry in a gain-loss-coupled open system elusive. Here, we challenge 
+this puzzle by exploiting twin beams produced from a nonlinear parametric process, one 
+undergoing phase-sensitive linear quantum amplification (PSA) and the other engaging 
+balanced loss merely. Unlike all previous studies involving phase-insensitive amplification 
+(PIA), our PSA-loss scheme allows one quadrature pair to experience PT symmetry, a unique 
+quantum effect without any classical counterpart. Such symmetry showcases many radical 
+noise behaviors beyond conventional quantum squeezing and inaccessible to any PIA-based 
+platform. Importantly, it is the only non-Hermitian system hitherto that enables the 
+emergence of non-Hermiticity-induced quantum-to-classical transition for the same quantum 
+observable when crossing exceptional point. Utilizing this quadrature-PT structure, we have 
+further studied its potential in quantum sensing by exploring the quantum Cramér-Rao bound 
+or Fisher information. Besides, the proposed quadrature PT symmetry also sheds new light 
+on protecting continuous-variable (CV) qubits from decoherence in lossy transmission, a 
+long-standing conundrum for various CV-based quantum technologies. 
+ 
+ 
+Introduction.—In canonical quantum mechanics, the 
+system Hamiltonian as a physical observable is required 
+to be Hermitian to ensure the realness of associated 
+eigenspectra. Yet, it has long been known that the 
+Hermiticity is just a sufficient but not necessary condition 
+for a Hamiltonian to have real eigenvalues. This radical 
+change of view stems from the seminal work by Bender 
+and Boettcher in 1998, where a large class of non-
+Hermitian quantum Hamiltonians enjoying the joint 
+parity-time (PT) symmetry was discovered to possess 
+entirely real eigenvalues below a phase-transition point or 
+exceptional point (EP) [1]. However, it remains elusive to 
+probe such a non-Hermitian but PT-symmetric quantum 
+Hamiltonian experimentally due to the lack of complex 
+quantum potential in reality. Nevertheless, the notion of 
+PT symmetry [2-9] has successfully survived in many 
+other physical branches. Thanks to the mathematical 
+equivalence between the quantum Schrödinger and 
+paraxial light propagation equation, classical optics was 
+first suggested to simulate the wave properties of PT-
+symmetric quantum mechanics in synthesized settings. 
+By incorporating linear gain and loss, optics has become 
+a fertile ground for exploring PT symmetry [2-17] with an 
+iconic feature of pair of eigenvalues phase transitioning 
+from purely real to complex conjugate when a parameter 
+crosses the EP. In this regard, a plethora of intriguing 
+
+phenomena have been uncovered by utilizing various 
+linear and nonlinear optical materials to control and 
+engineer light for practical applications [2-17]. 
+Despite the impressive progress, to date the PT studies 
+have been mostly limited to a mean-field approach that 
+encapsulates all quantum dissipation in an ‘effective 
+Hamiltonian’ [2-17]. This method treats light essentially 
+as a (semi) classical electromagnetic (EM) field and only 
+retains the minimum number of degrees of freedom to 
+describe an open system. As a result, it fails to yield valid 
+results when the nonclassicality of light are of interest. 
+Notably, if the EM field is quantized, one must introduce 
+the Langevin noise operator to preserve its corresponding 
+commutation relation [18], though the introduction of 
+quantum Langevin noise is generally thought to prevent 
+the system from approaching quantum optical PT 
+symmetry [19]. This is especially true when a phase-
+insensitive linear quantum amplifier (PIA) serves as an 
+optical gain resource. Unfortunately, so far the research 
+on PT optics has all been confined to PIA-based systems, 
+thereby rendering the observation of quantum signatures 
+highly challenging. The gain involvement also encounters 
+another fundamental issue from the famous quantum 
+noncloning theorem [20], namely how to maintain the 
+integrity of the signal state, let alone the limitation further 
+dictated by the Kramers-Kronig relation [21]. Therefore, 
+it becomes a legitimate question whether gain-loss-
+coupled PT symmetry is viable quantum optically [19]. 
+We notice that two alternative means have recently been 
+implemented by using either a passive scheme or a non-
+Hermitian subset Hamiltonian in a large Hermitian 
+system [22-27] to bypass the noise issues. These efforts 
+have unearthed some quantum features, but they are 
+incapable of providing a conclusive picture to the 
+problem. We are also aware that a distinct trajectory is 
+devoted to exploiting anti-PT symmetry [28-34], a 
+counterpart of PT, to avoid the adverse effect of Langevin 
+noise [35]. 
+By overcoming the aforementioned obstacles, here we 
+propose a novel and experimentally feasible platform 
+utilizing twin beams generated from a nonlinear optical 
+parametric process such as parametric down conversion 
+(PDC) and four-wave mixing (FWM) [36], with the signal 
+arm experiencing pure loss while the idler channel 
+undergoing phase-sensitive linear quantum amplification 
+(PSA). 
+Thanks 
+to 
+PSA-empowered 
+noiseless 
+amplification [37], our architecture not only makes the 
+observation of true quantum optical PT a reality, but also 
+displays distinctive features unapproachable to any 
+previous scheme. Opposite to PIA equally amplifying 
+paired quadratures with additive noise, PSA maximally 
+amplifies some of them but inversely attenuates the rest 
+without 
+adding 
+extra 
+noise. 
+This 
+asymmetric 
+amplification naturally gives rise to the so-called 
+quadrature PT symmetry, a unique quantum effect 
+without any classical counterpart. Given various never-
+before-seen attributes, our design further opens a door to 
+uncovering the stunning classical-to-quantum transition 
+for the same physical observable when a non-Hermitian 
+parameter passes through the EP. 
+Theoretical model.—For simplicity, let us focus on the 
+quadrature-PT setup schematic in Fig. 1, where under 
+perfect phase matching, nondegenerate paired signal-idler 
+waves are parametrically created from vacuum in a 
+counterpropagating geometry by driving an FWM 
+medium of length 𝐿. During their propagation, the idler 
+and signal channels are respectively subject to PSA and 
+loss with the rates of 𝑔 and 𝛾 . For nondepleted and 
+classical input pump lasers, the system evolves along the 
+∓𝑧 direction under the influence of the non-Hermitian 
+Hamiltonian 
+𝐻 =
+𝑖ℏ𝑔(𝑎𝑖
+2−𝑎𝑖
+†2)
+2
+− 𝑖ℏ𝛾𝑎𝑠
+†𝑎𝑠 + ℏ𝜅(𝑎𝑖
+†𝑎𝑠
+† + 𝑎𝑖𝑎𝑠) , (1) 
+and the signal-idler field operators (as, ai)  obey the 
+Heisenberg-Langevin equations, 
+Fig. 1. PT symmetry undergone by twin beams from a 
+backward-FWM process, where the −𝑧  idler mode 
+experiences PSA and the +𝑧 signal faces equal loss. 
+ 
+ 
+
+PSAdler
+Four-wave mixing
+Signal𝑑𝑎𝑖
+𝑑𝑧 = 𝑔𝑎𝑖
+† + 𝑖𝜅𝑎𝑠
+†,
+𝑑𝑎𝑠
+𝑑𝑧 = −𝛾𝑎𝑠 − 𝑖𝜅𝑎𝑖
+† + 𝑓𝑠 ,    (2) 
+with † denoting Hermitian conjugate, 𝜅 the parametric 
+conversion strength, and 𝑓𝑠 the quantum Langevin noise 
+of zero mean satisfying 〈𝑓𝑠(𝑧)𝑓𝑠
+†(𝑧′)〉 = 2𝛾𝛿(𝑧 − 𝑧′) 
+and 〈𝑓𝑠
+†(𝑧)𝑓𝑠(𝑧′)〉 = 0. At first glance, the dynamics (2) 
+seems PT-irrelevant even if 𝑔 = 𝛾 . Surprisingly, the 
+hidden PT arises if one transforms Eq. (2) into the 
+corresponding quadrature-operator evolution by defining 
+𝑞𝑗 = (𝑎𝑗
+† + 𝑎𝑗)/2  and 𝑝𝑗 = 𝑖(𝑎𝑗
+† − 𝑎𝑗)/2  (𝑗 = 𝑖, 𝑠) 
+with [𝑞𝑗, 𝑝𝑗] = 𝑖/2. That is 
+𝑑
+𝑑𝑧 [𝑞𝑖
+𝑝𝑠] = [ 𝑔
+𝜅
+−𝜅
+−𝛾] [𝑞𝑖
+𝑝𝑠] + [0
+𝑃𝑠] ,     (3a) 
+𝑑
+𝑑𝑧 [𝑝𝑖
+𝑞𝑠] = [−𝑔
+𝜅
+−𝜅
+−𝛾] [𝑝𝑖
+𝑞𝑠] + [ 0
+𝑄𝑠] ,    (3b) 
+where 𝑃𝑠 = 𝑖(𝑓𝑠
+† − 𝑓𝑠)/2 and 𝑄𝑠 = (𝑓𝑠
+† + 𝑓𝑠)/2 are the 
+Langevin-noise quadrature operators. The underlying 
+physics now becomes apparent. The 𝑔-𝛾 introduction 
+fundamentally intervenes the evolution of the usual two-
+mode squeezing. Specifically, for 𝑔 = 𝛾 , albeit the 
+impact of 𝑃𝑠 , {𝑞𝑖, 𝑝𝑠}  in Eq. (3a) become a PT-
+quadrature pair and adhere to PT-manifested noise 
+reduction with the advent of nontrivial phase transition at 
+the EP (𝛾 = 𝜅) for the pair of eigen-propagation constants 
+( 𝛽 = √𝜅2 − 𝛾2, −𝛽 ) transiting from purely real to 
+conjugate imaginary. In contrast, the conjugate pair 
+{𝑝𝑖, 𝑞𝑠} in Eq. (3b) simply follow 𝑄𝑠-mediated two-mode 
+quadrature squeezing, with their propagation decoupled 
+from {𝑞𝑖, 𝑝𝑠}. Such asymmetric and contrasting dynamics 
+are unavailable to any existing setting. More strikingly, 
+the system will facilitate dual opposing quadrature PT 
+symmetry–{𝑞𝑖, 𝑝𝑠} for active while {𝑝𝑖, 𝑞𝑠} for passive, 
+if without 𝑓𝑠 and 𝛾. The general solutions to Eqs. (3a) 
+and (3b) are 
+[𝑞𝑖(0)
+𝑝𝑠(𝐿)] = 𝑠𝑒𝑐(𝛽𝐿 − 𝜖) [
+𝑐𝑜𝑠 𝜖
+− 𝑠𝑖𝑛(����𝐿)
+− 𝑠𝑖𝑛(𝛽𝐿)
+𝑐𝑜𝑠 𝜖
+] [𝑞𝑖(𝐿)
+𝑝𝑠(0)] 
++𝑠𝑒𝑐(𝛽𝐿 − 𝜖) ∫ 𝑑𝑧𝑃𝑠(𝑧)
+𝐿
+0
+[−𝑠𝑖𝑛(𝛽(𝐿 − 𝑧))
+𝑐𝑜𝑠(𝛽𝑧 − 𝜖) ] , (4a) 
+[𝑝𝑖(0)
+𝑞𝑠(𝐿)] = 𝑠𝑒𝑐(𝜅𝐿) [
+𝑒𝛾𝐿
+− 𝑠𝑖𝑛(𝜅𝐿)
+− 𝑠𝑖𝑛(𝜅𝐿)
+𝑒−𝛾𝐿
+] [𝑝𝑖(𝐿)
+𝑞𝑠(0)] 
++𝑠𝑒𝑐(𝜅𝐿) ∫ 𝑑𝑧𝑄𝑠(𝑧) [−𝑒𝛾𝑧𝑠𝑖𝑛(𝜅(𝐿 − 𝑧))
+𝑒𝛾(𝑧−𝐿)𝑐𝑜𝑠(𝜅𝑧)
+]
+𝐿
+0
+ , (4b) 
+with 𝜖 = 𝑎𝑟𝑐𝑡𝑎𝑛(𝛾/𝛽). From these solutions, indeed, 
+{𝑞𝑖(0), 𝑝𝑠(𝐿)} but not {𝑝𝑖(0), 𝑞𝑠(𝐿)} carry on the PT-
+adjusted squeezing and anti-squeezing. 
+Homodyne detection.—To disclose quadrature PT 
+symmetry, one straightforward way is to analyze the noise 
+behaviors across the phase transition by homodyne 
+detecting quadrature variances in comparison with the 
+ideal squeezed vacuum. This in turn encourages us to look 
+at the following four variances: ⟨∆𝑞𝑗
+2⟩ = ⟨𝑞𝑗
+2⟩ − ⟨𝑞𝑗⟩
+2and 
+⟨∆𝑝𝑗
+2⟩ = ⟨𝑝𝑗
+2⟩ − ⟨𝑝𝑗⟩
+2 . Using Eqs. (4a) and (4b), after 
+some lengthy algebra, one reaches 
+ ⟨∆𝑞𝑖,0
+2 ⟩ =
+ℎ(𝐿)−2𝑠𝑖𝑛2𝜖−𝑠𝑒𝑐 𝜖𝑐𝑜𝑠(2𝛽𝐿−𝜖)
+8𝑐𝑜𝑠2(𝛽𝐿−𝜖)
+ ,        (5a) 
+⟨∆𝑝𝑠,𝐿
+2 ⟩ =
+ℎ(𝐿) 𝑐𝑜𝑠 𝜖−𝑐𝑜𝑠(2𝛽𝐿+𝜖)−2𝑠𝑖𝑛2𝜖 𝑐𝑜𝑠(2𝛽𝐿−𝜖)
+8 𝑐𝑜𝑠 𝜖𝑐𝑜𝑠2(𝛽𝐿−𝜖)
+ , (5b) 
+⟨∆𝑞𝑠,𝐿
+2 ⟩ =
+2+𝑐𝑜𝑠2𝜑𝑒−2𝛾𝐿−𝑐𝑜𝑠 𝜑 𝑐𝑜𝑠(2𝜅𝐿+𝜑)
+8 𝑐𝑜𝑠2(𝜅𝐿)
+ ,      (5c) 
+⟨∆𝑝𝑖,0
+2 ⟩ =
+(2+𝑐𝑜𝑠2𝜑)𝑒2𝛾𝐿−𝑐𝑜𝑠 𝜑 𝑐𝑜𝑠(2𝜅𝐿−𝜑)
+8 𝑐𝑜𝑠2(𝜅𝐿)
+ ,     (5d) 
+where ℎ(𝐿) = 3 + 2𝛾𝐿  and 𝜑 = tan−1(𝛾/𝜅) . As 
+expected, the PT-inherited variances (5a) and (5b) 
+differentiate themselves from the rest two (5c) and (5d). 
+A hallmark of such is the appearance of the argument 𝛽𝐿 
+in ⟨∆𝑞𝑖,0
+2 ⟩ and ⟨∆𝑝𝑠,𝐿
+2 ⟩. To have an intuitive picture, we 
+exemplify these variances in Figs. 2(a)–(d) for some 
+typical 𝛾/𝜅. From Figs. 2(a) and (b), we observe a few 
+extraordinary traits absent from all past studies on non-
+Hermitian physics as well as quantum squeezing. First, in 
+the PT-phase intact region (𝛾 < 𝜅), different from the 
+two-mode squeezed vacuum (TMSV) with an oscillation 
+period of 2𝜋 , both log4⟨∆𝑞𝑖,0
+2 ⟩  and log4⟨∆𝑝𝑠,𝐿
+2 ⟩ 
+generally display increased classical fluctuations with a 
+period 𝑇 ≈ 2𝜋𝜅/𝛽, except that the former shows a little 
+sub-vacuum-noise suppression at a very short distance 
+range due to the insufficient competition between PT and 
+squeezing. In contrast, both cease to oscillate in the phase 
+broken regime (𝛾 > 𝜅) and are upper bounded by their 
+respective variance curves at the EP. Moreover, 
+log4⟨∆𝑝𝑠,𝐿
+2 ⟩  always grows monotonically above the 
+vacuum-noise level; while 
+log4⟨∆𝑞𝑖,0
+2 ⟩  invariably 
+exhibits quantum squeezing, and the larger 𝛾/𝜅  the 
+larger the squeezing and 𝐿. What’s more, the completely 
+incompatible nature, quantum versus classical, of the 
+
+same physical observable 𝑞𝑖(0) before and after the PT 
+phase transition renders our system a unique candidate to 
+study the transition between these two different worlds, 
+whose boundary is physically defined by the EP curve. 
+Contrarily, since {𝑝𝑖(0), 𝑞𝑠(𝐿)}  are decoupled from 
+{𝑞𝑖(0), 𝑝𝑠(𝐿)}, their variances fluctuate periodically, akin 
+to the TMSV case. As shown in Figs. 2(c) and (d), though 
+notably affected by 𝑄𝑠  and 𝛾 , log4⟨∆𝑞𝑠,𝐿
+2 ⟩ resembles 
+the regular quadrature squeezing, but not log4⟨∆𝑝𝑖,0
+2 ⟩. 
+From the above analysis, we learned that for the same 
+single-mode quadrature, PT results in a nontrivial 
+fundamental transition from quantum to classical when 
+the non-Hermitian parameter 𝛾  oversteps a threshold. 
+One may wonder whether this exotic phenomenon can 
+also take place in a two-mode quadrature measurement. 
+The answer is affirmative. To see how this works, we pay 
+attention 
+to 
+𝑑1 = [𝑞𝑖(0) + 𝑞𝑠(𝐿)]/√2  and 
+𝑑2 =
+[𝑝𝑖(0) + 𝑝𝑠(𝐿)]/√2, which satisfy [𝑑1, 𝑑2] = 𝑖/2. For 
+the vacuum input, it is easy to check that their variances 
+are simply the sum of the single-mode ones (5a)–(5d), 
+〈∆𝑑1
+2〉 =
+⟨∆𝑞𝑖,0
+2 ⟩+⟨∆𝑞𝑠,𝐿
+2 ⟩
+2
+, 〈∆𝑑2
+2〉 =
+⟨∆𝑝𝑖,0
+2 ⟩+⟨∆𝑝𝑠,𝐿
+2 ⟩
+2
+ .    (6) 
+Based on Figs. 2(b) and (d), 〈∆𝑑2
+2〉 is expected to be 
+distributed above the vacuum noise all the time. 
+Moreover, because ⟨∆𝑝𝑠,𝐿
+2 ⟩ and ⟨∆𝑝𝑖,0
+2 ⟩ have different 
+fluctuation periods before the phase transition, we 
+envision that 〈∆𝑑2
+2〉 will exhibit interleaved dual periodic 
+oscillations but reduce to a single period after the phase 
+breaking. Though the situation becomes somewhat subtle 
+for 〈∆𝑑1
+2〉 , its layout can be deduced similarly by 
+compromising Figs. 2(a) and (c). To be specific, in the 
+PT-phase unbroken region, it is a double-cycle growth 
+fluctuation staggered on top of the vacuum noise (except 
+the very short distance case). When PT symmetry 
+spontaneously breaks down, counterintuitively, the 
+single-period oscillating 〈∆𝑑1
+2〉 will always return certain 
+squeezing at some effective distances, and these distances 
+will be extended for a bigger 𝛾/𝜅. Same as 𝑞𝑖(0), 𝑑1 
+can serve as another physical probe to visualize the 
+quantum-to-classical transition induced by quadrature PT 
+symmetry, too, with the boundary defined by the EP 
+curve. All these statements excellently agree with our 
+numerical simulations given in Figs. 3(a) and (b). 
+RISM.—Other than homodyne detection, there is one 
+additional means to explore quadrature PT, the so-called 
+relative intensity squeezing measurement (RISM). 
+Traditionally, this method enables the shot-noise of one 
+beam to be measured and subtracted from the other so as 
+to attain lower-noise differential measurement of a signal 
+of interest. To this end, we begin with our own relative-
+intensity 
+operator, 
+𝑁𝑖,0 − 𝑁𝑠,𝐿 = 𝑎𝑖
+†(0)𝑎𝑖(0) −
+𝑎𝑠
+†(𝐿)𝑎𝑠(𝐿) . The degree of squeezing is then 
+characterized by the noise figure (NF), which is 
+determined 
+by 
+the 
+relative-intensity 
+variance. 
+Mathematically, it takes the form [38] 
+Fig. 2. PT-manifested log4⟨∆𝑞𝑖,0
+2 ⟩ (a) and log4⟨∆𝑝𝑠,𝐿
+2 ⟩ 
+(b) in the presence of quantum Langevin noise. Non-PT-
+symmetric but loss-noise-mediated log4⟨∆𝑞𝑠,𝐿
+2 ⟩ (c) and 
+log4⟨∆𝑝𝑖,0
+2 ⟩ (d). As the references, the black solid and 
+dashed lines represent the regular TMSV (𝛾/𝜅 = 0) and 
+vacuum noise, respectively. 
+ 
+Fig. 3. PT-symmetric log4⟨∆𝑑1
+2⟩ (a) and log4⟨∆𝑑2
+2⟩ (b) 
+with account of quantum noise. Again, as the references, 
+the black solid and dashed curves are respectively the 
+ideal TMSV (𝛾/𝜅 = 0) and vacuum noise. 
+ 
+
+10
+10
+(a)
+(b)
+8
+8
+人人人
+人人人人人人
+log4(Aqi,o)
+4
+2
+2
+0
+0
+2
+0
+5
+10
+15
+20
+25
+0
+5
+10
+15
+20
+25
+2KL
+2KL
+8
+25
+(c)
+(d)
+6
+20
+vacuumnoise
+Y/x=O
+-y/x=1
+0.5
+y/x=0.6.y/x=1.4
+2
+1.5
+log4(
+10
+5
+0
+2
+5
+10
+15
+20
+0
+5
+10
+15
+20
+2KL
+2kL10
+25
+(a) 0.8
+(b)
+8
+从人人
+20
+vacuum noise
+6
+-y/x=0y/K=1
+《pv)
+15
+人人人
+Y/x = 0.6—/x = 1.4
+5
+10
+15
+20
+25
+0
+5
+10
+15
+20
+2K
+2KLNF =
+Var[𝑁𝑖,0−𝑁𝑠,𝐿]
+〈𝑁𝑖(0)〉+〈𝑁𝑠(𝐿)〉 .           (7) 
+Here, the average photon numbers are computed by 
+plugging Eqs. (4a) and (4b) to 〈𝑁𝑖,0〉 = ⟨𝑞𝑖
+2(0)⟩ +
+⟨𝑝𝑖
+2(0)⟩ − 1/2  and 〈𝑁𝑠,𝐿〉 = ⟨𝑞𝑠2(𝐿)⟩ + ⟨𝑝𝑠2(𝐿)⟩ − 1/2 . 
+In stark contrast to the quadrature variances discussed 
+earlier, the NF, while bringing about some alike 
+characteristics, clearly reveals some quite opposite 
+peculiarities. For 𝛾/𝜅 ≥ 1, as demonstrated in Fig. 4(a), 
+in addition to the incremental single-period fluctuation 
+log10(NF≥0 + 1)  grows along with the increment of 
+2𝜅𝐿, and the larger γ/κ is, the noisier it is. From the plot, 
+it is not difficult to conclude that in the PT-phase broken 
+region, NF is essentially occupied by the noise anti-
+squeezing. However, NF behaves highly complex as 
+𝛾/𝜅 < 1. Although it is still an interleaved double-period 
+oscillation within this range, the EP curve is no longer the 
+partition to separate the classical and quantum 
+fluctuations. In line with the numerical simulations, we 
+find that quantum squeezing materializes when 𝛾/𝜅 <
+0.52. Some representative examples are depicted in Fig. 
+4(b) by plotting −log10(NF<0 + 1) for different γ/κ. 
+Their comparison suggests that the smaller the value of 
+γ/κ, the more pronounced the achievable squeezing over 
+a longer distance 2𝜅𝐿. As a matter of fact, the RISM 
+obviously supplies certain sharp signatures unreachable to 
+the homodyne detection, regardless of the two highly 
+unbalanced channels. 
+Before proceeding, a few remarks are ready here. First, 
+even in the presence of Langevin noise, utilizing PSA 
+instead of PIA is practicable to accomplish quantum 
+optical PT under fair sampling measurement. Second, 
+contrary to PIA, PSA arouses the unusual quadrature PT 
+and licenses the singular quantum-to-classical transition 
+accompanied by the PT phase transition. Last but not 
+least, quadrature PT sheds new light on protecting 
+continuous-variable (CV) qubits from decoherence in 
+inevitable lossy transmission, a long-standing conundrum 
+for various CV-based quantum technologies [39]. 
+Quantum sensing.—Being a discipline of practical 
+application, quantum sensing [40-42] exploits quantum 
+properties, effects, or systems to fulfill high-resolution 
+and super-sensitive measurements of physical parameters 
+over the similar measurements performed within a 
+classical framework. For this, quantum squeezing has 
+long been recognized as one of the indispensable 
+nonclassical resources for ultra-precision estimations. 
+Among them, one far-reaching example is its recent 
+adoption by the Laser Interferometer Gravitational-Wave 
+Observatory (LIGO) for gravitational wave detection. 
+Nevertheless, the inevitable propagation loss often 
+degrades the available squeezing and compromises the 
+promised sensitivity. We note that in recent non-
+Hermitian studies, the abrupt change near EP has been 
+capitalized for enhanced sensing in classical settings [43-
+47]. Yet, its extension to the quantum level turns out to be 
+problematic because of quantum noise [48]. To avoid 
+such noise, one usually resorts to either ideal anti-PT 
+systems or post-selection measurement [25,34,35]. 
+Unlike these studies, here we directly confront Langevin 
+noise and explore the opportunity of quadrature PT in 
+quantum sensing under fair sampling measurement. We 
+are particularly interested to know whether the system 
+could have any advantage in improving sensitivity. As 
+shown below, the PT-quadrature observables can yield 
+the best performance of classical sensing before the phase 
+transition but departing far from the EP; while the non-
+PT-quadrature observables are capable of optimal 
+quantum sensing by noise-mediated squeezing for 𝛾/𝜅 
+less than 1. This distinguishes our work from the previous 
+anti-PT-, squeezing-, or EP-based proposals. Our analysis 
+is carried out by estimating 𝜅 (or 𝛾) and comparing the 
+achievable precision with the quantum Cramér-Rao 
+bound (set by the quantum Fisher information of the 
+Fig. 4. (a) PT-regulated noise figure (NF) via relative 
+intensity squeezing measurement. (b) Representative 
+examples of quantum PT-symmetric NF. 
+ 
+
+40
++
+(a)
+log10(NFzo+
+10
+(b)
+30
+-Y/x=0.8
+-/x=1
+20
+-Y/x=0.2
+-Y/x=2
+10
++1)
+0
+Y/x=0.2
++
+07
+107
+Y/x=0.1
+AN
+INF.
+/x=0.05
+10
+log10l
+5
+-l0g10()
+0
+10
+15
+20
+25
+30
+35
+0
+5
+10
+15
+20
+25
+30
+35
+2kL
+2kLquantum state). 
+Suppose that the two bosonic modes are initially 
+prepared in a coherent state |𝜓𝑖⟩ = |𝛼1, 𝛼2⟩. An optimal 
+homodyne measurement is then implemented on, say, 
+𝑞𝑖(0)after an evolution distance 𝐿. Using Eq. (4a), one 
+can readily have its mean value and variance, 
+⟨𝑞𝑖(0)⟩ =
+𝑐𝑜𝑠 𝜖⟨𝑞𝑖(𝐿)⟩−𝑠𝑖𝑛(𝛽𝐿)⟨𝑝𝑠(0)⟩
+𝑐𝑜𝑠(𝛽𝐿−𝜖)
+ ,  (8) 
+⟨∆𝑞𝑖,0
+2 ⟩ =
+3+𝑤[2𝛾𝐿−tan𝜖 sin(2𝛽𝐿)]−cos(2𝛽𝐿)−2sin2𝜖
+8cos2(𝛽𝐿−𝜖)
+ , (9) 
+where 𝑤 = 2𝑛𝑡ℎ + 1  and 𝑛𝑡ℎ  is the average thermal 
+boson number. From Eqs. (8) and (9), it is clear that the 
+Langevin noise shifts the peaks of ⟨∆𝑞𝑖,0
+2 ⟩ away from the 
+troughs of ⟨𝑞𝑖(0)⟩, so that they do not coincide at all. To 
+ease the subsequent derivations, hereafter we assume 
+𝛼𝑖 = 𝑖𝛼𝑠∗ = √2𝛼𝑒𝑖𝜋/4. The estimation precision relies on 
+measuring the change of ⟨𝑞𝑖(0)⟩  due to a tiny 
+perturbation 𝛿𝜅  on a preset 𝜅 . This alternatively 
+suggests to examine the system response to ⟨𝑞𝑖(0)⟩ for a 
+small variation 𝛿𝜅  around 𝜅 . We thus define the 
+susceptibility to capture such a response, 
+𝜒𝜅
+𝑞𝑖(0) ≡ 𝜕⟨𝑞𝑖(0)⟩
+𝜕𝜅
+= 
+𝛼{2𝛽𝐿[sin(𝛽𝐿)−1]+sin𝜖[2sin(𝛽𝐿)+cos(𝛽𝐿−𝜖)−cos𝜖]}
+2𝛽cos2(𝛽𝐿−𝜖)
+ . (10) 
+When 𝜅 → 𝛾  or 𝛽 → 0 , 𝜒𝜅
+𝑞𝑖(0) → 𝛼𝐿(3 + 𝜅2𝐿2)/
+[3(1 + 𝜅𝐿)2] is a curvetureless constant, implying the 
+loss of the sensing ability in the EP vicinity for the chosen 
+observable. On the other hand, the κ-estimation is jointly 
+determined by the variance and susceptibility, ∆𝜅,𝑞𝑖,0
+2
+=
+⟨∆𝑞𝑖,0
+2 ⟩/ [𝜒𝜅
+𝑞𝑖(0)]
+2
+, whose inverse dictates the accuracy, 
+∆𝜅,𝑞𝑖,0
+−2
+=
+2𝛼2{2𝛽𝐿[sin(𝛽𝐿)−1]+sin𝜖[2sin(𝛽𝐿)+cos(𝛽𝐿−𝜖)−cos𝜖]}2
+𝛽2cos2(𝛽𝐿−𝜖){3+𝑤[2𝛾𝐿− tan𝜖 sin(2𝛽𝐿)]−cos(2𝛽𝐿)−2sin2𝜖} .(11) 
+The sensing fulfillment is to compare ∆𝜅,𝑞𝑖,0
+−2
+,0with the 
+quantum Fisher information, 𝐹𝜅, which sets the ultimate 
+precision, i.e., the lower quantum Cram´ er-Rao bound for 
+any optimal measurement, 𝐹𝜅 ≥ ∆𝜅,𝑞𝑖,0
+−2
+. Fκcan be 
+accordingly derived from 
+𝐹𝜅 = 4𝐿2(⟨𝜓𝑓|𝜕𝜅𝐻†𝜕𝜅𝐻|𝜓𝑓⟩ −
+                        ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩) . (12) 
+for the final state |𝜓𝑓⟩ of the system in the Schrödinger 
+representation. 
+As 
+detailed 
+in 
+Supplementary 
+Information, one can perform similar sensing evaluations 
+to the rest three quadratures. Basing on the calculations 
+shown in Figs. 5(a) and (b), we note that in the current 
+arrangement, 𝑞𝑖(0) and 𝑝𝑠(𝐿) can only permit optimal 
+classical sensing for a moderate medium length in the 
+phase-unbroken regime but away from the EP, as revealed 
+by the ratios of ∆𝜅,𝑞𝑖,0
+−2
+/𝐹𝜅  and ∆𝜅,𝑝𝑠,𝐿
+−2
+/𝐹𝜅 . On the 
+contrary, 𝑞𝑠(𝐿) and 𝑝𝑖(0) behave distinctively in the 
+sense that, despite the impacts of the photon loss and 
+Langevin noise, they are still able to offer optimal 
+quantum sensing over a relatively longer distance for 
+smaller 𝛾/𝜅, as sketched in Figs. 5(c) and (d). 
+In short, fundamentally different from all previous 
+research, the PSA-induced quadrature PT enables a 
+unique way to observe a quantum-to-classical transition 
+for a physical observable at the breakdown of symmetry. 
+Such quadrature PT radically reshapes the dynamics of 
+two-mode squeezing with a striking phase transition 
+never seen before. Unfortunately, optical loss and 
+Langevin noise hinder the PT quadrature pair to confer a 
+quantum advantage in improving sensitivity, although the 
+non-PT pair can still offer optimal quantum sensing. 
+Besides of nonlinear wave mixing, our model can be also 
+Fig. 5. Quantum sensing in quadrature PT symmetry 
+characterized by the ratios of ∆𝜅,𝑞𝑖,0
+−2
+  (a), ∆𝜅,𝑝𝑠,𝐿
+−2
+  (b), 
+∆𝜅,𝑞𝑠,𝐿
+−2
+  (c), and ∆𝜅,𝑝𝑖,0
+−2
+  (d) to the quantum Fisher 
+information 𝐹𝜅 for the parameters {𝛼 = 2, 𝛾 = 0.2, 𝜅 =
+1 }  (blue), {2,1,1}  (red), and {2,2,1}  (orange), 
+respectively. 
+ 
+
+0.25
+0.25
+(a)
+(b)
+0.2
+X10-3
+0.2
+×10-3
+10
+10
+0.15
+-Y/x=0.2
+6
+Y/x=1
+6
+0.1
+4
+0.1
+Y/x=2
+4
+2
+0.05
+0.05
+0
+5
+1015
+202530
+0
+5
+1015202530
+0
+0
+0
+5
+10
+15
+20
+25
+30
+0
+5
+10
+15
+20
+25
+30
+2KL
+2KL
+0.25
+0.25
+(c)
+(d)
+X10-3
+X10-3
+0.2
+10
+0.2
+10
+8
+0.15
+6
+6
+2
+4
+AK
+0.1
+N
+0.05
+0
+0
+51015202530
+0.05
+0
+5
+1015202530
+0
+0
+5
+10
+15
+20
+25
+30
+0
+5
+10
+15
+20
+25
+30
+2KL
+2KLachieved in other platforms such as superconducting 
+circuits. Most importantly, our work forges a new avenue 
+to explore the long-sought, nontrivial quantum-to-
+classical transition utilizing non-Hermitian physics. 
+ 
+Acknowledgements.—This work was supported by NSF 
+1806519 and NSF EFMA-1741693. X.J. acknowledges 
+the support by the National Key R&D Program of China 
+(2021YF A1400803). D.L. was supported by the Nature 
+Science 
+Foundation 
+of 
+Guangdong 
+Province 
+(2019A1515011401). 
+ 
+AUTHOR CONTRIBUTIONS.—J.W. conceived the 
+theoretical scheme and supervised the whole project with 
+the help of D.L. and X.J. W.W., supervised by J.W., 
+carried out the whole calculations with the assistance of 
+Y.Z. and S.V.G. All authors contributed to the discussions 
+and writing of the manuscript. 
+ 
+* dmliu@scnu.edu.cn 
+† jxs@nju.edu.cn 
+‡ jianming.wen@kennesaw.edu 
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+ 
+
+1 
+ 
+Supplementary Information for 
+“Quantum-to-classical transition enabled by quadrature-PT symmetry” 
+Wencong Wang, Yanhua Zhai, Dongmei Liu*, Xiaoshun Jiang*,  
+Saeid Vashahri Ghamsari, and Jianming Wen* 
+Emails: dmliu@scnu.edu.cn; jxs@nju.edu.cn; jianming.wen@kennesaw.edu 
+I. Derivation of the Heisenberg-Langevin equations 
+In our previous work [1], we have theoretically proved that a forward parametric optical process 
+may lead to anti-PT symmetry while a backward parametric optical process can result in PT symmetry. 
+For this reason, as schematic in Fig. 1 in the main text [2], we are interested in a backward nonlinear 
+parametric optical process such as backward four-wave mixing, where the two counter-propagating 
+parametric modes, idler and signal, respectively experience balanced phase-sensitive linear quantum 
+amplification (PSA) and attenuation in their own channels within the medium of length 𝐿. For such 
+an open system, the evolution of the paired idler and signal field operators is effectively determined 
+by a non-Hermitian Hamiltonian, 
+ 
+𝐻 = 𝑖
+ℏ𝑔
+2 (𝑎𝑖
+†2 − 𝑎𝑖
+2) − 𝑖ℏ𝛾𝑎𝑠
+†𝑎𝑠 + ℏ𝜅(𝑎𝑖
+†𝑎𝑠
+† + 𝑎𝑖𝑎𝑠), 
+(S1.1) 
+where 𝑔  and 𝛾  respectively denote the PSA rate and loss rate. From Eq. (S1.1), one can readily 
+obtain the Heisenberg equations of the idler-signal field operators,  
+ 
+𝑖ℏ
+𝜕𝑎𝑖
+𝜕(−𝑧) = [𝑎𝑖, 𝐻], 
+(S1.2a) 
+ 
+𝑖ℏ
+𝜕𝑎𝑠
+𝜕𝑧 = [𝑎𝑠, 𝐻]. 
+(S1.2b) 
+Thanks to the noiseless amplification empowered by the PSA, the idler dynamics is not subject to the 
+additive noise and the commutation relation can be always satisfied throughout the whole process. 
+However, this is not true for the lossy signal propagation. To restore the commutation relation, one has 
+to introduce the quantum Langevin noise in the Heisenberg equation of the signal field operator. In 
+this way, we arrive at the following coupled Heisenberg-Langevin equations for the system of interest,  
+ 
+𝜕𝑎𝑖
+𝜕𝑧 = 𝑔𝑎𝑖
+† + 𝑖𝜅𝑎𝑠
+†, 
+(S1.3a) 
+ 
+𝜕𝑎𝑠
+𝜕𝑧 = −𝛾𝑎𝑠 − 𝑖𝜅𝑎𝑖
+† + 𝑓𝑠. 
+(S1.3b) 
+Though Eqs. (S1.3a) and (S1.3b) seem to have nothing to do with PT symmetry at first glance, as 
+pointed out in the main text, the hidden PT symmetry arises if transforming both equations into the 
+dynamics of the corresponding quadrature operators, 𝑞𝑗 = (𝑎𝑗
+† + 𝑎𝑗)/2  and 𝑝𝑗 = 𝑖(𝑎𝑗
+† − 𝑎𝑗)/2 
+(𝑗 = 𝑖, 𝑠) with [𝑞𝑗, 𝑝𝑗] = 𝑖/2. For simplicity, we concentrate on the case of the balanced PSA and loss, 
+𝑔 = 𝛾. With these preparations, one can easily attain the following sets of the coupled-quadrature 
+equations 
+ 
+𝑑
+𝑑𝑧 [𝑞𝑖
+𝑝𝑠] = [ 𝛾
+𝜅
+−𝜅
+−𝛾] [𝑞𝑖
+𝑝𝑠] + [0
+𝑃𝑠], 
+(S1.4a) 
+ 
+𝑑
+𝑑𝑧 [𝑝𝑖
+𝑞𝑠] = [−𝛾
+𝜅
+−𝜅
+−𝛾] [𝑝𝑖
+𝑞𝑠] + [ 0
+𝑄𝑠], 
+(S1.4b) 
+with 𝑃𝑠 = 𝑖(𝑓𝑠
+† − 𝑓𝑠)/2  and 𝑄𝑠 = (𝑓𝑠
+† + 𝑓𝑠)/2  being the Langevin-noise quadrature operators. 
+From Eq. (S1.4a), one can derive the effective Hamiltonian matrix for the quadrature pair (𝑞𝑖, 𝑝𝑠), 
+which reads 
+ 
+𝐻(𝑞𝑖,𝑝𝑠) = [ 𝑖𝛾
+𝑖𝜅
+−𝑖𝜅
+−𝑖𝛾] . 
+(S1.5) 
+It is straightforward to show that 𝐻(𝑞𝑖,𝑝𝑠) is indeed PT-symmetric, because it satisfies 𝑃𝑇𝐻(𝑞𝑖,𝑝𝑠) =
+𝐻(𝑞𝑖,𝑝𝑠)𝑃𝑇 for the combined PT operation with the parity operator being 𝑃 = [0
+1
+1
+0] and the time-
+reversal operator assuming the complex conjugation. For this reason, we call (𝑞𝑖, 𝑝𝑠)  the PT-
+
+2 
+ 
+quadrature pair. This is in a sharp contrast to the effective Hamiltonian matrix 𝐻(𝑝𝑖,𝑞𝑠) = [−𝑖𝛾
+𝑖𝜅
+−𝑖𝜅
+−𝑖𝛾] 
+in Eq. (S1.4b) for the other conjugate quadrature pair (𝑝𝑖, 𝑞𝑠), which is apparently irrelevant to PT 
+symmetry. The two eigenvalues of 𝐻(𝑞𝑖,𝑝𝑠) are 𝛽± = ±√𝜅2 − 𝛾2. Akin to the classical PT symmetry, 
+𝛾
+𝜅 < 1  corresponds to the quadrature PT-phase unbroken regime while for 
+𝛾
+𝜅 > 1 , quadrature PT 
+symmetry spontaneously breaks down. The quadrature PT phase transition occurs at the singular or 
+exceptional point (EP), 
+𝛾
+𝜅 = 1. In terms of the initial boundary conditions, the general solutions of Eqs. 
+(S1.4a) and (S1.4b) are readily found to be 
+ 
+[𝑞𝑖(0)
+𝑝𝑠(𝐿)] = sec(𝛽𝐿 − 𝜖) [
+cos 𝜖
+−sin(𝛽𝐿)
+−sin(𝛽𝐿)
+cos 𝜖
+] [𝑞𝑖(𝐿)
+𝑝𝑠(0)]
++ sec(𝛽𝐿 − 𝜖) ∫ 𝑑𝑧𝑃𝑠(𝑧)
+𝐿
+0
+[−sin(𝛽(𝐿 − 𝑧))
+cos(𝛽𝑧 − 𝜖) ], 
+(S1.6a) 
+ 
+[𝑝𝑖(0)
+𝑞𝑠(𝐿)] = sec(𝜅𝐿) [
+𝑒𝛾𝐿
+−sin(𝜅𝐿)
+−sin(𝜅𝐿)
+𝑒−𝛾𝐿
+] [𝑝𝑖(𝐿)
+𝑞𝑠(0)]
++ sec(𝜅𝐿) ∫ 𝑑𝑧𝑄𝑠(𝑧) [−𝑒𝛾𝑧sin(𝜅(𝐿 − 𝑧))
+𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧)
+]
+𝐿
+0
+, 
+(S1.6b) 
+with 𝜖 = arctan (
+𝛾
+𝛽). It is not difficult to prove that the dynamical solutions (S1.6a) and (S1.6b) well 
+maintain the commutation relations at all times,  
+ 
+[𝑞𝑖(0), 𝑝𝑖(0)] =
+𝑒𝛾𝐿cos 𝜖
+cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑞𝑖(𝐿), 𝑝𝑖(𝐿)]
++
+sin(𝛽𝐿)sin(𝜅𝐿)
+cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑝𝑠(0), 𝑞𝑠(0)]
++ ∫ 𝑑𝑧
+𝐿
+0
+𝑒𝛾𝑧sin(𝛽(𝐿 − 𝑧))sin(𝜅(𝐿 − 𝑧))
+cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿)
+[𝑃𝑠, 𝑄𝑠] = 𝑖
+2, 
+(S1.7a) 
+ 
+[𝑞𝑠(𝐿), 𝑝𝑠(𝐿)] =
+sin(𝛽𝐿)sin(𝜅𝐿)
+cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑝𝑖(𝐿), 𝑞𝑖(𝐿)]
++
+𝑒−𝛾𝐿 cos 𝜖
+cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿) [𝑞𝑠(0), 𝑝𝑠(0)]
++ ∫ 𝑑𝑧
+𝐿
+0
+𝑒𝛾(𝑧−𝐿)cos(𝛽𝑧 − 𝜖)cos(𝜅𝑧)
+cos(𝛽𝐿 − 𝜖)cos(𝜅𝐿)
+[𝑄𝑠, 𝑃𝑠] = 𝑖
+2. 
+(S1.7b) 
+Note that the quantum Langevin noise of zero mean satisfies 〈𝑓𝑠(𝑧)𝑓𝑠
+†(𝑧′)〉 = 2𝛾𝛿(𝑧 − 𝑧′) and 
+〈𝑓𝑠
+†(𝑧)𝑓𝑠(𝑧′)〉 = 0 . The quantumness of quadrature PT symmetry can be further approached by 
+analyzing the variances (or noise fluctuations) of 𝑞𝑗 and 𝑝𝑗 for the vacuum input state. After some 
+algebra, we have reached the following important results: 
+ 
+⟨∆𝑞𝑖,0
+2 ⟩ = ⟨𝑞𝑖
+2(0)⟩ − ⟨𝑞𝑖(0)⟩2 = ℎ(𝐿) − 2sin2𝜖 − sec 𝜖 cos(2𝛽𝐿 − 𝜖)
+8cos2(𝛽𝐿 − 𝜖)
+, 
+(S1.8a) 
+ 
+⟨∆𝑝𝑠,𝐿
+2 ⟩ = ⟨𝑝𝑠
+2(𝐿)⟩ − ⟨𝑝𝑠(𝐿)⟩2
+= ℎ(𝐿) cos 𝜖 − cos(2𝛽𝐿 + 𝜖) − 2sin2𝜖 cos(2𝛽𝐿 − 𝜖)
+8 cos 𝜖 cos2(𝛽𝐿 − 𝜖)
+ , 
+(S1.8b) 
+ 
+⟨∆𝑞𝑠,𝐿
+2 ⟩ = ⟨𝑞𝑠
+2(𝐿)⟩ − ⟨𝑞𝑠(𝐿)⟩2 = 2 + cos2𝜑𝑒−2𝛾𝐿 − cos 𝜑 cos(2𝜅𝐿 + 𝜑)
+8 cos2(𝜅𝐿)
+ , 
+(S1.8c) 
+ 
+⟨∆𝑝𝑖,0
+2 ⟩ = ⟨𝑝𝑖
+2(0)⟩ − ⟨𝑝𝑖(0)⟩2 =
+(2 + cos2𝜑)𝑒2𝛾𝐿 − cos 𝜑 cos(2𝜅𝐿 − 𝜑)
+8 cos2(𝜅𝐿)
+ . 
+(S1.8d) 
+where ℎ(𝐿) = 3 + 2𝛾𝐿  and 𝜑 = arctan (
+𝛾
+𝜅) . We notice from Eqs. (S1.8a)—(S1.8d) that although 
+
+3 
+ 
+these variances contain linear terms, they do not affect the periodic characteristics of the noise 
+fluctuations. As revealed in Fig. 2 in the main text, when taking 2𝜅𝐿 as the dimensionless variable, 
+we find that the oscillation period of the variances of the PT-quadrature pair (𝑞𝑖, 𝑝𝑠) is approximately 
+to be 
+2𝜅𝜋
+𝛽  while the fluctuation period of the variances of the non-PT quadrature pair (𝑝𝑖, 𝑞𝑠) simply 
+assumes 2𝜋 for the parameter space in the PT-phase intact region.  
+As emphasized in the main text, the ultimate novelty of our work is not just to find a system 
+capable of the observation of genuine quantum optical PT symmetry under fair sampling measurement, 
+but to unearth an extraordinary phenomenon that has never been discovered. That is, the PT-quadrature 
+observable enables one to witness a compelling quantum-to-classical transition perfectly coinciding 
+with the PT phase transition by varying the non-Hermitian parameter 𝛾, and the transition boundary 
+is physically defined by the EP curve. To the best of our knowledge, this is the first proposal on 
+exploring the untrivial transition between two incompatible worlds, classical and quantum, with a well-
+defined physical boundary by measuring the same quantum observable. We are aware that there exists 
+a parallel way in the literature that exploits some massive quantum systems such as cooled cavity 
+optomechanical structures to probe the quantum-to-classical transition by constantly checking the 
+decoherence of a quantum state when manipulating some system parameters. However, even if these 
+proposals are viable in the lab, they have to face an inescapable conundrum, that is, in these systems 
+it becomes extremely challenging to determine the exact transition boundary. In other words, observing 
+the sharp transition would be exceedingly difficult and even impossible for these protocols. In contrast, 
+these difficulties do not appear in our system. Moreover, our method aims to measure the expectation 
+of a quantum observable while the existing protocols concentrate on studying the state of the system. 
+This difference fundamentally distinguishes our work from all others. All in all, our work for the first 
+time presents a new way to explore the quantum-to-classical transition by taking advantage of non-
+Hermiticity and symmetry. 
+Before ending this part of discussion, here we would like to add a couple of additional comments 
+on the following important issues on quantum optical PT symmetry raised in the literature. One is in 
+response to the quantum noncloning theorem. In fact, the amplification won’t violate the quantum 
+noncloning theorem at all in our proposal, because the PSA here only acts on the single-mode idler 
+field. This also concurs with the well-established knowledge in the field of quantum optics, especially 
+in quantum squeezing. The second question concerns the law of causality. Although the gain may lead 
+to the fast-light or superluminal effect, since the quantum noise introduced by the transmission loss is 
+inseparable from the actual signal of interest, the causality proves not to be a problem when considering 
+PT symmetry at the quantum level (as we did here). 
+II. Derivation of quantum sensing 
+For the quantum sensor application, we consider the generation of the idler-signal bosonic modes 
+from a seeding two-photon coherent state |𝜓𝑖⟩ = |𝛼1, 𝛼2⟩. If taking into account the thermal reservoir 
+with an average thermal bosonic number 𝑛th, the quantum Langevin noise in Eq. (S1.3b) obeys the 
+following properties, 〈𝑓𝑠(𝑧)𝑓𝑠
+†(𝑧′)〉 = 2𝛾(𝑛th + 1)𝛿(𝑧 − 𝑧′) and 〈𝑓𝑠
+†(𝑧)𝑓𝑠(𝑧′)〉 = 2𝛾𝑛th𝛿(𝑧 − 𝑧′). 
+After an interaction length 𝐿, the mean values and variances of the quadrature measurements on 𝑞𝑗 
+and 𝑝𝑗 with respect to the final state |𝜓𝑓⟩ of the system can be obtained by using Eqs. (S1.6a) and 
+(S1.6b), 
+ 
+⟨𝑞𝑖(0)⟩ =
+cos 𝜖
+cos(𝛽𝐿 − 𝜖) ⟨𝑞𝑖(𝐿)⟩ −
+sin(𝛽𝐿)
+cos(𝛽𝐿 − 𝜖) ⟨𝑝𝑠(0)⟩, 
+(S2.1a) 
+ 
+⟨𝑝𝑠(𝐿)⟩ =
+− sin(𝛽𝐿)
+cos(𝛽𝐿−𝜖) ⟨𝑞𝑖(𝐿)⟩ +
+cos 𝜖
+cos(𝛽𝐿−𝜖) ⟨𝑝𝑠(0)⟩, 
+(S2.1b) 
+ 
+⟨𝑞𝑠(𝐿)⟩ = −
+sin(𝜅𝐿)
+cos(𝜅𝐿) ⟨𝑝𝑖(𝐿)⟩ +
+𝑒−𝛾𝐿
+cos(𝜅𝐿) ⟨𝑞𝑠(0)⟩, 
+(S2.1c) 
+
+4 
+ 
+ 
+⟨𝑝𝑖(0)⟩ =
+𝑒𝛾𝐿
+cos(𝜅𝐿) ⟨𝑝𝑖(𝐿)⟩ −
+sin(𝜅𝐿)
+cos(𝜅𝐿) ⟨𝑞𝑠(0)⟩, 
+(S2.1d) 
+and  
+ ⟨∆𝑞𝑖,0
+2 ⟩ = 3 + 𝑤[2𝛾𝐿 − tan𝜖 sin(2𝛽𝐿)] − cos(2𝛽𝐿) − 2sin2𝜖
+8cos2(𝛽𝐿 − 𝜖)
+, 
+(S2.2a) 
+ ⟨∆𝑝𝑠,𝐿
+2 ⟩ = 3 + 𝑤[2𝛾𝐿 + tan𝜖 sin(2𝛽𝐿 − 2𝜖)] − cos(2𝛽𝐿) + 4𝑛thsin2𝜖
+8cos2(𝛽𝐿 − 𝜖)
+ , 
+(S2.2b) 
+ ⟨∆𝑞𝑠,𝐿
+2 ⟩
+= 𝑤[cos(2𝑘𝐿) − cos 𝜑 cos(2𝑘𝐿 + 𝜑) + 1] + [cos2𝜑 − 2(1 + sin2𝜑)𝑛th]𝑒−2𝛾𝐿 + 2sin2(𝜅𝐿)
+8cos2(𝜅𝐿)
+ , 
+(S2.2c) 
+ ⟨∆𝑝𝑖,0
+2 ⟩
+= 𝑤{[cos2𝜑(𝑒2𝛾𝐿 − 1) − 2sin2𝜑] − 2sin 𝜑 cos(𝑘𝐿) sin(𝑘𝐿 − 𝜑)} + 2[𝑒2𝛾𝐿 + sin2(𝜅𝐿)]
+8cos2(𝜅𝐿)
+ . 
+(S2.2d) 
+Here, 𝑤 = 2𝑛th + 1  with the thermal average boson number 𝑛th = [Exp (
+ℎ𝑣𝜆
+𝑘𝐵𝑇) − 1]
+−1
+ . One can 
+easily check that the thermal photon number becomes infinitesimal at the room temperature (~300 K) 
+in the visible spectral range. Alternatively, the above Langevin noise properties reduce to the simpler 
+formats mentioned in Section I.  
+The ultimate precision of the parameter estimation of 𝜅 is essentially determined by the variance 
+∆𝜅
+2 in terms of a targeted physical observable. The performance of the proposed quadrature-PT sensing 
+scheme can be however evaluated by comparing the inverse variances ∆𝜅,𝑞𝑗
+−2 =
+(𝜒𝜅
+𝑞𝑗)
+2
+〈∆𝑞𝑗
+2〉   and ∆𝜅,𝑝𝑗
+−2 =
+(𝜒𝜅
+𝑝𝑗)
+2
+〈∆𝑝𝑗
+2〉   with the quantum Fisher information 𝐹𝜅  at the system’s final state |𝜓𝑓⟩ . Here, we have 
+introduced the susceptibilities 𝜒𝜅
+𝑞𝑗 = 𝜕𝜅⟨𝑞𝑗⟩  and 𝜒𝜅
+𝑝𝑗 = 𝜕𝜅⟨𝑝𝑗⟩  to capture the system response to 
+⟨𝑞𝑗⟩ and ⟨𝑝𝑗⟩ for a small perturbation 𝛿𝜅 about the preset 𝜅. With the help of Eqs. (S2.1a)—(S2.1d), 
+after some labor one can show that 𝜒𝜅
+𝑞𝑗 and 𝜒𝜅
+𝑝𝑗 (𝑗 = 𝑖, 𝑠) take the form of, 
+ 
+𝜒𝜅
+𝑞𝑖(0) = 𝛼{2𝛽𝐿[sin(𝛽𝐿) − 1] + sin 𝜖 [2sin(𝛽𝐿) + cos(𝛽𝐿 − 𝜖) − cos𝜖]}
+2𝛽cos2(𝛽𝐿 − 𝜖)
+, 
+(S2.3a) 
+ 
+𝜒𝜅
+𝑝𝑠(𝐿) = 𝜒𝜅
+𝑞𝑖(0), 
+(S2.3b) 
+ 
+𝜒𝜅
+𝑞𝑠(𝐿) = 𝛼𝐿 sec2(𝜅𝐿)[𝑒−𝛾𝐿sin(𝜅𝐿) − 1], 
+(S2.3c) 
+ 
+𝜒𝜅
+𝑝𝑖(0) = 𝛼𝐿 sec2(𝜅𝐿)[𝑒𝛾𝐿sin(𝜅𝐿) − 1]. 
+(S2.3d) 
+By plugging Eqs. (S2.2a)—(S2.2d) and Eqs. (S2.3a)—(S2.3d) into the inverse variances ∆𝜅,𝑞𝑗
+−2 =
+(𝜒𝜅
+𝑞𝑗)
+2
+〈∆𝑞𝑗
+2〉  
+and ∆𝜅,𝑝𝑗
+−2 =
+(𝜒𝜅
+𝑝𝑗)
+2
+〈∆𝑝𝑗
+2〉 , we arrive at the following key results: 
+ ⟨∆𝜅,𝑞𝑖,0
+−2
+⟩ = 2𝛼2{2𝛽𝐿[sin(𝛽𝐿) − 1] + sin𝜖 [2sin(𝛽𝐿) + cos(𝛽𝐿 − 𝜖) − cos𝜖]}2
+𝛽2cos2(𝛽𝐿 − 𝜖){3 + 𝑤[2𝛾𝐿 −  tan𝜖 sin(2𝛽𝐿)] − cos(2𝛽𝐿) − 2sin2𝜖} , 
+(S2.4a) 
+ ⟨∆𝜅,𝑝𝑠,𝐿
+−2
+⟩ =
+2𝛼2{2𝛽𝐿[sin(𝛽𝐿) − 1] + sin 𝜖 [2sin(𝛽𝐿) + cos(𝛽𝐿 − 𝜖) − cos𝜖]}2
+𝛽2cos2(𝛽𝐿 − 𝜖){3 + 𝑤[2𝛾𝐿 +  tan𝜖 sin(2𝛽𝐿 − 2𝜖)] − cos(2𝛽𝐿) + 4𝑛thsin2𝜖}, 
+(S2.4b) 
+ ⟨∆𝜅,𝑞𝑠,𝐿
+−2
+⟩
+=
+8𝛼2𝐿2sec2(𝜅𝐿)[𝑒−𝛾𝐿sin(𝜅𝐿) − 1]2
+𝑤[cos(2𝑘𝐿) − cos 𝜑 cos(2𝑘𝐿 + 𝜑) + 1] + [cos2𝜑 − 2(1 + sin2𝜑)𝑛th]𝑒−2𝛾𝐿 + 2sin2(𝜅𝐿) , 
+(S2.4c) 
+
+5 
+ 
+ 
+⟨∆𝜅,𝑝𝑖,0
+−2
+⟩
+=
+8𝛼2𝐿2sec2(𝜅𝐿)[𝑒𝛾𝐿sin(𝜅𝐿) − 1]2
+𝑤{[cos2𝜑(𝑒2𝛾𝐿 − 1) − 2sin2𝜑] − 2sin 𝜑 cos(𝑘𝐿) sin(𝑘𝐿 − 𝜑)} + 2[𝑒2𝛾𝐿 + sin2(𝜅𝐿)] . 
+(S2.4d) 
+After having the inverse variances (S2.4a)—(S2.4d), now let us turn our attention to the quantum 
+Fisher information 𝐹𝜅, i.e., the quantum Cramér-Rao bound, which demands the optimal measurement 
+to satisfy the inequality ∆𝜅
+−2≤ 𝐹𝜅. In this sensing protocol, we start with the initial system state to be 
+in a two-photon coherent state |𝜓𝑖⟩ = |𝛼1, 𝛼2⟩ for the sake of simplicity. Then, the final state of the 
+system evolves as |𝜓𝑓⟩ =
+𝑈
+√𝜇 |𝜓𝑖⟩, where 𝑈 = 𝑒−𝑖𝐻𝐿 is the evolution operator and 𝜇 = ⟨𝜓𝑓|𝜓𝑓⟩ is 
+the normalization coefficient. By working in the Schrödinger picture and treating the idler-signal field 
+operators 𝑎𝑖 = 𝑎𝑖(𝐿) and 𝑎𝑠 = 𝑎𝑠(0) as constant operators, the quantum Fisher information can be 
+calculated by the definition of 𝐹𝜅 = 4 (⟨𝜕𝜅𝜓𝑓|𝜕𝜅𝜓𝑓⟩ − |⟨𝜕𝜅𝜓𝑓|𝜓𝑓⟩|
+2)  for a parameter 𝜅  that 
+controls the strength of the system’s Hamiltonian 𝐻  (S1.1) with respect to a known physical 
+observable (in our case, it can be any of the four quadratures). To this end, let us give a detailed 
+examination on the first term in 𝐹𝜅: 
+ 
+⟨𝜕𝜅𝜓𝑓|𝜕𝜅𝜓𝑓⟩ 
+= ⟨𝜓𝑖|
+√𝜇(𝜕𝜅𝑈†) − 𝜕𝜅𝜇
+2√𝜇 𝑈†
+𝜇
+√𝜇(𝜕𝜅𝑈) − 𝜕𝜅𝜇
+2√𝜇 𝑈
+𝜇
+|𝜓𝑖⟩ 
+= 𝐿2⟨𝜓𝑓|𝜕𝜅𝐻†𝜕𝜅𝐻|𝜓𝑓⟩ − 𝑖𝐿
+𝜕𝜅𝜇
+2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩ + 𝑖𝐿
+𝜕𝜅𝜇
+2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩ +
+(𝜕𝜅𝜇)2
+4𝜇2 , 
+(S2.5) 
+where 𝜕𝜅𝑈 = −𝑖𝐿(𝜕𝜅𝐻)𝑈. Note that 𝜕𝜅𝐻 commutes with 𝑈. In the same way, we can also obtain 
+the exact expression for the second term as follows: 
+ 
+|⟨𝜕𝜅𝜓𝑓|𝜓𝑓⟩|
+2 
+= 𝐿2⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩ − 𝑖𝐿
+𝜕𝜅𝜇
+2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩ + 𝑖𝐿
+𝜕𝜅𝜇
+2√𝜇 ⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩ +
+(𝜕𝜅𝜇)2
+4𝜇2 . 
+(S2.6) 
+Substituting these two results into 𝐹𝜅  yields the concise and intuitive expression of the quantum 
+Fisher information, which is 
+ 
+𝐹𝜅 = 4𝐿2(⟨𝜓𝑓|𝜕𝜅𝐻†𝜕𝜅𝐻|𝜓𝑓⟩ − ⟨𝜓𝑓|𝜕𝜅𝐻†|𝜓𝑓⟩⟨𝜓𝑓|𝜕𝜅𝐻|𝜓𝑓⟩). 
+(S2.7) 
+Since 𝜕𝜅𝐻 = 𝜕𝜅𝐻† = ℏ(𝑎𝑖
+†𝑎𝑠
+† + 𝑎𝑖𝑎𝑠) = 2ℏ(𝑞𝑖𝑞𝑠 − 𝑝𝑠𝑝𝑖)  in the Schrödinger picture and the 
+expectation value of an operator does not change along with the picture transformation, we can 
+transform the above formulae of the quantum Fisher information into the Heisenberg representation to 
+ease the calculations. That is, 
+ 
+𝐹𝜅 = 16𝐿2 (⟨𝜓𝑓|(𝑞𝑖𝑞𝑠 − 𝑝𝑠𝑝𝑖)2|𝜓𝑓⟩ − (⟨𝜓𝑓|𝑞𝑖𝑞𝑠 − 𝑝𝑠𝑝𝑖|𝜓𝑓⟩)
+2) 
+ = 16𝐿2{⟨𝜓𝑖|[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]2|𝜓𝑖⟩
+− [⟨𝜓𝑖|[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]|𝜓𝑖⟩]2} 
+ = 16𝐿2{〈[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]2〉 − 〈𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)〉2}. 
+(S2.8) 
+From Eq. (S2.8), one can easily evaluate the term 𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0) using Eqs. (S1.6a) and 
+(S1.6b). After some lengthy derivations, we eventually get  
+ 
+𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0) 
+= 𝐴𝑞𝑖(𝐿)𝑝𝑖(𝐿) + 𝐵𝑝𝑠(0)𝑝𝑖(𝐿) + ∫ 𝑑𝑧𝐶𝑃𝑠𝑝𝑖(𝐿)
+𝐿
+0
++ 𝐷𝑞𝑖(𝐿)𝑞𝑠(0)
++ 𝐸𝑝𝑠(0)𝑞𝑠(0) + ∫ 𝑑𝑧𝐹𝑃𝑠𝑞𝑠(0)
+𝐿
+0
++ ∫ 𝑑𝑧𝐺𝑄𝑠𝑞𝑖(𝐿)
+𝐿
+0
++ ∫ 𝑑𝑧𝐽𝑄𝑠𝑝𝑠(0)
+𝐿
+0
++ ∫ 𝑑𝑧𝑅𝑃𝑠𝑄𝑠
+𝐿
+0
+, 
+(S2.9) 
+where all the involved coefficients are 
+
+6 
+ 
+ 
+𝐴 =
+𝑒𝛾𝐿sin(𝛽𝐿)−cos𝜖sin(𝜅𝐿)
+cos(𝛽𝐿−𝜖)cos(𝜅𝐿)
+, 
+(S2.10a) 
+ 
+𝐵 =
+sin(𝛽𝐿)sin(𝜅𝐿)−𝑒𝛾𝐿 cos𝜖
+cos(𝛽𝐿−𝜖)cos(𝜅𝐿)
+, 
+(S2.10b) 
+ 
+𝐶 =
+sin[𝛽(𝐿−𝑧)]sin(𝜅𝐿)−𝑒𝛾𝐿cos(𝛽𝑧−𝜖)
+cos(𝛽𝐿−𝜖)cos(𝜅𝐿)
+, 
+(S2.10c) 
+ 
+𝐷 =
+𝑒−𝛾𝐿 cos 𝜖−sin(𝛽𝐿)sin(𝜅𝐿)
+cos(𝛽𝐿−𝜖)cos(𝜅𝐿)
+, 
+(S2.10d) 
+ 
+𝐸 =
+cos𝜖sin(𝜅𝐿)−𝑒−𝛾𝐿sin(𝛽𝐿)
+cos(𝛽𝐿−𝜖)cos(𝜅𝐿)
+, 
+(S2.10e) 
+ 
+𝐹 =
+sin (𝜅𝐿)cos(𝛽𝑧−𝜖)−𝑒−𝛾𝐿sin(𝛽(𝐿−𝑧))
+cos(𝛽𝐿−𝜖)cos(𝜅𝐿)
+, 
+(S2.10f) 
+ 
+𝐺 =
+𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧) cos𝜖−𝑒𝛾𝑧sin[𝜅(𝐿−𝑧)]sin(𝛽𝐿)
+cos(𝛽𝐿−𝜖)cos(𝜅𝐿)
+, 
+(S2.10g) 
+ 
+𝐽 =
+𝑒𝛾𝑧sin[𝜅(𝐿−𝑧)]cos 𝜖−𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧)sin(𝛽𝐿)
+cos(𝛽𝐿−𝜖)cos(𝜅𝐿)
+, 
+(S2.10h) 
+ 
+𝑅 =
+𝑒𝛾𝑧sin[𝜅(𝐿−𝑧)]cos(𝛽𝑧−𝜖)−𝑒𝛾(𝑧−𝐿)cos(𝜅𝑧)sin[𝛽(𝐿−𝑧)]
+cos(𝛽𝐿−𝜖)cos(𝜅𝐿)
+. 
+(S2.10i) 
+With these results, we are ready to work out the following step,  
+ 
+〈[𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)]2〉 − 〈𝑞𝑖(0)𝑞𝑠(𝐿) − 𝑝𝑠(𝐿)𝑝𝑖(0)〉2
+= 𝐴2(⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩ − ⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩2)
++ 𝐵2(⟨𝑝𝑖
+2(𝐿)𝑝𝑠2(0)⟩ − ⟨𝑝𝑠(0)𝑝𝑖(𝐿)⟩2) + 𝐷2(⟨𝑞𝑖
+2(𝐿)𝑞𝑠2(0)⟩ − ⟨𝑞𝑠(0)𝑞𝑖(𝐿)⟩2)
++ 𝐸2(⟨𝑝𝑠(0)𝑞𝑠(0)𝑝𝑠(0)𝑞𝑠(0)⟩ − ⟨𝑝𝑠(0)𝑞𝑠(0)⟩2)
++ 𝐴𝐵(⟨𝑝𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩ + ⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑝𝑖(𝐿)⟩
+− 2⟨𝑝𝑖(𝐿)⟩⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩)⟨𝑝𝑠(0)⟩
++ 𝐴𝐷(⟨𝑞𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩ + ⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑞𝑖(𝐿)⟩
+− 2⟨𝑞𝑖(𝐿)⟩⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩)⟨𝑞𝑠(0)⟩
++ 𝐵𝐸(⟨𝑝𝑠(0)𝑞𝑠(0)𝑝𝑠(0)⟩ + ⟨𝑝𝑠(0)𝑝𝑠(0)𝑞𝑠(0)⟩
+− 2⟨𝑝𝑠(0)⟩⟨𝑝𝑠(0)𝑞𝑠(0)⟩)⟨𝑝𝑖(𝐿)⟩
++ 𝐷𝐸(⟨𝑝𝑠(0)𝑞𝑠(0)𝑞𝑠(0)⟩ + ⟨𝑞𝑠(0)𝑝𝑠(0)𝑞𝑠(0)⟩
+− 2⟨𝑞𝑠(0)⟩⟨𝑝𝑠(0)𝑞𝑠(0)⟩)⟨𝑞𝑖(𝐿)⟩
++ 𝐵𝐷(⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)𝑞𝑠(0)𝑝𝑠(0)⟩ + ⟨𝑝𝑖(𝐿)𝑞𝑖(𝐿)𝑝𝑠(0)𝑞𝑠(0)⟩
+− 2⟨𝑞𝑖(𝐿)⟩⟨𝑞𝑠(0)⟩⟨𝑝𝑖(𝐿)⟩⟨𝑝𝑠(0)⟩)
++ ∫ 𝑑𝑧[𝐶2⟨𝑝𝑖
+2(𝐿)⟩ + 𝐹2⟨𝑞𝑠2(0)⟩ + 2𝐶𝐹⟨𝑞𝑠(0)𝑝𝑖(𝐿)⟩]⟨𝑃𝑠
+2⟩
+𝐿
+0
++ ∫ 𝑑𝑧[𝐺2⟨𝑞𝑖
+2(𝐿)⟩ + 𝐽2⟨𝑝𝑠2(0)⟩ + 2𝐺𝐽⟨𝑝𝑠(0)𝑞𝑖(𝐿)⟩]⟨𝑄𝑠
+2⟩
+𝐿
+0
++ ∫ 𝑑𝑧𝐶𝐺[⟨𝑞𝑖(𝐿)𝑝𝑖(𝐿)⟩⟨𝑄𝑠𝑃𝑠⟩ + ⟨𝑝𝑖(𝐿)𝑞𝑖(𝐿)⟩⟨𝑃𝑠𝑄𝑠⟩]
+𝐿
+0
++ ∫ 𝑑𝑧𝐹𝐽[⟨𝑝𝑠(0)𝑞𝑠(0)⟩⟨𝑄𝑠𝑃𝑠⟩ + ⟨𝑞𝑠(0)𝑝𝑠(0)⟩⟨𝑃𝑠𝑄𝑠⟩]
+𝐿
+0
++ ∫ 𝑑𝑧𝑅2⟨𝑃𝑠𝑄𝑠𝑃𝑠𝑄𝑠⟩
+𝐿
+0
+− (∫ 𝑑𝑧𝑅⟨𝑃𝑠𝑄𝑠⟩
+𝐿
+0
+)
+2
+, 
+ 
+(S2.11) 
+
+7 
+ 
+in terms of the quadrature operators at the initial boundary conditions. By further simplification, we 
+finally approach the following resultant function for the quantum Fisher information, 
+ 
+𝐹𝜅 = 16𝐿2 [(𝐴2 + 𝐸2) (
+𝛼2
+2 +
+1
+8) + (𝐴𝐵 + 𝐴𝐷 + 𝐵𝐸 + 𝐷𝐸) (
+𝛼2
+2 ) + (
+1
+16 +
+𝛼2
+2 ) (𝐵2 + 𝐷2) −
+1
+8 𝐵𝐷 + (
+1
+4 + 𝛼2)
+𝛾
+2 (2𝑛th + 1) ∫ 𝑑𝑧(𝐶2 + 𝐹2 + 𝐺2 + 𝐽2)
+𝐿
+0
++ 𝛼2𝛾(2𝑛th + 1) ∫ 𝑑𝑧(𝐶𝐹 +
+𝐿
+0
+𝐽𝐺) +
+𝛾
+4 ∫ 𝑑𝑧(𝐽𝐹 − 𝐶𝐺)
+𝐿
+0
++
+𝛾
+2 (𝑛th +
+𝛾
+2) ∫ 𝑑𝑧𝑅2
+𝐿
+0
++
+𝛾2
+4 (∫ 𝑑𝑧𝑅
+𝐿
+0
+)
+2
+] .  
+(S2.12) 
+Obviously, the quantum Fisher information 𝐹𝜅 (S2.12) will feature different manifestations in 
+response to the contrasting PT domains of the system. As a representative example, in Fig. S1 we 
+accordingly present the quantum Fisher information log4𝐹𝜅  for three distinct scenarios: 
+𝛾
+𝜅 = 0.8 
+unbroken quadrature-PT phase), 
+𝛾
+𝜅 = 1 (EP point), and 
+𝛾
+𝜅 = 1.2 (breaking quadrature-PT phase).   
+FIG. S1.  The quantum Fisher information at different quadrature PT states. Blue, red, and 
+orange lines are, respectively, corresponding to 𝛾 𝜅
+Τ
+= 0.8  (unbroken quadrature-PT phase), 
+𝛾 𝜅
+Τ
+= 1 (EP point), and 𝛾 𝜅
+Τ
+= 1.2 (broken quadrature-PT phase). 
+FIG. S2.  Quantum sensing performance by comparing log4(Δ𝜅𝑞𝑖,0
+−2 )  (a), log4(Δ𝜅𝑝𝑠,𝐿
+−2 ) 
+(b), log4(Δ𝜅𝑞𝑠,𝐿
+−2 )  (c), and log4(Δ𝜅𝑝𝑖,0
+−2 )  (d) with log4 𝐹𝑘  in the quadrature-PT phase 
+unbroken region for the parameters (𝛼 = 2, 𝛾 = 0.2, 𝜅 = 1). 
+
+60
+50
+y
+y
+-
+0.8
+1
+1.2
+K
+K
+t
+40
+30
+20
+10
+0
+-10
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+2KL30
+30
+(a)
+(b)
+—log4((Fk)
+25
+25
+log4((Ago)
+log4((Akpz,))
+20
+20
+15
+15
+人儿从
+10
+10
+in
+0
+0
+-5
+-5
+10
+10
+0
+5
+10
+15
+20
+25
+30
+0
+5
+10
+15
+20
+25
+30
+30
+30
+(c)
+(d)
+25
+log4(Fk)
+25
+log4(Fk))
+log4((Akgz))
+20
+log4((Akpz0))
+20
+15
+15
+10
+10
+5
+5
+0
+0
+-5
+-5
+-10
+10
+0
+5
+10
+15
+20
+25
+30
+0
+5
+10
+15
+20
+25
+308 
+ 
+      
+ 
+ 
+ 
+ 
+Different from other existing (quantum) sensing protocols based on PT or EP enhancement, we find 
+that the PT-quadrature variables permit optimal classical sensor performance in the PT phase unbroken 
+regime but far away from the EP. This observation is strongly supported by analyzing the quantum 
+Fisher information with respect to the inverse variances across the parameter space. In the main text, 
+FIG. S4.  Evaluating quantum sensing performance (FIGs. S3(a)—(d)) near the exceptional 
+point by examining the ratios of ∆𝜅𝑞𝑖,0
+−2   (a), ∆𝜅𝑝𝑠,𝐿
+−2   (b), ∆𝜅𝑞𝑖,𝐿
+−2   (c), and ∆𝜅𝑝𝑖,0
+−2   (d) to 𝐹𝜅  for 
+𝛾
+𝜅 = 0.94 (red curve) and 
+𝛾
+𝜅 = 0.95 (black curve), respectively. The other parameters are 𝛼 =
+2 and 𝜅 = 1. 
+FIG. S3.  Quantum sensing performed near the exceptional point by comparing log4(Δ𝜅𝑞𝑖,0
+−2 ) 
+(a), log4(Δ𝜅𝑝𝑠,𝐿
+−2 )  (b), log4(Δ𝜅𝑞𝑠,𝐿
+−2 )  (c), and log4(Δ𝜅𝑝𝑖,0
+−2 )  (d) with log4 𝐹𝑘  for 
+𝛾
+𝜅 = 0.94 
+and 
+𝛾
+𝜅 = 0.95 for the parameters 𝛼 = 2 and 𝜅 = 1. 
+
+0.25
+0.25
+(a)
+(b)
+ 10-4
+×10-4
+0.2
+10
+0.2
+10
+8
+Y/K= 0.94
+8
+三 0.15
+0.15
+6
+—y/k=0.95
+6
+25
+4
+4
+0.1
+2
+0.1
+2
+0
+0.05
+2
+4
+6
+8
+0.05
+2
+4
+9
+8
+0
+0
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+2KL
+2KL
+0.25
+0.25
+(c)
+×10-4
+(d)
+×10-4
+10
+10
+0.2
+0.2
+8
+8
+6
+K
+6
+0.15
+0.15
+4
+4
+/T'S
+2
+2
+2
+20
+0.1
+0.1
+0
+6
+7
+8
+5
+6
+7
+8
+0.05
+0.05
+0
+0
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+2kL
+2KL60
+60
+(a)
+(b)
+50
+0.94
+0.95log4Fx
+50
+0.94
+0.95log4F
+K
+40
+Y
+=0.94
+Y
+40
+=0.94
+30
+30
+20
+20
+10
+10
+0
+0
+10
+10
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+2KL
+2KL
+60
+60
+(c)
+(d)
+50
+0.951og4Fx
+50
+0.951og4Fx
+=0.94
+K
+0.94
+K
+40
+Y
+=0.94
+Y
+=0.95 log4Axqz
+40
+Y
+=0.94
+Y
+=0.95
+30
+30
+20
+20
+10
+10
+0
+0
+-10
+10
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+2KL
+2KL9 
+ 
+we have shown the precision of the 𝜅 -parameter estimation by looking at the ratio of the inverse 
+variance ∆𝜅−2 to the quantum Fisher information 𝐹𝜅 in Figs. 5(a)—(d), which should be bounded in 
+the range of [0, 1]. In Fig. S2(a)—(d), we have further given these inverse variances in comparison 
+with 𝐹𝜅 . To make the point more straightforward and convincing, in Figs. S3(a)—(d) we have 
+particularly examined the measurement schemes implemented very close to the EP for 
+𝛾
+𝜅 = 0.94 and 
+FIG. S5.  Quantum sensing implemented exactly at the exceptional point by comparing  
+log4(Δ𝜅𝑞𝑖,0
+−2 )  (a), log4(Δ𝜅𝑝𝑠,𝐿
+−2 )  (b), log4(Δ𝜅𝑞𝑠,𝐿
+−2 )  (c), and log4(Δ𝜅𝑝𝑖,0
+−2 )  (d) with log4 𝐹𝑘  for 
+𝛼 = 2 and 𝜅 = 1. 
+FIG. S6.  Evaluating quantum sensing performance (FIG. S5(a)—(d)) at the exceptional point 
+by examining the ratios of ∆𝜅𝑞𝑖,0
+−2   (a), ∆𝜅𝑝𝑠,𝐿
+−2   (b), ∆𝜅𝑞𝑖,𝐿
+−2   (c), and ∆𝜅𝑝𝑖,0
+−2   (d) to 𝐹𝜅  for the 
+parameters 𝛼 = 2 and 𝜅 = 1. 
+
+40
+40
+(a)
+log4Fx
+(b)
+log4Fx
+30
+30
+20
+20
+10
+10
+0
+0
+-10
+10
+0
+5
+10
+15
+20
+25
+30
+0
+5
+10
+15
+20
+25
+30
+2KL
+2kL
+40
+40
+(c)
+log4Fx
+(d)
+log4Fk
+30
+30
+20
+20
+10
+10
+0
+0
+-10
+10
+0
+5
+10
+15
+20
+25
+30
+0
+5
+10
+15
+20
+25
+30
+2KL
+2KL0.25
+0.25
+(a)
+(q)
+×10-4
+× 10-4
+0.2
+10
+0.2
+10
+8
+K
+Y/K= 1
+8
+三 0.15
+0.15
+6
+6
+4
+2弘
+4
+AK
+0.1
+2
+0.1
+2
+0.
+0.05
+2
+4
+6
+8
+0.05
+2
+4
+6
+8
+0
+0
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+2KL
+2kL
+0.25
+0.25
+(c)
+× 10-4
+(d)
+× 10-4
+10
+0.2
+10
+0.2
+8
+8
+K
+6
+E
+K
+0.15
+6
+0.15
+4
+4 
+2
+2
+/0
+2
+0.1
+1 0.1
+2
+0
+AK
+0
+6
+7
+8
+5
+6
+7
+8
+0.05
+0.05
+0
+0
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+2kL
+2KL10 
+ 
+𝛾
+𝜅 = 0.95  by plotting log4 Δ𝜅𝑞𝑖,0
+−2  , log4 Δ𝜅𝑝𝑠,𝐿
+−2  , log4 Δ𝜅𝑞𝑠,𝐿
+−2  , and log4 Δ𝜅𝑝𝑖,0
+−2  . Similarly, the 
+quantitative sensing performance offered by each quadrature can be well assessed by evaluating the 
+corresponding ratio of ∆𝜅𝑞𝑖,0
+−2  (Fig. S4(a)), ∆𝜅𝑝𝑠,𝐿
+−2  (Fig. S4(b)), ∆𝜅𝑞𝑠,𝐿
+−2  (Fig. S4(c)), and ∆𝜅𝑝𝑖,0
+−2  (Fig. 
+S4(d)) to 𝐹𝜅 for the same parameters used in Figs. S3(a)—(d). By comparing these figures with Figs. 
+5(a)—(d) in the main text, it is not difficult to conclude that indeed, the presence of gain and loss in 
+gain-loss-coupled PT symmetry can substantially diminish the EP-based super-sensitivity promised in 
+the classical settings and make it unavailable in the quantum level. Moreover, even if one still insists 
+on performing any quantum sensing measurement in the vicinity of the EP (e.g., 
+𝛾
+𝜅 = 0.94 and 
+𝛾
+𝜅 =
+0.95), it would become highly challenging due to the vast difference between the peak values of ∆𝜅
+−2 
+and 𝐹𝜅 spanning over many orders of magnitude, regardless of whether the quadrature observables 
+are associated with the characteristics of PT symmetry. This is especially true if comparing with the 
+measurement carried out at 
+𝛾
+𝜅 = 0.2. 
+What happens if one attempts to fulfill the quantum sensing at the phase transition point? In such 
+a case, unfortunately, the paired PT quadratures will cease to showcase any response to the parameter 
+precision estimation, thereby making them fully unsuitable for quantum sensor applications when the 
+symmetry spontaneously breaks down. As demonstrated in Figs. S5(a) and (b), one can clearly see that 
+log4 Δ𝜅𝑞𝑖,0
+−2   and log4 Δ𝜅𝑝𝑠,𝐿
+−2   for the PT-symmetric quadrature pair (𝑞𝑖(0), 𝑝𝑠(𝐿))  become smooth 
+and curvatureless, indicating that they are completely insensitive to any perturbation on an unknown 
+parameter yet to be estimated. Alternatively, no gain on parameter estimation will be accessed at the 
+EP. On the other hand, we notice from Figs. S5(c) and (d) that the non-PT-symmetric quadrature pair 
+(𝑝𝑖(0), 𝑞𝑠(𝐿)) enables best sensing measurement only near the first peaks of the inverse variances 
+log4 Δ𝜅𝑞𝑠,𝐿
+−2 , and log4 Δ𝜅𝑝𝑖,0
+−2 , in accordance with the quantum Fisher information log4 𝐹𝑘. Obviously, 
+this behaves differently from the cases of 
+𝛾
+𝜅 = 0.2 , where the supersensitive measurements are 
+available near the first two peaks of ∆𝜅
+−2 and even more peaks (Figs. S2(c) and (d)). In fact, when PT 
+symmetry disappears, the quadrature pair (𝑞𝑖(0), 𝑝𝑠(𝐿)) lose to offer any sensing capabilities, despite 
+(sub)optimal sensing may be accessible to the other non-PT-symmetric conjugate pair (𝑝𝑖(0), 𝑞𝑠(𝐿)), 
+according to our numerical simulations. To have a more intuitive evaluation on the quantum sensing 
+performance exactly at the EP, it is better to look at the ratios of ∆𝜅𝑞𝑖,0
+−2  (Fig. S6(a)), ∆𝜅𝑝𝑠,𝐿
+−2  (Fig. 
+S6(b)), ∆𝜅𝑞𝑖,𝐿
+−2  (Fig. S6(c)), and ∆𝜅𝑝𝑖,0
+−2  (Fig. S6(d)) to 𝐹𝜅 in the same way as we did above. From 
+Figs. S6(a)—(d), we can easily find that these ratios quickly approach zero for the longer medium 
+length 𝐿, implying that the system loses its sensing ability at the EP. 
+ 
+References: 
+[1]  Jiang, Y., Mei, Y., Zuo, Y., Zhai, Y., Li, J., Wen, J. & Du, S. Anti-parity-time symmetry optical 
+four-wave mixing in cold atoms. Phys. Rev. Lett. 123, 193604 (2019). 
+[2]  Wang, W., Zhai, Y., Liu, D., Jiang, X., Ghamsari, S. V. & Wen, J. Quadrature parity-time 
+symmetry. (2022) 
+ 
+

89AyT4oBgHgl3EQfqPhE/content/tmp_files/2301.00538v1.pdf.txt

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+Topological Kondo Superconductors
+Yung-Yeh Chang,1, 2 Khoe Van Nguyen,2 Kuang-Lung Chen,2 Yen-Wen Lu,3 Chung-Yu Mou,4 and Chung-Hou Chung2
+1Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan Republic of China
+2Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan Republic of China
+3Department of Physics and Astronomy, University of California, Riverside, California 92511, U.S.A.
+4Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan Republic of China
+(Dated: January 3, 2023)
+Spin-triplet p-wave superconductors are promising candidates for topological superconductors. They have
+been proposed in various heterostructures where a material with strong spin-orbit interaction is coupled to a
+conventional s-wave superconductor by proximity effect. However, topological superconductors existing in na-
+ture and driven purely by strong electron correlations are yet to be studied. Here we propose a realization of
+such a system in a class of Kondo lattice materials in the absence of spin-orbit coupling and proximity effect.
+Therein, the odd-parity Kondo hybridization mediates ferromagnetic spin-spin coupling and leads to spin-triplet
+resonant-valence-bond (t-RVB) pairing between local moments. Spin-triplet p ± ip′-wave topological super-
+conductivity is reached when Kondo effect co-exists with t-RVB. We identify the topological nature by the
+non-trivial topological invariant and the Majorana fermions at edges. Our results offer a comprehensive under-
+standing of experimental observations on UTe2, a U-based ferromagnetic heavy-electron superconductor.
+I.
+INTRODUCTION
+Searching for topological superconductors (TSc) and the
+corresponding self-dual charge neutral Majorana zero modes
+associated with their excitations at edges has become one of
+the central problem in condensed matter physics [1, 2]. The-
+oretical proposals and experimental realizations of TSc are
+mostly heterostructure combining strong spin-orbit coupled
+materials and conventional superconductors by proximity ef-
+fect [3–5]. The emergence of the topological edge states in
+such systems can be explained in terms of the single-particle
+band structure without considering many-body electron corre-
+lations. Recently, the search for topological phases of matter
+has focused on a more intriguing class of materials that exist
+in nature. Their topological properties are driven by strong
+electron correlations instead of the proximity effect. Kondo
+effect, describing the screening of a local spin moment by con-
+duction electrons, is a well-known strong correlation between
+electrons existing in heavy electron compounds. The Kondo-
+mediated topological phases of matter have been studied in the
+context of topological Kondo insulators [6–8] and topological
+Kondo semi-metals [9], where the topological properties are
+driven by either the odd-parity Kondo hybridization or by the
+Kondo hybridization with strong spin-orbit coupling.
+Spin-triplet p-wave superconductors are known to be the
+prime candidates for TSc. However, they are scarce in na-
+ture. While it is still debatable for SrRu2O4 [10–12], more
+convincing evidence for p-wave triplet superconductivity was
+observed in noncentrosymmetric superconductor BiPd from
+phase-sensitive measurement [13]. More recently, signatures
+of triplet chiral p-wave superconductivity were observed in
+heavy-electron Kondo lattice compound UTe2 at the edge of
+ferromagnetism, possibly marking the first example of topo-
+logical superconductor induced by the strongly correlated
+Kondo effect [14–17].
+Motivated by these discoveries, in this paper, we propose a
+distinct class of triplet p-wave superconductors in the absence
+of spin-orbit coupling or proximity effect/heterostructure [18]
+in a two-dimensional Kondo lattice model driven by odd-
+parity Kondo hybridization. We start from the Anderson lat-
+tice model (ALM) with odd-parity hybridization, which oc-
+curs between d- and f-orbital electrons in various heavy-
+fermion compounds [6–8]. Via the Schrieffer-Wolff transfor-
+mation [19, 20], we derive an effective Kondo lattice model
+with odd-parity hybridization.
+Furthermore, by integrating
+out the conduction electron degrees of freedom, an effective
+ferromagnetic RKKY interaction is generated. We explore
+the mean-field phase diagram of this ferromagnetic Kondo-
+Heisenberg model. In the fermionic mean-field approach, the
+ferromagnetic RKKY coupling describes the p-wave (Sz =
+±1) t-RVB spin-liquid state. A time-reversal invariant topo-
+logical superconducting phase is reached when the Kondo ef-
+fect co-exists with the p-wave t-RVB order parameter. The
+topological nature of this superconducting phase is manifested
+by the non-trivial Z2 topological Chern number of the bulk
+band and by the existence of helical Majorana zero modes at
+the edges of a finite-sized ribbon. Our results offer a qualita-
+tive and some quantitative understanding of the spin-triplet su-
+perconductivity recently observed in UTe2 (see Discussions).
+II.
+MODEL
+A.
+Anderson lattice model with odd-parity hybridization
+We start with the odd-parity Anderson lattice model (ALM)
+on a two-dimensional (2D) square lattice, which has been
+shown to exhibit topologically non-trivial states [6–8]:
+HP AM = Hc + Hf + Hcf,
+(1)
+where Hc = �
+k,σ=↑,↓ εkc†
+kσckσ describes the hopping of
+electrons in the d orbits with orbital angular momentum l = 2
+and dispersion εk = −2t(cos kx + cos ky) − µ. The Hamil-
+tonian Hf of the more localized electron in the f orbits with
+arXiv:2301.00538v1  [cond-mat.str-el]  2 Jan 2023
+
+2
+●
+●
+●
+●
+●
+●
+●
+●●●●
+●
+●
+●
+●
+●
+●
+●
+●
+●●●●●●●●●●●●●
+●
+●
+●
+●●●●●●●●●●●
+●
+●
+●
+●
+●
+●●●
+●
+●
+●
+●●●●
+●
+●
+●
+●●●●
+●
+●
+●●●●●●●●
+●●
+■
+■
+■
+■
+■
+■
+■
+■
+■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
+■
+■
+■ ■ ■
+■
+■
+■ ■ ■ ■ ■
+■
+■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
+● μ/t = -1.5
+■ μ/t = -2.0
+0
+2
+4
+6
+-0.1
+-0.05
+0
+0.05
+R/a
+JH/JK
+2
+FIG. 1. The effective RKKY coupling JH (normalized with J2
+K) as a
+function of R/a for different chemical potentials µ. JH is computed
+by Eq. (7) with Rij ∥ (1, 1) and a = 1 being chosen.
+orbital angular momentum l = 3 is given by
+Hf =
+�
+i,σ
+�
+εff †
+iσfiσ + U
+2 nf
+iσnf
+i,−σ
+�
+,
+(2)
+where εf denote the energy level of the f-electron, and U
+is the repulsive on-site Coulomb potential (the Hubbard-U
+term). Hybridization of the local and conduction electrons is
+described by
+Hcf =
+�
+⟨i,j⟩
+�
+σ,σ′=↑↓
+V σσ′
+ij
+c†
+iσfjσ′ + H.c..
+(3)
+To conserve the parity symmetry of hybridization between
+electrons with their angular momentum quantum numbers dif-
+fering by one, V σσ′
+ij
+have to be odd under parity transforma-
+tion. This restriction results in the hybridization having to
+depend on sites and spins [6–8]:
+V σσ′
+ij
+≡ V σσ′
+ˆα
+= iV νˆασσσ′
+α
+,
+(4)
+distinct from the well-known onsite and spin-conserving An-
+derson hybridization. In Eq. (4), νij satisfies νij ≡ νˆα =
+−νji with ˆα ≡ i − j ∈ ˆx, ˆy (α ∈ x, y) on a 2D square lattice,
+and σα denotes the Pauli matrix of the α component.
+B.
+The effective odd-parity ferromagnetic Kondo lattice model
+In this paper, we focus on the competition of the Kondo
+and the magnetic interaction among impurities–the Doniach
+scenario [21].
+We, therefore, derive the effective Kondo-
+Heisenberg lattice Hamiltonian from ALM in the Kondo limit
+where the vacant and doubly-occupied states are projected out
+from the entire Hilbert space, namely 1 = �
+σ f †
+iσfiσ. The
+low-energy effective Kondo term from the odd-parity ALM
+of Eq. (1) can be derived by applying the Schrieffer-Wolff
+transformation (SWT) [19, 20, 22], yielding
+HK = (−JK)
+�
+i
+�
+σσ′
+�
+σ′′σ′′′
+�
+α,α′
+�
+iνˆασσσ′
+α
+c†
+i+ˆα,σfiσ′
+�
+×
+�
+iνˆα′σσ′′σ′′′
+α′
+f †
+iσ′′ci−ˆα′,σ′′′
+�
+(5)
+with JK =
+V 2
+U+εf −εF +
+V 2
+εF −εf > 0 (see Appendix A).
+The Kondo-like term of Eq. (5) describes the screening of
+an impurity by its neighboring conduction electrons, distinct
+from the conventional (on-site) Kondo term.
+Here, we go beyond the topological Kondo insulating phase
+by further deriving the magnetic RKKY interaction among
+the local f-fermions. By perturbatively expanding the Kondo
+term to second order [22–24], we obtain the effective RKKY-
+like interaction between the local f fermions fiσ,
+HJ =
+�
+i,j
+�
+σ,σ′
+Jijf †
+iσf †
+jσ′fjσfiσ′
+=
+�
+⟨i,j⟩
+Jij
+�
+f †
+i↑f †
+j↑fj↑fi↑ + f †
+i↓f †
+j↓fj↓fi↓
+�
++
+�
+⟨i,j⟩
+Jij
+2
+�
+f †
+i↑f †
+j↓ + f †
+i↓f †
+j↑
+�
+(fj↓fi↑ + fj↑fi↓)
+−
+�
+⟨i,j⟩
+Jij
+2
+�
+f †
+i↑f †
+j↓ − f †
+i↓f †
+j↑
+�
+(fj↓fi↑ − fj↑fi↓) , (6)
+where
+Jij ≡ JH(R) = 16J2
+K
+N 2s
+�
+εk<µ
+�
+εk′′>µ
+ei(k−k′′)·Rij
+εk − εk′′
+×
+�
+sin2 kx + sin2 ky
+� �
+sin2 k′′
+x + sin2 k′′
+y
+�
+(7)
+denotes the effective coupling of the spinons of sites i and
+j with R ≡ |Rij| ≡ |ri − rj|.
+The HJ term of Eq.
+(6) can be re-expressed as a linear combination of a spinon
+pair wave function with total spin S
+= 0 (spin-singlet)
+and S = 1 (spin-triplet).
+Note that the associated effec-
+tive spinon coupling of the spin-triplet channel is opposite
+to that of the spin-singlet. When HJ is expressed in terms
+of fermion pair with different spins, Eq.
+(6) is reminis-
+cent of the conventional Heisenberg interaction Si · Sj =
+− 1
+2
+�
+f †
+i↑f †
+j↓ − f †
+i↓f †
+j↑
+�
+(fi↓fj↑ − fi↑fj↓)+ 1
+4nf
+i nf
+j , except for
+the difference in the constant coefficients of the pair opera-
+tors. As expected, the RKKY coupling Jij in Eq. (7) shows
+an oscillatory behavior in R, accompanied by a decrease in
+its magnitude with increasing R, similar to the behavior of
+the conventional RKKY coupling. Due to the rapid attenua-
+tion of Jij, we only consider the dominated nearest-neighbor
+interaction and assume Jij to be spatially homogeneous, i.e.
+Jij → J(R = a) ≡ JH. Furthermore, when R = a, we find
+the effective RKKY coupling is attractive (or of the ferromag-
+netic type), i.e., JH < 0 (see Fig. 1), which energetically fa-
+vors the spin-triplet pairing of spinons. On the other hand, the
+effective RKKY coupling in the spin-singlet channel shows
+repulsive interaction and can be neglected here since it is not
+
+3
+◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
+◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
+●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
+●●●●●●●●●●●●●●●●●●●●
+◆ x
+● Δt
+0.5
+1
+1.5
+2.
+2.5
+0
+0.2
+0.4
+JH
+Mean-field parameters
+JK = 0.3, δ = -0.3
+FIG. 2. The zero-temperature mean-field solutions of t-RVB order
+parameter ∆t (brown) and the Kondo correlation x (black) as a func-
+tion of JH. We fix JK = 0.3 and doping of the conduction band
+δ = −0.3 (30 percent hole doping). Without loss of generality, we
+set t = 1. This plot reveals a (co-existing) superconducting ground
+state with x ̸= 0, ∆t ̸= 0 for 0 < JH ≲ 2.5 and a pure t-RVB
+phase where x = 0, ∆t ̸= 0 when JH ≳ 2.52. A pure Kondo phase
+(x ̸= 0, ∆t = 0) only exists at JH = 0.
+energetically favorable. Lastly, on a two-dimensional lattice,
+the triplet spin state | ↑↓⟩ + | ↓↑⟩ does not exist since the cor-
+responding structure factor is proportional to kz, and kz = 0
+is fixed here. Therefore, based on the above arguments, only
+the equal-spin states, | ↑↑⟩ and | ↓↓⟩, survive, and the HJ term
+is reduced to
+HJ ≈ − |JH|
+�
+⟨i,j⟩
+�
+f †
+i↑f †
+j↑fj↑fi↑ + f †
+i↓f †
+j↓fj↓fi↓
+�
+.
+(8)
+Combining HK and HJ of Eqs. (5), (6) and (8), the effec-
+tive Kondo-Heisenberg lattice model with odd-parity Kondo
+hybridization reads HF KH = H0 + Hλ + HK + HJ. Here,
+Hλ = − �
+i iλi
+��
+σ(f †
+iσfiσ) − 1
+�
+enforces the singly occu-
+pied local f-spinons with λi being the Lagrange multiplier.
+The Hamiltonian HF KH offers a platform for discovering a
+distinct class of topological superconducting states induced
+by electron correlations via collaboration between the ferro-
+magnetic RKKY coupling and the Kondo effect. To facilitate
+our numerical calculations of the mean-field phase diagram,
+we treat JK and JH as independent couplings here since it is
+more convenient to explore the phase diagram by tuning the
+ratio of JK/JH [25, 26]. In experiments, varying the non-
+thermal parameter can be expected to follow a certain trajec-
+tory of JK/JH in the phase diagram.
+III.
+MEAN-FIELD TREATMENT OF THE EFFECTIVE
+KONDO-HEISENBERG-LIKE MODEL
+We now employ a mean-field analysis on the above ef-
+fective Kondo-Heisenberg-like Hamiltonian with an effective
+ferromagnetic RKKY interaction and odd-parity Kondo hy-
+bridization.
+Via performing Hubbard-Stratonovich transformation, HK
+and HJ of Eqs. (5) and (6) can be factorized as
+HK →
+�
+i,α
+�
+σσ′
+�
+χ†
+i
+�
+iνˆασσσ′
+α
+f †
+iσci−ˆα,σ′
+�
++ H.c.
+�
++
+�
+i
+|χi|2
+JK
+,
+HJ →
+�
+⟨i,j⟩
+�
+∆↑
+t (i, j)f †
+i↑f †
+j↑ + ∆↓
+t (i, j)f †
+i↓f †
+j↓ + H.c.
+�
++
+�
+⟨i,j⟩
+���∆↑
+t (i, j)
+���
+2
++
+���∆↓
+t (i, j)
+���
+2
+JH
+(9)
+where the mean-field values of the bosonic Hubbard-
+Stratonovich fields, χi and ∆σ
+t (i, j) (σ =↑, ↓), represent the
+order parameters of the Kondo correlation and the Sz = ±1
+spin-triplet RVB bonds between two adjacent up/down spins,
+respectively.
+To describe the Kondo-screened Fermi-liquid state, we al-
+low the χi field to acquire uniformly Bose condensation over
+the real space; hence, χi can be expressed as χi → x + ˆχi
+with x = (−JK/Ns) �
+iσσ′α⟨iνˆασσσ′
+α
+f †
+iσci−ˆα,σ′⟩ being the
+Bose-condensed stiffness of χi while ˆχi represents its fluctua-
+tions. The mean-field order parameter of the tRVB is given by
+∆σ
+t = (−JH/4Ns) �
+⟨i,j⟩⟨fjσfiσ⟩. Since the ferromagnetic
+coupling is expected to favor spin-triplet p-wave pairing sim-
+ilar to superfluid helium-3 [27], we restrict ourselves to the
+p-wave pairing, i.e., ∆σ
+t (i, j) here is taken the p-wave form,
+see Eqs. (11) and (12) below. We further fix the Lagrange
+multiplier at the mean-field level via iλi → λ and neglect the
+fluctuations of λi, χi, and ∆σ
+t , leading to the following mean-
+field Kondo-Heisenberg-like Hamiltonian:
+HMF =
+�
+k,σ
+εkc†
+kσckσ +
+�
+kσ
+λf †
+kσfkσ
++
+�
+k
+�
+V1kf ∗
+k↑ck↓ + V2kf ∗
+k↓ck↑ + H.c.
+�
++
+�
+k
+�
+∆↑
+kf †
+k↑f †
+−k↑ + ∆↓
+kf †
+k↓f †
+−k↓ + H.c.
+�
++ 8Ns∆2
+t
+JH
++ Nsx2
+JK
+− Nsλ,
+(10)
+where
+V1k
+=
+2x (sin kx − i sin ky)
+and
+V2k
+=
+2x (sin kx + i sin ky).
+The Fourier transformation for
+the
+second-quantized
+operator
+is
+defined
+as
+ψiσ
+=
+1
+√Ns
+�
+k e−ik·riψkσ. Note that the mean-field Kondo term
+of Eq.
+(10) is reminiscent of the topological Kondo insu-
+lator shown in Ref. [28]. In Eq. (10), ∆σ
+t (k) represents
+the gap structure of the spin-triplet p-wave RVB pairing in
+the momentum space for the spin-σ sector, defined as ∆↑
+k =
+∆t (− sin ky − i sin kx) and
+∆↓
+k = ∆t (sin ky − i sin kx)
+with ∆t being denoted the mean-field pairing potential (see
+Appendix, Section II). This momentum-dependent gap struc-
+
+4
+ture for the up- and down-spin sectors correspond to the fol-
+lowing real-space patterns of ∆↑
+t (i, j) and ∆↓
+t (i, j) of Eq. (9):
+∆↑
+t (i, j) → ∆↑
+t (i, i + ˆx) = −∆↑
+t (i, i − ˆx) = −∆t,
+∆↑
+t (i, i + ˆy) = −∆↑
+t (i, i − ˆy) = i∆t,
+(11)
+and
+∆↓
+t (i, j) →∆↓
+t (i, i + ˆx) = −∆↓
+t (i, i − ˆx) = −∆t,
+∆↓
+t (i, i + ˆy) = −∆↓
+t (i, i − ˆy) = −i∆t.
+(12)
+Choosing Ψk = (φAk, φBk)T with the Nambu spinors
+defined by φAk =
+�
+ck↑, c†
+−k↑, fk↓, f †
+−k↓
+�T
+and φBk =
+�
+ck↓, c†
+−k↓, fk↑, f †
+−k↑
+�T
+,
+the
+mean-field
+Hamiltonian
+HMF = �
+k Ψ †
+kHkΨk + C can be expressed as a summation
+of two decoupled 4 × 4 matrices as follows
+HMF = HA + HB + C,
+HA(B) =
+�
+k
+φ†
+A(B)kHA(B)
+k
+φA(B)k
+(13)
+with C ≡ �
+k εk + 8Ns∆2
+t
+JH
++ Nsx2
+JK , and
+HA
+k =
+�
+�
+�
+�
+�
+εk
+2
+0
+V ∗
+2k
+2
+0
+0
+− εk
+2
+0
+V2k
+2
+V2k
+2
+0
+λ
+2
+∆↓
+k
+0
+V ∗
+2k
+2
+∆↓∗
+k
+− λ
+2
+�
+�
+�
+�
+� ,
+(14)
+HB
+k =
+�
+�
+�
+�
+�
+εk
+2
+0
+V ∗
+1k
+2
+0
+0
+− εk
+2
+0
+V1k
+2
+V1k
+2
+0
+λ
+2
+∆↑
+k
+0
+V ∗
+1k
+2
+∆↑∗
+k
+− λ
+2 .
+�
+�
+�
+�
+�
+(15)
+The Hamiltonian Eq. (13) possesses time-reversal symme-
+try: HA and HB constitute the time-reversal partner of each
+other, i.e. ΘHA(B)Θ−1 = HB(A) where the time-reversal
+operator Θ = ρ0 ⊗ (−iσy)K with σy being the y-component
+Pauli matrix on the spin subspace, ρ0 being a 2 × 2 identity
+matrix on the orbital subspace while K being the complex-
+conjugate operator. Under time-reversal transformation, the
+spin and quasi-momentum of conduction (c) and pseud-
+ofermion (f) operators are flipped: (ck↑, ck↓, fk↑, fk↓)
+Θ
+−→
+(c−k↓, −c−k↑, f−k↓, −f−k↑).
+Meanwhile, our Hamilto-
+nian respects charge-conjugation (particle-hole) symmetry:
+PHkP−1 = −H−k where P ≡ τ xK is the particle-hole op-
+erator with τx being the x-component of the Pauli matrices on
+the particle-hole basis. Due to the odd-parity p±ip′ RVB pair-
+ing of our model, the parity symmetry is broken here. Thus,
+our model Eq. (13) belongs to the DIII class of topological
+symmetry [29].
+IV.
+RESULTS
+A.
+Mean-field phase diagram
+The mean-field ground states are determined by minimiz-
+ing the mean-field free energy per site FMF
+=
+C
+Ns −
+kBT
+Ns
+�
+nk ln
+�
+1 + exp
+�
+− Enk
+kBT
+��
+with respect to the mean-
+field variables q = (λ, x, ∆t), i.e. ∂FMF /∂qi = 0.
+Here,
+Enk < 0 is the n-th band of Hk. The chemical potential µ is
+determined by the relation ∂FMF /∂µ = −(1 + δ) with δ be-
+ing the chemical doping of the c-electrons for which δ ⪋ 0 is
+for p/un/n− doped (half-filling corresponds to δ = 0). This
+leads to the following saddle-point equations at zero tempera-
+ture,
+1
+Ns
+�
+nk
+∂Enk
+∂x
++ 2x
+JK
+= 0,
+1
+Ns
+�
+nk
+∂Enk
+∂��t
++ 16∆t
+JH
+= 0,
+1
+Ns
+�
+nk
+∂Enk
+∂λ
+= 0,
+1
+Ns
+�
+nk
+∂Enk
+∂µ
++ δ = 0.
+(16)
+The ground-state phase diagram (Fig. 2) of our model is ob-
+tained by solving the saddle-point equations self-consistently.
+The phase diagram contains three distinct mean-field phases:
+a pure Kondo phase is found at JH = 0 where x ̸= 0, ∆t = 0.
+At the opposite limit where the RKKY interaction dominates,
+the ground state shows short-range magnetic correlation with
+p-wave spin-triplet RVB pairing (∆t ̸= 0, x = 0). In the
+intermediate range of 0 < JH/JK < (JH/JK)c, we find a
+Kondo-tRVB co-existing (superconducting) phase with x ̸= 0
+and ∆t ̸= 0, which can be explained via the mechanism of
+Kondo-stabilized spin liquid [26, 30]. The development of
+superconductivity in this co-existing phase requires higher-
+order processes involving both the Kondo and t-RVB terms:
+the mean-field t-RVB pairings of the local f fermions pro-
+vide preformed Cooper pairs. When the Kondo hybridization
+field χ gets Bose-condensed (x ̸= 0), the local fermions de-
+localize into the conduction band and make the preformed t-
+RVB Cooper pairs superconduct [31]. These processes can
+be described by the effective mean-field Hamiltonian Hsc =
+�
+k
+�
+¯
+∆↓∗
+k c−k↓ck↓ + ¯
+∆↑∗
+k c−k↑ck↑ + H.c.
+�
+, where the effec-
+tive gap functions take the form ¯
+∆↓∗
+k
+= V1kV1,−k∆↑∗
+k
+∼
+x2∆t(sin2 kx + sin2 ky)(sin kx − i sin ky) and
+¯
+∆↑∗
+k
+=
+V2kV2,−k∆↓∗
+k ∼ x2∆t(sin2 kx + sin2 ky)(sin kx + i sin ky)
+with the size of the superconducting gap being proportional
+to x2∆t. The superconducting gap function ¯
+∆↑
+k we obtained
+here shows a f-wave-like pairing symmetry on a generic
+anisotropic (non-circular) 2D Fermi surface.
+Nevertheless,
+as we are taking the continuous limit of the conduction band
+here, ¯
+∆↑
+k can be expressed as a product of s and p±ip′ pairing
+
+5
+(a)
+Γ
+X
+M
+(b)
+FIG. 3. Figures (a) (red curves) and (b) show the bulk energy spec-
+trum of the co-existing superconducting state near the Fermi level µ.
+The Fermi level locates at E(k) = 0. The coupling constants are
+JK = 0.3 and JH = 1.0. Inset of (a) displays the First Brillouin
+zone of a square lattice with indications of high-symmetry points
+Γ, X, M.
+orders, i.e., ¯
+∆↑/↓∗
+k
+∼ k2(kx±iky) with k2 ≡ k2
+x+k2
+y on a cir-
+cular Fermi surface but only the p±ip′ component plays a role
+here. Note that we find the co-existing superconducting state
+persists for an arbitrary small value of JH/JK → 0+. This is
+likely due to the overestimation of the co-existing phase at the
+mean-field level. Upon including fluctuations of the Kondo
+and t-RVB order parameters beyond the mean-field level, we
+expect a narrower co-existing superconducting phase. A first-
+order transition similar to the results found in Refs. [26, 32]
+is observed at the transition of the t-RVB and the co-existing
+superconducting phases (see Fig. 2). The bulk band structure
+in the co-existing superconducting state is shown in Fig. 3.
+B.
+Topological invariance
+We now address the topological properties of the coexisting
+superconducting state. Since this system is invariant under
+time-reversal transformation, the bulk topological properties
+of the coexisting Kondo-RVB superconducting state with p ±
+ip′ spin-triplet RVB pairing can be thus characterized by the
+Z2 Chern number cT (or time-reversal polarization) [33–35],
+kx
+E(kx)
+FIG. 4. The left figure displays the electronic band structure of the
+coexisting superconductor state for a strip with Ny = 81 described
+by HA at JK/t = 0.3 and JH/t = 1.0. Three pairs of edge states
+with Dirac spectra are observed near kx = 0 (the pink curves). The
+edge states at zero energy correspond to the Majorana zero modes.
+Due to the time-reversal symmetry of the model, the band structure
+for a strip for HB is identical to that of HA. The close-up band
+structures near three pairs of edge states (pink curves) on the top,
+middle and bottom bounded by the red squares are shown on the
+right figures.
+given by
+cT = cA − cB
+2
+(17)
+with cI (I ∈ A, B) being the Thouless-Kohmoto-Nightingal-
+den Nijs (TKNN) number [36] of HI, defined as
+cI = 1
+2π
+�
+k∈FBZ
+dSk ·
+�
+∇k × AI
+k
+�
+.
+(18)
+The Berry’s connection AI
+k for HI is given by AI
+k ≡
+i �
+n∈I⟨uI
+nk|∇k|uI
+nk⟩ with |uI
+nk⟩ being the normalized
+Bloch state of the n-th filled band for HI
+k. We numerically
+calculate the TKNN numbers [37], cA and cB, and find that
+cA = −cB = 1 in the co-existing phase, indicating a topolog-
+ically non-trivial Z2 Chern number cT = 1. By the bulk-edge
+correspondence, we expect this co-existing superconducting
+state to support a pair of counter-propagating Majorana zero
+modes at the edges of a finite-sized strip. Further band struc-
+ture calculations of our model on a strip in the following sub-
+section confirm our expectation.
+C.
+Edge states of the coexisting Kondo-RVB spin-triplet
+p ± ip′-wave superconducting state
+We now check whether our model would support helical
+Majorana zero modes at the edge of a finite-sized system.
+We shall examine our model’s band structures and edge-state
+wave functions on a finite-sized strip that extends infinitely
+along the x direction but contains a finite number of lattice
+sites in y. The results are shown in Figs. 4 to 6. As shown
+
+0.5
+E(k)
+-0.5
+π
+一π
+0
+0
+π一
+2
+2
+Ky
+kx
+π－
+2
+26
+E
+E
+x
+y
+yi = 1
+yi = Ny
+Ribbon
+γRA,k
+γL
+B,k
+γR
+B,k
+γL
+A,k
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+FIG. 5. Figures (a) and (d) show the Bogoliubov excitation spectra of HA and HB, respectively, near the chemical potential on a nano-strip with
+Ny = 81 chains. Figures (b), (c) and (e), (f) demonstrate the probability density of the Majorana edge state wave functions of HA and HB as
+a function of atom position yi,
+��γΓ
+I,kx(yi)
+��2 with I = A, B and Γ = R, L (pink curves in (a) and (d)), at a fixed energy E ≡ E(kx = ±0.03).
+The probability density is described by
+��γΓ
+I,kx(yi)
+��2 =
+���uΓ
+I,kx
+��2 ,
+��¯uΓ
+I,kx
+��2 ,
+��vΓ
+I,kx
+��2 ,
+��¯vΓ
+I,kx
+��2�
+(yi). The parameters are JK/t = 0.3,
+JH/t = 1.0, and doping δ = −0.3. The edge states are of the helical type, as schematically represented in (g).
+E
+E
+1
+2
+4
+3
+1
+3
+2
+4
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+FIG. 6. The finite-energy (E(kx) > 0) Bogoliubov excitation spectra of (a) HA (shown on top right in Fig. 5) and (d) HB. A pair of “helical”
+edge states is found to exist at finite energy [pink curve in (a) and (d)], and their probability densities are shown in (b) and (c), (e) and (f),
+respectively, at a fixed energy E(kx = ±0.22).
+in Fig. 4, gapless Dirac spectra of the Bogoliubov excitations
+around kx = 0 near zero energy are observed, exhibiting one
+of the typical features of topological edge states. The exci-
+tations can be effectively described by the linear-dispersed
+Hamiltonian ˜HI
+= �
+kx vx|kx|
+�
+γR †
+I,kxγR
+I,kx − γL †
+I,kxγL
+I,kx
+�
+with
+γΓ
+I,kx =
+�
+yi
+�
+uΓ
+I,kx(yi)ckx,yi,↑ + ¯uΓ
+I,kx(yi)c†
+−kx,yi,↑
++vΓ
+I,kx(yi)fkx,yi,↓ + ¯vΓ
+I,kx(yi)f †
+−kx,yi,↓
+�
+(19)
+with u, ¯u and v, ¯v being the coherent factors. In Eq. (19),
+I ∈ A, B, Γ ∈ R, L, and γR/L
+A/B,kx represents the right/left-
+moving Bogoliubov quasiparticle of ˜HA/B. Here, vx in ˜HI
+denotes the velocity.
+Due to time-reversal symmetry, HA
+is the time-reversal partner of HB, and thus their spectra
+are identical. The low-energy eigenstates with Dirac spectra
+near kx = 0 for both HA and HB exhibit the typical prop-
+erty of edge states, as their probability densities accumulate
+mostly at the edges of strip, as shown in Fig. 5. Combin-
+ing the directions of propagation inferred from the velocity
+vx ∼ ∂E(kx)/∂kx, we can classify these edge states into two
+
+7
+Tc
+Tonset
+Tc
+FIG. 7.
+Plot of the temperature-dependent mean-field order pa-
+rameters x(T)/t and ∆t(T)/t with kB = 1, JK/t = 0.3 and
+JH/t = 1.0 fixed.
+Inset shows the enlarged plot of ∆t(T).
+The single-impurity Kondo temperature occurs at Tonset/t ≈ 0.16
+while the transition of superconductivity takes places at temperature
+Tc/t ≈ 0.015.
+groups, each of them constitutes a pair of counter-propagating
+edge states (see Fig. 5), revealing the nature of helical Ma-
+jorana zero modes. The helical type of the Majorana zero
+modes is the consequence of time-reversal symmetry of our
+model, reminiscent of the well-known Kane-Mele model on a
+single-layered graphene [38, 39]. Remarkably, in addition to
+the Majorana fermions at zero energy, two pairs of counter-
+propagating edge-states are observed at finite energy, see Fig.
+6. The two pairs of edge states correspond to the edge states
+of the topological Kondo insulator, where the spin-triplet RVB
+order parameter is absent (∆t = 0) [6–8].
+V.
+DISCUSSIONS AND CONCLUSIONS
+We now discuss the application of our results for heavy-
+electron superconductors, particularly the Kondo lattice com-
+pound UTe2. Experimental evidence indicates that this com-
+pound does not show long-range magnetic order and is in
+the vicinity of the ferromagnetic quantum critical point, ex-
+hibiting both strong ferromagnetic fluctuations, possibly due
+to magnetic frustrations induced by sub-leading antiferromag-
+netic fluctuations [40, 41], and Kondo screening [14, 17, 42].
+The DFT+U calculations indicate that the dynamics of elec-
+tron bands and the physical properties of UTe2 are dominated
+by the electrons near the quasi-two-dimensional (cylindrical)
+Fermi surface with weak kz dependence despite its 3D crystal
+structure [40]. Superconductivity is reached at Tc = 1.6K,
+while the resistivity maximum observed at T ⋆ ≈ 15 ∼ 75 K
+reveals signature of coherent Kondo scattering [14, 43], indi-
+cating T ⋆/Tc ≈ 10 ∼ 50. The superconductivity can, in gen-
+eral, co-exist and compete with the Kondo effect [17]. When
+a magnetic field is applied along the hard-magnetic axis b of
+UTe2 and before entering the superconducting phase, a corre-
+lated paramagnetic phase is observed below the temperature
+at which the magnetic susceptibility shows a broad maximum
+[44]. Similar spin-liquid behavior has been observed in the
+magnetic susceptibility of another heavy fermion compound
+CePdAl [45]. This similarity suggests this correlated para-
+magnetic phase may feature short-range magnetic order. Our
+theoretical framework based on competition and collaboration
+between a Kondo-screened and a ferromagnetic t-RVB spin-
+liquid states on a two-dimensional Kondo lattice is consistent
+with the above observations in UTe2. It, therefore, consti-
+tutes a promising approach to account for its exotic phenom-
+ena. On the other hand, the chiral in-gap state, a signature of
+chiral topological superconductor, has been observed by scan-
+ning tunneling spectroscopy in the superconducting phase of
+UTe2 [17]. Combining with the ferromagnetic fluctuations
+that are known to induce spin-triplet pairing, people believe
+UTe2 is a promising candidate for the spin-triplet chiral topo-
+logical superconductor [14, 17]. Furthermore, the supercon-
+ducting phase co-existing with Kondo coherence in this ma-
+terial strongly suggests the role played by the Kondo effect
+in this possible topological superconductor.
+The topologi-
+cal Kondo superconducting state with equal-spin spin-triplet
+p-wave pairings we proposed here bears striking similarities
+to and strong relevance for the experimental observations on
+UTe2: (i) the d- and f-orbitals electrons with their angular
+momentum quantum number differing by 1 in the uranium
+atoms of UTe2 likely give rise to the odd-parity Kondo effect
+[6–8], (ii) the t-RVB state in our theory may be considered
+as one possible realization of the short-ranged ferromagnetic
+fluctuations in UTe2, (iii) the Kondo-t-RVB co-existing su-
+perconducting state we find here qualitatively agrees with the
+co-existence between superconductivity and Kondo effect ob-
+served in UTe2, (iv) the high upper critical field exceeding the
+Pauli limit [14, 46] implies that the superconducting state of
+UTe2 may have equal-spin Cooper pairs, and (v) the effective
+pairing ∆
+σ
+k formed in the conduction band mentioned in Sec-
+tion IV A shows characteristics of spin-triplet point-node gap
+structure [47]. Various characteristic temperature scales esti-
+mated from our mean-field calculations with JH/t = 1.0 and
+JK/t = 0.3 at finite temperatures agree reasonably well with
+experimental observations (see Fig. 7): The superconducting
+transition temperature Tc, theoretically determined from our
+mean-field analysis Tc = Min[T(x = 0), T(∆t = 0)], shows
+Tc ≈ 0.015t ≈ 2.3 K by taking estimated values of t = 150 K
+and half-bandwidth D = 1.25t [42]. The Kondo coherent
+scale can be obtained by T ⋆ = x2(T = 0)/D ≈ 17.4 K
+[48]. The ratio T ⋆/Tc ≈ 8 is in reasonable agreement with
+experimental observations. The onset temperature Tonset of
+Kondo hybridization, which occurs at x(T = Tonset) = 0,
+displays Tonset ≈ 0.16t ≈ 24 K, within the theoretically es-
+timated range 10K < Tonset < 100K by DMFT calculation
+[42]. Meanwhile, there have been evidences of TRS breaking
+in UTe2 from the observed two superconducting transitions
+and a finite polar Kerr effect at T < Tc [49], likely due to
+proximity to the ferromagnetic ordered phase. A number of
+theoretical attempts were proposed based on these observa-
+tions [50, 51]. However, the observed single superconducting
+transition near ambient pressure and zero field [44, 52, 53] as
+well as the theoretically proposed unitary triplet pairing [40]
+
+8
+suggest TRS may be preserved in UTe2. Though our results
+shown above are obtained in the presence of TRS, the chi-
+ral p-wave superconducting state with chiral Majorana zero
+mode at edges is expected to occur here once a time-reversal
+breaking magnetic field is applied [54]. Our distinct predic-
+tions with and without fields serve as theoretical guidance for
+future experiments to distinguish the time-reversal breaking
+from time-reversal preserving triplet pairing states in UTe2.
+Since the Kondo correlations stabilize the t-RVB spin liquid
+in the co-existing superconducting phase, it is expected to be
+robust against gauge-field fluctuations beyond the mean field.
+Our approach and results are distinct from the spin-triplet non-
+topological superconducting state recently proposed based on
+the Hund’s-Kondo coupling and Sz = 0 t-RVB state to ac-
+count for UTe2 [51].
+In conclusion, we propose a first realization of the topo-
+logical superconductivity in the Kondo lattice model, a dis-
+tinct class of topological superconductors due to purely strong
+electron correlations without employing spin-orbit coupling
+or proximity effect.
+A topological Kondo superconductor
+essentially constitutes of 1) itinerant c and localized f bands
+with different orbital quantum numbers, 2) strong Hubbard in-
+teraction of the f electrons, 3) odd-parity Kondo hybridization
+of the c and f bands, and 4) the attractive exchange interac-
+tion of the f electrons with spin-triplet correlations. Start-
+ing from the odd-parity Anderson lattice model, we obtain
+the unconventional type of Kondo hybridization and ferro-
+magnetic RKKY-like interaction via perturbation theory, lead-
+ing to spin-triplet resonating-valence-bond (RVB) pairing be-
+tween f-electrons with time-reversal invariant p ± ip′-wave
+gap symmetry. Via the mean-field approach, we find a Kondo
+triplet-RVB coexisting phase in the intermediate range of the
+Kondo to RKKY coupling ratio. This phase is shown as a
+time-reversal invariant topological superconducting state with
+a spin-triplet p ± ip′-wave RVB pairing gap. It exhibits non-
+trivial topology in the bulk band structure, and supports heli-
+cal Majorana zero modes at edges. Our prediction in the pres-
+ence of a time-reversal breaking field leads to chiral p-wave
+spin-triplet topological Kondo superconductor. Our results on
+the superconducting transition temperature, Kondo coherent
+scale, and onset temperature of Kondo hybridization not only
+qualitatively but also quantitatively agree with the observa-
+tions for UTe2. The theoretical framework we propose here
+opens up the search for topological superconductors induced
+by strongly electronic correlations on the Kondo lattice com-
+pounds.
+VI.
+ACKNOWLEDGEMENTS
+This work is supported by the Ministry of Science
+and Technology Grants 104-2112-M-009-004-MY3 and 107-
+2112-M-009-010-MY3, the National Center for Theoretical
+Sciences of Taiwan, Republic of China (to C.-H. C.).
+Appendix A: The Schrieffer-Wolff transformation (SWT)
+In this section, we provide derivations of the Kondo term
+via using the SWT. We first perform the SWT on an odd-parity
+single-impurity Anderson model where an impurity at an ar-
+bitrary site i hybridizes with the conduction electrons on the
+four nearest-neighbor sites of i. This result will be succes-
+sively generalized to the lattice version.
+The single-impurity Anderson model takes the following
+form
+H =
+�
+kσ
+εkc†
+kσckσ +
+�
+σ
+εff †
+iσfiσ + Unf
+i↑nf
+i↓
++
+�
+σσ′
+�
+α=x,y
+�
+iV νˆασσσ′
+α
+c†
+i+ˆα,σfiσ′ + H.c.
+�
+,
+(A1)
+where ˆα ≡ ±ˆx, ±ˆy denotes the nearest-neighbor vectors of a
+square lattice, and νˆα satisfies νˆα = −ν−ˆα and νˆx = νˆy = 1.
+The SWT aims at projecting out the empty and doubly oc-
+cupied states to generate the effective Hamiltonian Heff in
+the Kondo (singly-occupied) limit. Following Ref. [20], we
+first use the states of impurity occupation as the basis set,
+{|f 0⟩, |f 1⟩, |f 2⟩} with the superscripts being denoted as the
+occupation of the localized electrons, to expand the Hamilto-
+nian of Eq. (A1) in the following matrix form,
+H =
+�
+�
+H00 H01 H02
+H10 H11 H12
+H20 H21 H22
+�
+� .
+(A2)
+The matrix elements of Eq.
+(A2), denoted as Hij
+≡
+⟨f i|H|f j⟩ with i, j = 0, 1, 2, are
+H10 =
+�
+σσ′
+�
+α=±x,±y
+iV νˆασσσ′
+α
+f †
+iσci−ˆα,σ′ = H21,
+H01 = H†
+10 =
+�
+σσ′
+�
+α=±x,±y
+iV νˆασσσ′
+α
+c†
+i+ˆα,σfiσ′ = H12,
+H11 =
+�
+kσ
+εkc†
+kσckσ +
+�
+σ
+εff †
+iσfiσ, H00 =
+�
+kσ
+εkc†
+kσckσ,
+H22 =
+�
+kσ
+εkc†
+kσckσ +
+�
+σ
+εff †
+jσfjσ + Unf
+i↑nf
+i↓.
+(A3)
+We then project out |f 0⟩ and |f 2⟩ from the Hilbert space to
+obtain the effective Hamiltonian Heff at the Kondo limit sat-
+isfying Heff|f 1⟩ = E|f 1⟩ with E being the eigenenergy. Via
+Eq. (A2), Heff can be expressed as Heff = H11 + H′ with
+
+9
+H′ =H10(E − H00)−1H01 + H12(E − H22)−1H21
+=
+�
+α,α′=x,y
+�
+σσ′
+�
+σ′′σ′′′
+�
+V 2
+εF − εf − U
+�
+iνˆασσσ′
+α
+c†
+i+ˆα,σfiσ′
+� �
+iνˆα′σσ′′σ′′′
+α′
+f †
+iσ′′ci−ˆα′,σ′′′
+�
+(A4)
++
+V 2
+εf − εF
+�
+iνˆασσσ′
+α
+f †
+iσci−ˆα,σ′
+� �
+iνˆα′σσ′′σ′′′
+α′
+c†
+i+ˆα′,σ′′fiσ′′′
+��
+(A5)
+Here, we skip the derivations of H10(E − H00)−1H01 and
+H12(E − H22)−1H21 in Eq. (A5) as those are standard and
+can be found in a number of references. See, for example,
+Refs. [19, 20]. H′ can be further cast into the form simi-
+lar to the conventional single-impurity Kondo term, with the
+following antiferromagnetic Kondo coupling
+JK =
+V 2
+U + εf − εF
++
+V 2
+εF − εf
+> 0,
+(A6)
+plus a potential scattering term. Eq. (A5) can be generalized
+to the lattice version by summing over all lattice sites, as de-
+scribed by Eq. (5).
+Appendix B: Derivation of the effective ferromagnetic
+RKKY-like interaction
+In the section, we derive the RKKY-like interaction by per-
+turbatively expanding HK of Eq. (5) to second order.
+The unperturbed state is described as
+|0, f⟩ = |k1m1, k2m2, · · · , kNmN⟩ |f⟩ ,
+(B1)
+where conduction electrons do not interact with the impuri-
+ties. In Eq. (B1), |k1m1, k2m2, · · · , kNmN⟩ represents the
+Fermi sea with all wave vectors lying below the Fermi wave
+vector, namely ki < kF . After imposing perturbation, the un-
+perturbed state acquires correction and the corrected eigenen-
+ergy is expressed in powers of JK, E = E0 + ∆E(1) +
+∆E(2) + O(J3
+K) with E0 being the eigenenergy of the un-
+perturbed state.
+The first and second order energy corrections take the form
+∆E(1) = ⟨0, f| HK |0, f⟩ ,
+∆E(2) =
+�
+(0,f)̸=(A,f ′)
+|⟨0, f| HK |A, f ′⟩|2
+E0 − EA
+,
+(B2)
+where |A, f ′⟩ denotes the excited state which can be expressed
+as a direct product of the building blocks |k′′
+i , m′′
+i ⟩, with part
+of wave vectors lying above the Fermi surface, i.e. k′′
+i > kF .
+Here, we first derive the effective interaction of the f
+fermions for a simpler two-impurity model and generalize the
+results to the lattice version.
+∆E(1) can be evaluated by summing over the subspace of
+the conduction electron, yielding
+∆E(1) = ⟨0, f| HK |0, f⟩
+= 4nfJK
+Ns
+�
+k<kF
+�
+sin2 kx + sin2 ky
+�
++ C,
+(B3)
+where nf = �
+i=1,2,σ ⟨f| f †
+iσfiσ |f⟩ and C is a constant. HK
+in Eq. (B3) denotes the two-impurity Kondo term. It turns
+out that ∆E(1) only introduces a constant energy shift for the
+bare energy level of the f fermions.
+∆E(2) is given by
+∆E(2) = 1
+N 4s
+�
+f ′′
+�
+k′′
+1
+�
+m′′
+1
+· · ·
+�
+k′′
+N
+�
+m′′
+N
+�
+J2
+K
+E0 − EA
+�
+2
+�
+i=1
+�
+σσ′
+�
+σ′′σ′′′
+�
+α,α′
+�
+k,k′
+�
+�
+2
+�
+j=1
+�
+ττ ′
+�
+τ ′′τ ′′′
+�
+β,β′
+�
+q,q′
+�
+�
+×
+�
+eik·(ri+ˆα)−ik′·(ri−ˆα′)iνˆαiνˆα′
+� �
+eiq·(rj+ ˆβ)−iq′·(rj− ˆβ′)iν ˆβiν ˆβ′
+�
+× σσσ′
+α
+σσ′′σ′′′
+α′
+⟨f| fiσ′f †
+iσ′′ |f ′′⟩ ⟨k1m1, k2m2, · · · , kNmN|
+�
+c†
+kσck′σ′′′
+�
+|k′′
+1m′′
+1, k′′
+2m′′
+2, · · · , k′′
+Nm′′
+N⟩
+× σττ ′
+β
+στ ′′τ ′′′
+β′
+⟨f ′′| fjτ ′f †
+jτ ′′ |f⟩ ⟨k′′
+1m′′
+1, k′′
+2m′′
+2, · · · , k′′
+Nm′′
+N|
+�
+c†
+qτcq′τ ′′′�
+|k1m1, k2m2, · · · , kNmN⟩ .
+(B4)
+An annihilation operator acts on |A, f ′⟩ can be obtained as
+cqσ′ |k′′
+1m′′
+1, k′′
+2m′′
+2, · · · , k′′
+Nm′′
+N⟩
+=
+N
+�
+α=1
+(−1)pαδq,k′′
+αcσ′
+�����
+�α−1
+�
+l=1
+k′′
+l m′′
+l
+�
+m′′
+α
+�
+N
+�
+l=α+1
+k′′
+l m′′
+l
+��
+,
+(B5)
+
+10
+we can thus obtain
+⟨k1m1, k2m2, · · · , kNmN|
+�
+c†
+kσck′σ′′′
+�
+|k′′
+1m′′
+1, k′′
+2m′′
+2, · · · , k′′
+Nm′′
+N⟩
+=
+N
+�
+α=1
+N
+�
+β=1
+(−1)pα(−1)pβδk,kαδk′,k′′
+β ⟨mα| c†
+σcσ′′′ ��m′′
+β
+�
+��α−1
+�
+l=1
+klml
+� �
+N
+�
+l=α+1
+klml
+������
+�β−1
+�
+l=1
+k′′
+l m′′
+l
+� �
+�
+N
+�
+l=β+1
+k′′
+l m′′
+l
+�
+�
+�
+.
+(B6)
+The above matrix element is nonzero only if the momentum
+is restricted by certain constraints and (ki, mi) = (k′′
+i , m′′
+i ),
+signifying pα = pβ:
+⟨k1m1, · · · , kNmN|
+�
+c†
+kσck′σ′′′
+�
+|k′′
+1m′′
+1, · · · , k′′
+Nm′′
+N⟩
+=Θ(kF − |k|)Θ (|k′| − kF )
+×
+N
+�
+α=1
+�
+�δk,kαδk′,k′′
+α ⟨mα| c†
+σcσ′′′ |m′′
+α⟩
+�
+l̸=α
+δklk′′
+l δmlm′′
+l
+�
+�
+(B7)
+Plugging this into ∆E(2), we have
+∆E(2) = 1
+N 4s
+�
+f ′′
+N
+�
+a=1
+�
+k′′
+a
+�
+m′′
+a
+�
+J2
+K
+E0 − EA
+�
+2
+�
+i=1
+�
+σσ′
+�
+σ′′σ′′′
+�
+α,α′
+�
+�
+2
+�
+j=1
+�
+ττ ′
+�
+τ ′′τ ′′′
+�
+β,β′
+�
+q,q′
+�
+�
+× Θ(kF − |ka|)Θ (|k′′
+a| − kF )
+�
+eika·(ri+ˆα)−ik′′
+a ·(ri−ˆα′)iνˆαiνˆα′
+� �
+eiq·(rj+ ˆβ)−iq′·(rj− ˆβ′)iν ˆβiν ˆβ′
+�
+× σσσ′
+α
+σσ′′σ′′′
+α′
+σττ ′
+β
+στ ′′τ ′′′
+β′
+⟨f| fiσ′f †
+iσ′′ |f ′′⟩ ⟨f ′′| fjτ ′f †
+jτ ′′ |f⟩ ⟨ma| c†
+σcσ′′′ |m′′
+a⟩
+×
+��a−1
+�
+l=1
+klml
+�
+k′′
+am′′
+a
+�
+N
+�
+l=a+1
+klml
+������
+�
+c†
+qτcq′τ ′′′�
+|k1m1, k2m2, · · · , kNmN⟩
+(B8)
+The matrix element of c†
+qτcq′τ ′′′ in the fourth line of Eq. (B8)
+can be evaluated as
+��a−1
+�
+l=1
+klml
+�
+k′′
+am′′
+a
+�
+N
+�
+l=a+1
+klml
+������
+�
+c†
+qτcq′τ ′′′�
+|k1m1, k2m2, · · · , kNmN⟩ = δq,k′′
+a δq′,kα⟨m′′
+a|c†
+τcτ ′′′|ma⟩.
+(B9)
+Hence, the energy correction ∆E(2) can be further simplified
+as (sum over f ′′, m′′
+a, q, q′ and suppress the subscript a below)
+∆E(2) = 1
+N 4s
+2
+��
+i,j=1
+�
+α,α′
+�
+εk<µ
+�
+εk′′>µ
+�
+m,τ=±
+�
+β,β′
+�
+J2
+K
+εk − εk′′
+� �
+iνˆαiνˆα′iν ˆβiν ˆβ′
+�
+× eik·(ri+ˆα)−ik′′·(ri−ˆα′)eik′′·(rj+ ˆβ)−ik·(rj− ˆβ′)σm,−m
+α
+σ−τ,τ
+α′
+στ,−τ
+β
+σ−m,m
+β′
+⟨f| fi,−mf †
+i,−τfj,−τf †
+j,−m |f⟩
+(B10)
+
+11
+The effective interacting term among the f fermions can be
+obtained by removing the bracket ⟨f| · · · |f⟩. This result can
+be simply generalized to the lattice version by extending the
+summation of i and j over the entire lattice, as shown in Eqs.
+(6) and (7).
+Appendix C: The mean-field Kondo-Heisenberg Hamiltonian on
+a strip
+In this section, we provide the details of the matrix elements
+of the Kondo-Heisenberg Hamiltonian on a nano-strip with
+Ny chains along y-axis. We choose the basis of the Kondo-
+Heisenberg strip as
+φA,k =
+�
+ck1↑, ck2↑, · · · , ckNy↑, c†
+−k1↑, c†
+−k2↑, · · · , c†
+−kNy↑, fk1↓, fk2↓, · · · , fkNy↓, f †
+−k1↓, f †
+−k2↓, · · · , f †
+−kNy↓
+�T
+,
+φB,k =
+�
+ck1↓, ck2↓, · · · , ckNy↓, c†
+−k1↓, c†
+−k2↓, · · · , c†
+−kNy↓, fk1↑, fk2↑, · · · , fkNy↑, f †
+−k1↑, f †
+−k2↑, · · · , f †
+−kNy↑
+�T
+,
+(C1)
+where we take kx → k. The total Hamiltonian H is repre-
+sented as a summation of two decoupled Hamiltonians, HA
+and HB, each of which is 4Ny × 4Ny in size, given by
+H =
+�
+k
+φ†
+A,kHA(k)φA,k +
+�
+k
+φ†
+B,kHB(k)φB,k.
+(C2)
+Below, we provides the matrix elements of HA(k) and HB,
+respectively:
+1.
+HA
+The matrix elements of the hopping term for HA are
+HA(yi, yi) = −t cos k − µ
+2 ,
+HA(yi + Ny, yi + Ny) = t cos k + µ
+2
+(C3)
+for yi = 1, 2, · · · , Ny while
+HA(yi, yi + 1) = − t
+2,
+HA(yi + 1, yi) = − t
+2,
+HA(Ny + yi + 1, Ny + yi) = t
+2,
+HA(Ny + yi, Ny + yi + 1) = t
+2
+(C4)
+for yi = 1, 2, · · · , Ny − 1.
+For Hf, we have for yi = 1, 2, · · · , Ny
+HA(2Ny + yi, 2Ny + yi) = λ/2,
+HA(3Ny + yi, 3Ny + yi) = −λ/2.
+(C5)
+The Kondo term HK for HA describes the Kondo interaction
+with the following matrix form: the Kondo hybridization of c
+and f with the same y chain are
+HA(2Ny + yi, yi) = x sin k,
+HA(Ny + yi, 3Ny + yi) = x sin k,
+HA(yi, yi + 2Ny) = x sin k,
+HA(3Ny + yi, Ny + yi) = x sin k
+(C6)
+for yi = 1, · · · , Ny. The matrix elements of the Kondo term
+for yi = 1, · · · , Ny − 1 are
+HA(2Ny + yi + 1, yi) = −x
+2 ,
+HA(Ny + yi, 3Ny + yi + 1) = x
+2 ,
+HA(2Ny + yi, yi + 1) = x
+2 ,
+HA(Ny + yi + 1, 3Ny + yi) = −x
+2 ,
+HA(yi, 2Ny + yi + 1) = −x
+2 ,
+HA(3Ny + yi + 1, Ny + yi) = x
+2 ,
+HA(yi + 1, 2Ny + yi) = x
+2 ,
+HA(3Ny + yi, Ny + yi + 1) = −x
+2 ,
+(C7)
+which corresponds to the hybridization of c and f with the
+nearest-neighboring y chains.
+The RVB pairing term HJ on a nano-strip is described by
+the following matrix elements: for yi = 1, · · · , Ny,
+HA
+∆(2Ny + i, 3Ny + i) = −i∆t sin k,
+HA
+∆(3Ny + i, 2Ny + i) = i∆t sin k
+(C8)
+are the matrix elements for the pairing of spinons with the
+same yi. For yi = 1, · · · , Ny − 1, we have
+HA(2Ny + yi, 3Ny + yi + 1) = − i
+2∆t,
+HA(2Ny + yi + 1, 3Ny + yi) = i
+2∆t.
+HA(3Ny + yi + 1, 2Ny + yi) = i
+2∆t,
+HA(3Ny + yi, 2Ny + yi + 1) = − i
+2∆t.
+(C9)
+
+12
+2.
+HB
+The matrix elements for the hopping term in HB are
+HB(yi, yi) = −t cos k − µ
+2 ,
+HB(yi + Ny, yi + Ny) = t cos k + µ
+2
+(C10)
+for yi = 1, 2, · · · , Ny. While, for for yi = 1, 2, · · · , Ny − 1,
+we obtain
+HB(yi, yi + 1) = − t
+2,
+HB(yi + 1, yi) = − t
+2,
+HB(Ny + yi + 1, Ny + yi) = t
+2,
+HB(Ny + yi, Ny + yi + 1) = t
+2.
+(C11)
+The matrix elements for Hf are
+HB(2Ny + yi, 2Ny + yi) = λ/2,
+HB(3Ny + yi, 3Ny + yi) = −λ/2
+(C12)
+with yi = 1, 2, · · · , Ny.
+The matrix elements of the Kondo term for c and f lying
+on the same chain yi are
+HB(2Ny + yi, yi) = x sin k,
+HB(Ny + yi, 3Ny + yi) = x sin k,
+HB(yi, 2Ny + yi) = x sin k,
+HB(3Ny + yi, Ny + yi) = x sin k,
+(C13)
+where yi = 1, · · · , Ny. For Kondo term where the hybridiza-
+tion is happening between nearest-neighboring chains, we
+have
+HB(2Ny + yi + 1, yi) = x
+2 ,
+HB(Ny + yi, 3Ny + yi + 1) = −x
+2
+HB(2Ny + yi, yi + 1) = −x
+2 ,
+HB(Ny + yi + 1, 3Ny + yi) = x
+2 ,
+HB(yi, 2Ny + yi + 1) = x
+2
+HB(3Ny + yi + 1, Ny + yi) = −x
+2 ,
+HB(yi + 1, 2Ny + yi) = −x
+2 ,
+HB(3Ny + yi, Ny + yi + 1) = x
+2
+(C14)
+for yi = 1, · · · , Ny − 1.
+The matrix elements for the RVB spinon-pairing term are
+HB(2Ny + yi, 3Ny + yi) = −i∆t sin k,
+HB(3Ny + yi, 2Ny + yi) = i∆t sin k
+(C15)
+for yi = 1, · · · , Ny, and
+HB(2Ny + yi, 3Ny + yi + 1) = i
+2∆t,
+HB(2Ny + yi + 1, 3Ny + yi) = − i
+2∆t,
+HB(3Ny + yi + 1, 2Ny + yi) = − i
+2∆t,
+HB(3Ny + yi, 2Ny + yi + 1) = i
+2∆t
+(C16)
+for yi = 1, · · · , Ny − 1.
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@@ -0,0 +1,1141 @@
+Transition from chemisorption to physisorption of H2 on Ti 
+functionalized [2,2,2]paracyclophane: A computational 
+search for hydrogen storage. 
+Rakesh K. Sahoo, Sridhar Sahu* 
+ 
+Computational Materials Research Lab, Department of Physics, Indian Institute of Technology 
+(Indian School of Mines) Dhanbad, India 
+Abstract 
+In this work, we studied the hydrogen adsorption-desorption properties and storage capacities of Ti 
+functionalized [2,2,2]paracyclophane (PCP222) using density functional theory and molecular dynamic 
+simulation. The Ti atom was bonded strongly with the benzene ring of PCP222 via Dewar interaction. 
+Subsequently, the calculation of the diffusion energy barrier revealed a significantly high energy barrier of 
+5.97 eV preventing the Ti clustering over PCP222 surface. On adsorption of hydrogen, the first H2 molecule 
+was chemisorbed over PCP222 with a binding energy of 1.79 eV with the Ti metals. Further addition of 
+H2 molecules, however, exhibited their physisorption over PCP222-Ti through the Kubas-type 
+H2 interaction. Charge transfer mechanism during the hydrogen adsorption was explored by the Hirshfeld 
+charge analysis and electrostatic potential map, and the PDOS, Bader’s topological analysis revealed the 
+nature of the interaction between Ti and H2. The PCP222 functionalized with three Ti atoms showed a 
+maximum hydrogen uptake capacity of up to 7.37 wt%, which was fairly above the US-DOE criterion. The 
+practical H2 storage estimation revealed that at ambient conditions, the gravimetric density of up to 6.06 
+wt% H2 molecules could be usable, and up to 1.31 wt% of adsorbed H2 molecules were retained with the 
+host. The ADMP molecular dynamics simulations assured the reversibility by desorption of adsorbed 
+H2 and the structural integrity of the host material at sufficiently above the desorption temperature (300K 
+and 500K). Therefore, the Ti-functionalized PCP222 can be considered as a thermodynamically viable and 
+potentially reversible H2 storage material. 
+Keywords: Hydrogen storage, DFT, ADMP, [2,2,2]paracyclophane, PCP222, ESP, 
+Chemisorption, Physisorption 
+1 Introduction 
+Extensive use of fossil fuels not only results in the depletion of those energy resources but also 
+leads the world towards an alarming environmental catastrophe in terms of pollution and global 
+
+warming. These consequences have motivated researchers across the globe to search for alternative 
+sustainable and environment-friendly energy resources. Therefore, hydrogen drew the attention 
+because it is considered as an ideal, pollution-free, and sustainable energy carrier, which can 
+replace fossil fuels by fulfilling the energy need of the world, and thus can resolve the pollution 
+due to fossil fuels[1, 2]. However, the major difficulty in hydrogen energy as fuel for domestic and 
+vehicular application is its efficient storage and delivery at ambient conditions. Hydrogen can be 
+stored mainly in two ways: system-based and material-based. System-based storage methods 
+which is being adopted by few industries require huge volume vessels which should be made of 
+composite material to withstand high pressure (~70 MPa) making the process quite expensive. 
+However, compressed hydrogen storage systems are reported to have low volumetric densities, 
+even at high pressure [3], and hydrogen storage in liquid state requires a very low temperature (~ 
+-253oC) under high pressure (~ 250-350 atm) which is highly prone to safety concerns. On the 
+other hand, the solid-state material-based hydrogen storage method is substantiated as efficient 
+alternative to use hydrogen energy provided it adsorbs and desorb a desirable amount of H2 at 
+ambient conditions [4]. In solid-state materials, hydrogen is usually adsorbed by the physisorption 
+or chemisorption process. In the physisorption process, the adsorbed hydrogen binds in molecular 
+to the surface of host materials through weak interaction (adsorption energy ~ 0.1-0.8 eV/H2). 
+However, in the chemisorption process, the H2 molecules dissociate into individual H atoms and 
+migrate to the host materials by producing a strong chemical bond (with a binding energy of >1 
+eV/H2) with the host atoms. Another type of adsorption process observed is similar to the 
+physisorption, in which the inter-atomic H-H bond in the H2 molecule is elongated but not 
+dissociated and adsorbed by Kubas-type orbital interactions[2]. It enhances the H2 adsorption 
+energy and makes most of the H2 storage capacities that fulfil the target of the US department of 
+energy (DOE-US) [5, 6]. 
+Since last few years, researchers are engaged extensively to study various materials, including 
+carbon nanostructures [7, 8], metal hydrides [9, 10], graphene [11, 12], metal alloys[13, 14], 
+metal-organic frameworks (MOF)[15, 16], and covalent-organic frameworks [17], etc. for the 
+reversible hydrogen storage at ambient condition. However, it has been reported that these 
+materials often have several limitations, including poor storage capacity, instability at significantly 
+high temperatures, and low reversibility at normal temperatures. For example, Mg-based metal 
+hydrides showed a high storage capacity of up to 7.6 wt% under ambient condition, however; it 
+
+could be used only for 2-3 cycles [18].Similarly, metal alloys have very poor reversibility when 
+used as hydrogen storage materials [19]. Using MOFs as H2 storage materials, researchers could 
+attain up to 15 wt% of storage capacity at temperatures and pressures of 77 K and 80 bar. However, 
+under normal environmental condition its gravimetric and volumetric storage capacity remained 
+very low [20]. To address the aforesaid issue and to develop commercially effective hydrogen 
+storage materials, the experimentally synthesized organic compounds functionalized with 
+transition metals (TMs), such as TM-doped organometallic buckyballs, TM-ethylene, etc., were 
+introduced and investigated extensively [21, 22]. Early reports show that the TM atoms form a 
+strong bond with the π -electron delocalized compounds through the Dewar mechanism and adsorb 
+hydrogen molecules via Kubas interaction[23, 24]. For example, Chakraborty et al. studied the 
+hydrogen storage in Ti-doped Ψ-graphene and reported an H2 uptake capacity of up to 13.1 wt% 
+with an average adsorption energy of -0.30 eV/H2 [25]. Dewangan et al. predicted up to 10.52 wt% 
+of H2 adsorption in Ti-functionalized holey graphyne via the Kubas mechanism with adsorption 
+energy and desorption temperature of 0.38 eV/H2 and 486 K, respectively[26]. 
+Numerous theoretical and experimental studies revealed that metal-adorned small organic 
+molecules like CnHn could capture a large number of H2 molecules. For example, Zhou et al. 
+estimated hydrogen uptake capacity up to 12 wt% in TiC2H4 with H2 binding energy of 0.24 
+eV/H2[27, 28]. High capacities of H2 storage in TMC2H4 (M = Ti, Sc, V, Ni, Ce, Nb) complexes 
+was reported by Chaudhari et al.[29, 30, 31]. At low benzene pressure (35 millitorrs) and ambient 
+temperature, TiC6H6 was experimentally shown to absorb up to 6 wt% hydrogens [32]. Phillips et 
+al. obtained an H2 uptake of up to 14 wt% and quick kinetics at room temperature on TiC6H6 by 
+laser ablation; however, the experiments did not discuss the desorption process[33]. Recently, 
+Ma et al. theoretically studied an interesting combination of chemisorption and physisorption in 
+Ti-doped C6H6 and reported an uptake capacity of 6.02 wt % with complete desorption at 935 
+K [34]. Mahamiya et al. revealed the H2 storage capacities of 11.9 wt % in K and Ca decorated 
+biphenylene with an average adsorption energy of 0.24-0.33 eV [35]. Y atom doped zeolite showed 
+high capacity adsorption of H2 with binding energy 0.35 eV/H2 and the desorption energy of 437K 
+for fuel cells[36]. 
+Macrocyclic compounds, like paracyclophane (PCP), a subgroup derivative of cyclophanes, 
+comprises aromatic benzene rings with number of -CH2- moieties linking the subsequent benzene 
+
+rings [37]. The PCPs are easier to synthesize in the laboratory; they can be functionalized with 
+metal atoms due to the presence of aromatic benzene rings in the geometry, making them a feasible 
+alternative for hydrogen storage prospects. For instance, Sathe et al. studied the Sc and Li 
+decorated PCP and reported the molecular H2 physisorbed via Kubas-Niu-Jena interaction 
+resulting in up to 10.3 wt% H2 uptake capacity [38]. The hydrogen storage transition metal (Sc, 
+Y) functionalized [1,1]paracyclophane was investigated by Sahoo et al. and reported a storage 
+capacity of 6.33-8.22 wt%, with an average adsorption energy of 0.36 eV/H2 and desorption 
+temperature of 412 K - 439 K[39]. The H2 storage on Li and Sc functionalized [4,4]paracyclophane 
+shows an uptake capacity of 11.8 wt% and 13.7 wt%, as estimated by Sathe et al. [40]. Kumar et 
+al. revealed the combination of physisorption and chemisorption of hydrogen on Sc and Ti 
+functionalized BN-analogous [2.2]PCP[41]. They showed the first hydrogen molecule 
+chemisorbed on the host material followed by physisorption of other H2, resulting in a storage of 
+~8.9 wt% via Kubas interaction. Numerous other metal-decorated macrocyclic compounds have 
+been explored as hydrogen storage possibilities, with storage capacities above the DOE 
+requirement; however, only a few have shown practical H2 capacity at varied thermodynamic 
+conditions. Though few PCP-based hydrogen storage systems are available in the literature, the 
+[2,2,2]paracyclophane, which is experimentally synthesized by Tabushi et al.[42] is yet to be 
+explored as a hydrogen storage material. 
+In the present work, we investigated the chemisorption and physisorption properties of hydrogen 
+molecules on [2,2,2]paracyclophane (PCP222) functionalized with Ti atoms and estimated their 
+hydrogen uptake capacity at varied thermodynamics. In paracyclophane, there are many molecules 
+in the group and are named after their pattern of arene substitution. The preceding square bracket 
+number, “[2,2,2]” in [2,2,2]paracyclophane, indicates that the consecutive benzene rings (3 
+benzene rings) in paracyclophane are linked with two (-CH2-) moieties. The linking bridges are 
+relatively short; thus, the separation between consecutive benzene rings is small, which develops 
+a strain in the aromatic rings. This strain in the rings can be utilized for Ti functionalization over 
+the aromatic benzene ring. Due to the strain and metal functionalization, the aromatic benzene 
+rings lose their inherent planarity. We choose to functionalize Ti metal atoms over the PCP222, as 
+the d- block transition metal elements are well known for reversible hydrogen adsorption and could 
+bind the H2 molecules via Kubas interaction[25, 26]. Though there are few reports available based 
+on hydrogen storage in macrocyclic organic compounds and other Ti-doped nanostructures, our 
+
+work is the first to investigate the efficiency of Ti-functionalized PCP222 using the atomistic MD 
+simulation, practical storage capacity, and diffusion energy barrier estimation 
+2 Theory and Computation 
+We have performed the theoretical calculations on [2.2.2] paracyclophane (PCP222) and their 
+hydrogenated structures within the framework of density functional theory (DFT)[43]. In the 
+computation, the advanced hybrid ωB97Xd functional is used, and molecular orbitals (MO) are 
+expressed as the linear combination of atom-centered basis function for which the valence diffuse 
+and polarization function 6-311+G(d,p) basis set is used for all atoms. ωB97Xd includes the long-
+range and Grimme’s D2 dispersion correction which is a range-separated version of Becke’s 97 
+functional[44, 45]. It is important to note that the ωB97Xd technique is a trustworthy method for 
+studying the non-covalent interactions, Organometallic complexes, and their thermochemistry. To 
+ensure the studied structures are in true ground state on the potential surface, the harmonic 
+frequencies of all the systems are determined and are found to be positive. All the theoretical 
+computations are performed with the computational program Gaussian 09[43]. 
+In order to investigate the binding strength of titanium (Ti) atoms on the PCP222, we have 
+calculated the average binding energy of decorated Ti atoms by using the following equation. 
+𝐸𝑏 =
+1
+𝑚 [𝐸𝑃𝐶𝑃222 + 𝑚𝐸𝑇𝑖 − 𝐸𝑃𝐶𝑃222+𝑚𝑇𝑖]      
+ 
+ 
+ (1) 
+Where EPCP222, ETi, and EPCP222+mTi is the total energy of PCP222, Ti atom and Ti-decorated 
+PCP222 respectively. m is the number of Ti atoms added PCP222. 
+The average adsorption energy of molecular hydrogen with metal atoms is calculated as[46]. 
+𝐸𝑎𝑑𝑠 =
+1
+𝑛 [𝐸𝑃𝐶𝑃222+𝑚𝑇𝑖 + 𝑛𝐸𝐻2 − 𝐸𝑃𝐶𝑃222+𝑚𝑇𝑖+𝑛𝐻2]     
+  (2) 
+Where EPCP222+mTi, EH2, and EPCP222+mTi+nH2 is the total energy of host material, hydrogen molecule 
+and hydrogen trapped complexes respectively. n is the number of H2 molecules adsorbed in each 
+complex. 
+The global reactivity descriptors such as hardness (η), electronegativity (χ), and electrophilicity 
+(ω) were estimated and used to study the stability and reactivity of Ti functionalized PCP222 and 
+their hydrogen adsorbed derivatives [47, 48]. The energy gap between the highest occupied 
+
+molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) is computed to 
+assure the kinetic stability of the studied systems. Further, to understand the electronic charge 
+transfer properties, the Hirshfeld charge and electrostatic potential map (ESP) were explored. 
+Moreover, partial density of states (PDOS) investigation was also carried out to further understand 
+the process of hydrogen interaction. The topological parameters were studied using Bader’s theory 
+of atoms in molecules (AIM) to analyze more about the nature of the interaction between metal on 
+PCP222 and adsorbed hydrogen molecules. 
+To obtained the hydrogen uptake capacity, gravimetric density (wt%) of hydrogen is calculated 
+using the following equation[49]: 
+𝐻2(𝑤𝑡%) =
+𝑀𝐻2
+𝑀𝐻2+𝑀𝐻𝑜𝑠𝑡 × 100 
+ 
+ 
+ 
+   (3) 
+Here MH2represent the mass of the total number of H2 molecules adsorbed and MHost represent the 
+mass of metal-doped PCP222. 
+3 Results and Discussion 
+3.1 Structural properties of PCP222 
+The optimized geometrical structure of PCP222 is depicted in Figure 1(a). PCP222 has three 
+benzene rings connected by two -CH2- moiety as a bridge between the consecutive rings. The 
+distance between the two consecutive -CH2- moiety and the -CH2- across the benzene ring are 
+found to be 1.54 Å and 5.84 Å respectively, which is consistent with the earlier experimentally 
+reported value by Cohen-Addad et al. [50]. To validate the π aromaticity of the optimized 
+molecule, we computed the Nucleus Independent Chemical Shift (NICS) of PCP222 before 
+functionalizing by any metal atom. The NICS values are determined with 1 Å increment from the 
+center to 3 Å above the three benzene rings. NICS(1) is found to be negative maximum (-10.1 
+ppm), suggesting the aromatic nature of PCP222. This indicates that the benzene rings of PCP222 
+are π electron-rich and can bind a metal atom outside the benzene rings. 
+3.2 Functionalization of Ti atom on PCP222 
+
+ 
+Figure 1: (a) Optimized structure of PCP222 with all possible marked adsorption site marked, (b) Ti 
+functionalized PCP222 
+ 
+        
+Next, we explore different possible adsorption sites of pristine PCP222, such as C-C bridge of 
+benzene ring (B1), CH2 moiety and benzene bridge (B2), CH2 - CH2 bridge (B3), and above the 
+center of benzene (Rc) which are depicted in Figure 1(a). To design the host material for hydrogen 
+adsorption, a single Ti atom is positioned about 2 Å above at the regioselective sites of PCP222, 
+and the resulting structure is re-optimized. The binding energy between Ti and PCP222 calculated 
+using Equation 1 at different adsorption sites shows that the Ti atom is stable at two positions, B3 
+and Rc sites of PCP222 with binding energies of 0.37 eV and 2.20 eV, respectively which fairly 
+agree with the previously reported value of Ti on CNT by Yildirim et al. [51]. Hence, the most 
+favourable site for Ti atom functionalization is at the Rc site above the benzene ring of PCP222. 
+3.2.1 Bonding mechanism of Ti on PCP222 
+To understand the binding mechanism of Ti on PCP222, we analyzed the partial density of state 
+(PDOS), electrostatic potential map (ESP), Hirshfeld charge, and Bader’s topological parameters 
+of the Ti functionalized PCP222 system as discussed below. 
+Density of states 
+
+5.84
+B2
+5.87
+1.543
+(a)
+(b) 
+Figure 2: Density of states plot on Ti and C atom on PCP222 
+ 
+The Ti atom is functionalized on PCP222 via the Dewar mechanism in which π-electron gets 
+transferred from the highest occupied molecular orbitals (HOMO) of the substrates to the vacant 
+d-orbital of Ti followed by the back-donation of charges from the partially filled d-orbital of Ti to 
+empty π*-anti-bonding of the benzene ring of PCP222[26]. To understand the orbital interaction 
+between the Ti and C atom of PCP222, we have performed the partial density of states (PDOS) 
+calculation of PCP222-Ti and the result is plotted in Figure 2. Figure 2 clearly shows that the 
+electronic states of the Ti atom and the C atom of PCP222 overlap below and above the Fermi 
+level (E = 0). The transferred electrons partially fill the unoccupied states of PCP222, as seen by 
+the intense peaks near the Fermi level. This infers an orbital interaction between Ti and C atom of 
+PCP222 mediated by charge transfer. The fact is also obvious because Ti has the relatively lower 
+ionization potential than the C atom. 
+ESP and Hirshfeld charges 
+To get a picture of electronic charge distribution over the PCP222 during Ti functionalization, we 
+plotted the electrostatic potential (ESP) map over the total electron density, as shown in Figure.S1. 
+The variation of electron density in the ESP map is shown by using different colour codes, which 
+follows the pattern of accumulation and reduction of electron density as; red (maximum electron 
+density) >orange > yellow > green > blue (minimum electron density). In the ESP plot (Figure.S1), 
+the red region over the benzene ring of PCP222 implies the aggregation of electron density. After 
+
+22
+c
+PCP222-Ti
+20.
+Ti
+18-
+Total
+16-
+14.
+12 -
+DOS
+10-
+8.
+6.
+4.
+2
+0
+-18
+-16
+-14
+-12
+-10
+-8
+-6
+2
+0
+4
+Energy (eV)the functionalization of the Ti atom, the region changed to dark blue, indicating the deficiency of 
+electron density around the metal making it susceptible to bind with the guest molecules. 
+Moreover, region around the carbon atoms of PCP222 turns from red to green supporting the 
+charge transfer as discussed above. The estimated Hirshfeld charge on C and Ti atoms is computed 
+to be -0.121 e.u and +0.511 e.u, respectively, which makes the Ti atom nearly ionic, opening the 
+possibility for H2 adsorption. 
+3.2.2 Diffusion energy barrier calculation 
+ 
+Figure 3: Ti diffusion energy barrier over the PCP222 
+ 
+According to earlier reports, the aggregation of transition metal atoms on the substrate may lower 
+the ability of the host material for hydrogen adsorption. So, before hydrogen adsorption on the 
+surface of PCP222, it is necessary to study the possibility of metal clustering on the substrate. If 
+the Ti atom is displaced from its stable adsorption position on PCP222 due to an increase in 
+temperature, there is a strong possibility of metal clustering. Since the Ti binding energy on 
+PCP222 (2.2 eV/Ti) is lower than the cohesive energy of an isolated single Ti atom (4.85 eV), we 
+evaluated whether or not there is an energy barrier for Ti atom diffusion on PCP222. The diffusion 
+energy barrier is calculated by displacing Ti to a finite neighbourhood (δr) over the adsorption site 
+of PCP222. as shown in Figure 3. The difference in energy calculated between the initial and that 
+of the close neighbourhood is then plotted with the diffusion coordinates as shown in Figure 3. 
+The figure illustrates the diffusion energy barrier to be 5.97 eV, which is sufficient to prevent the 
+diffusion of the Ti atom over PCP222 and therefore avoid Ti-Ti clustering which is also supported 
+
+AE= 5.97 eV
+6
+5
+4
+ev
+4
+2
+1
+0
+2
+3
+5
+0
+1
+4
+Diffusion coordinatesby the works of Dewangan et al. [26] and Chakraborty et al. [25]. Therefore, the present Ti 
+functionalized PCP222 can be considered a suitable candidate for hydrogen adsorption. 
+3.3 Adsorption of H2 molecules on PCP222-Ti 
+ 
+ 
+ 
+ 
+ 
+Figure 4: Optimized geometry of hydrogenated Ti functionalized PCP222, (a) PCP222-Ti-1H2, (b) 
+PCP222-Ti-2H2, (c) PCP222-Ti-3H2, (d) PCP222-Ti-4H2, (e) PCP222-Ti-5H2, (f) PCP222-Ti-6H2 
+ 
+Figure 5: Optimized geometry of hydrogenated Ti functionalized PCP222, (a) PCP222-Ti-2H, (b) 
+PCP222-Ti-2H-1H2, (c) PCP222-Ti-2H-2H2, (d) PCP222-Ti-2H-3H2, (e) PCP222-Ti-2H-4H2. 
+
+(a)
+(b)
+(c)
+d)
+e
+f)a
+b
+(c)Table 1: Average bond distance between carbon bridge (C-C), center of PCP222 benzene ring (Rc) 
+and Titanium atom (Rc-Ti), Titanium and hydrogen molecules (Ti-H2), and hydrogen Hydrogen 
+(H-H) in Å. Average adsorption energy of H2 on PCP222-Ti. 
+ 
+Name of complex 
+Bridge C-C 
+Rc -Ti 
+Ti-H 
+H-H 
+Eads (eV) 
+PCP222-Ti 
+1.542 
+1.566 
+ 
+ 
+ 
+PCP222-Ti-2H 
+1.540 
+1.800 
+1.750 
+2.796 
+1.797 
+PCP222-Ti-2H2 
+1.540 
+1.765 
+1.770 
+0.884 
+0.953 
+PCP222-Ti-3H2 
+1.540 
+1.798 
+1.830 
+0.852 
+0.784 
+PCP222-Ti-4H2 
+1.540 
+1.818 
+1.905 
+0.806 
+0.672 
+PCP222-Ti-5H2 
+1.540 
+1.842 
+2.332 
+0.816 
+0.554 
+PCP222-Ti-6H2 
+1.540 
+1.842 
+2.633 
+0.804 
+0.467 
+ 
+ 
+ 
+ 
+ 
+ 
+PCP222-Ti-2H-1H2 
+1.540 
+1.822 
+1.926 
+0.800 
+0.480 
+PCP222-Ti-2H-2H2 
+1.540 
+1.837 
+1.868 
+0.803 
+0.474 
+PCP222-Ti-2H-3H2 
+1.540 
+1.851 
+1.899 
+0.801 
+0.406 
+PCP222-Ti-2H-4H2 
+1.540 
+1.837 
+2.840 
+0.774 
+0.256 
+ 
+To investigate the hydrogen adsorption on the surface of Ti functionalized PCP222, we added the 
+H2 molecules sequentially to PCP222-Ti. First, we added a single H2 molecule at about 2 Å above 
+the Ti atom functionalized on PCP222 and allowed the system to relax. It is observed that the 
+H2 molecule dissociates into two fragments of H atoms and forms chemical bond with the Ti atom. 
+The Ti-H bond length is found to be 1.75 Å which is close to the experimental result for titanium 
+monohydride [52]. The H-H bond distance is noted to be about 2.8 Å (Figure 4(a)). The binding 
+energy between Ti and H is calculated to be 1.79 eV which lies in the range of chemisorption 
+mechanized by Kubas’s interaction [2, 38]. Similar result was also reported by Ciraci et al. for the 
+adsorption of a single H2 molecule on Ti-decorated SWNT8 ( and SWBNT ) where the 
+H2 molecules dissociate into individual H atoms with a binding energy of 0.83 eV/H (0.93 eV/H) 
+and H-H- distance of 2.71Å (3.38 Å)[51, 53]. However, when two H2 molecules are 
+simultaneously added to the sorption center, the calculated average adsorption energy is reduced 
+to 0.95 eV/H2, with the average H-H bond length stretching from 0.74 Å to 0.8 Å. This result 
+
+clearly indicates the adsorption process to be physisorptive. This is because of reduced interaction 
+strength between Ti atoms and H2 molecules caused due to screening effect. From the ESP analysis 
+(7) it is obvious that simultaneous presence of two H2 molecules reduces the charge densities of 
+Ti and H2 thereby inducing a weak charge polarization which causes the physisorption of hydrogen 
+on the surface of Ti functionalized PCP222. Another way of generating similar isomeric 
+configuration is chemisorption induced physisorption of H2 molecules on Ti functionalized 
+PCP222 in which one H2 molecule is adsorbed over n PCP222-Ti-2H (Figure 5(b)). Interestingly, 
+this configuration is 0.37 eV lower in energy than that of PCP222-Ti-2H2, and the H2 adsorbed 
+with lower adsorption energy (0.48 eV). Therefore, we proceed with both configurations for 
+further hydrogen adsorption. Sequential adsorption of H2 molecules on PCP222-Ti results in the 
+maximum adsorption up to 6H2 molecules. The adsorption of 3rd, 4th, 5th, and 6th H2 molecules 
+to PCP222-Ti reduces the average H2 adsorption energy to 0.784, 0.68, 0.554, and 0.467 eV/H2, 
+respectively. On the other hand, successive addition of H2 molecules to PCP222-Ti-2H leads to 
+maximum adsorption of four hydrogen molecules. More addition of H2 molecules beyond maxima 
+in both the cases causes them to fly away from the sorption center. It is observed that the average 
+adsorption energy decreases with an increase in the number of H2 molecules in the system which 
+is due to the steric hindrance among the adsorbed H2 crowed and the increase in distances between 
+the H2 and sorption centers. The estimated data of adsorption energy and geometrical parameters 
+of all the bare hydrogenated systems and presented in Table 1. 
+3.3.1 Partial density of states 
+
+ 
+Figure 6: Partial density of state on Ti and H atoms of (a) PCP222-Ti-2H, (b) PCP222-Ti-2H-1H2, (c) 
+PCP222-Ti-2H2, and (d) PCP222-Ti-6H2 
+ 
+The partial density of states (PDOS) of Ti and H atoms of the hydrogen adsorbed PCP222-Ti with 
+the chemisorbed, and physisorbed hydrogen is plotted in Figure 6. The adsorption of 1H2 to the 
+host resulting in chemisorption is contributed from the strong overlapping of H and Ti orbital near 
+-9 eV. Upon adsorption of another H2 molecule over PCP222-Ti-2H, the peaks of σ-orbital 
+(HOMO) of hydrogen and Ti orbital appears at around -15.7 eV below the Fermi level 
+and σ* (LUMO) of hydrogen interacts with the orbital of Ti and chemisorbed H above the Fermi 
+level (figure 6(b)) which can be explained by the Kubas mechanism in which a small charge 
+transfer occurs from the σ(HOMO) orbital of H2 to the vacant 3d orbital of the Ti atom, followed 
+by a back-donation of charges in the other direction from the partially filled 3d orbitals of Ti 
+to σ* (LUMO) of H2 molecules. When two H2 molecules are introduced simultaneously to the 
+PCP222-Ti, similar DOS peaks are observed, suggesting the H2 adsorption via the Kubas 
+
+2.5
+Ti
+(a) PCP222-Ti-2H
+2.0
+H (chemisorbed)
+1.5
+1.0
+0.5 -
+V
+0.0
+2
+-18
+-16
+-14
+-12
+-10
+-8
+-6
+-4
+-2
+0
+4
+-
+2.4
+Ti
+(b) PCP222-Ti-2H-1H2
+2.0
+1.6
+H (chemisorbed)
+1.2
+H (physisorbed)
+0.8 
+0.4 
+人
+PDOS
+0.0
+1
+.
+-18
+-16
+-14
+-12
+-10
+-8
+-6
+-4
+-2
+0
+4
+2.5
+Ti
+(c) PCP222-Ti-2H,
+2.0
+H (physisorbed)
+1.5
+1.0 -
+人
+0.5 -
+0.0
+-16
+-14
+-12
+-10
+-8
+-6
+-
+-18
+-4
+-2
+2
+4
+2.5
+Ti
+(d) PCP222-Ti-6H2
+2.0
+H (physisorbed)
+1.5
+1.0-
+0.5
+0.0
+1
+T2
+-18
+-16
+-14
+-12
+-10
+-8
+-6
+-4
+-2
+0
+4
+Energy (eV)mechanism. However, here the σ orbital of H2 splits into several peaks in the range of -15.2 to -
+6.2 eV and moves closer to the Fermi level inferring lower in the interaction strength. On 
+adsorption of 6H2 molecules to Ti functionalized PCP222, the σ orbitals split into numerous peaks 
+in a broad range of -16.3 eV to -6.1 eV with enhanced intensity. This signifies that the adsorption 
+strength gets weaker with an increase in the quantity of H2 molecules in the host systems. 
+3.3.2 Electrostatics potential and Hirshfeld charges 
+ 
+Figure 7: Electrostatics potential map of (a) PCP222-Ti, (b) PCP222-Ti-2H, (c) PCP222-Ti-2H2, (d) 
+PCP222-Ti-3H2, (e) PCP222-Ti-4H2, (f) PCP222-Ti-5H2, (f) PCP222-Ti-6H2. 
+ 
+To obtain a qualitative depiction of electronic charge distribution over the bare and hydrogenated 
+PCP222-Ti, we generated and plotted the electrostatic potential (ESP) map on the total electron 
+density as shown in Figure 7 The charge distribution is used to determine the active adsorption 
+region for the guest hydrogen molecules. The dark blue zone above the Ti atom on PCP222-Ti 
+(Figure 7(a)) and the dark red region over the first adsorbed hydrogen atom indicates a strong 
+interaction between them leading to chemisorption of hydrogen atom. Upon adsorption of two 
+H2 molecules simultaneously, the region over Ti turns from dark blue to light blue, suggesting the 
+fact that, positive charge get transferred from the Ti atom to the adsorbed H2 and C atom of 
+PCCP222 thereby inducing charge polarization which causes physisorption of the second 
+H2 molecule. Further addition of H2 molecules to PCP222-Ti, the region over Ti atom turns to 
+bluish-green and then to green inferring further charge transfer (depletion of electron density near 
+Ti ) and the yellow region over the adsorbed H2 represents a little accumulation of electron density 
+at hydrogen molecules[26]. 
+
+4.000e-2
++ 4.000 e-2
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+Sideview
+Topview 
+Figure 8: Hirshfeld charges before and after hydrogen adsorption on PCP222-Ti 
+ 
+Figure 8 shows the average Hirshfeld charges on the Ti atom, the adsorbed H2 molecules, and the 
+C atoms of the benzene ring (Ti functionalized site) as a function of the number of H2 adsorbed on 
+the host. The average charges on the C atom of the benzene ring are initially computed to be -
+0.031 e which then raises to -0.121 e with the functionalization of the Ti atom. The charge on the 
+Ti atom of PCP222-Ti is found to be +0.511 e, indicating the transfer of electronic charges from 
+the Ti atom to the C atom of the benzene ring. On chemisorption of the first hydrogen on PCP222-
+Ti, the electronic charges on the Ti and H atoms are +0.41 a.u and -0.24 a.u implying a strong 
+attractive interaction between them as discussed above. Adding more H2 molecules gradually 
+lessen the Hirshfeld charges over the Ti and H atoms implying polarization induced weak 
+interaction between them. (Figure 8). 
+3.3.3 Bader’s topological analysis 
+The topological analysis at the bond critical point (BCP) is used to investigate the nature of 
+interactions between the Ti-functionalized PCP222 and the adsorbed H2 molecules employing 
+Bader’s quantum theory of atoms in molecules (QTAIM). The topological descriptors associated 
+with the electronic distribution, such as electron density (), Laplacian (2), and total energy 
+density (ℌ) (calculated as the sum of local kinetic G() potential energy density V() ), at BCPs 
+
+0.8
+- Ring C before Ti decoration
+0.7
+ Ring C after Ti decoration
+0.6
+ Ti atom
+0.5.
+H atom
+Hirshfeld Charges (eu)
+0.4
+0.3
+0.2
+0.1
+0.0
+-0.2
+0.3
+-0.4
+-0.5
+0
+2
+5
+6
+Number of H, molecules, nare presented in Table S1. Kumar et al. reported that the positive value of the Laplacian of electron 
+density (2>0) at BCP indicates a decrease in  at the bonding region, suggesting an interaction 
+of closed-shell (non-covalent) type [56]. For PCP222-Ti-6H2, the value of  and 2 at BCP of Ti 
+and adsorbed H2 are found to be 0.057 a.u and 0.208 a.u, respectively which infers a closed-shell 
+interaction between Ti and H2. Moreover, the negative value of ℌBCP and −
+G() 
+V() > 1 at BCP of 
+Ti and H2 confirm the closed-shell interaction among sorption center and H2 as proposed by 
+Koch et al. (Table S1) [57]. For C–C and C-Ti bond, the average  value shows very nominal 
+changes after the hydrogen adsorption which suggests the post-adsorption chemical stability of the 
+host material. Additionally, the average  on BCP of the H-H bond in PCP222-Ti-6H2 is 0.231 a.u 
+which is almost the same as on isolated bare H2 molecule (0.263 a.u). This implies that the 
+adsorbed hydrogens are in quasi-molecular form during the adsorption which also reflected in H-
+H bond elongation by 0.06-0.14 Å. 
+3.4 Thermodynamically usable H2 capacity 
+3.4.1 Storage capacity 
+ 
+Figure 9: Optimized geometry of hydrogen saturated 3Ti functionalized PCP222 
+ 
+To examine the maximum H2 gravimetric storage capacity of the system, we have 
+functionalized the Ti atom on each benzene ring of PCP222 resulting in the structure of 
+PCP222-3Ti as shown in Figure 9 and S3. Further, we added H2 molecules to each Ti 
+
+atom functionalized on PCP222 sequentially as discussed in previous section (3.3). The 
+calculated average H2 adsorption energy and the change in geometrical parameters are 
+presented in Table 2. The adsorption of H2 on PCP222-3Ti is observed to behave similar 
+to that of on single Ti atom on PCP222. On saturation of the H2 uptake capacity of 
+PCP222-3Ti, each sorption center is found holding a maximum of 6H2 molecules with 
+a gravimetric storage capacity of 7.37 wt%. Since the first H2 molecule on each Ti atom 
+dissociate into two H atom and bonded strongly with Ti atoms, 1.31 wt% of hydrogen 
+adsorbed via the chemisorption process is difficult to desorb. However, the concurrent 
+addition of two or more H2 molecules to each Ti atom over PCP222, results in 
+physisorption kind of adsorption. Further, to confirm the stability of maximum 
+hydrogenated systems, the energy gap (Eg) (gap between HOMO-LUMO) and global 
+reactivity parameters such as η, χ, and ω were estimated using the Koopmans 
+theorem[58]. Notwithstanding, the studied system follow the “maximum hardness and 
+minimum electrophilicity principle,” ensuring their chemical stability (Figure S4)[59]. 
+ 
+     Table 2: Average bond distance between carbon bridge (C-C), center of PCP222 benzene ring (Rc) 
+and Titanium atom (Rc-Ti), Titanium and hydrogen molecules (Ti-H2), and hydrogen-hydrogen 
+(H-H) in Å. Average adsorption energy and successive desorption energy of PCP222-3Ti-
+nH2 (n=3,6,9,12,15,18) 
+Name of complex 
+Bridge C-C 
+Rc-Ti 
+Ti-H 
+H-H 
+Eads (eV) 
+Edes (eV) 
+PCP222_3Ti 
+1.543 
+1.590 
+- 
+- 
+- 
+- 
+PCP222_3Ti-3H2 
+1.537 
+1.799 
+1.747 
+2.824 
+1.824 
+1.824 
+PCP222_3Ti-6H2 
+1.537 
+1.756 
+1.776 
+0.880 
+0.988 
+0.152 
+PCP222_3Ti-9H2 
+1.537 
+1.790 
+1.832 
+0.849 
+0.813 
+0.464 
+PCP222_3Ti-12H2 
+1.536 
+1.824 
+1.801 
+0.821 
+0.700 
+0.360 
+PCP222_3Ti-15H2 
+1.535 
+1.825 
+2.332 
+0.806 
+0.570 
+0.050 
+PCP222_3Ti-18H2 
+1.536 
+1.838 
+2.622 
+0.803 
+0.482 
+0.043 
+ 
+ 
+
+ 
+Figure 10: Hydrogen occupation number for PCP222-3Ti at various T and P. 
+ 
+For a practically usable hydrogen medium, a substantial amount of H2 molecules should be 
+adsorbed by the host material at attainable adsorption conditions and the adsorbed H2 molecules 
+should be desorbed effectively at a suitable temperature (T) and pressure (P). Thus, it is essential 
+to estimate the number of hydrogen molecules usable at a wide variety of T and P. We have 
+estimated the usable hydrogen gravimetric density of the studied system by calculating the number 
+of H2 molecules stored in PCP222-3Ti at different T and P using the empirical value of H2 gas 
+chemical potential (μ). The H2 gravimetric density is estimated from the occupation number (N) 
+by the following equation and plotted with various T and P in Figure 10[60]. 
+𝑁 =
+∑
+𝑛𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇]
+𝑁𝑚𝑎𝑥
+𝑛=0
+∑
+𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇]
+𝑛𝑚𝑎𝑥
+𝑛=0
+ 
+ 
+ 
+ 
+(4) 
+Here Nmax is the maximum number of H2 molecules adsorbed on each Ti atom on 
+PCP222, n and gn represents the number of H2 molecules adsorbed and configurational 
+degeneracy for a n respectively. kB is the Boltzmann constant and -Eads (>0) indicates the average 
+adsorption energy of H2 molecules over PCP222-3Ti. μ is the empirical value of chemical potential 
+of H2 gas at specific T and P, obtained by using the following expression [61]. 
+𝜇 = 𝐻0(𝑇) − 𝐻0(0) − 𝑇𝑆0(𝑇) + 𝐾𝐵𝑇 ln (
+𝑃
+𝑃0)      
+ 
+ (5) 
+Here H0(T), S0(T) are the enthalpy and entropy of H2 at pressure P0 (1 bar). 
+
+7.380
+6.772
+7
+6.164
+6
+5.556
+4.948
+5-
+4.340
+wt%
+3.732
+4
+工
+3.124
+3-
+2.516
+1.908
+2
+1.300
+50
+100
+150
+30
+e(bar)
+200
+250
+Pressure
+300
+400From the Figure 10 it is clear that, the PCP222-3Ti can store 18H2 molecules at temperatures up 
+to 80 K and 10-60 bar pressure. Up-to these thermodynamic conditions, the maximum H2 storage 
+capacity of the studied system is estimated as 7.37 wt%, which is consistent the experimentally 
+reported value for Pd functionalized carbon nanotubes [62] and is fairly above the target set by 
+US-DOE (5.5 wt% by 2025). On raising the temperature above 80 K, the H2 molecules start to 
+desorb from the PCP222-3Ti and retain >5.5 wt% of H2 till the temperature of 120 K under 30-60 
+bar. Further, rise in temperature, the system maintains an H2 gravimetric density of 5 wt% (close 
+to the target of US-DOE) throughout a temperature range of 120-300 K and a pressure range of 3-
+60. This thermodynamic condition may be treated as an ideal storage condition for H2 on PCP222-
+3Ti. At the temperature of 400 K and pressure of 1-10 bar, the system retains 1.31 wt% of 
+hydrogen, that are adsorbed via the chemisorption process and may be desorbed at very high 
+temperatures. Thus, a total gravimetric density of 6.06 wt% (difference in G.D at 80 K and 400 K) 
+H2 molecules are usable under ambient conditions, which is fairly higher than the US-DOE target. 
+This result justifies that the Ti functionalization over PCP222 can be used as a potential reversible 
+hydrogen storage material. 
+3.5 Molecular dynamics simulations 
+ 
+Figure 11: (a) Potential energy trajectories of hydrogenated PCP222-3Ti and (b) Time evolution 
+trajectory of average bond length between the Ti atom and C atoms of PCP222 at 300K and 500K. 
+ 
+
+3498.14
+300K
+(Hartree)
+-3498.16
+500K
+-3498.18
+-3498.20
+Potentialenergy
+3498.22
+3498.24
+3498.26
+-3498.28
+3498.30
+3498.32
+-3498.34
+0
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+Time (fs)
+2.8
+2.7
+C-Tidistance@300K
+2.6
+C-Tidistance@500K
+2.5
+2.4
+2.3
+2.2
+2.1
+2.0
+1.9
+1.8
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+Time (fs)We have performed molecular dynamic (MD) simulations using the atom-centered density matrix 
+propagation (ADMP) to check the desorption of hydrogen from the PCP222-3Ti-nH2and the 
+structural integrity of the host. During the simulations, the temperature was maintained by the 
+velocity scaling method, and the temperature was checked and corrected at every time step of 10 
+fs. Figure 11(a) and S5, show the time variation potential energy trajectories and system snapshots, 
+respectively. The MD simulations at 300K and 1 ps reveal that 2H2 molecules from each Ti atom 
+fly away, and each Ti continues to hold three physisorbed H2 molecules and two chemisorbed 
+hydrogen atoms. When the temperature is elevated to 500 K, almost all the H2 molecules get 
+desorbed and each sorption center hold one physisorbed H2 and two chemisorbed H atoms. Since 
+the first physisorbed H2 is bound strongly with the host material, it may desorb at a higher 
+temperature and time scale. This indicates that the system PCP222-3Ti is not complete reversible 
+at normal temperatures and may show 100% desorption at a higher temperature. 
+For a practical hydrogen storage material, it is necessary that the host material must keep the 
+structural integrity above the average desorption temperature. To examine the structural integrity 
+of the host material (PCP222-3Ti), we carried out the MD simulations with the host material at 
+300 K and significantly above the room temperature (500 K) using ADMP. With a time step of 1 
+fs, the ADMP-MD simulations are carried out for 1 ps. Figure 11(b) depicts the time variation 
+trajectory of the average distance between the Ti atom and the carbon atoms of PCP222 benzene 
+rings. We observe that the PCP222-3Ti maintains its structural stability at 500 K, and the distances 
+between the C-C and C-H bonds essentially remain unchanged. The time evolution trajectories of 
+the average distance between the Ti and C atom of PCP222 were noticed to oscillate about the 
+mean value (2.32 Å) with little variance. This illustrates that the host material’s structural stability 
+is maintained significantly above room temperature. In light of this, we believe that PCP222-3Ti 
+can be a viable option for hydrogen storage material. 
+4 Conclusion 
+In this study, we investigated the thermodynamical stability and hydrogen storage properties of 
+Ti-functionalized [2,2,2]paracyclophane, using the density functional theory. The Ti atoms are 
+strongly bonded to the PCP222 via Dewar mechanism, and no clustering of Ti atoms over PCP222 
+was noticed. The first H2 molecule is chemisorbed with binding energy of 1.797 eV, while the 
+
+remaining H2 molecules are physisorbed with an average H2 adsorption energy in the range of 
+0.467 - 0.953 eV/H2. On saturation with the H2, the Ti atom on PCP222 could adsorb up to 
+6H2 molecules, while the Ti-2H on PCP222 could adsorb up to 4H2. The average H-H bond 
+distance elongated by 0.06-0.14 Å during the adsorption process which implied that the adsorbed 
+H2 molecules were in quasi-molecular form and the fact is supported by the Hirshfeld charge 
+distribution analysis. . When three Ti atoms were functionalized on PCP222, the H2 gravimetric 
+capacity of the system was up to 7.37 wt%, which was fairly above the US-DOE requirements for 
+practical hydrogen applications. During saturation of H2 adsorption, the host material displayed no 
+significant change in geometry. The thermodynamic usable hydrogen capacity was found to be up 
+to 5 wt% throughout a temperature range of 120-300 K and a pressure range of 3-60 bar. At the 
+temperature of 400 K and pressure of 1-10 bar, the system retains 1.31 wt% of hydrogen which 
+could be desorbed at very high temperatures. A total gravimetric density of up to 6.06 wt% 
+H2 molecules are usable under ambient conditions which is fairly higher than the US-DOE target. 
+MD simulations at 500 K revealed the structural integrity and reversibility of the host and also 
+showed that chemisorbed hydrogens are retained at this temperature. Since, there is no 
+experimental works reported on Ti-functionalized PCP222 for hydrogen storage, we hope our 
+computational work will contribute significantly to the research of hydrogen storage in 
+macrocyclic compounds and provide supporting reference for the future experiments. 
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+Robust Surface Reconstruction from  
+Orthogonal Slices 
+ 
+Radek Sviták1, Václav Skala2 
+Department of Computer Science and Engineering,  
+University of West Bohemia in Pilsen, Univerzitní 8, 306 14 Plzeň, Czech Republic 
+E-mail: rsvitak@kiv.zcu.cz 
+ 
+Abstract 
+The surface reconstruction problem from sets of planar parallel slices representing cross sections 
+through 3D objects is presented. The final result of surface reconstruction is always based on the correct 
+estimation of the structure of the original object. This paper is a case study of the problem of the structure 
+determination. We present a new approach, which is based on considering mutually orthogonal sets of slices. 
+A new method for surface reconstruction from orthogonal slices is described and the benefit of orthogonal 
+slices is discussed too. The properties and sample results are presented as well. 
+ 
+ 
+                                                           
+This work is was supported by the Ministry of Education of the Czech Republic – projects:  
+1FRVŠ 1348/2004/G1 
+2MSM 235200005 
+1. Introduction 
+ 
+The crucial task of the surface reconstruction 
+from slices is a correct estimation of the 
+original object structure, i.e. the solution of the 
+contour correspondence problem. Most of the 
+existing methods simply consider the overlap 
+of contours in a pair of consecutive parallel 
+slices as the only correspondence criterion. 
+Therefore, they produce unacceptable structure 
+estimation when the angle between the axis of 
+the object and the normal of the slices 
+increases.  
+Higher density of slices can help to 
+solve this problem, but it is not always 
+possible because of the resolution limit of the 
+scanning device, etc. It is obvious that other 
+slices in non-parallel planes offer an additional 
+information. In this paper we will concentrate 
+on the benefit of orthogonal slices for the 
+reconstruction process. In comparison to the 
+existing methods, our currently achieved 
+results show, that for a set of objects the 
+resultant surface is significantly more accurate 
+with respect to the similarity to the original 
+surface. 
+The concept of the new proposed 
+method 
+is 
+presented 
+and 
+results 
+of 
+comparisons with the existing methods are 
+discussed as well. 
+ 
+ 
+A 
+C 
+B 
+ 
+D 
+Figure 1: Problematic cases when solving the contour 
+correspondence problem. Expected problems using the 
+overlapping 
+criterion: 
+A, B, C, D; 
+generalized 
+cylinders: B, D; 
+MST: C, D; 
+Reeb graph 
+based 
+methods: D. 
+Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004
+
+2. Brief survey of existing methods 
+ 
+ 
+Several methods for surface reconstruction 
+from slices have been developed since about 
+1970. In this section we will classify them 
+according to their approach to solving the 
+contour correspondence problem. For more 
+extensive study of the existing methods from 
+the other  viewpoints, see [2, 4, 6, 7]. 
+The simplest methods estimate the 
+contour correspondence locally between each 
+consecutive pair of contours. Typically, 
+contours 
+that 
+overlap 
+each 
+other 
+are 
+considered as correspondent. This works if the 
+density of slice is high, i.e. the distance 
+between slices is low, and the axis of the input 
+object is nearly perpendicular to the slices 
+planes.  
+A 
+more 
+advanced 
+method 
+uses 
+generalized elliptical cylinder to solve the 
+correspondence problem [1, 11, 12]. Contours 
+are first classified as elliptical or complex by 
+determining how well the vertices of their 
+perimeter can be fit by an ellipse. If the fit is 
+too poor, a contour is classified as complex, 
+and can not be incorporated into an elliptical 
+cylinder. Then the ellipses are grouped to the 
+cylinders. When as many contours as possible 
+have been organized into cylinders, then the 
+algorithm uses the geometric relationship 
+between cylinders to group them into objects. 
+This method is most useful for elongated 
+smooth objects with roughly elliptical cross 
+section. 
+Apparently 
+the 
+best 
+existing 
+approaches that have been published are two 
+graph-based methods. The first of them 
+presented by Skinner [10] computes a 
+minimum spanning tree based on contour 
+shape and position. In the first step a graph is 
+constructed by representing each contour as a 
+node and connecting each node to all nodes 
+representing contours in adjacent sections. The 
+best fitting ellipse is computed for each 
+contour. The cost of an edge of the graph 
+relies on the mutual position and size of two 
+ellipses: 
+ 
+( )
+2
+2
+2
+2
+)
+b
+(b
+)
+a
+(a
+)
+y
+(y
+)
+x
+(x
+i,j
+c
+j
+i
+j
+i
+j
+i
+j
+i
+−
++
+−
++
+−
++
+−
+=
+, 
+ 
+where (xi, yi, zi), (xj, yj, zj) represent the 
+centers of the ellipses of contours i, j, 
+respectively, and ai, bi, aj, bj are their major 
+and minor axis lengths. 
+The minimum spanning tree computed 
+for the graph represents the solution to the 
+correspondence problem. The method works 
+well for naturally tree-structured objects, the 
+main limitation is its inability to solve the 
+correspondence problem correctly for general 
+graph topologies, e.g. genus > 0. 
+ 
+ 
+A 
+ 
+ 
+ 
+ 
+ 
+B 
+Figure 2: A) Data set of slices of the cochlea.  Using the 
+Reeb graph it is possible to detect and represent the 
+right contour correspondence. The advantage consists in 
+the possibility of considering the correspondence 
+among contours of one slice (B). Taken from [8]. 
+ 
+The second graph based method  
+presented by Shinagawa [8, 9] uses surface 
+coding based on Morse Theory to construct a 
+Reeb graph [14] representing the contour 
+connectivity. Each contour represents a node 
+in the graph, edges of the graph represent the 
+contour correspondence relation. Edges are 
+added to the graph in the manner to avoid 
+making connections that would result in a 
+surface that is not a 2-manifold. For each pair 
+of contours that can be legally connected, a 
+weight function is evaluated, and its value is 
+used to establish a priority for connecting that 
+pair of contours. The algorithm proceeds by 
+making the highest priority connections in 
+regions where the number of contours in each 
+section does not change, and then adds 
+connections in order of decreasing priority 
+with respect to the a priori knowledge of the 
+number of connected components and the 
+topological genus. 
+Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004
+
+2
+0
+3It is necessary to note that all the 
+existing solutions just estimate the contours 
+correspondence, i.e. the structure of the 
+original object, should be emphasized. In 
+Figure 1 there are some typical example data 
+sets to illustrate capabilities of the approaches 
+mentioned in this section. 
+ 
+3. Orthogonal slices 
+ 
+One set of parallel planar slices is one of the 
+well-known boundary representations of a 3D 
+object. Usually the planes of such slices set are 
+perpendicular to the z-axis, and thus called z-
+slices.  
+If we slice an object by more then one 
+set of parallel slices and moreover when these 
+sets 
+are 
+mutually 
+orthogonal, 
+we 
+get 
+orthogonal sets of slices.  Consider now that 
+we have z-slices, x-slices and y-slices of an 
+object, see Figure 3. Note, that the contours 
+are supposed to be polygonal, oriented  the 
+way that when looking from the positive 
+direction of the given slices set axis, the 
+contours have the interior on its left side and 
+the exterior on the right side, see Figure 4. 
+ 
+3.1 Contour correspondence 
+ 
+The main advantage of orthogonal slices 
+consists in the approach how the contour 
+correspondence can be determined. It is 
+important to emphasize that two orthogonal 
+contours which intersect each other comes 
+aparently from one and the same surface 
+component of the input object. It means that 
+the intersection of contours is very important 
+since it provides accurate information about 
+the correct structure, see Figure 5. 
+It is obvious that if the slices in the 
+orthogonal sets sample the object sufficiently, 
+then the intersections of contours from the 
+orthogonal slices identify the correspondence 
+relation accurately, i.e. the correct structure of 
+the original object. 
+ 
+4. The algorithm 
+ 
+The planes of slices divide space into a set of 
+spatial cells of a spatial grid M. In Figure 3 
+can be seen three mutually orthogonal planes 
+of grid M. We distinguish two kinds of cells of 
+M, the surface-crossing and the surface-
+passing cells. There are parts of contours on 
+some sides of a surface-crossing cell, which 
+means that the resultant surface intersects the 
+cell, see Figure 6. 
+ 
+ 
+Figure 3: An example of three orthogonal slices sets. 
+ 
+ 
+Figure 4: Correct contour orientation. 
+ 
+ 
+Figure 5: The mutual crossings of orthogonal contours 
+define the correspondence relation. 
+ 
+ 
+Figure 6: A surface-crossing cell. Parts of contours on 
+the sides of the cell together with node points form 
+spatial polygons. Node points are denoted as white 
+circles. Each edge of G is adjacent with two cells of M. 
+Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004
+
+The intersection of two orthogonal 
+slices consisting of curvilinear contours is a set 
+of points and we call them node points, see 
+Figure 7. Now we focus on surface-crossing 
+cell. An important observation is that parts of 
+input contours and the node points form 
+spatial polygons. Each such polygon is 
+enclosed in a surface-crossing cell, its patch is 
+part of the resultant surface, see Figure 6.  
+ 
+4.1 The correspondence problem 
+ 
+At this moment we suppose that the 
+correspondence of contours is identified 
+sufficiently by the intersections of orthogonal 
+contours as it has been discussed in 
+section 3.1.  
+Consider the intersection of two 
+contours as the relation of correspondence. 
+Note that the number of components of a 
+graph 
+constructed 
+of 
+such 
+a 
+relation 
+corresponds to the number of disjoint 
+components of the resultant surface. 
+ 
+4.2 Node points computation 
+ 
+A node point is geometrically the intersection 
+of two contours. Topologically it is the 
+representation of a contour correspondence 
+relation. It holds that each node point must lie 
+on the edge of the grid M. Since the contours 
+are supposed as polygonal curves, we cannot 
+compute the intersections of two orthogonal 
+sections directly. We obtain them in two 
+phases.  
+ 
+ 
+A 
+B 
+Figure 7: A) An input contour, the lattice represents 
+positions of orthogonal slices planes. B) The contour 
+formed by its node points (black spots). 
+ 
+In the first step intersections of each 
+contour and the grid M are computed. These 
+intersections are added among the current 
+contour vertices on the appropriate position. 
+They are registered on the corresponding edge 
+of the grid M simultaneously. Our algorithm 
+works on the same principle as the Cohen-
+Sutherland’s line clipping algorithm [3]. 
+An intersection of a slice plane and all 
+other orthogonal slice planes forms a lattice 
+ 1
+ 10
+ 100
+1 000
+10 000
+100 000
+2
+3
+4
+5
+6
+7
+8
+9
+10
+frequency of 
+occurence
+ 
+Figure 8: Polygon size (number of edges) histogram. 
+Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004
+
+with cells, see Figure 7. Each node point arises 
+as the intersection of a contour and a side of a 
+cell. Since the contour is supposed to be 
+polygonal, a node point is simply computed as 
+an intersection of two segments. Singular 
+cases when a contour crosses a cell at its 
+corner are handled separately [13]. 
+In the second step the node point 
+construction is completed. The correspondent 
+vertex, which is a member of the orthogonal 
+contour and also a member of the same edge 
+of M, must be found. As it was said before it is 
+done very fast searching the auxiliary 
+registrations of contour intersections on the 
+appropriate edge of M. Each two nearest 
+intersections coming from orthogonal contours 
+registered on an edge of M are qualified as 
+correspondent vertices building together a 
+node point. 
+ 
+4.3. Constructing the surface 
+ 
+Now suppose graph G, whose set of vertices 
+consists of a set of the node points and whose 
+edges represent the parts of contours between 
+two node vertices. Note that the geometrical 
+shape of the edges still corresponds to the 
+appropriate parts of contours. Now the task is 
+to find such cycles of graph G, which have the 
+property that their geometrical representation 
+lies within one cell of M. Those cycles 
+represents spatial polygons that lie on the 
+surface.  
+We suppose each edge e of G is 
+adjacent with cells B1 and B2, see Figure 6. 
+Each cell from {B1, B2} includes one cycle c 
+of our interest, which is adjacent with e (that 
+results from the consideration of 2-manifold 
+objects). The circle c represents the spatial 
+polygon being searched. Thus for each e two 
+cycles 
+e
+B
+c
+1 , 
+e
+B
+c
+2  must be searched and then 
+polygons 
+1cp , 
+2
+cp  correspondent to those 
+cycles are constructed. 
+As soon as all polygons are obtained, 
+we can start to patch them. We can use any 
+arbitrary patching technique. Note that the 
+number of sides of such polygon can be high, 
+but in cases of our data sets it is in range 2 – 
+10, see the graph in Figure 8.  
+The proposed method starts with 
+finding a suitable point in the center of each 
+polygon. Then using the center point each 
+polygon is divided into set of quadrilaterals, 
+which are easier to patch, see Figure 9. 
+ 
+ 
+Figure 9: Partition of a generic polygon in the set of 
+quadrilaterals. 
+Requires 
+the 
+central 
+point 
+C 
+determination. 
+ 
+ 
+A 
+ 
+B 
+ 
+C 
+ 
+D 
+Figure 10: Results of the surface reconstruction. A) An 
+input data set (courtesy of Martin Čermák), B) VTK 
+surface reconstruction from slices class, C) A common 
+volume based method, D) Proposed method for surface 
+reconstruction from orthogonal slices. 
+Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004
+
+5. Results 
+ 
+All the problematic data sets mentioned in 
+section 1 and many more have been processed 
+using: 
+- 
+surface reconstructing from slices class 
+from VTK, 
+- 
+a common volume based method; see 
+[5] for more details, 
+- 
+our proposed method for surface 
+reconstruction from orthogonal slices.  
+The results of the reconstruction of one data 
+set are illustrated in Figure 10, the complete 
+documentation and experimental results can be 
+found at http://herakles.zcu.cz/research/slices.  
+ 
+6. Conclusion and further research 
+ 
+Our 
+current 
+research 
+proves 
+that 
+the 
+advantages of orthogonal slices in the process 
+of surface reconstruction are significant. There 
+is a set of objects for which the orthogonal 
+slices are almost the only way to reconstruct 
+them correctly. 
+The proposed method supposes that the 
+object is sampled well enough, so that the 
+number of components of the correspondence 
+graph G equals to the number of disjoint 
+components of the original surface. 
+ 
+The main point of our further research 
+is the solution of problems caused by under-
+sampling, i.e. to deal with data sets that do not 
+sample 
+the 
+input 
+object 
+sufficiently. 
+Furthermore we would like to study the 
+influence of contour inaccuracy on the node 
+point computation. 
+ 
+ 
+References 
+ 
+[1] Bresler, Y., Fessler, J.A., Macovski, A.: A 
+Bayesian approach to reconstruction from 
+incomplete projections of a multiple object 3D 
+domain. IEEE Trans. Pat. Anal. Mach. Intell., 
+11(8):840-858, August 1989. 
+[2] Cong, G., Parvin, B.: Robust and efficient 
+surface reconstruction from contours. The 
+Visual Computer, (17):199-208, 2001 
+[3] Foley, J. D., van Dam, A., Feiner, S. K. 
+and Hughes, J. F., Computer Graphics: 
+Principles and Practice, Addison-Wesley, 
+1990. 
+[4] Jones, M., Chen, M.: A new approach to 
+the construction of surfaces from contour data. 
+Computer Graphics Forum (13): 75-84, 1994 
+[5] Klein, R., Schilling, A.: Fast Distance 
+Interpolation for Reconstruction of Surfaces 
+from 
+Contours. 
+In 
+proceedings 
+of 
+Eurographics '99, Short Papers and Demos, 
+September 1999. 
+[6] Meyers, D.: Multiresolution tiling. In 
+Proceedings, Graphics Interface '94, pages 25-
+32, Banff, Alberta, May 1994. 
+[7] Meyers, D.: Reconstruction of Surfaces 
+From Planar Contours. PhD thesis, University 
+of Washington, 1994. 
+[8] Shinagawa, Y., Kunii, T.L.: Constructing a 
+Reeb graph automatically from cross sections. 
+IEEE Comuter Graphics and Applications, 
+11(6): 44-51, November 1991. 
+[9] Shinagawa, Y., Kunii, T.L., Kergosien, 
+Y.L.: Surface coding based on Morse theory. 
+IEEE Comuter Graphics and Applications, 
+11(5): 66-78, September 1991. 
+[10] Skinner, 
+S.M.: 
+The 
+correspondence 
+problem: Reconstruction of objects from 
+contours in parallel sections. Master’s thesis, 
+Department 
+of 
+Computer 
+Science 
+and 
+Engineering, University of Washington, 1991. 
+[11] Soroka, B.I.: Understanding Objects From 
+Slices: 
+Extracting 
+Generalised 
+Cylinder 
+Descriptions From Serial Sections. PhD thesis, 
+University of Kansas Dept of Computer 
+Science, March 1979. TR-79-1. 
+[12] Soroka, B.I.: Generalized cones from 
+serial sections. Computer Graphics and Image 
+Processing, (15): 54-166, 1981. 
+[13] Svitak, 
+R., 
+Skala, 
+V.: 
+Surface 
+Reconstruction 
+from 
+Orthogonal 
+Slices, 
+ICCVG 2002, Zakopane, Poland, 2002 
+[14] Wood, Z. J.: Computational Topology 
+Algorithms 
+For 
+Discrete 
+2-Manifolds. 
+California Institute of Techology, PhD Thesis, 
+May 2003 
+ 
+Machine Graphics and Vision, Vol.13, No.3, Polish Academy of Sciences, Vol.13, No.3, pp.221-233, ISSN 1230-0535, 2004
+

AdAzT4oBgHgl3EQfvv7L/content/tmp_files/load_file.txt

@@ -0,0 +1,246 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf,len=245
+page_content='Robust Surface Reconstruction from  Orthogonal Slices  Radek Sviták1, Václav Skala2 Department of Computer Science and Engineering, University of West Bohemia in Pilsen, Univerzitní 8, 306 14 Plzeň, Czech Republic E-mail: rsvitak@kiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='cz  Abstract The surface reconstruction problem from sets of planar parallel slices representing cross sections through 3D objects is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The final result of surface reconstruction is always based on the correct estimation of the structure of the original object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' This paper is a case study of the problem of the structure determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' We present a new approach, which is based on considering mutually orthogonal sets of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' A new method for surface reconstruction from orthogonal slices is described and the benefit of orthogonal slices is discussed too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The properties and sample results are presented as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  This work is was supported by the Ministry of Education of the Czech Republic – projects: 1FRVŠ 1348/2004/G1 2MSM 235200005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Introduction  The crucial task of the surface reconstruction from slices is a correct estimation of the original object structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' the solution of the contour correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Most of the existing methods simply consider the overlap of contours in a pair of consecutive parallel slices as the only correspondence criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Therefore, they produce unacceptable structure estimation when the angle between the axis of the object and the normal of the slices increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Higher density of slices can help to solve this problem, but it is not always possible because of the resolution limit of the scanning device, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' It is obvious that other slices in non-parallel planes offer an additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' In this paper we will concentrate on the benefit of orthogonal slices for the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' In comparison to the existing methods, our currently achieved results show, that for a set of objects the resultant surface is significantly more accurate with respect to the similarity to the original surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The concept of the new proposed method is presented and results of comparisons with the existing methods are discussed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  A  C  B  D Figure 1: Problematic cases when solving the contour correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Expected problems using the overlapping criterion: A, B, C, D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' generalized cylinders: B, D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' MST: C, D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Reeb graph based methods: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Machine Graphics and Vision, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='221-233, ISSN 1230-0535, 2004  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Brief survey of existing methods  Several methods for surface reconstruction from slices have been developed since about 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' In this section we will classify them according to their approach to solving the contour correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' For more extensive study of the existing methods from the other  viewpoints, see [2, 4, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The simplest methods estimate the contour correspondence locally between each consecutive pair of contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Typically, contours that overlap each other are considered as correspondent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' This works if the density of slice is high, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' the distance between slices is low, and the axis of the input object is nearly perpendicular to the slices planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' A more advanced method uses generalized elliptical cylinder to solve the correspondence problem [1, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Contours are first classified as elliptical or complex by determining how well the vertices of their perimeter can be fit by an ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' If the fit is too poor, a contour is classified as complex, and can not be incorporated into an elliptical cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Then the ellipses are grouped to the cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' When as many contours as possible have been organized into cylinders, then the algorithm uses the geometric relationship between cylinders to group them into objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' This method is most useful for elongated smooth objects with roughly elliptical cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Apparently the best existing approaches that have been published are two graph-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  The first of them presented by Skinner [10] computes a minimum spanning tree based on contour shape and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' In the first step a graph is constructed by representing each contour as a node and connecting each node to all nodes representing contours in adjacent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The best fitting ellipse is computed for each contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The cost of an edge of the graph relies on the mutual position and size of two ellipses:  ( )  2  2  2  2  )  b  (b  )  a  (a  )  y  (y  )  x  (x  i,j  c  j  i  j  i  j  i  j  i  −  +  −  +  −  +  −  =  ,  where (xi, yi, zi), (xj, yj, zj) represent the centers of the ellipses of contours i, j, respectively, and ai, bi, aj, bj are their major and minor axis lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The minimum spanning tree computed for the graph represents the solution to the correspondence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The method works well for naturally tree-structured objects, the main limitation is its inability to solve the correspondence problem correctly for general graph topologies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' genus > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  A  B Figure 2: A) Data set of slices of the cochlea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Using the Reeb graph it is possible to detect and represent the right contour correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The advantage consists in the possibility of considering the correspondence among contours of one slice (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Taken from [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  The second graph based method presented by Shinagawa [8, 9] uses surface coding based on Morse Theory to construct a Reeb graph [14] representing the contour connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Each contour represents a node in the graph, edges of the graph represent the contour correspondence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Edges are added to the graph in the manner to avoid making connections that would result in a surface that is not a 2-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' For each pair of contours that can be legally connected, a weight function is evaluated, and its value is used to establish a priority for connecting that pair of contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The algorithm proceeds by making the highest priority connections in regions where the number of contours in each section does not change, and then adds connections in order of decreasing priority with respect to the a priori knowledge of the number of connected components and the topological genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Machine Graphics and Vision, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='221-233, ISSN 1230-0535, 2004  2 0 3It is necessary to note that all the existing solutions just estimate the contours correspondence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' the structure of the original object, should be emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' In Figure 1 there are some typical example data sets to illustrate capabilities of the approaches mentioned in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Orthogonal slices  One set of parallel planar slices is one of the well-known boundary representations of a 3D object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Usually the planes of such slices set are perpendicular to the z-axis, and thus called z- slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' If we slice an object by more then one set of parallel slices and moreover when these sets are mutually orthogonal, we get orthogonal sets of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Consider now that we have z-slices, x-slices and y-slices of an object, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Note, that the contours are supposed to be polygonal, oriented  the way that when looking from the positive direction of the given slices set axis, the contours have the interior on its left side and the exterior on the right side, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='1 Contour correspondence  The main advantage of orthogonal slices consists in the approach how the contour correspondence can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' It is important to emphasize that two orthogonal contours which intersect each other comes aparently from one and the same surface component of the input object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' It means that the intersection of contours is very important since it provides accurate information about the correct structure, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' It is obvious that if the slices in the orthogonal sets sample the object sufficiently, then the intersections of contours from the orthogonal slices identify the correspondence relation accurately, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' the correct structure of the original object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The algorithm  The planes of slices divide space into a set of spatial cells of a spatial grid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' In Figure 3 can be seen three mutually orthogonal planes of grid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' We distinguish two kinds of cells of M, the surface-crossing and the surface- passing cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' There are parts of contours on some sides of a surface-crossing cell, which means that the resultant surface intersects the cell, see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  Figure 3: An example of three orthogonal slices sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  Figure 4: Correct contour orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  Figure 5: The mutual crossings of orthogonal contours define the correspondence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  Figure 6: A surface-crossing cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Parts of contours on the sides of the cell together with node points form spatial polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Node points are denoted as white circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Each edge of G is adjacent with two cells of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Machine Graphics and Vision, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='221-233, ISSN 1230-0535, 2004  The intersection of two orthogonal slices consisting of curvilinear contours is a set of points and we call them node points, see Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Now we focus on surface-crossing cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' An important observation is that parts of input contours and the node points form spatial polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Each such polygon is enclosed in a surface-crossing cell, its patch is part of the resultant surface, see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='1 The correspondence problem  At this moment we suppose that the correspondence of contours is identified sufficiently by the intersections of orthogonal contours as it has been discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Consider the intersection of two contours as the relation of correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Note that the number of components of a graph constructed of such a relation corresponds to the number of disjoint components of the resultant surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='2 Node points computation  A node point is geometrically the intersection of two contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Topologically it is the representation of a contour correspondence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' It holds that each node point must lie on the edge of the grid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Since the contours are supposed as polygonal curves, we cannot compute the intersections of two orthogonal sections directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' We obtain them in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  A B Figure 7: A) An input contour, the lattice represents positions of orthogonal slices planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' B) The contour formed by its node points (black spots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  In the first step intersections of each contour and the grid M are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' These intersections are added among the current contour vertices on the appropriate position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' They are registered on the corresponding edge of the grid M simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Our algorithm works on the same principle as the Cohen- Sutherland’s line clipping algorithm [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' An intersection of a slice plane and all other orthogonal slice planes forms a lattice 1 10 100 1 000 10 000 100 000 2 3 4 5 6 7 8 9 10 frequency of occurence  Figure 8: Polygon size (number of edges) histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Machine Graphics and Vision, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='221-233, ISSN 1230-0535, 2004  with cells, see Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Each node point arises as the intersection of a contour and a side of a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Since the contour is supposed to be polygonal, a node point is simply computed as an intersection of two segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Singular cases when a contour crosses a cell at its corner are handled separately [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' In the second step the node point construction is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The correspondent vertex, which is a member of the orthogonal contour and also a member of the same edge of M, must be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' As it was said before it is done very fast searching the auxiliary registrations of contour intersections on the appropriate edge of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Each two nearest intersections coming from orthogonal contours registered on an edge of M are qualified as correspondent vertices building together a node point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Constructing the surface  Now suppose graph G, whose set of vertices consists of a set of the node points and whose edges represent the parts of contours between two node vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Note that the geometrical shape of the edges still corresponds to the appropriate parts of contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Now the task is to find such cycles of graph G, which have the property that their geometrical representation lies within one cell of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Those cycles represents spatial polygons that lie on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' We suppose each edge e of G is adjacent with cells B1 and B2, see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Each cell from {B1, B2} includes one cycle c of our interest, which is adjacent with e (that results from the consideration of 2-manifold objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The circle c represents the spatial polygon being searched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Thus for each e two cycles e B c 1 , e B c 2  must be searched and then polygons 1cp , 2 cp  correspondent to those cycles are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' As soon as all polygons are obtained, we can start to patch them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' We can use any arbitrary patching technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Note that the number of sides of such polygon can be high, but in cases of our data sets it is in range 2 – 10, see the graph in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The proposed method starts with finding a suitable point in the center of each polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Then using the center point each polygon is divided into set of quadrilaterals, which are easier to patch, see Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  Figure 9: Partition of a generic polygon in the set of quadrilaterals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Requires the central point C determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  A  B  C  D Figure 10: Results of the surface reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' A) An input data set (courtesy of Martin Čermák), B) VTK surface reconstruction from slices class, C) A common volume based method, D) Proposed method for surface reconstruction from orthogonal slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Machine Graphics and Vision, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='221-233, ISSN 1230-0535, 2004  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Results  All the problematic data sets mentioned in section 1 and many more have been processed using: - surface reconstructing from slices class from VTK, - a common volume based method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' see [5] for more details, - our proposed method for surface reconstruction from orthogonal slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The results of the reconstruction of one data set are illustrated in Figure 10, the complete documentation and experimental results can be found at http://herakles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='cz/research/slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Conclusion and further research  Our current research proves that the advantages of orthogonal slices in the process of surface reconstruction are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' There is a set of objects for which the orthogonal slices are almost the only way to reconstruct them correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' The proposed method supposes that the object is sampled well enough, so that the number of components of the correspondence graph G equals to the number of disjoint components of the original surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='  The main point of our further research is the solution of problems caused by under- sampling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' to deal with data sets that do not sample the input object sufficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
+page_content=' Furthermore we would like to study the influence of contour inaccuracy on the node point computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfvv7L/content/2301.01713v1.pdf'}
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+Inside the Black Box: Detecting and Mitigating Algorithmic Bias across 
+Racialized Groups in College Student-Success Prediction 
+ 
+Denisa Gándara 
+The University of Texas at Austin 
+1912 Speedway, Stop D5000; Austin, Texas 78712 
+denisa.gandara@austin.utexas.edu   
+Hadis Anahideh* 
+University of Illinois Chicago 
+1200 West Harrison St., Chicago, Illinois 60607 
+hadis@uic.edu 
+Matthew P. Ison 
+Northern Illinois University 
+1425 W. Lincoln Hwy., DeKalb, Illinois 60115 
+mison@niu.edu 
+Anuja Tayal 
+University of Illinois Chicago 
+1200 West Harrison St., Chicago, Illinois 60607 
+atayal4@uic.edu 
+ 
+Acknowledgments: The research reported here was supported, in whole or in part, 
+by the Institute of Education Sciences, U.S. Department of Education, through 
+grant R305D220055 to the University of Illinois Chicago and by grant, 
+P2CHD042849 awarded to the Population Research Center at The University of 
+Texas at Austin by the Eunice Kennedy Shriver National Institute of Child Health 
+and Human Development. The content is solely the responsibility of the authors. 
+Abstract: Colleges and universities are increasingly turning to algorithms that 
+predict college-student success to inform various decisions, including those 
+related to admissions, budgeting, and student-success interventions. Because 
+predictive algorithms rely on historical data, they capture societal injustices, 
+including racism. A model that includes racial categories may predict that racially 
+minoritized students will have less favorable outcomes. In this study, we explore 
+bias in education data by modeling bachelor’s degree attainment using various 
+machine-learning modeling approaches. We also evaluate the utility of leading 
+bias-mitigating techniques in addressing unfairness. Using nationally 
+representative data from the Education Longitudinal Study of 2002, we 
+demonstrate how models incorporating commonly used features to predict 
+college-student success produce racially biased results. 
+ 
+*Corresponding author 
+
+ 
+ 
+1 
+ 
+Since the emergence of “big data” in the 1990s, efforts to use advanced 
+statistical techniques to predict outcomes of interest have proliferated across 
+various social domains, education notwithstanding (Baker et al., 2019; 
+Government Accountability Office [GAO], 2022). The suite of techniques used to 
+forecast outcomes and inform decision-making within organizations is broadly 
+known as “predictive analytics.” Although largely unseen, predictive analytics 
+fuel myriad decisions within educational institutions, from college admissions 
+(Hutt et al., 2019) and student retention interventions (Baker et al., 2019), to fiscal 
+health and resource allocation (Wayt, 2019; Yanosky & Arroway, 2015).  
+A key component within the vast array of predictive statistical techniques 
+is the predictive model, a computational tool that maps the input set of attributes 
+of individuals (e.g., high school GPA and demographic features) to their 
+outcomes (e.g., college credits accumulated) in order to identify underlying 
+associations and patterns in the data. The predictive model is especially useful 
+with large datasets, where it is impossible or inefficient to identify associations 
+and patterns manually.  
+In recent years, observers have raised concerns that predictive models in 
+education may perpetuate social disparities, especially when they ignore how 
+extant societal injustices can bias historical data (GAO, 2022). For instance, a 
+model that includes socially relevant attributes, such as race, gender, and income, 
+will often predict that students from socially disadvantaged categories (e.g., 
+
+ 
+ 
+2 
+women in STEM) will have less favorable outcomes. Such a model will be 
+extrapolating from prior relationships between socially relevant attributes (e.g., 
+race) and educational outcomes (e.g., graduation) that are partly shaped by 
+societal injustices, such as racism, sexism, and classism (e.g., López et al., 2018).  
+In this study, we appraise predictive models within the higher education 
+context. We begin by modeling bachelor’s degree attainment to explore biases in 
+educational data. We then assess the utility of bias-mitigating techniques in 
+addressing unfairness. This analysis focuses on disparities in college-student 
+success predictions across racialized groups, since educational attainment rates 
+across racial/ethnic groups remain markedly unequal (U.S. Department of 
+Education, 2021). Given these inequities in educational attainment levels, 
+predictive models that are agnostic to racial bias may penalize groups that have 
+been subject to racialized social disadvantages.  
+We situate our statistical analyses within relevant historical and social 
+contexts (Zuberi, 2001), recognizing that racially minoritized groups are 
+disadvantaged in the educational context through various interlocking social 
+systems of oppression (Reskin, 2012). Although an exhaustive review is beyond 
+the scope of this paper, we refer readers to examples of systems, structures, and 
+practices that penalize racially minoritized groups. In the education domain, 
+oppressive barriers to educational success include educational tracking (Oakes, 
+1985), deepening school segregation (Orfield et al., 2012), teacher racial bias 
+
+ 
+ 
+3 
+(Gershenson & Papageorge, 2018), racial disparities in school funding that track 
+with levels of segregation (Weathers & Sosina, 2019), and disparate punishment 
+of Black and Latinx students (Davison et al., 2019). Racially minoritized students’ 
+educational success is also conditioned by racialized barriers outside education, 
+including constraints on wealth accumulation and income, which limit students’ 
+ability to pay for higher education (Mitchell et al., 2019).  
+It is important to understand this background since the state of the world, 
+which is rooted in various societal injustices, affects the data distribution. These 
+historical injustices condition educational opportunities and experiences for 
+racially minoritized students. Subsequently, when predictive models make 
+predictions on students who are racially minoritized, they may be predicted to fail, 
+reinforcing historical biases. Amidst this backdrop, this study addresses the 
+following questions: 
+1. To what extent are college student-success predictions biased across 
+racial/ethnic groups? 
+2. How effective are computational strategies in mitigating racial/ethnic 
+bias? 
+Predictive models warrant greater attention in education not only because 
+they are ubiquitous, but also because they have the potential to reinforce and 
+legitimize societal inequities. Decisions grounded in biased predictions can yield 
+significant societal consequences. For instance, college admission may unfairly be 
+
+ 
+ 
+4 
+denied to racially minoritized students if the model shows they have lower 
+predicted likelihoods of success (Hutt et al., 2019). With course 
+recommendations, predictions could lead to educational tracking, encouraging 
+students from racially minoritized groups to pursue courses or majors that are 
+perceived as less challenging (Ekowo & Palmer, 2017). Such consequences may 
+go undetected since automated sorting mechanisms remain both obfuscated (due 
+to their invisibility to educational stakeholders) and legitimized through 
+perceptions that statistical models are objective (Hirschman & Bosk, 2020).  
+Literature on Fairness in College-Student Success Prediction 
+ 
+In recent years, educational researchers and data scientists have begun to 
+develop insights into fairness and bias within various stages of the machine 
+learning (ML) process. Among the most important discernments from these 
+studies are the importance of representation of socially relevant groups in training 
+datasets (Riazy et al., 2020),1 and novel statistical techniques intended to measure 
+and enhance predictive fairness between groups (Gardner et al., 2019; Hutt et al., 
+2019). A small number of studies have examined algorithmic fairness in college-
+student success (Anderson et al., 2019; Hu & Rangwala, 2020; Hutt et al., 2019; 
+Lee & Kizilcec, 2020; Yu et al., 2020). Most of these studies have detected bias in 
+existing data, particularly with models using institutional (college or university) 
+administrative data (see Hutt et al., 2019 for an exception). For instance, 
+Anderson and colleagues (2019), who used administrative data from a single 
+
+ 
+ 
+5 
+institution, found that their predictive models advantaged White students (e.g., 
+higher rates of predicted success for students who failed and lower rates of 
+predicted failure for students who succeeded) and disadvantaged Hispanic/Latinx 
+and male students. Yu and colleagues (2020) examined how the data source (e.g., 
+learning management system [LMS], institutional data, or survey data) affected 
+predictions on college outcomes, concluding that institutional data were more 
+likely to be biased against disadvantaged groups.  
+Gardner and colleagues (2019) also used institutional data to examine the 
+fairness of models used to predict course success in higher education. Their 
+analysis, which focused on gender, showed that model fairness varied according 
+to the algorithm used, the variables included in models, the specific course 
+examined, and the gender imbalance ratio in a given course. Importantly, they did 
+not identify a meaningful tradeoff between fairness and accuracy. These results 
+contradict arguments and other evidence signaling that as predictive algorithms 
+implement bias-mitigation efforts, the predictive accuracy of the algorithm 
+declines (e.g., Lee & Kizilcec, 2020).  
+Expanding upon this prior work, the present study offers a more holistic 
+picture of bias in college-student success predictions. Most research on this topic 
+is situated in the Learning Analytics literature, predicting outcomes within 
+courses (Gardner et al., 2019; Hu & Rangwala, 2020; Lee & Kizilcec, 2020; 
+Riazy et al., 2020; Yu et al., 2020). In this study, we adopt a broader 
+
+ 
+ 
+6 
+conceptualization of college-student success by modeling an educational 
+attainment outcome. Predictions of attainment-related outcomes are more likely to 
+be used to inform practices related to admissions and campus-wide student-
+success interventions (Ekowo & Palmer, 2017; Wayt, 2019). We extend prior 
+work by using nationally representative data instead of course data (e.g., Hu & 
+Rangwala, 2020), single-institution data (e.g., Anderson et al. 2019), or non-
+representative national data (e.g., Hutt et al., 2019). Moreover, beyond exploring 
+bias in the data, we test various approaches for mitigating bias, both in data 
+preparation (preprocessing) and in models (in-processing). Finally, we bolster our 
+empirical contribution by exploring various notions of fairness, presenting 
+conceptual models that can be used for further exploration of bias in educational 
+data.  
+Data Sources 
+Data come from the Education Longitudinal Study of 2002 (ELS), a 
+nationally representative, longitudinal study of students who were 10th graders in 
+2002. Given our focus on bachelor’s degree attainment, the dataset is filtered 
+based on the institution type to only include students who attended four-year 
+postsecondary institutions. The outcome variable captures the students’ highest 
+level of education as of the third follow-up interview (eight years after expected 
+high-school graduation). To construct a binary classification problem, we label 
+
+ 
+ 
+7 
+students with a bachelor’s degree and higher as the favorable outcome (label=1), 
+and all others as the unfavorable outcome (label=0).  
+Predictive variables include features commonly used for student-success 
+prediction, including student demographic characteristics, socioeconomic traits, 
+grades, college preparation, and school experience. Since category labels are not 
+ordinal, we create binary variables for each level of the categorical variables 
+following National Center for Education Statistics (NCES) documentation 
+(NCES, n.d.). The complete list of variables appears in Supplementary Materials 
+(Appendix A). While our dataset does not include all possible variables that could 
+be incorporated in a model predicting college-student success, our dataset has the 
+advantage of being large (n = 15,244) and nationally representative, and including 
+the most commonly used features (p = 29) based on our review of literature on 
+college-student success prediction.  
+Since we have a high number of missing values, we ran the models 
+separately with multiple imputation (Rubin, 1996) and without imputation 
+(listwise deleted rows with missing data).2 To avoid the confounding impact of 
+imputation on both unfairness and model performance, we stratified on the 
+response variable (bachelor’s degree attainment) and racial groups for the 
+training-testing splits, retaining the distribution of the historical data in both 
+partitions. For simplicity, we present results without imputation in our main 
+results. Results with imputation appear in supplementary materials (Appendix B). 
+
+ 
+ 
+8 
+Those results indicate that imputation is inconsequential for all models except for 
+support vector machine (SVM), where it reduces the variance of unfairness, 
+resulting in a more robust model. A deeper investigation of how imputation 
+affects the unfairness of the prediction outcome appears elsewhere (Anahideh et 
+al., 2021). 
+First, we randomly split the dataset into training and testing subsets with 
+an 80:20 ratio (80% training, 20% testing). The ML models are trained on the 
+training data and evaluated on the testing data to demonstrate their 
+generalizability. To evaluate the fairness of the prediction outcome using various 
+fairness notions (described below), we stratified the training and testing datasets 
+by the outcome variable class labels (1, 0) and racial/ethnic categories, ensuring 
+that we have enough observations from each group. The results are averaged over 
+30 different splits of the data. Table 1 presents the distribution of the outcome 
+variable by racial/ethnic category after dropping observations with missing 
+values. 
+<<Table 1 about Here>> 
+Analysis Methods 
+Evaluating Unfairness 
+We employed the most widely used ML models in higher education, 
+including Decision Tree (Hamoud et al., 2018), Random Forest (Pelaez, 2018), 
+Logistic Regression (Thompson et al., 2018), and SVM (Agaoglu, 2016). Each 
+
+ 
+ 
+9 
+ML model has predefined parameters known as hyperparameters that must be 
+provided before the training phase (e.g., depth of the tree in Decision Trees). 
+Since the optimal values of such hyperparameters are data-dependent, we 
+performed a five-fold cross-validation (CV) for each model to determine the best 
+set of hyperparameters. In this process the dataset was divided into five partitions, 
+four of which were utilized for training and one for validation. Cross-validation 
+repeats this process and selects a different partition for validation each time. A 
+grid of feasible hyperparameters was assessed based on the CV schema described 
+above to choose the optimum set. Under 30 distinct random splits of training and 
+testing datasets, we obtained the best set of hyperparameters before we performed 
+model training. To evaluate model performance, we report the average and 
+variance of the accuracy as well as unfairness towards different racial/ethnic 
+groups using various notions of unfairness. 
+Fairness Notions. We consider four different conceptions of fairness 
+commonly used in algorithmic fairness: statistical parity, equal opportunity, 
+predictive equality, and equalized odds (Barocas et al., 2017). In practice, users 
+can select the measure of fairness that is preferred based on context, knowledge 
+of social disparities, use case, and regulations. We briefly describe each fairness 
+notion in turn; the probabilistic definitions of these notions appear in the 
+supplemental materials (Appendix C). 
+
+ 
+ 
+10 
+Statistical Parity is achieved by having equal favorable outcomes (degree 
+attainment) received by the unprivileged group (e.g., Black) and the privileged 
+group (e.g., White). Said differently, under the notion of statistical parity, we 
+consider a model fair if being a member of a racially minoritized group is not 
+correlated with the probability of bachelor’s degree attainment.  
+The next three fairness measures build on the statistical notions of 
+true/false positives/negatives (for a visual, see Confusion Matrix in Appendix D). 
+Specifically, 
+• A true positive result would correctly predict success for a student who 
+succeeds (in our case, attains a bachelor’s degree).  
+• A true negative result would correctly predict failure for a student who 
+does not succeed. 
+• A false positive result (Type I error) would incorrectly predict success for 
+a student who does not succeed.  
+• A false negative result (Type II error) would incorrectly predict failure for 
+a student who does succeed.  
+Building on these statistical notions, Equal Opportunity represents equal 
+false negative rates between groups. This fairness notion requires that each group 
+receive the negative outcome at equal rates, conditional on their success. In other 
+words, under this notion, the model should (incorrectly) predict failure for those 
+who succeeded (attained at least a bachelor’s degree) at the same rate for 
+
+ 
+ 
+11 
+students across racial/ethnic groups. This notion assumes knowledge of the true 
+outcome values (whether a student attained at least a bachelor’s degree) and aims 
+to satisfy parity across socially relevant groups, subject to the true values.  
+A third fairness notion is Predictive Equality, which represents equal 
+false positive rates. To satisfy this criterion, positive predictions (that a given 
+student will attain a bachelor’s degree) for students who do not actually attain a 
+bachelor’s degree should be the same across racial/ethnic groups.  
+Finally, Equalized Odds represents the average difference in false 
+positive and true positive rates between groups. To achieve fairness under this 
+notion, both the false positive rate (wrongly predicting success) and the true 
+positive rate (correctly predicting success) should be the same across 
+racial/ethnic groups. We use these notions to evaluate fairness in college-student 
+success predictions.  
+Mitigating Bias 
+In addition to evaluating unfairness, we implement statistical techniques 
+to mitigate bias. Literature on bias-mitigation techniques for ML models is 
+burgeoning. Such techniques can be categorized into three groups: preprocessing, 
+in-processing, and post-processing approaches (Pessach et al., 2022). 
+Preprocessing techniques entail fairness evaluation in the data-preparation step, 
+which should, in turn, mitigate bias for downstream tasks. We apply two 
+
+ 
+ 
+12 
+preprocessing techniques: Reweighting (Kamiran & Calders, 2012) and 
+Disparate Impact Remover (DIR) (Feldman et al., 2015).  
+Reweighting assigns different weights to the training samples in each 
+combination of racial/ethnic group and outcome-variable class label (e.g., Black 
+X outcome label=1). It does so before training a model to adjust the bias across 
+groups. Because individual observations from the unprivileged 
+groups with positive outcomes are underrepresented in the training data (see 
+Table 1), classifiers are susceptible to bias. In this preprocessing approach, the 
+data points representing successful outcomes for unprivileged groups are 
+identified and upweighted, so they have a larger influence on model training.  
+In contrast to Reweighting, DIR changes the distributions of other 
+features in the model (not race/ethnicity) to force distributions to overlap at the 
+group level. This process removes the ability to distinguish between group 
+membership from a feature that otherwise offers a good indication of which 
+group a data point may belong to. 
+In-processing techniques generally involve modifying the ML algorithms 
+to account for fairness during the training process, such that the parameter 
+estimation of the classifier forces the prediction outcome to be fair toward all 
+(racial/ethnic) groups. The enforcement is accomplished in the optimization 
+subproblem by adding a fairness metric as a constraint similar to the 
+Exponentiated Gradient Descent approach (Agarwal, 2018).  
+
+ 
+ 
+13 
+We also employ a second in-processing technique, Meta Fair Classifier 
+(Celis et al., 2018), which takes a large class of fairness metrics as inputs and 
+returns an optimal classifier that is fair with respect to constraints on the given 
+set of metrics. This approach works for various fairness criteria and provides 
+theoretical guarantees by developing a general form of constrained optimization 
+problems, which encompasses many existing fair classification problems. This 
+stands in contrast to earlier work on fair classification, which focused on 
+constructing classifiers that are constrained with respect to a single fairness 
+metric (e.g., Zafar et al., 2017). 
+Post-processing techniques for mitigating bias adjust the prediction 
+outcome after training a regular ML model, changing the values across different 
+groups. We exclude these techniques since post-processing mechanisms are 
+implemented at a later phase in the learning process, often producing inferior 
+results (Woodworth et al., 2017). These approaches are also more controversial 
+than in-process and preprocess strategies in the domains of “affirmative action” 
+and are thus less likely to be used in education settings (Hirschman & Bosk, 
+2020). 
+In the results section, we refer to Reweighting as ReW, Disparate Impact 
+Remover as DIR, Exponentiated Gradient Reduction as ExGR, and Metaclassifier 
+as MetaC. For comparison, we also consider the baseline classification scenario, 
+where no mitigation strategy is used. 
+
+ 
+ 
+14 
+Comparisons 
+We use two comparison approaches to appraise model unfairness and test 
+mitigation techniques, namely, at 1) the subgroup level (i.e., each racial/ethnic 
+group versus the rest), and 2) the aggregate level (i.e., privileged versus 
+unprivileged). First, we compared each racial/ethnic group against all others and 
+consider 1 for a certain group (e.g., Black) and 0 for every other group (e.g., 
+White, Asian, Hispanic, and Two or More Races) to calculate gaps as discussed 
+previously. 
+To evaluate the limitations of aggregation, which is common in this type 
+of work, we also aggregate White and Asian groups in the privileged category 
+and Black, Hispanic, and Two or More Races (2+) groups in the unprivileged 
+category. These comparisons represent an extension over prior work as they 
+allow us to investigate the impact of existing mitigation techniques at both the 
+subgroup and aggregate levels. Most existing techniques only work with binary 
+sensitive attributes (e.g., “White” and “Non-White”), requiring the researcher to 
+specify the privileged group and forcing other subgroups to be aggregated as the 
+unprivileged group (Pessach et al., 2022).  
+Although some existing unfairness mitigation techniques have the 
+potential to incorporate non-binary sensitive attributes, such extension has not 
+been implemented in the literature. Binarizing sensitive attributes (1: privileged, 
+0: unprivileged) for the mitigation processes may not reduce fairness gaps for 
+
+ 
+ 
+15 
+each group. This is important in educational settings where research shows that 
+students from different racial/ethnic groups have distinct experiences and 
+outcomes (e.g., López et al., 2018). Hence, it is critical to evaluate unfairness 
+after applying mitigation techniques at the subgroup levels, as there might be 
+significant differences between unprivileged subgroups.  
+Results 
+We find no significant difference between the performance (accuracy) of 
+different ML classifiers, although there are some differences in levels of 
+unfairness across fairness notions and models. To facilitate comparison, Figure 1 
+presents results for all ML models. We discuss the main findings for our 
+assessments of unfairness and the effectiveness of bias-mitigation techniques in 
+turn.  
+<<Figure 1 about Here>> 
+Evaluating Unfairness 
+Subgroup Level: Each Group Versus the Rest.  Figure 1 shows a 
+comparison of unfairness levels using all four fairness notions and ML models 
+for the baseline (without bias mitigation). The testing accuracy across these 
+models is 78%, on average. These results indicate that Black and Hispanic groups 
+are treated unfairly across models. Generally, the SVM model yields less unfair 
+results, across fairness notions, compared to the other ML models. Under the 
+fairness notions of Statistical Parity (Figure 1a), Predictive Equality (Figure 1c), 
+
+ 
+ 
+16 
+and Equalized Odds (Figure 1d), the boxes for Black and Hispanic students are at 
+a lower level across all ML models, indicating that these students receive 
+favorable outcomes (i.e., bachelor’s attainment or higher) at a lower rate than 
+students in other categories. For the notion of Equal Opportunity (Figure 1b), 
+higher levels in the box plots, which we observe for Black and Hispanic groups, 
+represent more unfairness.  
+For a concrete example of unfairness with respect to Statistical Parity, in 
+one of the test splits, students in the Asian and White categories have a 91% 
+probability of attainment, while those in the Black and Hispanic categories have 
+63% and 68% probabilities, respectively. Without correcting for bias, predictive 
+models will be more likely to predict that students categorized as Black and 
+Hispanic are less likely to attain a bachelor’s degree or higher when compared to 
+more privileged peers.  
+Findings for Predictive Equality further illustrate bias in the predictions. 
+Among the students who did not complete their degree (y=0), the probability of 
+attainment is estimated as 78% for White and 83% for Asian, while it is 
+estimated as 33% for Hispanic, and 0% for Black. 3 As illustrated in Figure 1b, 
+the models are also more likely to falsely predict failure for Black and Hispanic 
+students than for White and Asian students. Illustratively, for a single split, 
+among the students who completed their degree (y=1), the probability of failure 
+
+ 
+ 
+17 
+is estimated as 4.6% for White and 5.5% for Asian, while it is estimated as 20% 
+for Hispanic and 8% for Black. 
+Moreover, the plots show that the variation of values for the White and 
+Asian groups is minimal, especially for the White group, whereas the variation of 
+unfairness gaps for the other groups is significantly larger. Variation for the 
+category of two or more races is especially large, suggesting this is not a 
+meaningful category and should be used with caution in student-success 
+prediction. Differences in variation across racial/ethnic groups indicate that 
+models for minoritized groups are more sensitive to the train/test splits. Due to 
+the population bias across different racial/ethnic groups in the ELS dataset (i.e., 
+statistical underrepresentation of Black and Hispanic students), the train/test 
+splits can significantly change the distribution and presence of underrepresented 
+individuals in each partition, significantly impacting the unfairness of the model 
+for each split scenario. In practice, this will result in less stable and fair model 
+performance for predicting the success of an unobserved individual from a 
+statistically underrepresented group.  
+<<Figure 2 about Here>> 
+Aggregate Level: Privileged vs. Unprivileged. Figure 2 presents the box 
+plots for all four unfairness notions at the aggregated level of privileged (Asian 
+and White) versus unprivileged (Black, Hispanic, and two or more racial/ethnic 
+categories) for all prediction models. The first evident pattern from all four plots 
+
+ 
+ 
+18 
+is the mean difference between the two groups. Similar to results at the subgroup 
+level, we observe higher false negative rates for the unprivileged group. In other 
+words, the models are more likely to predict failure for Black and Hispanic 
+students who succeed compared to White and Asian students.  
+Comparing findings at the subgroup and aggregate levels, we observe that 
+aggregate results mask substantial differences we can glean from the subgroup 
+analysis. For instance, in Figure 1a, the DT and RF models show similar levels of 
+unfairness for Black and Hispanic groups, but LR and SVM are more unfair for 
+the Black group than the Hispanic group. At the aggregate level of analysis, this 
+variation cannot be observed (all models are unfair toward the unprivileged 
+group). We now turn to results for bias-mitigation techniques.  
+Mitigating Bias 
+Given space constraints and for ease of interpretability, we present 
+mitigation results using one predictive model, RF, which is a non-linear classifier 
+commonly used in the education literature. These results appear in Figure 3 
+(findings from other ML models are in Appendix E). Our first observation is that 
+the preprocessing and in-processing mitigation methods only minimally decrease 
+accuracy (by 1% to 2%). One technique, MetaC, significantly improves accuracy 
+(by 10-to-17-points over the baseline model without bias mitigation).  
+The bias-mitigation techniques we used required us to specify the 
+privileged and unprivileged groups and to treat the sensitive attribute as binary. 
+
+ 
+ 
+19 
+The results demonstrate that the mitigation techniques are generally not effective 
+at reducing bias at the aggregate level. At the subgroup level, we do not find a 
+mitigation technique that improves fairness across all racial/ethnic subgroups; 
+when a technique reduces unfairness for one subgroup, it harms another. We first 
+present findings for the preprocessing mitigation techniques, ReW and DIR.  
+The results (in Figure 3) indicate that the ReW technique does not 
+effectively reduce bias for unprivileged groups when compared to the baseline 
+model. If the goal of the education data analyst is to reduce unfairness in student-
+success predictions, it is not enough to increase the influence of datapoints that 
+represent successful students from unprivileged groups (e.g., Black students who 
+succeed) in the training process. This finding suggests that the 
+underrepresentation of successful students from unprivileged groups in the 
+training data is not a key source of bias in student-success predictions. 
+The second preprocessing mitigation technique we employed, DIR, 
+decreases unfairness for the Black group but leads to more unfair predictions for 
+the Hispanic group. This approach modifies the distributions of other features in 
+the model (e.g., students’ native language and family composition) to reduce 
+their correlation with racial/ethnic categorizations. A feature can provide a strong 
+hint as to which group a data point might belong to. DIR aims to eliminate this 
+capacity to distinguish between group membership. In addition to reducing 
+unfairness for the Black group, DIR diminishes the advantage of the Asian group 
+
+ 
+ 
+20 
+relative to that of other groups. However, the advantage for the White group is 
+actually exacerbated in two of the fairness notions (Statistical Parity and Equal 
+Opportunity). Contrary to expectations, applying DIR increases the Equal 
+Opportunity gap between the White group and all other groups, indicating a 
+decrease in the number of successful White students who are falsely predicted to 
+be unsuccessful.  
+Note that the DIR approach corrects the dataset measuring and 
+considering the Statistical Parity notion at the aggregate level. Hence, it is 
+expected to observe equal proportions of positive prediction from each group at 
+the aggregated level of privileged versus unprivileged. Our results show that DIR 
+cannot effectively achieve statistical parity for each subgroup using ELS data. 
+Even at the aggregate level (Figure 4a), DIR slightly removes the advantage for 
+the privileged group but does not improve fairness for the unprivileged group. 
+These findings also highlight differences between two groups that are often 
+considered privileged (Asian and White) and two groups that are often 
+considered unprivileged (Black and Hispanic), underscoring the importance of 
+disaggregation. 
+Turning to the in-processing techniques, ExGR did not significantly alter 
+the privilege of the White group or diminish the unfairness of the Black or 
+Hispanic groups for any of the four notions. Instead, both in-processing 
+techniques (ExGR and MetaC) result in greater variation, which indicates that the 
+
+ 
+ 
+21 
+repaired model is less robust to data splits. The MetaC technique effectively 
+reduces all four types of biases for the Hispanic group but is more unfair for 
+Black students with respect to Statistical Parity and Equalized Odds, again 
+highlighting the need to disaggregate education data across racialized groups.  
+The results confirm that even at the aggregated level, unfairness is not 
+mitigated significantly and the only technique that is slightly effective is MetaC. 
+Even then, MetaC only works for Hispanic students and is ineffective at reducing 
+bias for Black students. These preprocessing and in-processing techniques do not 
+significantly reduce demographic bias, demonstrating the need for better bias-
+mitigation techniques. Future work should examine bias-mitigation when both 
+preprocessing and in-processing techniques are applied simultaneously. 
+Discussion 
+ 
+The ubiquity of predictive analytics in higher education demands greater 
+attention to the “black box” of student-success-prediction models. This work 
+shows how such models produce unfair outcomes across various notions of 
+fairness. Further, we illustrate the limitations of existing techniques to reduce 
+bias. Using a nationally representative dataset with student-level data, we 
+demonstrate that across notions of fairness and various common ML models, 
+Black and Hispanic groups are treated unfairly. Not only are they more likely to 
+predict success for White and Asian groups (Statistical Parity) but they are also 
+significantly more likely to predict failure for Black and Hispanic students who 
+
+ 
+ 
+22 
+succeeded. This work illustrates how, without correcting for bias, Black and 
+Hispanic students may be offered fewer opportunities (e.g., admission) as a result 
+of student-success prediction models. We also show how bias-mitigation 
+techniques—both those that correct the dataset before modeling and those that 
+apply fairness constraints in the modeling process—generally fail to improve 
+fairness across subgroups.  
+One such technique, Reweighting, increases the influence of observations 
+representing racially minoritized students who are successful (e.g., Black students 
+who graduate). This technique is ineffective at reducing bias, indicating that the 
+main source of bias is not statistical underrepresentation but underlying, 
+unobservable sources of systemic and historical discrimination.  
+ 
+In evaluating student-success prediction models, it is important to 
+understand the use case. While bias-agnostic models may reproduce social 
+inequities in college-admissions use cases, they may lead to greater support for 
+students when used to inform student-success interventions. Even then, 
+practitioners must take care not to produce deficit narratives of minoritized 
+students, treating them as though they have a lower likelihood of success.  
+ 
+Despite widespread perceptions that statistical analysis is independent of 
+human judgment and error, this work demonstrates myriad decisions researchers 
+must make that have significant consequences for fairness, including which ML 
+model to use and which bias-mitigation techniques to employ. For example, if a 
+
+ 
+ 
+23 
+predictive algorithm closes the gap between a comparison group while benefiting 
+the majoritized group (rising tide metaphor), should such an algorithm be 
+considered fair (Kizilcec & Lee, in press)?  
+As higher education institutions strive to better serve students by 
+becoming more data-informed (Gagliardi & Turk, 2017), it is imperative that 
+predictive models are designed with attention to their potential social 
+consequences. It is critical to be aware of historical discrimination embedded in 
+the data and consider fairness measures to reduce bias in the outcomes of the 
+models. This paper demonstrates that more work is needed to develop fairness 
+measures to reduce bias across racialized groups. Future research should also 
+examine the influence of training/testing splits in the data. Another important 
+avenue for future work is understanding how feature selection (which variables 
+to include in the model) affects predictions and fairness across racialized groups. 
+Such work could expand on existing and conflicting recommendations 
+concerning the inclusion of race/ethnicity variables in student-success prediction 
+models (Hu & Rangwala, 2020). Finally, while we demonstrate the importance of 
+disaggregating beyond privileged/unprivileged, the ELS categories are severely 
+limited. Future work should disaggregate further to lead us toward more racially 
+just student-success practices in higher education.  
+ 
+ 
+
+ 
+ 
+24 
+Notes 
+1. In ML, a training dataset includes the data you use to train the model or 
+algorithm to predict the outcome of interest.  
+2. In all versions, we avoid imputing socially relevant (sensitive) attributes and 
+outcome variables, hence observations with missing values for these variables are 
+always dropped before imputation. 
+3. These estimated probabilities are based on RF modeling on a single train/test 
+split. 
+ 
+ 
+ 
+ 
+
+ 
+ 
+25 
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+cuts have pushed costs to students, worsened inequality. Center on Budget 
+and Policy Priorities, 24, 9-15. 
+
+ 
+ 
+29 
+National Center for Education Statistics (NCES) (n.d.) Education longitudinal 
+study of 2022 (ELS:2002). 
+https://nces.ed.gov/surveys/els2002/avail_data.asp 
+Oakes, J. (1985). How schools structure inequality. Keeping track. New Haven: 
+Yale University Press.  
+Orfield, G., Kucsera, J., & Siegel-Hawley, G. (2012). E pluribus... separation: 
+Deepening double segregation for more students. 
+https://civilrightsproject.ucla.edu/research/k-12-education/integration-and-
+diversity/mlk-national/e-pluribus...separation-deepening-double-
+segregation-for-more-
+students/orfield_epluribus_revised_omplete_2012.pdf  
+Pelaez, K. (2018). Latent class analysis and random forest ensemble to identify at-
+risk students in higher education (Doctoral dissertation, San Diego State 
+University). 
+Pessach, D., and Erez, S. (2022). A review on fairness in machine learning. ACM 
+Computing Surveys (CSUR) 55.3 (2022): 1-44. 
+Reskin, B. (2012). The race discrimination system. Annual review of 
+sociology, 38(1), 17-35. 
+Riazy, S., Simbeck, K., & Schreck, V. (Eds.). (2020). Fairness in learning 
+analytics: Student at-risk prediction in virtual learning environments. In 
+
+ 
+ 
+30 
+Proceedings of the 12th international conference on computer supported 
+education (CSEDU 2020). 15-25. DOI: 10.5220/0009324100150025 
+Rubin, D. B. (1996). Multiple imputation after 18+ years. Journal of the 
+American Statistical Association, 91(434), 473-489. 
+Thompson, E. D., Bowling, B. V., & Markle, R. E. (2018). Predicting student 
+success in a major’s introductory biology course via logistic regression 
+analysis of scientific reasoning ability and mathematics scores. Research 
+in Science Education, 48(1), 151-163. 
+U.S. Department of Education. (2021). Table 104.10. Rates of high school 
+completion and bachelor's degree attainment among persons age 25 and 
+over, by race/ethnicity and sex: Selected years, 1910 through 2021. 
+https://nces.ed.gov/programs/digest/d21/tables/dt21_104.10.asp 
+Wayt, L. (2019). 2019 NACUBO study of analytics. National Association of 
+College and University Business Officers. 
+https://www.nacubo.org/News/2019/11/2019-NACUBO-Study-of-
+Analytics-Released 
+Weathers, E. S., & Sosina, V. E. (2019). Separate remains unequal: Contemporary 
+segregation and racial disparities in school district revenue. American 
+Educational Research Journal, 00028312221079297. 
+
+ 
+ 
+31 
+Woodworth, B., Gunasekar, S., Ohannessian, M. I., & Srebro, N. (2017, June). 
+Learning non-discriminatory predictors. In Conference on Learning 
+Theory (pp. 1920-1953). PMLR.  
+Yanosky, R., & Arroway, P. (2015). The analytics landscape in higher education, 
+2015. EDUCAUSE.  
+Yu, R., Li, Q., Fischer, C., Doroudi, S., & Xu, D. (2020). Towards accurate and 
+fair prediction of college success: Evaluating different sources of student 
+data. International Educational Data Mining Society. 
+https://eric.ed.gov/?id=ED608066 
+Zafar, Muhammad Bilal, Isabel Valera, Manuel Gomez Rogriguez, and Krishna 
+P. Gummadi. (2017)."Fairness constraints: Mechanisms for fair 
+classification." In Artificial intelligence and statistics, pp. 962-970. 
+PMLR, 2017. 
+Zuberi, T. (2001). Thicker than blood: How racial statistics lie. U of Minnesota 
+Press. 
+
+ 
+ 
+32 
+Table 1: Distribution of bachelor’s degree or higher variable by racial/ethnic 
+category 
+Race 
+Bachelor’s or Higher 
+% of data 
+Asian, Hawaiian/Pacific Islander 
+1 
+0.73984 
+  
+0 
+0.26016 
+Black or African American 
+1 
+0.62766 
+  
+0 
+0.37234 
+Hispanic 
+1 
+0.67470 
+  
+0 
+0.32530 
+More than one race 
+1 
+0.69697 
+  
+0 
+0.30303 
+White 
+1 
+0.76005 
+  
+0 
+0.23995 
+ 
+ 
+ 
+ 
+
+ 
+ 
+33 
+Figure 1: Baseline with all ML models for all racial/ethnic groups 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 1d Equalized Odds of baseline for different ML models 
+Figure 1c Predictive Equality of baseline for different ML models 
+Figure 1b Equal Opportunity of baseline for different ML models 
+Figure 1a Statistical Parity of baseline for different ML models 
+
+0.6
+LR
+SVM
+0.4
+DT
+RF
+0.2
+QQ
+0.0
+百白
+-0.2
+-0.4
+-0.6
+Asian
+Black
+Hispanic
+2+
+White0.6
+LR
+SVM
+0.4
+DT
+RF
+0.2
+0.0
+广
+-0.2
+0.4 
+0.6
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+0.0
+-0.2
+LR
+-0.4
+SVM
+DT
+0.6
+RF
+Asian
+Black
+Hispanic
+2+
+White1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+-0.50
+LR
+SVM
+0.75
+DT
+RF
+-1.00
+Asian
+Black
+Hispanic
+2+
+White 
+ 
+34 
+Figure 2: Baseline with all ML models for privileged vs. unprivileged groups  
+ 
+ 
+ 
+Figure 2d Equalized Odds of baseline for different ML models 
+Figure 2c Predictive Equality of baseline for different ML models 
+Figure 2b Equal Opportunity of baseline for different ML models 
+Figure 2a Statistical Parity of baseline for different ML models 
+
+0.6 
+LR
+SVM
+0.4
+DT
+RF
+0.2 
+0°0
+0.2
+0.4 
+0.6 
+privileged
+unprivilegec0.6 
+LR
+SVM
+0.4 
+DT
+RF
+0.2
+0.0 
+0.2
+0.4
+0.6
+privileged
+unprivileged1.00
+LR
+0.75
+SVM
+DT
+0.50
+RF
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+privileged
+unprivileqed0.6
+LR
+SVM
+0.4
+DT
+RF
+0.2
+0.0 
+0.2
+0.4
+0.6
+privileged
+unprivileged 
+ 
+35 
+Figure 3: Mitigation for all racial/ethnic groups 
+ 
+Figure 3d Equalized Odds of RF using bias-mitigation techniques 
+Figure 3c Predictive Equality of RF using bias mitigation techniques 
+Figure 3b Equal Opportunity of RF using bias mitigation techniques 
+Figure 3a Statistical Parity of RF using bias mitigation techniques 
+
+0.6
+Baseline
+ReW
+0.4 -
+DIR
+ExGR
+0.2
+MetaC
+oo
+8
+白白
+0.0
+8
+0.2
+0.4 
+0.6
+Asian
+Black
+Hispanic
+2+
+White0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+88
+0.0
+0.2
+0.4
+0.6
+Asian
+Black
+Hispanic
+2+
+White1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+Baseline
+-0.50
+ReW
+DIR
+0.75
+ExGR
+MetaC
+-1.00
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+白白
+0.0
+-0.2
+Baseline
+ReW
+0.4 
+DIR
+ExGR
+-0.6
+MetaC
+Asian
+Black
+Hispanic
+2+
+White 
+ 
+36 
+Figure 4: Mitigation for privileged vs. unprivileged groups 
+Figure 4d Equalized Odds of RF using bias mitigation techniques 
+Figure 4c Predictive Equality of RF using bias mitigation techniques  
+Figure 4a Statistical Parity of RF using bias mitigation techniques 
+Figure 4b Equal Opportunity of RF using bias mitigation techniques 
+
+0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+0.4
+0.6
+privileged
+unprivileged0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+白臣
+0.0
+0.2
+0.4
+0.6
+privileged
+unprivileged0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+0.4
+0.6
+privileged
+unprivileged1.00
+Baseline
+0.75
+ReW
+DIR
+0.50
+ExGR
+0.25
+MetaC
+0.00
+0.25
+0.50
+0.75
+1.00
+privileged
+unprivileged 
+1 
+Supplementary Materials 
+Appendix A: Variable List 
+Student's native language-
+composite 
+Student’s perception of teacher-student 
+relationships in the school 
+Family Composition 
+BY highest level of participation in 
+interscholastic athletics 
+Generational status (immigration) 
+BY-F1 high school attendance pattern: by school 
+control 
+Parents Education 
+Number of school-sponsored activities 
+participated in 01-02 
+Current (2006) marital-parental 
+status 
+Transcript: GPA in first year of known attendance 
+Sex-composite 
+Transcript: First year known enrollment: credits 
+earned 
+Student's race/ethnicity-composite 
+F1 hours worked per week during 03-04 school 
+year 
+Total family income from all 
+sources -composite 
+Offered scholarship/grant for first year at first PS 
+institution 
+Sector of first postsecondary 
+institution 
+College entrance exam scores relative to average 
+scores at 1st PS institution 
+School Urbanicity 
+Units in mathematics (SST) - categorical 
+% of full-time teachers are 
+Hispanic 
+F1 math standardized score 
+
+ 
+2 
+% of full-time teachers are Black 
+Transfer student 
+% of full-time teachers are White 
+GPA for all courses taken in the 9th - 12th grades 
+- categorical 
+% of full-time teachers are 
+Hawaiian 
+Standardized test composite score-math/reading 
+% of full-time teachers are Indian 
+Highest level of education earned as of F3 
+
+ 
+3 
+Appendix B: Results with Imputation 
+ 
+ 
+  
+Statistical Parity of RF using bias mitigation techniques 
+Equal Opportunity of RF using bias mitigation techniques 
+Equalized Odds of RF using bias mitigation techniques 
+Predictive Equality of RF using bias mitigation techniques 
+Statistical Parity of SVM using bias mitigation techniques 
+Equal Opportunity of SVM using bias mitigation techniques 
+Equalized Odds of SVM using bias mitigation techniques 
+Predictive Equality of SVM using bias mitigation techniques 
+
+0.6
+Baseline
+ReW
+0.4
+DIR
+0
+ExGR
+0.2
+0
+.8
+MetaC
+08
+0.0
+7
+T
+0.2
+-0.4
+0.6
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+0.0
+。
+0.2
+Baseline
+ReW
+0.4 
+DIR
+ExGR
+0.6
+MetaC
+Asian
+Black
+Hispanic
+2+
+White0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+F
+0.2
+0.4
+0.6
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+白
+0.0
+百
+白白
+Tr
+亨
+-0.2
+Baseline
+ReW
+0.4 
+DIR
+ExGR
+0.6
+Metac
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+0.0
+0.2
+Baseline
+ReW
+0.4 
+DIR
+ExGR
+0.6
+MetaC
+Asian
+Black
+Hispanic
+2+
+White1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+Baseline
+-0.50
+ReW
+DIR
+0.75
+ExGR
+MetaC
+-1.00
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+1 T.
+0.0
+-0.2
+01
+Baseline
+ReW
+0.4 
+DIR
+ExGR
+0.6
+MetaC
+Asian
+Black
+Hispanic
+2+
+White1.00
+0.75
+0.50
+0.25
+白
+8
+0.00
+白
+0.25
+Baseline
+-0.50
+ReW
+DIR
+0.75
+ExGR
+MetaC
+-1.00
+Asian
+Black
+Hispanic
+2+
+White 
+4 
+Statistical Parity of LR using bias mitigation techniques 
+Equal Opportunity of LR using bias mitigation techniques 
+Equalized Odds of LR using bias mitigation techniques 
+Predictive Equality of LR using bias mitigation techniques 
+Statistical Parity of DT using bias mitigation techniques 
+Equal Opportunity of DT using bias mitigation techniques 
+Equalized Odds of DT using bias mitigation techniques 
+Predictive Equality of DT using bias mitigation techniques 
+
+0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+-0.2
+0.4 
+0.6
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+0.0
+-0.2
+Baseline
+ReW
+0.4 
+DIR
+ExGR
+0.6
+MetaC
+Asian
+Black
+Hispanic
+2+
+White0.6
+Baseline
+ReW
+0.4
+DIR
+o
+ExGR
+0.2
+8
+MetaC
+0.0
+8
+0.2
+0.4 
+0.6
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+00
+0.0
+口
+0.2
+Baseline
+8
+ReW
+0.4
+DIR
+ExGR
+0.6
+MetaC
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+白日eE
+0.0
+「白臣
+-0.2
+Baseline
+ReW
+0.4 
+DIR
+ExGR
+0.6
+MetaC
+Asian
+Black
+Hispanic
+2+
+White1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+Baseline
+-0.50
+ReW
+DIR
+0.75
+ExGR
+MetaC
+-1.00
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+0.0
+白白
+-0.2
+Baseline
+ReW
+-0.4
+DIR
+ExGR
+0.6
+MetaC
+Asian
+Black
+Hispanic
+2+
+White1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+Baseline
+-0.50
+ReW
+DIR
+0.75
+ExGR
+MetaC
+-1.00
+Asian
+Black
+Hispanic
+2+
+White 
+5 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ Statistical Parity of SVM using bias mitigation techniques 
+Equal Opportunity of SVM using bias mitigation techniques 
+Equalized Odds of SVM using bias mitigation techniques 
+Predictive Equality of SVM using bias mitigation techniques 
+Statistical Parity of LR using bias mitigation techniques 
+Equal Opportunity of LR using bias mitigation techniques 
+Equalized Odds of LR using bias mitigation techniques 
+Predictive Equality of LR using bias mitigation techniques 
+
+0.6 
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+0.4
+0.6
+privileged
+unprivileged0.6 
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0 
+0.2
+0.4
+0.6
+privileged
+unprivileged0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+0.4
+0.6
+privileged
+unprivileged0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2 
+MetaC
+0.0
+白
+0.2
+0.4
+0.6
+privileged
+unprivileged0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+0.4
+0.6
+privileged
+unprivileged1.00
+Baseline
+0.75
+ReW
+DIR
+0.50
+ExGR
+0.25
+MetaC
+0.00
+0.25
+0.50
+0.75
+1.00
+privileged
+unprivileged0.6 
+Baseline
+ReW
+0.4 
+DIR
+ExGR
+0.2
+MetaC
+0.0 
+0.2
+0.4
+0.6
+privileged
+unprivileged0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+0.4
+0.6
+privileged
+unprivileged1.00
+Baseline
+0.75
+ReW
+DIR
+0.50
+ExGR
+0.25
+MetaC
+0.00
+0.25
+0.50
+0.75
+1.00
+privileged
+unprivileged0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+-0.4
+0.6
+privileged
+unprivileged 
+6 
+ 
+ 
+Statistical Parity of RF using bias mitigation techniques 
+Equal Opportunity of RF using bias mitigation techniques 
+Equalized Odds of RF using bias mitigation techniques 
+Predictive Equality of RF using bias mitigation techniques 
+Statistical Parity of DT using bias mitigation techniques 
+Equal Opportunity of DT using bias mitigation techniques 
+Equalized Odds of DT using bias mitigation techniques 
+Predictive Equality of DT using bias mitigation techniques 
+
+0.6 
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0 
+0.2
+0.4
+0.6
+privileged
+unprivileged0.6 
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+0.4
+0.6
+privileged
+unprivileged1.00
+Baseline
+0.75
+ReW
+DIR
+0.50
+ExGR
+0.25
+MetaC
+0.00
+0.25
+0.50
+0.75
+1.00
+privileged
+unprivileged0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+0.4
+0.6
+privileged
+unprivileged0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+0.4
+0.6
+privileged
+unprivileged0.6 
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+8
+0.2
+~0.4
+0.6
+privileged
+unprivileged1.00
+Baseline
+0.75
+ReW
+DIR
+0.50
+ExGR
+0.25
+MetaC
+0.00
+0.25
+0.50
+0.75
+1.00
+privileged
+unprivileged0.6
+Baseline
+ReW
+0.4
+DIR
+ExGR
+0.2
+MetaC
+0.0
+0.2
+~0.4
+0.6
+privileged
+unprivileged 
+7 
+ 
+ 
+Appendix C: Probabilistic Definitions of Fairness Notions 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fairness Notion 
+Formulation 
+Statistical Parity (SP) 
+|𝑃 (𝑌̂ =  1|𝑆 =  1) −  𝑃 (𝑌̂  =  1|𝑆 =  0)| 
+Equal Opportunity (EoP) 
+|𝑃 (𝑌̂   =  0|𝑌  =  1,𝑆 =  1) −  𝑃 (𝑌̂  =  0|𝑌  =  1, 𝑆 =  0)| 
+Predictive Equality (PE) 
+|𝑃 (𝑌̂ =  1|𝑌  =  0, 𝑆 =  1) −  𝑃 (𝑌̂   =  1|𝑌  =  0, 𝑆 =  0)| 
+Equalized Odds (EO) 
+|𝑃 (𝑌̂ =  1|𝑌  =  𝑦, 𝑆 =  1) −  𝑃 (𝑌̂  =  1|𝑌  =  𝑦, 𝑆 =  0)|, ∀𝑦 ∈  {0, 1} 
+
+ 
+8 
+Appendix D: Confusion Matrix 
+ 
+  
+Predicted Response 
+True Response 
+  
+𝑌̂ = 1 
+𝑌̂ = 0 
+𝑌 = 1 True Positive 
+False Negative 
+𝑌 = 0 False Positive True Negative 
+ 
+
+ 
+9 
+Appendix E: Plots with Mitigation for Alternative ML Models (No Imputation) 
+ 
+ 
+Statistical Parity of SVM using bias mitigation techniques 
+Equal Opportunity of SVM using bias mitigation techniques 
+Equalized Odds of SVM using bias mitigation techniques 
+Predictive Equality of SVM using bias mitigation techniques 
+Statistical Parity of LR using bias mitigation techniques 
+Equal Opportunity of LR using bias mitigation techniques 
+Equalized Odds of LR using bias mitigation techniques 
+Predictive Equality of LR using bias mitigation techniques 
+
+0.6
+Baseline
+ReW
+0.4
+：
+DIR
+ExGR
+0.2
+MetaC
+0.0
+工
+600
+0.2
+0.4 
+0.6
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0
+0.2
+0.0
+-0.2
+Baseline
+ReW
+0.4 
+DIR
+ExGR
+0.6
+MetaC
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+0.0
+百白
+-0.2
+Baseline
+ReW
+0.4 
+DIR
+ExGR
+0.6
+MetaC
+Asian
+Black
+Hispanic
+2+
+White1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+Baseline
+-0.50
+ReW
+DIR
+0.75
+ExGR
+MetaC
+-1.00
+Asian
+Black
+Hispanic
+2+
+White0.6
+Baseline
+ReW
+0.4
+DIR
+0
+ExGR
+0.2
+MetaC
+0.0
+-0.2
+-0.4
+0.6
+Asian
+Black
+Hispanic
+2+
+White0.6
+0.4
+0.2
+0.0
+百
+-0.2
+Baseline
+ReW
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+Matrix Multiplication:
+Verifying Strong Uniquely Solvable Puzzles⋆
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+Department of Computer Science
+Union College
+Schenectady, New York, USA
+{andersm2, jiz, xua}@union.edu
+Abstract. Cohn and Umans proposed a framework for developing fast
+matrix multiplication algorithms based on the embedding computation
+in certain groups algebras [12]. In subsequent work with Kleinberg and
+Szegedy, they connected this to the search for combinatorial objects
+called strong uniquely solvable puzzles (strong USPs) [11]. We begin
+a systematic computer-aided search for these objects. We develop and
+implement constraint-based algorithms build on reductions to SAT and
+IP to verify that puzzles are strong USPs, and to search for large strong
+USPs. We produce tight bounds on the maximum size of a strong USP for
+width k ≤ 5, construct puzzles of small width that are larger than previ-
+ous work, and improve the upper bounds on strong USP size for k ≤ 12.
+Although our work only deals with puzzles of small-constant width, the
+strong USPs we find imply matrix multiplication algorithms that run
+in O(nω) time with exponent ω ≤ 2.66. While our algorithms do not
+beat the fastest algorithms, our work provides evidence and, perhaps, a
+path to finding families of strong USPs that imply matrix multiplication
+algorithms that are more efficient than those currently known.
+Keywords: matrix multiplication · strong uniquely solvable puzzle ·
+arithmetic complexity · integer programming · satisfiability · satisfiability
+benchmark · upper bounds · reduction · application
+1
+Introduction
+An optimal algorithm for matrix multiplication remains elusive despite substan-
+tial effort. We focus on the square variant of the matrix multiplication problem,
+i.e., given two n-by-n matrices A and B over a field F, the goal is to com-
+pute the matrix product C = A × B. The outstanding open question is: How
+many field operations are required to compute C? The long thought-optimal
+na¨ıve algorithm based on the definition of matrix product is O(n3) time. The
+groundbreaking work of Strassen showed that it can be done in time O(n2.808)
+[30] using a divide-and-conquer approach. A long sequence of work concluding
+with Coppersmith and Winograd’s algorithm (CW) reduced the running time
+⋆ An extended abstract of this paper appeared in the Proceedings of SAT 2020 [5].
+arXiv:2301.00074v1  [cs.CC]  30 Dec 2022
+
+2
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+Fig. 1: The leftmost diagram is a width-4 size-5 puzzle P. The middle three diagrams are
+the three sets of subrows of P. The rightmost diagram is the puzzle P ′ resulting from
+reordering the subrows of P as indicated by the arrows and then recombining them.
+Since P can be rearranged as P ′ ̸= P without overlap, P is not uniquely solvable.
+to O(n2.376) [26,28,31,13]. Recent computer-aided refinements of CW by others
+reduced the exponent to ω ≤ 2.3728639 [16,32,22].
+Approach Cohn and Umans [12] introduced a framework for developing faster
+algorithms for matrix multiplication by reducing this to a search for groups
+with subsets that satisfy an algebraic property called the triple-product property,
+which allows matrix multiplication to be embedded in the group algebra. Their
+approach takes inspiration from the O(n log n) algorithm for multiplying degree-
+n univariate polynomials by embedding into the group algebra of the fast Fourier
+transform, c.f., e.g., [14, Chapter 30]. Subsequent work [11] elaborated on this
+idea and developed the notion of combinatorial objects called strong uniquely
+solvable puzzles (strong USPs). These objects imply a group algebra embedding
+for matrix multiplication, and hence give a matrix multiplication algorithm as
+well.
+A width-k puzzle P is a subset of {1, 2, 3}k, and the cardinality of P is the
+puzzle’s size. Each element of P is called a row of P, and each row consists
+of three subrows that are elements of {1, ∗}k, {2, ∗}k, {3, ∗}k respectively. In-
+formally, a puzzle P is a uniquely solvable puzzle (USP) if there is no way to
+permute the subrows of P to form a distinct puzzle P ′ without cells with num-
+bers overlapping. Figure 1 demonstrates a puzzle that is not a USP. A uniquely
+solvable puzzle is strong if a tighter condition for non-overlapping holds (see
+Definition 3). For a fixed width k, the larger the size of a strong USP, the faster
+matrix multiplication algorithm it gives [11]. In fact, Cohn et al. show that there
+exist an infinite family of strong USPs that achieves ω < 2.48.
+We follow Cohn et al.’s program by developing: (i) verification algorithms
+and heuristics to determine whether a puzzle is a strong USP, (ii) search algo-
+rithms to find large strong USPs, (iii) practical implementations1 of these
+1 Source code available here: https://bitbucket.org/paraphase/matmult
+
+3232
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+１132
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+1123
+1213
+1
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+1312
+3113
+11
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+1
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+1
+1
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+１231Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+3
+algorithms, and (iv) new upper bounds on the size of strong USPs. The most
+successful of our verification algorithms work by reducing the problem through
+3D matching to the satisfiability (SAT) and integer programming (IP) prob-
+lems that are then solved with existing tools. The algorithms we develop are not
+efficient—they run in worst-case exponential time in the natural parameters.
+However, the goal is to find a sufficiently large strong USP that would provide
+a faster matrix multiplication algorithm, and the resulting algorithm’s running
+time is independent of the running time of our algorithms. The inefficiency of
+our algorithms limit the search space that we can feasibly examine.
+Results Our theoretical results and implementation produces new bounds on
+the size of the largest strong USP for small-width puzzles. For small-constant
+width, k ≤ 12, we beat the largest sizes of [11, Proposition 3.8]. Our lower
+bounds on maximum size are witnessed by strong USPs we found via search.
+For k ≤ 5 we give tight upper bounds determined by exhaustively searching all
+puzzles after modding out common symmetries. For k ≤ 12, we improve the
+upper bounds on the size of strong USPs. Although our current results do not
+beat [11] for unbounded k, they give evidence that there may exist families of
+strong USPs that give matrix multiplication algorithms that are more efficient
+than those currently known. The best strong USP we can produce imply matrix
+multiplication algorithms with ω ≤ 2.66.
+We also create a benchmark data set of SAT/UNSAT instances based on our
+reductions from strong-USP verification and examine the performance of solvers
+from the 2021 SAT Competition [6].
+Related Work For background on algorithms matrix multiplication problem,
+c.f, e.g., [9]. There are also a number of negative results known. Na¨ıvely, the
+dimensions of the output matrix C implies that the problem requires at least
+Ω(n2) time. Slightly better lower bounds are known in general and also for
+specialized models of computation, c.f., e.g., [29,20]. There are also lower bounds
+known for a variety of algorithmic approaches to matrix multiplication. Ambainis
+et al. showed that the laser method cannot alone achieve an algorithm with
+ω ≤ 2.3078 [4]. A recent breakthrough on arithmetic progressions in cap sets [15]
+combined with a conditional result on the Erd¨os-Szemeredi sunflower conjecture
+[3] imply that Cohn et al.’s strong USP approach cannot achieve ω = 2 + ϵ for
+some ϵ > 0 [10]. Subsequent work has generalized this barrier [1,2] to a larger
+class of algorithmic techniques. Despite this, we are unaware of a concrete lower
+bound on ϵ implied by these negative results. There remains a substantial gap in
+our understanding between what has been achieved by the positive refinements
+of LeGall, Williams, and Stothers, and the impossibility of showing ω = 2 using
+the strong USP approach.
+Recently Fawzi et al. showed how reinforcement learning techniques can be
+used to develop new matrix multiplication algorithms [17]. Their work produces
+matrix multiplication algorithms with ω < 2.77, which is faster than Strassen’s
+
+4
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+original algorithm (ω < 2.81), but far from the refinements of Coppersmith-
+Winograd (ω < 2.372) or the results achieved in this work.
+Organization Section 2 begins with the formal definition of a strong USP and
+the Cohn-Umans framework. Sections 3 & 4, respectively, discuss our algorithms
+and heuristics for verifying that and searching for a puzzle that is a strong USP.
+Section 5 describes several upper bounds on the size of strong USPs. Sections 6
+& 7 discuss our implementation and experimental results.
+2
+Preliminaries
+For an integer k, we use [k] to denote the set {1, 2, . . . , k}. For a set Q, SymQ de-
+notes the symmetric group on the elements of Q, i.e., the group of permutations
+acting on Q. Cohn et al. introduced the idea of a puzzle [11].
+Definition 1 (Puzzle). For s, k ∈ N, an (s, k)-puzzle is a subset P ⊆ [3]k with
+|P| = s. We call s the size of P, and k the width of P.
+We say that an (s, k)-puzzle has s rows and k columns. The columns of a puzzle
+are inherently ordered and indexed by [k]. The rows of a puzzle have no inherent
+ordering, however, it is often convenient to assume that they are ordered and
+indexed by the set of natural numbers [s].
+Cohn et al. establish a particular combinatorial property of puzzles that
+allows one to derive group algebras that matrix multiplication can be efficiently
+embedded into. Such puzzles are called strong uniquely solvable puzzles. However,
+to give some intuition we first explain a simpler version of the property called
+uniquely solvable puzzles.
+Definition 2 (Uniquely Solvable Puzzle (USP)). An (s, k)-puzzle P is
+uniquely solvable if for all π1, π2, π3 ∈ SymP : Either (i) π1 = π2 = π3, or
+(ii) there exists r ∈ P and c ∈ [k] such that at least two of the following hold:
+(π1(r))c = 1, (π2(r))c = 2, (π3(r))c = 3.
+Informally, a puzzle is not uniquely solvable if each row of the puzzle can be
+broken into ones, twos, and threes pieces and then the rows can be reassembled
+in a different way so that each new row is a combination of a ones, a twos, and a
+threes piece where there is exactly one element of [3] for each column. Observe
+that uniquely solvable puzzles can have at most 2k rows because each ones piece,
+twos piece, and threes piece must be unique, as otherwise the duplicate pieces
+can be swapped making the puzzle not uniquely solvable.
+The definition of strong uniquely solvable puzzle is below, it is nearly the same
+except that it requires that there be a collision on a column between exactly two
+pieces, not two or more pieces like in the original definition.
+Definition 3 (Strong USP (SUSP)). An (s, k)-puzzle P is strong uniquely
+solvable if for all π1, π2, π3 ∈ SymP : Either (i) π1 = π2 = π3, or (ii) there exists
+r ∈ P and c ∈ [k] such that exactly two of the following hold: (π1(r))c = 1,
+(π2(r))c = 2, (π3(r))c = 3.
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+5
+Finally, Cohn et al. defined a strengthening of SUSP which requires that every
+triple of rows witness the necessary overlap.
+Definition 4 (Local SUSP). A local strong uniquely solvable puzzle is an
+(s, k)-puzzle where for each triple of rows u, v, w ∈ P with u, v, w not all equal,
+there exists c ∈ [k] such that (uc, vc, wc) is an element of
+L = {(1, 2, 1), (1, 2, 2), (1, 1, 3), (1, 3, 3), (2, 2, 3), (3, 2, 3)}.
+Every SUSP P corresponds to a much larger local SUSP P ′, which, informally,
+is the result of concatenating and duplicating the rows of P to explicitly demon-
+strate the ∀π1, π2, π3 part of Definition 3.
+Proposition 1 ([11, Proposition 6.3]). Let P be a (s, k)-SUSP, then there
+is a local (s!, s · k)-SUSP P ′.
+Note that in all of the definitions, local, strong, uniquely solvability is invariant
+to the ordering of the rows of the puzzle, because P is a set—we use this fact
+implicitly.
+Cohn et al. show the following connection between the existence of strong
+USPs and upper bounds on the exponent of matrix multiplication ω.
+Lemma 1 ([11, Corollary 3.6]). Let ϵ > 0, if there is a strong uniquely solv-
+able (s, k)-puzzle, there is an algorithm for multiplying n-by-n matrices in time
+O(nω+ϵ) where
+ω ≤ min
+m∈N≥3
+�
+3 log m
+log(m − 1) −
+3 log s!
+s · k log(m − 1)
+�
+.
+This result motivates the search for large strong USPs that would result in faster
+algorithms for matrix multiplication. In the same article, the authors also demon-
+strate the existence of an infinite family of strong uniquely solvable puzzles, for
+width k divisible by three, that achieves a non-trivial bound on ω.
+Lemma 2 ([11, Proposition 3.8]). There is an infinite family of strong uniquely
+solvable puzzles that achieves ω < 2.48.
+Finally, they conjecture that strong uniquely solvable puzzles provide a route to
+achieving quadratic-time matrix multiplication. Unfortunately, as mentioned in
+the introduction, this conjecture was shown to be false.
+Lemma 3 ([10]). Strong uniquely solvable puzzles cannot show ω < 2 + ϵ, for
+some ϵ > 0.
+That said, there remains hope that the uniquely solvable puzzle approach could
+beat the refinements of Coppersmith-Winograd even if it cannot reach ω = 2.
+
+6
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+Algorithm 1 : Brute Force Verification
+Input: An (s, k)-puzzle P.
+Output: YES, if P is a strong USP and NO otherwise.
+1: function VerifyBruteForce(P)
+2:
+for π2 ∈ SymP do
+3:
+for π3 ∈ SymP do
+4:
+if π2 ̸= 1 ∨ π3 ̸= 1 then
+5:
+found = false.
+6:
+for r ∈ P do
+7:
+for i ∈ [k] do
+8:
+if δri,1 + δ(π2(r))i,2 + δ(π3(r))i,3 = 2 then found = true.
+9:
+if not found then return NO.
+10:
+return YES.
+3
+Verifying Strong USPs
+The core focus of this article is the problem of verifying strong USPs, i.e., given
+an (s, k)-puzzle P, output YES if P is a strong USP, and NO otherwise. In this
+section we discuss the design of algorithms to solve this computational problem
+as a function of the natural parameters s and k.
+All of the exact algorithms we develop in this section have worst-case expo-
+nential running time. However, asymptotic worst-case running time is not the
+metric we are truly interested in. Rather we are interested in the practical per-
+formance of our algorithms and their capability for locating new large strong
+USPs. The algorithm that we ultimately develop is a hybrid of a number of
+simpler algorithms and heuristics.
+We begin by discussing a na¨ıve brute force algorithm based on the defini-
+tion of strong USP (Subsection 3.1), see how it motivations a reduction to the
+3D matching problem (Subsection 3.2), and then how we might formulate a re-
+duction to the satisfiability and integer programming problems (Subsections 3.4
+& 3.5). We then describe several verification heuristics based on properties of
+strong USP (Subsection 3.6) and combine them with the verification algorithms
+to produce a hybrid algorithm Verify (Subsection 3.7). As we discuss in Sub-
+section 7.2, our hybrid algorithm is quickly able to check whether a given puzzle
+is a strong USP and aid in the search for strong USP.
+3.1
+Brute Force
+The obvious algorithm for verification comes directly from the definition of a
+strong USP. Informally, we consider all ways of permuting the twos and threes
+pieces relative to the ones pieces and check whether the non-overlapping con-
+dition of Definition 3 is met. A formal description of the algorithm is found in
+Algorithm 1.
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+7
+The ones in Line 4 of Algorithm 1 denote the identity in SymP , and δa,b is
+the Kronecker delta function which is one if a = b and zero otherwise. Observe
+that Algorithm 1 does not refer to the π1 of Definition 3. This is because the
+strong USP property is invariant to permutations of the rows and so π1 can be
+thought of as an arbitrary phase. Hence, we fix π1 = 1 to simplify the algorithm.
+Seeing that |SymP | = s!, we conclude that the algorithm runs in time O((s!)2 ·
+s · k · poly(s)) where the last factor accounts for the operations on permutations
+of s elements. The dominant term in the running time is the contribution from
+iterating over all pairs of permutations. Finally, notice that if P is a strong USP,
+then the algorithm runs in time Θ((s!)2·s·k·poly(s)), and that if P is not a strong
+USP the algorithm terminates early. The algorithm’s poor performance made it
+unusable in our implementation, however, its simplicity and direct connection to
+the definition made its implementation a valuable sanity check against later more
+elaborate algorithms (and it served as effective onboarding to the undergraduate
+students collaborating on this project).
+Although Algorithm 1 performs poorly, examining the structure of a seem-
+ingly trivial optimization leads to substantially more effective algorithms. Con-
+sider the following function on triples of rows a, b, c ∈ P: f(a, b, c) = ∨i∈[k](δai,0+
+δbi,1+δci,2 = 2). We can replace the innermost loop in Lines 7 & 8 of Algorithm 1
+with the statement found = found ∨ f(r, π1(r), π2(r)). Observe that f neither
+depends on P, r, nor the permutations, and that Algorithm 1 no longer depends
+directly on k. To slightly speed up Algorithm 1 we can precompute and cache f
+before the algorithm starts and then look up values as the algorithm runs. We
+precompute f specialized to the rows in the puzzle P, and call it fP .
+3.2
+Strong USP Verification to 3D Matching
+It turns out to be more useful to work with fP than with P. It is convenient
+to think of fP as a function fP : P × P × P → {0, 1} that is the complement
+of the characteristic function of the relations of a tripartite hypergraph HP =
+⟨P ⊔ P ⊔ P, ¯
+fP ⟩ where the vertex set is the disjoint union of three copies of P
+and fP indicates the edges that are not present in HP .
+Let H = ⟨P ⊔ P ⊔ P, E ⊆ P 3⟩ be a tripartite 3-hypergraph. We say H has
+a 3D matching (3DM) iff there exists a subset M ⊆ E with |M| = |P| and for
+all distinct edges e1, e2 ∈ M, e1 and e2 are vertex disjoint, i.e., e1 ∩ e2 = ∅.
+Determining whether a hypergraph has a 3D matching is a well-known NP-
+complete problem (c.f., e.g., [18]). We say that a 3D matching is non-trivial if
+it is not the set {(r, r, r) | r ∈ P}. Figure 2 demonstrates a 3-hypergraph with a
+non-trivial 3D matching.
+The existence of non-trivial 3D matchings in HP is directly tied to whether
+P is a strong USP.
+Lemma 4. A puzzle P is a strong USP iff HP has no non-trivial 3D matching.
+Proof. We first argue the reverse. Suppose that Hp has a non-trivial 3D matching
+M. We show that P is not a strong USP by using M to construct π1, π2, π3 ∈
+
+8
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+Fig. 2: An example hypergraph G with edges E = {(r1, r1, r2), (r1, r3, r3), (r2, r2, r1),
+(r2, r3, r1), (r3, r2, r3)}. The highlighted edges are a non-trivial 3D matching M =
+{(r1, r1, r2), (r2, r3, r1), (r3, r2, r3)} of G.
+SymP that witness this. Let π1 be the identity permutation. For each r ∈ P,
+define π2(r) = q where (r, q, ∗) ∈ M. Note that q is well defined and unique
+because M is 3D matching and so has vertex disjoint edges. Similarly define
+π3(r) = q where (r, ∗, q) ∈ M. Observe that by construction
+M = {(π1(r), π2(r), π3(r)) | r ∈ P}.
+Since M is a matching of HP , M ⊆ ¯
+fP . Because M is a non-trivial matching at
+least one edge in (a, b, c) ∈ M has either a ̸= b, a ̸= c, or b ̸= c. This implies,
+respectively, that as constructed π1 ̸= π2, π1 ̸= π3, or π2 ̸= π3. In each case we
+have determined that π1, π2, and π3 are not all identical. Thus we determined
+permutations such that for all r ∈ P, f(π1(r), π2(r), π3(r)) = 0. This violates
+Condition (ii) of Definition 3, hence P is not a strong USP.
+The forward direction is symmetric. Suppose that P is not a strong USP. We
+show that HP has a 3D matching. For P not to be a strong USP there must exist
+π1, π2, π3 ∈ SymP not all identical such that Condition (ii) of Definition 3 fails.
+Define e(r) = (π1(r), π2(r), π3(r)) and M = {e(r) | r ∈ P}. Since Condition (ii)
+fails, we have that fP (e(r)) = false for all r ∈ P. This means that for all r ∈ P,
+e(r) ∈ ¯
+fP and hence M ⊆ ¯
+fP . Since π1 is a permutation, |M| = |P|. Observe
+that M is non-trivial because not all of the permutations are identical and there
+must be some r ∈ P with e(r) having non-identical coordinates. Thus M is a
+non-trivial 3D matching.
+⊓⊔
+As a consequence of Definition 3, strong-USP verification is in coNP. Note
+that although 3D matching is an NP-complete problem, Lemma 4 does not im-
+mediately imply that verification of strong USPs is coNP-complete because HP
+is not an arbitrary hypergraph. It remains open whether strong-USP verification
+is coNP-complete. Lemma 4 implies that to verify P is a strong USP it suffices to
+determine whether HP has a non-trivial 3D matching. In the subsequent subsec-
+tions we examine algorithms for the later problem. We can, in retrospect, view
+Algorithm 1 as an algorithm for solving 3D matching.
+We note that the parameters s and k are not fully independent. First, s ≤ 3k
+because the maximum number of rows in a puzzle of width k is |[3]k| = 3k. Sec-
+ond, we eliminate the dependence on k entirely by transforming an (s, k)-puzzle
+
+GMatrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+9
+Algorithm 2 : Bidirectional Dynamic Programming Verification
+Input: An (s, k)-puzzle P.
+Output: YES, if P is a strong USP and NO otherwise.
+1: function VerifyDynamicProgramming(P)
+2:
+Let T = ∅.
+3:
+Construct 3D matching instance HP .
+4:
+function SearchHalf(ℓ, Q, ℓQ, R, ℓR, δ, t)
+5:
+if ℓ = t then
+6:
+if δ = 1 then
+▷ Forward Base Case
+7:
+Insert (Q, R) into T.
+8:
+return false.
+9:
+else
+▷ Reverse Base Case
+10:
+if (P − Q, P − R) ∈ T then
+11:
+return true.
+12:
+else
+13:
+return false.
+14:
+res = false.
+▷ Recursive Case
+15:
+for ℓ′
+Q = ℓQ + 1 to s do
+16:
+for ℓ′
+R = ℓR + 1 to s do
+17:
+if (pℓ, pℓ′
+Q, pℓ′
+R) ∈ HP ∧ ¬res then
+18:
+res = SearchHalf(ℓ + δ, Q ∪ {pℓ′
+Q}, ℓ′
+Q, R ∪ {pℓ′
+R}, ℓ′
+R, δ, t).
+19:
+return res.
+20:
+SearchHalf(1, ∅, 0, ∅, 0, 1, ⌊s/2⌋ + 1).
+21:
+return SearchHalf(s, ∅, 0, ∅, 0, −1, ⌊s/2⌋).
+into a 3D matching instance on the vertex set [s]3. However, this transformation
+is not without cost, because the size of HP is a function of the cube of s rather
+than linear in the size of the puzzle s · k.
+3.3
+Dynamic Programming
+The realization that the verification of strong USPs is a specialization of 3D
+matching leads to a dynamic programming algorithm for verification that runs
+in linear-exponential time O(22spoly(s)+poly(s, k)). The reduction allows us to
+replace the permutations from SymP with subsets of P and effectively reduce
+the cost of the outer loops of Algorithm 1 from s! = Θ(2s log s) to 2s.
+Algorithm 2 describes a recursive bidirectional dynamic programming al-
+gorithm for strong-USP verification that uses the 3D matching instance. The
+algorithm consists of two phases. Let t = ⌊s/2⌋. The first phase determines all
+possible sets Q, R ⊆ P with |Q| = |R| = t such that there is 3D matching M1
+of HP when restricted to the vertices {p1, p2, . . . , pt} ⊔ Q ⊔ R. The sets Q, R
+satisfying the requirement are stored in a table T during the first phase on Line
+7. The second phase determines all possible sets Q, R ⊆ P with |Q| = |R| = s−t
+
+10
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+such that there is a 3D matching M2 of HP when restricted to the vertices
+{pt+1, pt+2, . . . , ps} ⊔ Q ⊔ R. For each pair (Q, R) the algorithm considers in the
+second phase, it checks whether (P − Q, P − R) was inserted into T during the
+first phase. If the pair is present, it means that there is a 3D matching of HP
+which is M = M1 ∪ M2. This works because, by Line 10, M1 and M2 are partial
+3D matchings on {p1, . . . , pt} ⊔ (P − R) ⊔ (P − Q) and {pt+1, . . . ps} ⊔ R ⊔ Q,
+respectively, which implies that M1 and M2 are vertex disjoint. The first phase
+always returns false, which is ignored, and the second phase returns whether
+a complete matching could be found, and, hence, by Lemma 4, whether P is a
+strong USP.
+The running time of this algorithm is dominated by the number of pairs of
+sets (Q, R) it examines. Observe that rows of P are considered in order in Lines
+15 & 16. Further, the algorithm tracks the index of the last elements added to Q
+and R in ℓQ and ℓR, respectively. The algorithm only adds new elements to Q or
+R that have higher indexes than ones previously added. Altogether this implies
+that each pair of sets (Q, R) is only considered at most once during a phase. Since
+Q, R ⊆ P, there are at most �t
+i=0
+�s
+i
+�
+·
+�s
+i
+�
+≤ (�t
+i=0
+�s
+i
+�
+)2 ≤ (2s)2 = 4s pairs
+(Q, R). This means that SearchHalf is called at most 4s times during each
+phase. Hence the running time of the algorithm is O(4s·s2·poly(s)+T3DM(s, k))
+where s2 factor comes from the inner loops, poly(s) time to manipulate the sets
+and track the contents of T as a hash table, and T3DM(s, k) accounts for the
+time to construct HP . The memory requirements of Algorithm 2 are similarly
+high—the first phase uses O(4s · s) bits to store T.
+Note that Algorithm 2 does not early terminate on P that are strong USP,
+because it must search through all pairs before determining that none can be
+found. The algorithm could be modified to allow early termination when P is
+not a strong USP by causing the second phase of search to immediately return
+in Line 18 once the first 3D matching witness has been located. However, this
+still requires the first phase to run to completion. A remedy for this would be to
+run both phases in parallel and have them check against each other. We chose
+not to because it would substantially complicate the implementation and would
+be unlikely to ultimately improve the performance of our combined algorithms.
+For comparison, more advanced techniques like those of Bj¨orklund et al. can
+achieve a better asymptotic time of O(2spoly(s)) [8]. We chose not to implement
+their algorithm, because we judged that it would not substantially increase the
+domain for which verification was possible.
+3.4
+3D Matching to Satisfiability
+By Lemma 4, one can determine whether a puzzle P is a strong USP by con-
+structing the graph HP and deciding whether it has a non-trivial 3D matching.
+Here we reduce our 3D matching problem to the satisfiability (SAT) problem on
+conjunctive normal form (CNF) formulas and then use a state-of-the-art SAT
+solver to resolve the reduced problem. To perform the reduction, we convert
+the graph HP into a CNF formula ΨP , a depth-2 formula that is the AND of
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+11
+ORs of Boolean literals. We construct ΨP so that ΨP is satisfiable iff HP has a
+non-trivial 3D matching.
+Let HP = ⟨V = P ⊔ P ⊔ P, E ⊆ P 3⟩ be the 3D matching instance associated
+with the puzzle P. Our goal is to determine whether there is a non-trivial 3D
+matching M ⊆ E. A na¨ıve reduction would be to have variables Mu,v,w indicating
+inclusion of each edge (u, v, w) ∈ P 3 in the matching. This results in a formula
+ΨP with s3 variables and size Θ(s5) because including an edge e ∈ P 3 excludes
+the Θ(s2) edges e′ with e∩e′ ̸= ∅. To decrease the size of ΨP we instead use sets of
+variables to indicate which vertices in the second and third part of V are matched
+with each vertex in the first part. In particular we have Boolean variables M 1
+u,v
+and M 2
+u,w for all u, v, w ∈ P, and these variable map to assignments in the na¨ıve
+scheme in the following way: M 1
+u,v ∧ M 2
+u,w ⇔ Mu,v,w.
+We now write our CNF formula for 3D matching. First, we have clauses that
+prevents non-edges from being in the matching:
+Ψ non-edge
+P
+=
+�
+(u,v,w)∈E
+(¬M 1
+u,v ∨ ¬M 2
+u,w).
+(1)
+Second, we add clauses require that every vertex in HP is matched with some
+edge:
+Ψ ≥1
+P
+=
+� �
+u∈P
+(∨v∈P M 1
+u,v) ∧ (∨w∈P M 2
+u,w)
+�
+∧
+� �
+v∈P
+(∨u∈P M 1
+u,v)
+�
+∧
+� �
+w∈P
+(∨u∈P M 2
+u,w)
+�
+.
+(2)
+Third, we require that each vertex be matched with at most one edge and so
+have clauses that exclude matching edges that overlap on one or two coordinates.
+Ψ ≤1
+P
+=
+�
+i∈{1,2}
+�
+(u,v),(u′,v′)∈P 2
+(u = u′ ∨ v = v′) ∧ (u, v ̸= u′, v′) ⇒ ¬M i
+u,v ∨ ¬M i
+u′,v′.
+(3)
+Fourth, we exclude the trivial 3D matching by requiring that at least one of the
+diagonal edges not be used: Ψ non-trivial
+P
+= �
+u∈P ¬M 1
+u,u∨¬M 2
+u,u. Finally, we AND
+these into the overall CNF formula: ΨP = Ψ non-edge
+P
+∧ Ψ ≤1
+P
+∧ Ψ ≥1
+P
+∧ Ψ non-trivial
+P
+.
+The size of the CNF formula ΨP is Θ(s3), has 2s2 variables, and is a factor of s2
+smaller than the na¨ıve approach. Thus we reduce 3D matching to satisfiability
+by converting the instance HP into the CNF formula ΨP .
+3.5
+3D Matching to Integer Programming
+In parallel to the previous subsection, we use the connection between verifica-
+tion of strong USPs and 3D matching to reduce the former to integer program-
+ming, another well-known NP-complete problem (c.f., e.g., [21]) and then apply
+
+12
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+a state-of-the-art solver to resolve it. Again, let HP = ⟨V, E⟩ be the 3D match-
+ing instance associated with P. We construct an integer program QP over {0, 1}
+that is infeasible iff P is a strong USP. Here the reduction is simpler than the
+previous one because linear constraints naturally capture matching.
+We use Mu,v,w to denote a variable with values in {0, 1} to indicate whether
+the edge (u, v, w) ∈ P 3 is present in the matching. To ensure that M is a subset
+of E we add the following edge constraints to QP : ∀u, v, w ∈ P, ∀(u, v, w) ̸∈
+E, Mu,v,w = 0. We also require that each vertex in each of the three parts
+of the graph is incident to exactly one edge in M. This is captured by the
+following vertex constraints in QP : ∀w ∈ P, �
+u,v∈P Mu,v,w = �
+u,v∈P Mu,w,v =
+�
+u,v∈P Mw,u,v = 1. Lastly, since we need that the 3D matching be non-trivial
+we add the constraint: �
+u∈P Mu,u,u < |P|.
+To check whether P is a strong USP we determine whether QP is not feasible,
+i.e., that no assignment to the variables M satisfy all constraints. We note that
+reduction from 3D matching to IP is polynomial time and that there are s3
+variables in QP , and that the total size of the constraints is s3 · Θ(1) + 3s ·
+Θ(s2) + 1 · Θ(s3) = Θ(s3), similar to size of ΨP in the SAT reduction.
+3.6
+Heuristics
+Although the exact algorithms presented in the previous sections make sub-
+stantial improvements over the brute force approach, the resulting performance
+remains impractical. To resolve this, we also develop several fast verification
+heuristics that may produce the non-definitive answer MAYBE in place of YES
+or NO. Then, to verify a puzzle P we run this battery of fast heuristics and
+return early if any of the heuristics produce a definitive YES or NO. When all of
+the heuristics result in MAYBE, we then run one of the slower exact algorithms
+that were previously discussed. The heuristics have different forms, but all rely
+on the structural properties of strong uniquely solvable puzzles.
+Downward Closure The simplest heuristics we consider is based on the fact
+that strong USPs are downward closed.
+Lemma 5. If P is a strong USP, then so is every subpuzzle P ′ ⊆ P.
+Proof. Let P be a strong USP and P ′ ⊆ P. By Definition 3, for every (π1, π2, π3) ∈
+Sym3
+P not all identity, there exist r ∈ P and i ∈ [k] such that exactly two of the
+following hold: (π1(r))i = 1, (π2(r))i = 2, (π3(r))i = 3. Consider restricting the
+permutations to those that fix the elements of P\P ′. For these permutations it
+must be the case that r ∈ P ′ because otherwise r ∈ P\P ′ and there is exactly
+one j ∈ [3] for which (πj(r))i = j holds. Thus we can drop the elements of P\P ′
+and conclude that for every tuple of permutations in SymP ′ the conditions of
+Definition 3 hold for P ′, and hence that P ′ is a strong USP.
+⊓⊔
+This leads to a polynomial-time heuristic that can determine that a puzzle is
+not a strong USP. Informally, the algorithm takes an (s, k)-puzzle P and s′ ≤ s,
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+13
+Algorithm 3 : Downward-Closure Heuristic
+Input: An (s, k)-puzzle P, and size s′ ≤ s.
+Output: NO, if P has a set of s′ rows that do not form a strong USP, and
+MAYBE otherwise.
+1: function HeuristicDownwardClosed(P, s′)
+2:
+for P ′ ⊆ P, |P ′| = s′ do
+3:
+if P ′ is not a strong USP then return NO.
+4:
+return MAYBE.
+and verifies that all subsets P ′ ⊆ P with size |P ′| = s′ are strong USPs. If any
+subset P ′ is not a strong USP, the heuristic returns NO, and otherwise it returns
+MAYBE. For completeness, this algorithm is described in Algorithm 3.
+This algorithm runs in time O(
+� s
+s′
+�
+·T(s′, k)) where T(s′, k) is the runtime for
+verifying an (s′, k)-puzzle. In practice we did not apply this heuristic for s′ larger
+than 3. When s′ is some constant d, the running time becomes O(sd · T(d, k)) =
+O(sdk) using the brute force algorithm (Algorithm 1) for verification of the
+puzzle P ′.
+Unique Pieces Every strong uniquely solvable puzzle is a uniquely solvable
+puzzle. A necessary condition for a puzzle to be a USP is that for each element
+in [3], the collection of subrows contains no duplicates.
+Lemma 6 (Implicit in [11]). If P is a USP, then for all e ∈ [3], and distinct
+rows r1, r2 ∈ P, there is a column c ∈ [k] were one of the rows r1 or r2 has an
+e and the other one does not.
+Proof. Suppose, for the sake of contradiction, that this is not the case, and dis-
+tinct rows r1, r2 ∈ P have e in exactly the same columns for some e ∈ [3]. We
+show that P is not a USP. Choose πe = (r1r2), i.e., the permutations that trans-
+poses the subrows for e in rows r1 and r2. Choose the other two permutations
+for the elements of [3]\{e} to be the identity. Since the permutations are not all
+the identity, the second half of Definition 2 applies. However, the puzzle that
+results from the permutations is identical to P and for all c ∈ [k] and each row
+r ∈ P there exists exactly on i ∈ [3] where (πi(r))c = i. Hence the definition of
+uniquely solvable is not satisfied and we have a contradiction.
+⊓⊔
+Note that the reverse direction of Lemma 6 does not hold. The puzzle in Figure 1
+is an example of this: It is not uniquely solvable, but the subrows for each element
+are distinct.
+We can make Lemma 6 effective as via a linear-time heuristic capable of
+ruling out puzzles that are not (strong) USPs. Although straightforward, for
+completeness we formalize our approach in Algorithm 4. When the sets are
+implemented as hash tables, the expected running time of this algorithm is O(s·
+k) time, which is linear in the size of the puzzle P. An alternative worst-case
+O(s · k) time implementation uses radix sort to sort the characteristic sequences
+
+14
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+Algorithm 4 : Unique Pieces Heuristic
+Input: An (s, k)-puzzle P.
+Output: NO, if a witness is found for P not being a (strong) USP, and MAYBE
+otherwise.
+1: function HeuristicUniquePieces(P)
+2:
+Initialize empty sets S1, S2, S3.
+3:
+for r ∈ P do
+4:
+for e ∈ [3] do
+5:
+Let h = {c ∈ [k] | rc = e}.
+6:
+if h ∈ Se then return NO.
+7:
+Se = Se ∪ {h}.
+8:
+return MAYBE.
+of the subrows as binary numbers and then scans adjacent rows to to detect
+duplication.
+The unique pieces heuristic is equivalent to the downward-closure heuristic
+for subpuzzles of size two.
+Lemma 7. Let P be an (s, k)-puzzle, then HeuristicUniquePieces(P) =
+HeuristicDownwardClosed(P, 2).
+Proof. We show both directions.
+Suppose that P fails the unique pieces heuristic for, w.l.o.g., e = 1, then there
+are distinct rows r1, r2 ∈ P where the cells that contain 1 are all in the same
+columns. This means we can swap those 1’s subrows without causing overlap
+or changing the puzzle. This implies that P ′ = {r1, r2} is not a (strong) USP.
+Since |P ′| = 2 and P ′ ⊆ P, the downward closure heuristic for s′ = 2 will also
+conclude that P is not a (strong) USP.
+Suppose that P fails the downward-closure heuristic for s′ = 2. Then there
+is a pair of distinct rows r1, r2 ∈ P for which P ′ = {r1, r2} is not a strong
+USP. Suppose there is no columns were r1 and r2 differ, then the subrows of
+r1, r2 are the same for all elements, and so P fails the unique pieces heuristic.
+For the other case, suppose there is a least one column c ∈ [k] where r1 and r2
+differ. W.l.o.g., let that column be ((r1)c, (r2)c) = (1, 2). Because P ′ is not an
+SUSP and this column is (1, 2), there can be other no columns that are in from
+the set {(1, 3), (2, 3), (3, 2), (3, 1)} otherwise they would form an SUSP with the
+column (1, 2). This means the only columns that P ′ contains are from the set
+{(1, 2), (2, 1), (1, 1), (2, 2), (3, 3)}. Therefore, the columns which contain 2 must
+match and the subrows for 2 in r1 and r2 are identical. Thus, P ′, and so P, fails
+the unique pieces heuristic.
+⊓⊔
+A corollary of this proof is that for size-two puzzles, every USP is also a strong
+USP.
+Corollary 1. Let P be a (2, k)-puzzle, if P is a uniquely solvable puzzle, then
+P is a strong uniquely solvable puzzle.
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+15
+Since the unique pieces heuristic is equivalent to the downward-closure heuristic
+for s′ = 2 and the running time of unique pieces is linear in the puzzle size,
+O(s·k), and the running time of downward closed is O(s2 ·k), we use the unique
+pieces heuristic in place of downward closed for s′ = 2.
+Greedy This heuristic attempts take advantage of Lemma 4 and greedily search
+for a 3D matching for the instance HP . The heuristic proceeds iteratively, de-
+termining the vertex of the first part of the 3D matching instance with the
+least edges and randomly selecting an edge of that vertex to put into the 3D
+matching. If the heuristic successfully constructs a 3D matching it returns NO
+indicating that the input puzzle P is not a strong USP. If the heuristic reaches a
+point were prior commitments have made the matching infeasible, the heuristic
+starts again from scratch. This process is repeated some number of times be-
+fore it gives up and returns MAYBE. In our implementation we use s2 attempts
+because it is similar to the running time of the reductions and it empirically re-
+duced the number of instances requiring full verification in the domain of puzzles
+with k = 6, 7, 8 while not increasing the running time by too much. The greedy
+heuristic is formalized in Algorithm 5.
+The array cts is used to store the number of edges cts[u] that remain associ-
+ated with vertex u along the first coordinate. Much of the algorithm is devoted
+to maintaining this invariant. The sets U, V, W store the vertices along the three
+coordinates, respectively, that have already been incorporated into the partial
+3D matching. Like in Algorithm 2 we do not store the matching itself, only the
+vertices involved. The break at Line 10 triggers when the partial 3D matching
+is a dead end and cannot be extended into a full 3D matching. The condition
+of Line 23 is true when a full 3D matching has been constructed and causes the
+algorithm to return that P is not a strong USP.
+The running time of this algorithm is O(s3t+T3DM(s, k)), where T3DM(s, k)
+is the time required to construct 3D matching instances from (s, k)-puzzles.
+This algorithm has the potential to be considerably slower than the downward-
+closure heuristic, and in practice we set t = s2. However, the main loop can
+terminate early at Line 10 when it fails to extend the 3D matching, this permits
+the expected time to much less than the worst case. For a puzzle P that is a
+strong USP, the heuristic takes the full Ω(s3t + T3DM(s, k)) time.
+Compared to the downward-closure and unique pieces heuristics this heuristic
+is much less efficient. As a result we only run it when when the other heuristics
+have failed. See Subsection 7.2 for a comparison of effectiveness these heuristics
+in our experiments.
+3.7
+Hybrid Algorithm
+Our final verification algorithm (Algorithm 6) is a hybrid of several exact al-
+gorithms and heuristics. The size thresholds for which algorithm and heuristic
+to apply were determined experimentally for small k and are focused on the
+values where our strong USP search algorithms are tractable k ≤ 6 (or nearly
+
+16
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+Algorithm 5 : Greedy Heuristic
+Input: An (s, k)-puzzle P, and iteration bound t.
+Output: NO, if a witness is found for P not being a strong USP, and MAYBE
+otherwise.
+1: function HeuristicGreedy(P)
+2:
+Construct 3D matching instance HP .
+3:
+for i = 1 to t do
+4:
+for u ∈ P do
+5:
+cts[u] = �
+v,w∈P HP (u, v, w).
+▷ Number of edges incident vertex u.
+6:
+Let U, V, W = ∅.
+7:
+Let m = 0.
+▷ Number of edges in matching.
+8:
+while m < s do
+9:
+Select u ∈ {w ∈ ¯U | cts[w] = maxv∈ ¯U cts[v]} uniformly at random.
+10:
+if cts[u] = 0 then break.
+11:
+Let D = {(v, w) ∈ ¯V × ¯W | HP (u, v, w) = 1}.
+12:
+Select (v, w) ∈ D uniformly at random.
+13:
+for v′ ∈ P do
+▷ Update edge counts.
+14:
+for w′ ∈ P do
+15:
+if (v′, w′) ∈ ¯V × ¯W and HP (u, v′, w′) = 1 then
+16:
+cts[u]--.
+17:
+if (v′, w′) ∈ ¯U × ¯W and HP (v′, v, w′) = 1 and v′ ̸= u then
+18:
+cts[v′]--.
+19:
+if (v′, w′) ∈ ¯U × ¯V and HP (v′, w′, w) = 1 and v′ ̸∈ {u, v} then
+20:
+cts[v′]--.
+21:
+U, V, W = U ∪ {u}, V ∪ {v}, W ∪ {w}.
+▷ Add edge to matching.
+22:
+m = m + 1.
+23:
+if m ≥ s then return NO.
+▷ 3D matching found so not SUSP, halt.
+24:
+return MAYBE.
+tractable k ≤ 8). We decide to run both of the reductions to SAT and IP in
+parallel because it is not clear which algorithm performs better in general. Since
+verification halts when either algorithm completes, the wasted effort is within a
+factor of two of what the better algorithm could have done alone. We also chose
+to do this because we experimentally observed that there were many instances
+that one of the algorithms struggled with that the other did not—this resulted
+in a hybrid algorithm that out performed the individual exact algorithms on
+average. We show in Subsection 7.2 that our hybrid algorithm and heuristics
+perform well in practice at quickly verifying strong USPs for small width k. Fur-
+ther, Subsection 7.3 contains a discussion of the relative performance of the SAT
+and IP approaches on different instance types from our benchmark experiments.
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+17
+Algorithm 6 : Hybrid Verification
+Input: An (s, k)-puzzle P.
+Output: YES, if P is a strong USP, and NO otherwise.
+1: function Verify(P)
+2:
+if s ≤ 2 then return VerifyBruteForce(P).
+3:
+Return result if HeuristicUniquePieces(P) is not MAYBE.
+4:
+if s ≤ 7 then return VerifyDynamicProgramming(P).
+5:
+Return result if HeuristicDownwardClosed(P, 3) is not MAYBE.
+6:
+Return result if HeuristicGreedy(P) is not MAYBE.
+7:
+Run VerifySAT(P) and VerifyIP(P) in parallel and return first result.
+4
+Searching for Strong USPs
+With a practical verification algorithm in hand, we consider the problem of
+searching for large strong USPs. Because the set of strong USPs is downward
+closed, a natural search strategy is: Start with the empty set and repeatedly
+consider adding rows while maintaining the strong-USP property. However, while
+this strategy will lead to a maximal-size strong USP, it is not guaranteed to
+produce a maximum-size strong USP. This is because the set of strong USPs
+does not form a matroid, rather it is only an independence system (c.f., e.g.,
+[25]).
+In particular, (i) the empty puzzle is a strong USP and (ii) the set of strong
+USP are downward closed by Lemma 5. The final property required to be a
+matroid, the augmentation property, requires that for every pair of strong USPs
+P1, P2 with |P1| ≤ |P2| there is a row of r ∈ P2\P1 such that P1 ∪ {r} is also a
+strong USP. For a simple counterexample consider the strong USPs P1 = {32}
+and P2 = {12, 23}. Using Lemma 6, we see that neither P1 ∪ {12} = {12, 32}
+nor P1 ∪{23} = {23, 32} are strong USPs, and hence the augmentation property
+fails. One consequence is that na¨ıve greedy algorithms will likely be ineffective
+for finding maximum-size strong USPs. Furthermore, we do not currently know
+of an efficient algorithm that can take a strong USP P and determine a row r
+such that P ∪ {r} is a strong USP.
+Despite that, we have had some success in applying general-purpose tree-
+search techniques with pruning based on the symmetries of strong USPs together
+with our practical verification algorithm to construct maximum-size strong USPs
+for small k.
+4.1
+Puzzle Symmetry
+Since puzzles are defined as sets of rows, the ordering of the rows of a puzzle
+P does not affect the SUSP property. Similarly, but slightly less obviously, the
+SUSP property is invariant to reordering the columns of the puzzle, because the
+required existential condition ∃c ∈ [k] st. (...) from Definition 3 is independent
+
+18
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+of the ordering of the columns. Lastly, the alphabet [3] typically used to repre-
+sent the elements of a puzzle is completely arbitrary, any set of three distinct
+values would suffice. These values are not interpreted mathematically, aside from
+their convenience in expressing the SUSP definition concisely. This logic can be
+formalized into the following lemma.
+Lemma 8. Let ρ ∈ Sym[k], δ ∈ Sym[3]. A (s, k)-puzzle P is a strong USP iff
+{(δ(rρ(c)))c∈[k] | r ∈ P} is a strong USP.
+Proof. Follows immediately from Definition 1 and Definition 3.
+⊓⊔
+This lemma implies that the SUSP property is invariant with respect to these
+kinds of puzzle transformations. We call two puzzles P, P ′ that are related in this
+way isomorphic, and use the notation P ∼= P ′ to denote this. The relation ∼= is
+an equivalence relation, because permutations are invertable, and so it partitions
+the set of puzzles into equivalence classes.
+This notion of isomorphism is naturally related to the same notion in graphs.
+For each (s, k)-puzzle P we can define a colored, undirected graph GP . This
+graph consists of vertices that are partitioned into four sets of different colors:
+V = {rowr}r∈[s]⊔{colc}c∈[k]⊔{ei}i∈[3]⊔{vr,c}(r,c)∈[s]×[k]. There are s+k+3+s·k
+vertices in GP . The first three parts are vertices representing the rows and
+columns of P, and the elements of [3], respectively, and the fourth part are
+vertices for each of the s·k cells in the P. The edge relation of GP is straightfor-
+ward: Each vertex vr,c is connected to three vertices corresponding to the row,
+columns and element that the cell indexed (r, c) contains in P. In particular, the
+three edges attached to vr,c are (vr,c, rowr), (vr,c, colc), (vr,c, eltP (r,c)). In total,
+GP has 3·s·k edges. Because the vertex sets for rows, columns, and elements are
+each uniquely colored and each cell of P is connected to vertices representing its
+row, column, and element, the automorphisms of GP are in 1-1 correspondence
+to the automorphisms of P under permutations of rows, columns, and elements.
+This implies that for two (s, k)-puzzles P, P ′, if GP ∼= GP ′ then there exists per-
+mutations of the rows, columns, and elements of P which results in P ′. Further
+by Lemma 8, if GP ∼= GP ′, then P ∼= P ′, and P is an SUSP iff P ′ is an SUSP.
+4.2
+Symmetry-Pruned Tree Search
+A natural way to search for strong USPs is based on breadth-first search and
+uses the fact that strong USP are downward closed (Lemma 5): To find the
+largest possible width-k strong USP, (i) start with all possible first rows – the 3k
+(1, k)-puzzles, (ii) attempt to extend the resulting puzzles with all possible rows
+keeping only the puzzles that are strong USPs and which are not isomorphic to
+the strong USPs that have been seen before to form the new search frontier, and
+(iii) repeat Step (ii) until the search frontier is empty.
+To ensure the algorithm does not revisit isomorphic puzzles, we use canonical
+graph representations [Gp] of the puzzle graphs GP . A canonical graph represen-
+tation is a binary encoding of a graph with the property that for any two graphs
+G1, G2, [G1] = [G2] iff G1 ∼= G2 (c.f., e.g., [24]). As the search algorithm runs we
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+19
+Algorithm 7 : Symmetry-Pruned Breadth-First Search
+Input: An integer k ≥ 0.
+Output: The number b, which is the size of the largest width-k strong USP.
+1: function SP-BFS(k)
+2:
+Let Q be an empty queue.
+3:
+Let I be an empty set.
+4:
+Let b = 0.
+5:
+enqueue(Q, ∅).
+6:
+while Q is not empty do
+7:
+P = dequeue(Q).
+8:
+for r ∈ [3]k\P do
+9:
+Let P ′ = P ∪ {r}.
+10:
+if Verify(P ′) and [G′
+P ] ̸∈ I then
+11:
+enqueue(Q, P ′).
+12:
+I = I ∪ {[G′
+P ]}.
+13:
+b = |P ′|.
+14:
+return b.
+record the set I of canonical graph representations [GP ] of each distinct puzzle
+P that has been added to the search frontier. Each time a puzzle P ′ is consid-
+ered for being added to the search frontier we first check whether its canonical
+graph representation [GP ′] ∈ I, if it is, we do not add P ′ to the frontier. The use
+of canonical representations of puzzles dramatically shrinks the search space by
+searching from [P] rather than every P ′ ∼= P and by not allowing duplicates of
+[P] to be enqueued. This algorithm SP-BFS is formalized in Algorithm 7.
+We argue the correctness of this algorithm.
+Lemma 9. For k ∈ N, SP-BFS(k) returns the maximum integer s for which
+there exists an (s, k)-SUSP.
+Proof. Ignoring the pruning that I performs for a moment, it is routine to argue
+that SP-BFS behaves like a generic breadth-first search algorithm over the tree of
+all strong USPs. This is because of the downward-closure property of strong USP
+(Lemma 5), which makes any strong USP P reachable from the trivial strong
+USP ∅ using a series of row inclusions. SP-BFS(k) results in an exhaustive
+search of all strong USPs of width k and return the maximum size b of such
+SUSPs.
+We argue that when considering the pruning that I contributes to, SP-
+BFS(k) enqueues exactly one element of each equivalence class of puzzles that
+are SUSPs. Then, as a consequence of Lemma 8, the algorithm must explore ev-
+ery equivalence class of width-k SUSPs. Hence, it explores an equivalence class
+with SUSPs of maximum size and subsequently returns that size, which is the
+expected output.
+To complete the argument and show that the symmetry pruned search covers
+the entire search space of equivalence classes, suppose, for the sake of contradic-
+
+20
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+tion, that there is some smallest s such that there is an (s, k)-puzzle P that does
+not have its equivalence class [P] searched. We know that s > 1, because the
+algorithm starts by considering all possible (1, k)-puzzles. Let P ′ be the (s−1, k)-
+puzzle created from P by removing one of its rows r, P ′ has as least one row
+because s > 1. By hypothesis, the equivalence class of [P ′] has been visited by
+SP-BFS because P ′’s size is s − 1 < s. Consider [P] and remove the row that
+corresponded to r to form [P]′. It must be the case that [P ′] ∼= [P]′. This isomor-
+phism extends to [P] in that there must be a row r′ such that ([P ′]∪{r′}) ∼= [P],
+where r′ is replaces the row remove from [P]. Therefore, since [P ′] is searched,
+the algorithm must consider all possible rows to extend by, including r′. This is
+means that the equivalence class of [P] is searched, a contradicting our assump-
+tion. Therefore every equivalence class of SUSPs is searched by SP-BFS.
+⊓⊔
+This approach reduces the size of the search space, improving both the run-
+ning time of the search and the space required to keep track of the frontier puz-
+zles. The worst case running time of SP-BFS is O(3k·#EQUIV (k)·(TVerify(sk+
+1, k)+TCanonize(sk, k)), where #EQUIV (k) is the number equivalence classes of
+strong USP of width k, TVerify(sk + 1, k) is the time to verify the maximum size
+(sk +1, k)-puzzles examined by the algorithm, and TCanonize(sk, k) is the time to
+compute the canonical graph representation of each puzzle P considered by the
+algorithm (assuming TVerify and TCanonize are monotone in their parameters).
+See Subsection 7.1 for the experimental results of running SP-BFS and a
+discussion of implementation issues.
+5
+Upper Bounds
+Although the main focus of this research line is to construct sufficiently large
+strong USP that would imply faster matrix multiplication algorithms, our tech-
+niques and approach can also be applied to search for tighter upper bounds on
+the size of strong USP. We describe several SUSP-size upper bounds in this
+section.
+ω Bound. Prior work explicitly discusses bounds on the capacity of infinite
+families of USP (c.f., [11, Lemma 3.2, Theorem 3.3]). Since every SUSP is a
+USP, these bounds also apply to SUSP and can be restated to apply to individual
+puzzles. The first bound, which we denote as the “ω bound”, results from (i)
+Lemma 1, which is monotone non-increasing for fixed k, and (ii) the fact that ω ≥
+2. To compute this bound we evaluate the inequality of Lemma 1 on increasingly
+large s until just before the consequence implies ω < 2 which is in contradiction
+with ω ≥ 2.
+Unique Pieces Bound. The second bound, which we denote as the “unique pieces
+bound”, following directly from Lemma 6. Since that lemma requires that each
+row of a (strong) USP have a unique ones, twos, and threes piece, the total
+number of rows in a strong USP cannot be more than 2k.
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+21
+USP Bound. The third bound, which we denote as the “USP bound”, results
+from the proof of [11, Lemma 3.2]. Although not spelled out in that article, the
+proof relies on the following subclaim that directly bounds s as a function of k.
+Proposition 2. Let P be a (s, k)-USP, then
+s ≤
+k
+�
+c1=0
+k−c1
+�
+c2=0
+min
+�� k
+c1
+�
+,
+� k
+c2
+�
+,
+�
+k
+k − (c1 + c2)
+��
+= O
+�
+k2 ·
+� 3
+22/3
+�k�
+.
+Note that the USP bound is asymptotically tighter than the unique pieces bound
+as
+3
+22/3 ≈ 1.8899 < 2.
+Clique Bound. The fourth bound, which we denote as the “clique bound”, results
+from the fact that SUSPs are downward closed (Lemma 5). In particular if P
+is an SUSP, then for every P ′ ⊆ P with 2 rows must also be an SUSP. Fix
+k ∈ N and consider a graph Gk whose vertices correspond to the possible rows
+of a width-k puzzle, i.e., strings in [3]k, and where there is an edge between
+r1, r2 ∈ [3]k if {r1, r2} is an SUSP. Observe that by downward closure, each
+(s, k)-SUSP corresponds to a clique of size s in Gk. This approach naturally
+generalizes from the Clique problem to h-HypergraphClique problem where the
+graph Gh
+k consists the same 3k vertices as Gk = G2
+k, but instead has the arity-h
+edges {r1, r2, . . . , rh} which are (h, k)-SUSPs.
+Proposition 3. Let P be an (s, k)-SUSP and 2 ≤ h ≤ s. Then for
+Gh
+k = ⟨V = [3]k, E = {P ′ ⊆ V | P ′ is a strong USP and |P ′| = h}⟩,
+(Gh
+k, s) ∈ h-HypergraphClique.
+Therefore, the size of a maximum hypergraph clique in Gh
+k is an upper bound of
+size of width-k SUSP. We use “clique bound” to denote the specific instantiation
+of this bound for h = 2.
+Exhaustive Bound. For fifth bound, which we denote as the “exhaustive bound”,
+we consider the results of Algorithm 7 when run in the domain of k where the
+full search space can be feasibly explored. Because these bounds are based on
+exhaustive search they are inherently tight.
+Downward-Closure Bound. The final bound we consider follows from the downward-
+closure property of SUSPs.
+Proposition 4. Let P be an (s, k)-SUSP with k > 1, then there exists an
+(⌈ s
+3⌉, k − 1)-SUSP.
+Proof. Fix any c ∈ [k] and consider the cth column of P, then, by averaging,
+there must be an element of e ∈ [3] that appears at least ⌈ s
+3⌉ times in that
+column. Let P ′ ⊂ P be the subpuzzle of P whose rows have e in the cth column.
+P ′ is a strong USP, because P is a strong USP and strong USPs are downward
+
+22
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+closed (Lemma 5). Form P ′′ by removing the cth column of P ′. P ′′ is a strong
+USP, because P ′ is a strong USP and the strong-USP property is invariant to
+addition or removal of constant columns. By construction, P ′′ is a (⌈ s
+3⌉, k − 1)-
+SUSP.
+⊓⊔
+This bound is not as independently applicable like the others, but it can lift
+upper bounds of s ≤ u at k to s ≤ 3u at k + 1.
+See Subsection 7.1 for the results of evaluating the above bounds for small
+width and a discussion of issues involved in concretely calculating them.
+6
+Implementation
+We implemented our verification algorithms, heuristics, and search algorithms,
+along with various utilities and appropriate datastructures to represent under-
+lying information such as puzzles in C++. The source code for our implementa-
+tion is available under a MIT License at https://bitbucket.org/paraphase/
+matmult.
+We use a number of external libraries with subroutines that are key to the
+functioning of our algorithms. Our IP-based verifier and Clique bound calcula-
+tor both use the commercial, closed-source mixed-integer programming solver
+Gurobi to solve the integer programs produced by our reductions [19]. Our SAT-
+based verifier uses, by default, the kissat-sc2021-sat solver from the 2021
+SAT Competition by A. Biere, M. Fleury, and M. Heisinger [6, page 10]. Note
+that the conference version of this article used the MapleCOMSPS solver—see
+Subsection 7.3 for a discussion of solver benchmarks, comparisons, and choice.
+We implemented Algorithm 7 using our hybrid verifier, and the graph automor-
+phism library Nauty [24] as a subroutine to perform the required graph canon-
+ization on GP . The original versions of our SP-BFS implementation targeted a
+high-performance computing cluster environment, because our brute force and
+dynamic programming implementations were not efficient enough. Subsequent
+improvements to our verification algorithms made this unnecessary. Despite this,
+our SP-BFS implementation is still in MPI and uses a MapReduce framework
+[27] to maintain a distributed search frontier.
+Our code base also contains multiple implementations of depth-first-search-
+inspired algorithms for locating strong USPs. These algorithms use our hybrid
+verification implementation and puzzle symmetry pruning technique discussed
+in Section 4. For brevity and to keep this article focused on strong-USP verifi-
+cation, we elect not to discuss these algorithms and defer them to a subsequent
+article. That said, some of the concrete puzzles we found and report in the next
+section were generated by such algorithms. These puzzles once found were ex-
+perimentally verified as strong USPs using the techniques discussed in detail in
+Section 3.
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+23
+k
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+[11]
+s ≥
+1
+2
+3
+4
+4
+10
+10
+16
+36
+36
+36
+136
+ω ≤ 3.00 2.88 2.85 2.85
+2.80
+2.74
+2.70
+This work s ≥
+1
+2
+3
+5
+8
+14
+21
+30
+42
+64
+112
+196
+ω ≤ 3.00 2.88 2.85 2.81 2.78 2.74 2.73 2.72 2.72 2.71 2.68 2.66
+Table 1: Comparison with [11] of lower bounds on the maximum of size of width-k
+strong USPs and the upper bounds on ω they imply. Bold font indicates tight results
+for that k.
+7
+Experimental Results
+Our experimental results come in several flavors for small-constant width k:
+(i) constructive lower bounds on the maximum size of width-k strong USPs
+witnessed by found puzzles, (ii) upper bounds on the maximum size of width-k
+strong USPs, (iii) the number of SUSPs and SUSP equivalence classes for width
+k, (iv) experimental data comparing the run times of our verification algorithms
+and distinguishing likelihood of our heuristics, and (v) a benchmark data set of
+SAT/UNSAT instances that we use to compare the effectiveness of competitive
+SAT solvers as subroutines for the SAT-based part of our verifier.
+All of the results in this section were produced by running our algorithm
+implementations on the same Ubuntu 20.04 PC with a 3.00 GHz Intel Core
+i9-10980XE CPU and 128 GB of RAM.
+7.1
+New Upper and Lower Bounds on the Size of Strong USPs
+New Lower Bounds. Table 1 summarizes new lower bounds for maximum SUSP
+size in comparison with [11]. The lower bounds of [11] are from the constructions
+in their Propositions 3.1 and 3.8, which give families of strong USPs for even
+k or k divisible by three. For k’s which are not divisible by two or three, we
+extrapolate their construction by adding a new column, this preserves the SUSP
+property. The upper bounds on ω in this table are computed by plugging s and
+k into Lemma 1 and optimizing over m. For clarity we omit ω’s that would be
+larger than previous columns. Our results in this table we produced by running
+SP-BFS and other search algorithms which verify that the final result is a strong
+USP. Our bounds are tight for all k ≤ 5, because of the exhaustive nature of
+SP-BFS, and constructively improve the known lower bounds for 4 ≤ k ≤ 12.
+Figure 3 contains representative examples of maximal-size strong USPs we
+found for k ≤ 6. The strong uniquely solvable (14, 6)-puzzles we found represent
+the greatest improvement in ω versus the construction of [11] for small k. Further,
+our puzzle for k = 12 is the result of taking the Cartesian product of two copies
+of a strong uniquely solvable (14, 6)-puzzles. Note that Proposition 3.8 of [11]
+
+24
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+Fig. 3: Representative maximal-size strong USPs found for width k = 1, 2, . . . , 6.
+gives an infinite family of strong USPs that achieves ω < 2.48 as k goes to
+infinity, which is stronger than our results are directly able to achieve.
+New Upper Bounds. Table 2 summarizes the results of evaluating the bounds
+from Section 5 for puzzles of width k ≤ 12. The calculations were routine except
+for the clique bound that required constructing Gk, converting it into a mixed
+integer program, and solving that program using Gurobi [19]. This was feasible
+on our test system up to k = 11. We also experimented with calculating the upper
+bounds for the 3-HypergraphClique bound, but found it infeasible to compute
+for k ≥ 5 and so have omitted the results. The final row of the table contains the
+best upper bounds we achieved, including applying the downward-closure bound
+to lift adjacent bounds at k = 6 and k = 12. These upper bounds are stronger
+than those immediately implied by [11].
+Observe that exhaustive search produced the best and tightest bounds, and
+that the clique bound is considerably stronger than the unique pieces, USP, and ω
+bounds. The unique pieces bounds appears to be stronger than the USP bound,
+but we know that that is an artifact of the small value of k. As k increase,
+the USP bound will become tighter than the unique pieces bound. Based on
+the processing time we spent on k = 6, we conjecture that s = 14 is tight
+for k = 6 and that our lower bounds for k > 6 are not. Our results suggests
+there is considerable room for improvement in the construction of strong USPs,
+and that it is possible that there exist large puzzles for k = 7, 8, 9 that would
+beat [11]’s constructions and perhaps come close to the Coppersmith-Winograd
+refinements. That said, it seems that new insights into the SUSP search problem
+are required to proceed for k > 6.
+Counting Strong USP. Table 3 shows the number of strong USPs and equiva-
+lence classes of SUSP exhaustively calculated using SP-BFS with and without
+symmetric pruning. Observe that the number of strong USPs is many orders of
+magnitude more than the number of equivalence classes of strong USPs, even for
+(3, 3)-SUSPs. Exhaustive search became infeasible even with puzzle symmetry
+
+312
+331213
+2.2)
+3.3Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+25
+k
+Bound
+1 2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+ω
+3 7 15 31 62 120 230 438 831 1,575 2,890 5,637
+Unique
+2 4
+8 16 32
+64 128 256 512 1,024 2,048 4,096
+USP
+3 6 12 24 45
+87 168 312 597 1,140 2,112 4,023
+Clique
+1 3
+5
+9 17
+30
+55 105 186
+348
+654
+Exhaustive 1 2
+3
+5
+8
+Best
+1 2
+3
+5
+8
+24
+55 105 186
+348
+654 1,962
+Table 2: Upper bounds on the size of SUSPs for widths k ≤ 12. Bold font indicates the
+bound is tight, and blanks indicate the calculation for this puzzle width was infeasible.
+pruning for k ≥ 6 as the memory usage of Algorithm 7 for storing the search
+frontier exceeds the 128GB available on our test system.
+7.2
+Algorithm Performance
+To measure the performance of our verification algorithms and heuristics we ran
+them on 10,000 random puzzles at each point on a sweep through parameter
+space for widths k = 5 . . . 12 and sizes s = 1 . . . 60. We chose to test performance
+via random sampling because we do not have access to a large set of solved
+instances. This domain coincides with the frontier of our search space, and we
+tuned the parameters of the heuristics and algorithms in the hybrid algorithm to
+perform well in this domain. We did not deeply investigate performance charac-
+teristics outside of this domain. In Figures 4, 5, & 6 we plot results, for brevity,
+that are representative of the parameter space only for k ∈ {6, 9}.
+Running Time. Figure 4 shows the average running times of our verification
+algorithms in seconds. The brute force and dynamic programming algorithms
+perform poorly except for very small size, s ≤ 8, and their curves loosely match
+the exponential-time bounds we expect. The plots for the two reduction-based
+algorithms (SAT and IP) behave similarly to each other. They are slower than
+brute force and dynamic programming for small values of s, and their behavior
+for large s is quite a bit faster. We speculate that the former is due to the cost of
+constructing the reduced instance and overhead of the third party tools. Further
+observe that the SAT reduction handily beats the IP reduction on large size for
+k = 6, but as k increases, the gap decreases. We also note that across the settings
+of k the IP reduction has effectively the same running time and is independent
+of k. This is likely because the size of the IP instance depends only on s. The
+hybrid algorithm generally performs best or close to best at small values of s
+and is clearly faster for large values of s. Notice that it matches the dynamic
+programming algorithm closely for small values of s and then diverges when the
+
+26
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+k
+s
+1
+2
+3
+4
+5
+6
+1 1 3
+2
+9
+3
+27
+4
+81
+5
+243
+7
+729
+2
+2 24
+9
+408
+33
+4,848
+91
+50,160
+229
+486,024
+3
+9 1,800
+240
+182,304
+2,429
+8,361,000
+16,971 291,347,280
+4
+728 2,445,120
+59,149 992,377,400
+1,611,648
+?
+5
+190 3,248,640
+707,029
+?
+?
+?
+6
+2,337,715
+?
+?
+?
+7
+1,359,649
+?
+?
+?
+8
+89,196
+?
+?
+?
+9
+?
+?
+Table 3: Number of equivalence classes (bold face, left) versus total number of encoded
+SUSPs (normal face, right) by (s, k)-puzzle dimensions. Computed using Algorithm 7.
+Empty cells indicate that the number of SUSPs and equivalence classes is zero. ?’s
+indicate unknown values that were infeasible to compute.
+reduction-based algorithms and heuristics are activated at larger s. Observe that
+the hybrid algorithm is effectively constant time for large s, though the size for
+which this happens increases as a function of k. We expect this is because the
+density of strong USPs decreases rapidly with s, and that the randomly selected
+puzzles are likely far from satisfying Definition 3 and, hence, they are quickly
+rejected by the unique pieces heuristics. Further evidence of this is that running
+time of the hybrid algorithm converges to the running time of the unique pieces
+heuristic for large k.
+Heuristic Effectiveness. Figure 5 shows the probability that each individual
+heuristic distinguishes a random puzzle in our benchmark. Observe that the
+distinguishing power of the downward closure heuristic for s′ = 2 and unique
+pieces heuristics coincide, demonstrating experiment consistency with Lemma 7.
+Further, and for the same reason, the downward closure heuristic for s′ = 3 has
+at least as high a distinguishing likelihood as the unique pieces heuristic. In the
+plots, these three heuristics achieve almost 100% probability of distinguishing
+random puzzles by size s = 30. The greedy heuristic perform less well than the
+others and get substantially worse as k increases. We do not plot the running
+times of the heuristics here, but they behave as expected by the earlier analysis.
+As we noted earlier, unique pieces is linear time in the size of the puzzle and
+the fastest of the heuristics. Figure 4 shows how the running time of the hybrid
+algorithm and unique pieces converges as essentially all random puzzles of large
+size, which the benchmark examined, are verified as non-SUSPs by this heuristic.
+Variation in Running Time. Finally, we look at the variation in the running
+times of the hybrid algorithm in Figure 6. For small s, the running time dis-
+tribution is far from a normal distribution–the average is far above the median
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+27
+Fig. 4: Log plots of the average running times for verifying 10,000 random (s, k)-puzzles
+for each s ∈ [50], k ∈ {6, 9}. The plots describe the behavior of five verification algo-
+rithms brute force (BF), dynamic programming (DP), reduction to satisfiability (SAT),
+reduction to integer programming (IP), and our hybrid algorithm (Hybrid). The run-
+ning time of the unique pieces heuristic is also included.
+and middle 50% of running times. This effect becomes even more pronounced
+as k increases. However, we find that as s increases, the median running time
+converges with the median running time of the unique pieces heuristic, and then
+for larger s, the average running time converges as well. This is a consequence
+of the hybrid algorithm having to run the orders of magnitude slower reduction-
+based algorithms when the fast heuristics fail to resolve the instance. Although
+not plotted here, we found that the range of the distribution of running times
+for the SAT-based verifier was larger than for the IP-based verifier, even though
+the IP-based verifier was slower on average.
+Overall, our hybrid verification algorithm performs reasonably well in prac-
+tice on random instances, despite reductions through NP-complete problems.
+7.3
+Choice of SAT Solver
+In the conference version of this article we examined only one SAT solver for
+use in our implementation, MapleCOMSPS, a conflict-driven solver that uses a
+learning rate branching heuristic, and that was a top performer at the 2016 SAT
+Competition [7,23,5]. In this article we create a set of benchmark satisfiability
+instances, using the SUSP verification reduction on a variety of puzzles (recall
+
+Average Verification Time (sec) vs Puzzle Size
+k=6
+k=9
+100
+1
+0
+4
+A
+10-1
+口
+(sec)
+00
+10-2
+A
+Time
+Hybrid
+10-3
+BF
+10-4
+DP
+8
+SAT
+10-5
+IP
+Unique
+10-6
+胰*★*
+*★★
+10
+20
+30
+40
+50
+10
+20
+30
+40
+50
+Puzzle size
+Puzzle size28
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+Fig. 5: Plots of the likelihood that each of the heuristics produces a definitive
+results on 10,000 random (s, k)-puzzles for each size s ∈ [50] and width k
+∈
+{6, 9}. Here “row pairs” is HeuristicDownwardClosed(P, 2) and “row triples” is
+HeuristicDownwardClosed(P, 3). The row pairs points are plotted, but are hard
+to see, because the unique pieces points coincides with them.
+Subsection 3.4), and examined the performance of 352 solvers submitted to the
+main track of the 2021 SAT Competition [6].
+We select benchmark instances consisting of (s, k)-puzzle with sizes from the
+set
+{(2, 2), (3, 3), (5, 4), (8, 5), (14, 6), (21, 7), (30, 8), (42, 9)}.
+We choose these sizes, because we want positive and negative instances and these
+sizes represent the largest strong USPs of each width we have been able to locate
+through search. For each size we created ten puzzles that are strong USPs and
+ten puzzles that are not. To create the ten non-SUSPs we randomly generated a
+puzzle of that size and verified it was not a strong USP. To create the ten strong
+USPs we for each size we used the results of our search algorithms. Then we ran
+all of the puzzles through our SAT reduction to create .dimacs files for each
+instance. Note that the SUSPs correspond to UNSAT instances and non-SUSPs
+correspond to SAT instances. In total there are 160 instances in this benchmark.
+We then ran each of the 35 solvers on each the 160 instance files and check the
+output of each run against the expected result. For each trial, we record the user
+CPU time reported by the Linux time command, or a timeout if the program
+runs more than 5000 seconds without halting (mimicking the rules of the real
+SAT competition). For comparison, we also run the MapleCOMSPS solver (from
+2 There were 39 SAT solvers submitted to the main track. We use the default build
+configuration for each submission. We were unable to build three of them, and one
+that builds repeatedly crashed on all benchmarks without producing a result. We
+tested the remaining 35.
+
+Heuristic Definitive Result Likelihood vs Puzzle Size
+k=6
+k=9
+100
+Definitive Result
+80
+60
+40
+Greedy
+Row Pairs
+%
+20
+Row Triples
+Unique
+0
+5
+10
+15
+20
+25
+30
+5
+10
+15
+20
+25
+30
+Puzzle size
+Puzzle sizeMatrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+29
+Fig. 6: Log box plots of the distribution of the running times of the hybrid verification
+algorithm on 10,000 random (s, k)-puzzles for each s ∈ [50], k ∈ {6, 9}. The blue
+circles denote the average running times of the hybrid algorithm. The dark blue blocks
+indicates the median times. The thick vertical lines indicate the middle 50% of times,
+and the thin vertical lines indicate the full range of running times at each s.
+earlier version of this article), our MIP-based verifier (recall Subsection 3.5) and
+our final hybrid verification algorithm on the same set of benchmark puzzles.
+To compare the results of each solver we calculate the maximum time to
+complete each instance across all of the runs, which is 5000 seconds if a run
+timed out, and then divide by that maximum time to normalize all of the running
+times to the interval [0, 1]. We calculate a benchmark score for each solver by
+summing their relative running times across all instances. Table 4 contains the
+benchmark scores for each solver.
+MapleCOMSPS, the solver we used in the conference version of this article,
+performs similarly to the best scoring solvers from the 2021 competition. The
+recorded timeouts across all solvers come almost exclusively from the UNSAT
+instances derived from (30, 8)-SUSPs and (42, 9)-SUSPs. The Gurobi-based ver-
+ifier performs substantially worse than the best performing satisfiability solvers
+on SAT instances (non-SUSPs), but dramatically better on UNSAT instances
+(SUSPs).
+Figure 7 shows the performance of the Gurobi-based verifier against the five
+solvers with the best SAT scores. In this plot the instance completion times
+for each solver are sorted in increasing order, so that curves further to the left
+are better. If this were not a log-plot, the area to the left of the curve would
+be proportional to the benchmark scores from Table 4. Observe that for SAT
+instances, the SAT solvers, including MapleCOMSPS, follow similar trajectories.
+Gurobi performs an order of magnitude worse across all SAT instances. The
+hybrid algorithm, although plotted, is not visible because of how effective the
+heuristics are at identifying random SAT (non-SUSP) instances. For UNSAT
+
+Hybrid Running Time (sec) vs Puzzle Size
+k=6
+k=9
+10
+10-3
+10-5
+10-6
+10
+20
+30
+40
+50
+10
+20
+30
+40
+50
+Puzzle size
+Puzzle size30
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+instances, the situation is different. Gurobi performs relatively more slowly for
+small, easier instances, but substantially better than the SAT solvers for larger,
+harder instances. The performance of the solvers on easier UNSAT instances is
+more varied than the corresponding case for SAT instances, but this does not
+translate into much of a difference in benchmark score because the magnitude
+of the relative completion time is low.
+For UNSAT instances, the benchmark score is dominated by the number of
+timeouts, each of which effectively adds one to the score. Indeed, the plots for
+the SAT solver cut off between instance numbers 60 to 70, because the remaining
+instances cause timeouts. Finally, notice that hybrid algorithm out performs the
+others for small UNSAT instances – these are instances of the sort where the
+brute force and bi-directional search algorithms are applied. For larger instances
+the hybrid algorithm tracks an order of magnitude worse than the Gurobi-based
+verifier. This is because our algorithm is tuned to encounter many more SAT
+instances (non-SUSPs) than UNSAT instances (SUSPs). Further, because the
+one-sided heuristics rule out SAT instances quickly in practice, on UNSAT in-
+stances the hybrid algorithm runs these heuristics first, but then has to fall back
+on the Gurobi-based verifier causing some overhead.
+Ultimately, the results of these benchmarking experiments suggest that there
+is not a substantial difference between using the 2016 MapleCOMSPS and the
+best solvers from the 2021 competition. Even so, we choose kissat-sc20221-sat
+as the default solver in our implementation, because it performed the best on our
+benchmark of SAT instances. Using our current approach, Gurobi is essential to
+the feasible verification of SUSPs.
+The benchmark instances and puzzles, and the entirety of the raw timing
+data can be found in our repository3.
+8
+Conclusions
+We initiated the first study of the verification of strong USPs and developed
+practical software for both verifying and searching for them. We give tight results
+on the maximum size of width-k strong USPs for k ≤ 5 and improved upper and
+lower bounds on maximum strong-USP size for k ≤ 12. We prove a number of
+properties of strong USPs related the verification and search. We also produce
+a new set of benchmark instances for SAT solvers.
+Although our results do not produce a new upper bound on the running
+time of matrix multiplication, they demonstrate there is promise in this ap-
+proach. There are a number of open questions. Is strong-USP verification coNP-
+complete? What is the maximum strong-USP capacity? Is there a way to bridge
+the apparent gap between the values of ω implied by single SUSPs and the values
+implied by infinite families of SUSPs? What are tight bounds on maximum-size
+strong USPs for k ≥ 6 and do these bound lead to asymptotically faster algo-
+rithms for matrix multiplication?
+3 https://bitbucket.org/paraphase/matmult/src/main/data_set/
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+31
+The main bottleneck in our work is the size of the search space—new insights
+seem to be required to substantially reduce it. Are there subclasses of strong
+USPs that can be more effectively searched? Are there search strategies that
+would be more effective on this space?
+Acknowledgments
+The authors thank the anonymous reviewers for their detailed and thoughtful
+suggestions for improving this work.
+The second and third authors thank Union College for the Undergraduate
+Summer Research Fellowships funding their work. The first author thanks the
+many undergraduate students that have contributed in some form to this project
+over the years, including: Jonathan Kimber, Akriti Dhasmana, Jingyu Yao, Kyle
+Doney, Quoc An, Harper Lyon, Zachary Dubinsky, Talha Mushtaq, Jing Chin,
+Diep Vu, Hung Duong, Vu Le, Siddhant Deka, Baibhav Barwal, Aavasna Ru-
+pakheti.
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+
+34
+Matthew Anderson, Zongliang Ji, and Anthony Yang Xu
+Solver
+SAT
+UNSAT
+Total
+Timeouts
+cadical-hack-gb
+17.51
+15.97
+33.48
+15
+cadical-less-UP
+19.81
+16.14
+35.95
+15
+cadical-PriPro
+19.49
+15.62
+35.11
+15
+cadical-PriPro no bin
+16.55
+15.73
+32.28
+15
+cadical-rp
+19.08
+15.78
+34.85
+15
+cadical-sc2021
+18.82
+16.80
+35.62
+16
+Cadical SCAVEL01
+33.49
+16.73
+50.23
+15
+Cadical SCAVEL02
+40.97
+27.28
+68.26
+15
+cleanmaple
+30.44
+18.93
+49.37
+17
+CleanMaple PriPro
+30.70
+20.18
+50.87
+18
+hCaD
+19.70
+16.52
+36.22
+16
+hKis
+13.15
+17.30
+30.45
+16
+kissat bonus
+13.04
+16.59
+29.63
+15
+kissat cf
+12.06
+16.19
+28.26
+14
+kissat gb
+12.52
+17.27
+29.79
+17
+kissat-MAB
+15.28
+16.07
+31.36
+15
+kissat-sat crvr gb
+13.37
+16.64
+30.01
+16
+kissat-sc2021
+12.32
+16.08
+28.40
+14
+kissat-sc2021-sat
+12.02
+16.06
+28.08
+14
+kissat-sc2021-sweep
+12.82
+16.24
+29.07
+16
+lstech maple
+15.13
+14.83
+29.96
+12
+Maple MBDR BJL6 Tier2
+19.46
+16.02
+35.47
+14
+Maple MBDR BJL7 Local
+19.98
+15.49
+35.47
+13
+Maple MBDR Cent PERM 10K
+25.20
+15.96
+41.16
+12
+Maple MBDR Cent PERM 75K
+25.07
+16.00
+41.06
+12
+Maple simp21
+12.53
+16.72
+29.26
+15
+MapleSSV
+15.56
+16.68
+32.24
+16
+parafrost-nomdm-sc2021
+18.11
+15.56
+33.67
+14
+parafrost-sc2021
+24.15
+15.61
+39.76
+14
+Relaxed LCFTP
+12.80
+17.55
+30.35
+16
+Relaxed LCFTP V2
+13.97
+16.17
+30.14
+12
+Relaxed LCMDCBDL BLB
+15.38
+15.95
+31.33
+14
+Relaxed LCMDCBDL SCAVEL01
+13.95
+16.08
+30.03
+15
+Relaxed LCMDCBDL SCAVEL02
+25.45
+79.43
+104.88
+17
+slime
+17.26
+14.73
+31.99
+13
+MapleCOMSPS
+12.98
+17.42
+30.40
+16
+Gurobi
+30.20
+0.00
+30.20
+0
+Hybrid
+0.00
+0.01
+0.01
+0
+Table 4: Scores for solvers on our SUSP verification benchmark. The SAT and UNSAT
+score are out of 80, the total score and timeouts are out of 160. Lower scores are better
+and minimum values for each SAT solver are bold in each column. The top part of the
+table includes the SAT solvers we tested from the 2021 SAT Competition [6].
+
+Matrix Multiplication: Verifying Strong Uniquely Solvable Puzzles
+35
+Fig. 7: Plots of the sorted relative completion times for SAT and UNSAT instances on
+the five best-scoring solvers for that instance type.
+
+Sorted Instance # vs Relative Completion Time
+SAT
+80
+kissat cf = 12.06
+kissat_gb = 12.52
+kissat-sc2021 = 12.32
+F OZ
+kissat-sc2021-sat = 12.02
+MapleCOMSPS = 12.98
+60 
+Maple simp21 = 12.53
+#
+hybrid = 0.00
+50 
+gurobi = 30.20
+40 
+orted
+30
+S
+20 -
+10 -
+0 :
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Relative Completion Time
+UNSAT
+80 -
+70 -
+60 -
+#
+Instance
+50
+40
+orted
+lstech maple = 14.83
+30
+MapleCOMSPS = 17.42
+Maple MBDR BJL7 Local = 15.49
+20 -
+parafrost-nomdm-sc2021 = 15.56
+parafrost-sc2021 = 15.61
+10 -
+slime = 14.73
+hybrid = 0.01
+gurobi = 0.00
+0
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Relative Completion Time

BtE2T4oBgHgl3EQfngj6/content/tmp_files/2301.04010v1.pdf.txt

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+Energy deposition and formation of nanostructures in the interaction of highly charged
+xenon ions with gold nanolayers
+I. Stabrawaa, D. Bana´sa,∗, A. Kubala-Kuku´sa, Ł. Jabło´nskia, P. Jagodzi´nskia, D. Sobotaa, K. Szarya, M. Pajeka, K. Skrzypiecb, E.
+Mendykb, M. Borysiewiczc, M. D. Majki´cd, N. N. Nedeljkovi´ce
+aInstitute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland
+bDepartament of Chemistry, Maria Curie-Skłodowska University, Plac M. Curie-Skłodowskiej 3, 20-031 Lublin, Poland
+cInstitute of Electron Technology, aleja Lotnik´ow 32/46, 02-668 Warszawa, Poland
+dFaculty of Technical Sciences, University of Priˇstina in Kosovska Mitrovica, Knjaza Miloˇsa 7, 38220 Kosovska Mitrovica, Serbia
+eFaculty of Physics, University of Belgrade, P.O. Box 368, 11001 Belgrade, Serbia
+Abstract
+The effect of the deposition of kinetic energy and neutralization energy of slow highly charged xenon ions on the process of the
+nanostructures creation at the surface of gold nanolayers is investigated. The nanolayers of thickness of 100 nm were prepared by
+e-beam evaporation of gold on crystalline silicon Si(100) substrate. The samples were irradiated at the Kielce EBIS facility of the
+Jan Kochanowski University (Kielce, Poland), under high vacuum conditions. The irradiations were performed for constant kinetic
+energy 280 keV and different ions charge states (Xeq+, q = 25, 30, 35, 36 and 40) and for constant charge state Xe35+ and different
+kinetic energies: 280 keV, 360 keV, 420 keV and 480 keV. The fluence of the ions was on the level of 1010 ions/cm2. Before and
+after irradiation the nanolayer surfaces were investigated using the atomic force microscope.
+As the result, well pronounced modifications of the nanolayer surfaces in the form of craters have been observed. A systematic
+analysis of the crater sizes (diameter on the surface and depth) allowed us to determine the influence of the deposited kinetic and the
+neutralization energy on the size of the obtained nanostructures. The results are theoretically interpreted within the micro-staircase
+model based on the quantum two-state vector model of the ionic Rydberg states population. The charge dependent ion-atom
+interaction potential inside the solid is used for the calculation of the nuclear stopping power. According to the model the formation
+of the nanostructures is governed by the processes of the ionic neutralization in front of the surface and the kinetic energy loss
+inside the solid. The interplay of these two types of processes in the surface structure creation is described by the critical velocity.
+Using the proposed theoretical model, the neutralization energy, deposited kinetic energy and critical velocities were calculated and
+compared qualitatively with the experimental results. The results are consistent (after normalization) with previous experimental
+data and molecular dynamics simulations for single ionized Xe and crystalline gold surface.
+1. Introduction
+Modification of metal, semiconductor and insulator surfaces
+by the ion irradiation is of great importance for developing
+new technologies for manufacturing a small functional elec-
+tronics systems with nanometer dimensions, and has the po-
+tential to introduce novel nanostructures and material proper-
+ties not achievable by any other material processing methods
+[1]. Modification of materials by swift (high kinetic energy)
+heavy ion (SHI) irradiation is already used in many industrial
+processes, such as: the generation of nanopores in polymers [2],
+controlled drug delivery in biomedicine [3], precise band gaps
+modification [4, 5], modification of high temperature supercon-
+ductors [6], and others [7, 8]. It has also been demonstrated that
+with SHI beams regular patterns (usually in amorphic form [9])
+of lateral dimensions in the order of several tens of nanometers
+can be created.
+One of the promising alternatives for creation of surface
+nanostructures is modification of surface by an impact of a
+∗Corresponding author
+Email address: d.banas@ujk.edu.pl (D. Bana´s )
+single (i.e. each ion creates nanostructure) low-energy (slow)
+highly charged ions (HCI). The term slow HCI usually refers
+to impact velocities v ≪ 1 a.u., corresponding to 25 keV/amu
+(nuclear stopping power regime). HCI are characterized by an
+additional (to the kinetic energy) high potential energy, result-
+ing from the removal of many of the electrons from the neutral
+atom. For example, for Xe50+ ion the potential energy is around
+100 keV, i.e. 8400 times higher than that of a single charged
+xenon Xe+ ion. The neutralization energy of the HCI in the
+interaction with solid surface is also large for very slow ions
+(in keV energy range). As a consequence, the interaction of
+slow single HCI with a surface is also governed by the potential
+(neutralization) energy of the ion [10, 11, 12]. This energy is
+deposited on a small surface area along the first few nanometers
+below the target surface [13]. Recent research on 2D materials
+shows [14], that potential energy deposition of highly charged
+ion (Xe38+) is limited to only up to two layers within multilayer
+MoS2 (on graphene). For very low ionic velocities (down to
+v = 0.03 a.u.) the deposited potential energy (close to the ionic
+neutralization energy) can lead [15] to creation of various sur-
+face nanostructures, so far mainly observed on insulators such
+Preprint submitted to Vacuum
+January 11, 2023
+arXiv:2301.04010v1  [physics.atom-ph]  10 Jan 2023
+
+as alkali and alkaline earth halides, oxides and polymers, but
+also on highly oriented pyrolytic graphite (HOPG), sapphire
+and gold crystals, and silicon semiconductor [16]. On the other
+hand, for moderate ionic velocities (v ≈ 0.25 a.u.) both the
+neutralization and the deposited kinetic energy participate in
+the surface modification [10, 12]. Nanostructures created us-
+ing HCI can have a form of hillocks, craters (called also pits)
+or caldera-like structures, with diameter of about 5-20 nm and
+a few nanometers vertical extension [15, 16, 11]. It is known
+from experiments, that different parameters of ion beams, type
+of irradiated materials and processing conditions lead to differ-
+ent characteristic of the modifications obtained on a material
+surface, including defect production, sputtering of material and
+changes in material surface topology.
+The recent studies of nanostructures formation on surfaces by
+HCI concentrate mainly on the basic characterization of nanos-
+tructures and fundamental understanding of the mechanisms re-
+sponsible for the surface modifications [15, 10, 12]. Moreover,
+most of the experimental observations were performed for in-
+sulator while for semiconductors (pure Si) and metals (Ti, Au)
+only single experiments were carried out, which due to the lack
+of the systematic studies did not allow for a detailed exami-
+nation of the mechanism of nanostructures production on such
+surfaces [16]. The reason for the small interest in this type
+of studies were the earlier experiments with swift heavy ions
+(SHI), which suggested that in the interaction of such ions with
+materials of high thermal conductivity, the production of nanos-
+tructures is unlikely due to the rapid outflow of energy from the
+area of impact. However, the results of experiments performed
+by Pomeroy et al. [17] and our recent results [18] showed that
+different nanostructures can be produced by slow single HCI
+also on metallic surfaces. Unfortunately in both of these exper-
+iments potential and kinetic energies of the ions were simulta-
+neously changed, which made it difficult to separate their influ-
+ence on the produced nanostructures.
+Systematic experimental studies of the interaction mecha-
+nism are very important also from the theoretical point of view
+because there is still no unified picture of the nanostructure
+creation process. Up to now, the proposed theoretical models
+of the nanostructure production by slow single HCI, includ-
+ing Coulomb explosion [19, 20, 21], molecular dynamics sim-
+ulations [22, 23], inelastic thermal spike model [24, 25], and
+plasma model [26, 27] describe the mechanism only in a qual-
+itative manner and agree quantitatively only with the results
+of selected experiments, mainly for insulators. For the metal-
+lic surface modifications, the micro-staircase model of the HCI
+neutralization accompanied by the charge dependent model of
+the kinetic energy loss has been proposed [28, 12].
+The aim of the present study is the systematic experimental
+and theoretical investigation of the mechanism of energy depo-
+sition and nanostructures creation in collisions of a single HCI
+with metallic surfaces. We consider the moderate ionic veloc-
+ity region, characterized by the interplay of the neutralization
+energy and the deposited kinetic energy. We performed the ex-
+periment with Xeq+ ions (q = 25, 30, 35, 36 and 40) impinging
+upon a gold nanolayer at kinetic energy 280-290 keV and with
+Xe35+ ions at kinetic energies: 280 keV, 360 keV, 420 keV and
+480 keV. As the results, we obtained the well pronounced mod-
+ification of the surface in the form of craters. In the present
+paper, the results are interpreted within the prediction of the
+micro-staircase model [28, 12] and molecular dynamics simu-
+lations for single ionized xenon hitting crystalline gold surface
+[29]. According to the micro-staircase model, simultaneously
+with the ion cascade neutralization above the surface, the neu-
+tralization energy deposits into the solid inducing the first desta-
+bilization of the target as a consequence of the high free elec-
+tron density characteristic for conducting surfaces. Below the
+surface, the kinetic energy loss is governed by the elastic colli-
+sions between the ion (carrying the information about the ionic
+initial charge and velocity) and target atoms.
+This article is organized as follows. In Section 2 we dis-
+cuss the energy deposition process during the interaction of HCI
+with surface and current status of the experimental and theoret-
+ical studies for single and highly ionized xenon atoms interact-
+ing with metallic (gold) surface. In this section we also intro-
+duce the micro-staircase model of the HCI-metal interaction.
+Section 3 is devoted to the present experiment. We characterize
+samples, describe the experimental conditions and the atomic
+force microscope (AFM) system used for the sample imaging.
+In Section 4 we present examples of the AFM images and ex-
+tracted diameters and depths of the observed craters. In Sec-
+tion 5 we discuss the results and compare them with theoretical
+predictions and available experimental data for single ionized
+xenon [30]. The concluding remarks are given in Section 6.
+2. HCI - surface interaction
+2.1. Overview of the HCI interaction with metallic surfaces
+Up to now, nanometer-sized structures produced by individ-
+ual HCI impact on conductive surfaces were reported for a crys-
+talline Au(111) by Pomeroy et al. [17]. In this experiment,
+the samples were irradiated with 200 keV Xe25+ and 350 keV
+Xe44+ ions, which have significantly different potential ener-
+gies, 8 keV and 51 keV, respectively. After irradiation the sam-
+ples were analyzed in situ with scanning tunneling microscope
+(STM). The STM images showed many different features on
+the gold surface, such as isolated hexagons, hexagonal rings
+with craters in the center and hexagonal islands with pits, with
+the features density approximately equal to the ions fluence.
+It is worth to note that previous sputtering measurements [31]
+with gold did not report a measurable increase in sputter yield
+with increasing of the HCI charge and thus probability for a
+nanostructure formation on gold was assumed to be negligible.
+Finally, Pomeroy et al. concluded that the primary formation
+mechanism of the features they observed on Au(111), is related
+to the kinetic energy (nuclear energy loss) and seems weakly
+dependent on the potential energy of the HCI, they emphasized
+the simultaneous change in the potential and kinetic energy of
+the ions used in the experiment, which complicated to isolate
+their contribution to the created nanostructures. Subsequent at-
+tempt to repeat Pomeroy et al. experiment using 440 keV Xe44+
+ions has failed, probably due to too high surface roughness [32].
+A similar experiment, but at lower velocities, was also carried
+out by our group [18]. In this experiment, nanolayers of gold
+2
+
+Q=Q
+Rmin
+D
+Q R
+( )
+q-1
+micro-staircase model of the cascade  neutralization
+and surface destabilization
+final charge
+state
+elastic collisions
+in solid
+nanocrater formation
+Q = q
+fin
+q-2
+e
+-
+macro steps
+e
+-
+e
+-
+intermediate Rydberg state population
+Q = q
+fin
+Z
+q+
+initial charge
+state
+surface
+Figure 1: Schematic description of the micro-staircase model of the cascade neutralization with intermediate Rydberg state population followed by rapid deexcitation
+(both presented by dashed curves) and the nanocrater formation processes during the interaction of HCI with solid surface [12].
+and titanium, were irradiated with low-energy (50-120 keV)
+highly charged xenon ions. The samples were prepared at In-
+stitute of Electronic Materials Technology, Warsaw, Poland, by
+sputtering of gold (50 nm) and titanium (75 nm) nanolayers on
+polished crystalline quartz SiO2(100) 4-inch diameter wafers,
+and titanium (25 nm - 75 nm) nanolayers on crystalline sili-
+con Si (100) wafers. The samples were irradiated at the Kielce
+EBIS facility (Institute of Physics, Jan Kochanowski Univer-
+sity, Kielce, Poland) [33]. As a result of irradiation, we were
+able to create nanohillocks on both titanium and gold surfaces
+and perform statistical analysis of their heights and volumes us-
+ing AFM images [18]. In this experiment the kinetic energy of
+the ions have been charge dependent (because of the ion source
+configuration) and thus it was difficult to extract separately the
+potential or the kinetic energy influence.
+A systematic analysis of the Xeq+ ion interaction with gold
+nanolayers at moderate velocities (craters formation) will be
+presented in Section 3.
+2.2. Overview of the single charged ions interactions with
+metallic surfaces
+A many of scientists have discovered small craters on metal-
+lic surfaces bombarded with single ionized high-energy heavy
+ions which they attribute to the effect of spikes. The concept of
+thermal spikes resulting from single ion impacts was discussed
+for the first time in the literature in the 1950s by researchers
+such as Brinkmann [34], Seeger [35], and Seitz and Koehler
+[36, 37]. In particular, Merkle and J¨ager used transmission elec-
+tron microscopy (TEM) to examine Au surfaces irradiated with
+single ionized Bi and Au ions, in the energy range of 10-500
+keV and discovered craters on the irradiated surfaces for the
+ion energies above 50 keV with fewer than 1% of collisions
+causing the crater formation [38]. Average crater sizes were
+typically about 5 nm. Although the authors conclude that spike
+effects were responsible for the crater formation they attribute
+the effect mainly to sublimation of surface atoms from the sur-
+face [38]. Following this experiment, Birtcher and Donelly ir-
+radiated Au(110) films with Xe+ ions at energies of 50 keV,
+200 keV or 400 keV. They found, using in situ TEM, that single
+xenon ion impacting on gold forms crater with size as large as
+12 nm and that approximately 2-5% of impinging ions produce
+craters [39, 30]. Authors concluded that crater formation re-
+sults from ion-induced sudden melting (and volume expansion)
+of the material associated with localized energy deposition (sur-
+face energy spikes) and explosive outflow of material from the
+hot molten core. The later experiments of Donelly and Birtcher
+on surfaces of Ag, In, and Pb led them to the same conclusions
+[40].
+The results of Donelly and Birtcher experiment [30] were ex-
+amined using classical molecular-dynamics (MD) simulations
+by Bringa et al. [29]. They performed simulations of crater for-
+mation during 0.4-100 keV single charged Xe+ bombardment
+of Au target. The simulations confirmed that the craters are
+built by liquid flow of atoms from the interaction zone. They
+also found that energy density needed for crater production
+strongly depends on the heat spike lifetime and that for xenon
+energies higher than 50 keV cratering can results from lower
+energy densities due to long lifetime of the heat spike. MD
+simulated cratering probability was always higher than 50% in
+the studied energy range [29].
+2.3. Nanostructure formation on metallic surface:
+micro-
+staircase model
+Recently, in the article [12] we discussed the nanohillocks
+formation by the impact of Xeq+ ions on titanium and gold
+nanolayers [18] using the micro-staircase model for the cascade
+neutralization based on the quantum two-state vector model
+(TVM). The model takes into account both the ionic neutraliza-
+tion energy and the kinetic energy deposition inside the solid
+3
+
+[28, 12]. The similar model can be used for the analysis od the
+craters formation.
+According to the model, the process of the cascade neutral-
+ization of the ion, Q = q → q−1 → ...Q(R) → ...qfin, is mainly
+localized in front of the surface (see Fig. 1). At ion-surface
+distance R, the electron is captured from the metal into the in-
+termediate high-n (Rydberg) state of the ion almost in a ground
+state. The population of each macro step consists of several
+micro-steps (population of the low-l Rydberg states nQ with the
+probabilities PnQ). For example, considering the Xe25+ ion im-
+pinging upon the metal surface at moderate velocity v = 0.25
+a.u. we have the following populate scheme [28]: at ion-surface
+distances R, in the range from R = 28 a.u. to R = 10 a.u., the
+Rydberg states corresponding to n = 23 to n = 15 of the ion
+with charge Q = q − 1 = 24 (core charge 25) are populated
+with probabilities Pn25 = 0.01 → 0.2, � Pn25 = 1. After the first
+macro-step is finished at R = 10 a.u, the population of the ion
+of the charge Q = q − 2 with core charge 24 begins in the range
+from R = 9 a.u. to R = 6 a.u. The states n = 14 to n = 12 are
+populated with probabilities Pn24 = 0.3 → 0.35, � Pn24 = 1, and
+so on. In Fig. 1) we present the final stages of the macrosteps
+Q = q, Q = q − 1, ... Q = qfin. At each macro step, the rapid
+deexcitation could be via radiative process and closer to the sur-
+face via Auger type processes with secondary electron emission
+in interplay with the described population process [28]. The
+neutralization cascade finishes when the HCI arrives into the in-
+teraction region at minimal ion-surface distance R = Rmin [12]
+with the final charge qfin (Rmin is the distance from the jellium
+edge [41]). The corresponding neutralization energy W(q,nMV)
+within the nanolayer-metal-vacuum (nMV) system is deposited
+into the first nanometers of the surface [42] very fast (few fs for
+metal targets [11, 17]), increasing the energy density in the im-
+pact region [43] and inducing the destabilization of the surface
+[12].
+The neutralization energy W(q,nMV) is defined as a difference
+between the potential energy Ep ≡ Wq,pot (which describes the
+state of the ion before the beginning of the neutralization) and
+the potential energy in front of the solid surface WqnMV
+fin ,pot [41,
+44, 12]:
+W(q,nMV) = Wq,pot − WqnMV
+fin ,pot.
+(1)
+The energy W(q,nMV) can be calculated using the results valid
+for the metal-vacuum MV-system [12], which is supported by
+the experimental fact that the lattice structure of the nanolayer
+is very similar to the bulk material for the layers thickness [45]
+considered in the present article.
+Although many elementary charge exchange processes are
+possible at solid surface, we assume that the ion with Q = qfin
+penetrates the surface. Within the framework of the model, we
+also assume that neutralization inside the solid in the process
+of the nanostructure formation can be neglected (has negligible
+influence on the total deposited energy). The last assumption
+is based on the fact that the nanocraters are formed in the nar-
+row region of the depth ∆x smaller compared to the penetration
+depth necessary for the ionic charge to be significantly changed
+[12]. Therefore, we use Q ≈ qfin as the ionic charge for the
+analysis of the ionic motion and the corresponding processes
+inside the target (see Fig.2). For more accurate description of
+the overall neutralization process, the analysis of the neutraliza-
+tion below the surface can be added to our model.
+Below the surface, the ions constantly lose their kinetic en-
+ergy due to the elastic collisions with the target nuclei (nu-
+clear stopping power dEn/dx) [11, 46, 47] and the inelastic in-
+teraction with the target electrons (electronic stopping power
+dEe/dx). Simultaneously, the damage of the local atomic struc-
+ture and the surface modification due to the ionic kinetic energy
+loss take place. On the overall ionic trajectory the ionic kinetic
+energy is deposited into the solid. However, analyzing the ex-
+perimentally obtained surface nanostructures, relevant is only
+the near surface region of the length ∆x [12], so that
+Ek,dep = (dE/dx) · ∆x.
+(2)
+For low to moderate ionic velocities electronic stopping power
+can be neglected, i.e., dE/dx = dEn/dx = NS n, where N is the
+atomic density of the target. The corresponding nuclear stop-
+ping cross section S n we calculate using the classical scattering
+theory with charge dependent ion-target atom interaction poten-
+tial [48, 12]:
+Vint(r) = (Z1 − Q)Z2
+r
+ϕ( r
+au
+) + QZ2
+r ϕ( r
+as
+),
+(3)
+where Z1 and Z2 are the nuclear charges of the projectile and the
+target atom, respectively. Other quantities are explicitly given
+in [12]. We note that, the charge dependence of the energy
+loss has been firstly theoretically introduced by Biersack [49].
+Further, the charge dependent kinetic energy transfer for HCI
+interacting with C nanomembrane and C foil target was elabo-
+rated in [50]. The time-dependent interatomic potential energy
+was used to get the more accurate model for the calculation of
+the kinetic energy loss in [51].
+3. Experiment
+The studies presented in this paper are continuation of our
+research [18, 52] related to the nanostructure formation in in-
+teractions of highly charged xenon ions with metallic surfaces.
+The aim of the experiments carried out for the purposes of cur-
+rent work is to separate, as far as it is possible, the influence of
+kinetic and potential energies of the HCI xenon ions on the pro-
+duced surface nanostructures. In the measurements we use gold
+nanolayers of the thickness of 100 nm deposited on Si (110)
+wafers. The structure and properties of such nanolayers were
+expected to be similar to the bulk metal, but with possible lower
+density due to nanolayer structure [45] and an influence of the
+substrate cannot be completely excluded [10].
+3.1. Samples
+The 100 nm Au nanolayers used in this experiment were
+prepared at Institute of Electron Technology (Warsaw, Poland)
+using high vacuum (13·10−8 hPa) e-beam evaporation VST
+TFDS-462U deposition system. The metallic nanolayers were
+evaporated on Topsil (Warsaw, Poland) Si (110) polished prime
+4
+
+wafers type N 4-inch diameter. The thickness of the silicon
+wafer was 0.635 mm ± 0.015 mm. The deposition rate was
+0.4 nm/s. The thickness of the gold was set and controlled by
+a crystal oscillator. Just after preparing, the wafers were cut
+into rectangles of dimensions of 0.5 cm x 1 cm. The rough-
+ness of the samples surface was checked by AFM technique.
+Root-mean-squared (RMS) roughness determined by the AFM
+technique (UMCS, Lublin, Poland) for a few randomly selected
+(1 µm x 1 µm) areas of the gold nanolayers using NanoScope
+Analysis ver.1.40 (Veeco, USA) program were on the level of
+0.45±0.1 nanometers. The crystalline structure of the substrate
+was confirmed based on the measurements carried out using
+the XRD technique. Using the GIXRD technique, it was deter-
+mined that the 100 nm gold nanolayers have a polycrystalline
+structure homogeneous in depth. Additionally using the XRR
+technique, the thickness and density of the nanolayers were
+measured. Thickness turned to be consistent with the declared
+one (100 nm), but the density was slightly lower (but within
+the uncertainties) than the bulk density. The XRR, XRD and
+GIXRD measurements were performed with X’Pert Pro MPD
+reflectometer/diffractometer (for details see [53, 54]), placed at
+Institute of Physics of UJK (Kielce, Poland). The 100 nm Au
+nanolayers were irradiated at the Kielce EBIS facility of the
+Jan Kochanowski University (Kielce, Poland) [33], under high
+vacuum conditions. After irradiation the samples were again
+checked with AFM technique (UMCS, Lublin).
+3.2. Kielce EBIS facility
+The Kielce EBIS facility, built by the Dreebit (Dresden, Ger-
+many), is equipped with electron beam ion trap (EBIS-A) [55].
+The source supplies a wide range of slow HCI from bare ions of
+light elements to Ne-like and Ar-like ions of high-Z elements.
+The maximum electron energy and current available for ioniza-
+tion of the trapped ions are equal 25 keV and 200 mA, respec-
+tively. The ions produced in the EBIS-A source can be extracted
+both in a pulse mode (pulse width from 2 µs up to 40 µs) and
+leaky mode (DC mode) by applying an acceleration voltage up
+to 30 kV. Highly charged ions extracted from the EBIS-A ion
+source are guided by ion beam optical elements (einzel lens and
+X-Y deflectors) of the first straight section of the facility to the
+double focusing analyzing magnet separating the ions accord-
+ing to their mass to charge ratio. The first section of the beam-
+line includes a quadrupole section with pressure gauge, 4-jaw-
+slit collimation system and a Faraday cup. The ions separated
+in the analyzing magnet are directed to the second straight sec-
+tion of the EBIS-A facility. In this section, a pressure gauge,
+X-Y deflectors, a Faraday cup and an einzel lens are mounted.
+Finally, the highly charged ions collide with a sample mounted
+on a 5-axis universal manipulator placed in the experimental
+chamber. The manipulator allows for x, y, z linear movements,
+polar and azimuthal rotations of a sample and variation of its
+temperature in the range of 100-1000 K. The beamline can be
+biased with positive or negative high voltage allowing ion ac-
+celeration or deceleration. For current facility configuration the
+ion energies can be set from 2.5 keV x q up to 30 keV x q, with
+q denoting the ion charge state. All components of the EBIS-A
+facility fulfill the UHV standards and after baking of the sys-
+tem at 150◦C the pressure is in the few 10−10 mbar range (in
+the beamline). One of the unique features of the EBIS facility
+is the ability to prepare, irradiate by highly charged ions and
+characterize the studied samples in the UHV conditions.
+3.3. Measurements
+In the measurements isotopically pure highly charged Xeq+
+ions were extracted from the EBIS-A and, after selecting given
+ion charge state in the dipole magnet, were used to irradiate the
+nanolayers. The ion beam current was measured with a mov-
+able Faraday cup mounted in front of the sample. The spot ra-
+dius of the ion beam on the sample was around 1.5 mm ± 15%
+as it was determined by moving the Faraday cup across the ion
+beam (from the beam profile). The ion fluence was estimated
+on the level 1010 ions/cm2 (with uncertainty of the 10-15%).
+The samples were first placed in a loading chamber pumped
+to about 10−7 mbar, and then transmitted to the experimental
+chamber. The vacuum in the experimental chamber was around
+(2 − 5) × 10−8 mbar. After irradiation, the sample was trans-
+ferred back to the loading chamber and was stored there until it
+was removed for atomic force microscopy investigations, which
+were performed in the air. The measurements were performed
+for two configuration: constant kinetic energy of the ions equal
+to 280-290 keV and different charge states of the xenon ions
+Xeq+, where q = 25, 30, 35, 36, 40 and constant charge state
+(Xe35+) of the ions and different kinetic energies 280 keV, 360
+keV, 420 keV and 480 keV.
+3.4. AFM system
+The topographic modifications of the samples surface in-
+duced by Xeq+ ions were investigated using atomic force mi-
+croscopy in the Analytical Laboratory of Faculty of Chem-
+istry, UMCS, Lublin, Poland. AFM measurements of the stud-
+ied samples were performed using Multimode 8 (Bruker) AFM
+equipped with NanoScope software (Bruker-Veeco, USA). The
+AFM was operated in SCANASYST-HR fast scanning mode
+using SCANASYST-AIR-HR probe (Silicon Tip on Nitride
+Lever) (Bruker) with the cantilever of force constant k = 0.4
+N/m. The lateral and vertical resolutions were 4 nm and 0.1
+nm for the 1 µm x 1 µm, and 2 nm and 0.1 nm for the 500 nm
+x 500 nm images. The obtained images were analyzed with
+Nanoscope Analysis ver. 1.40 software (Veeco, USA).
+4. Results
+4.1. AFM images
+The AFM images of the nanolayers before irradiation (left
+panel) and after irradiation with 280-290 keV Xe30+, Xe36+,
+Xe40+, and Xe35+ of different kinetic energies are presented in
+the Fig. 2. The images were analyzed using the NanoScope
+Analysis software. The size of presented area is 500 nm x 500
+nm. In the images of the irradiated samples, we can clearly see
+the modifications caused by the ion impact. We would like to
+stress here, that such excellent images of metallic surface mod-
+ification caused by HCI impact, to our knowledge, have never
+5
+
+Xe36+ Ekin= 290 keV
+Xe30+ Ekin= 280 keV
+Xe40+ Ekin= 290 keV
+Xe35+ Ekin= 280 keV
+Xe35+ Ekin= 360 keV
+Xe35+ Ekin= 420 keV
+Xe35+ Ekin= 480 keV
+before irradiation
+Figure 2: Topographic AFM 3D images of Au 100 nm nanolayer deposited on Si surface before and after irradiation with HCI Xeq+. Top row: images of the
+nanolayers before irradiation (left panel) and after irradiation with 280-290 keV Xe30+, Xe35+, Xe40+. Bottom row: images of the nanolayers after irradiation with
+Xe35+ of different kinetic energies. The images were analysed using the software NanoScope Analysis ver. 1.40 (Veeco, USA).
+been registered. The same modifications were observed for all
+irradiated samples, with surface density of the nanostructures
+approximately equal to the ion fluence, i.e. one nanostructure
+per one HCl ion impact. Analogous efficiency of the nanostruc-
+ture creation was observed by Pomeroy [17].
+The measured modifications have the form of craters, which
+is confirmed by enlarged AFM 3D images of the individual
+Figure 3: Upper panel: examples of the 3D AFM images of the nanostructures
+on Au 100 nm/Si nanolayer surface irradiated by 280-290 keV Xe30+, Xe35+
+and Xe40+. Lower panel: upside-down 3D AFM images of the nanostructures
+on Au 100 nm/Si nanolayer surface irradiated by 360, 420 and 480 keV Xe35+.
+The images were analysed using the software NanoScope Analysis ver. 1.40
+(Veeco, USA).
+nanostructures which are presented in the Fig. 3. In the upper
+panel the example of the 3D AFM images of the nanostructures
+on Au 100 nm/Si nanolayer surface irradiated by 280-290 keV
+Xe30+, Xe35+, Xe40+ are presented. All observed structures had
+a similar crater-like shape, i.e. a cavity, sometimes with a ring
+around it (check the middle image). Merkle and Jager [38] and
+Bringa et al. [29] postulate that these rings around the cavity
+arise from the sputtering (or rather an outflow) of the original
+atoms being at the place of the structure formation. This was
+confirmed by MD simulations presented in the article of Bringa
+et al. [29]. Similar shape of the crater formed on a Si(100)
+surface by bombardment of a Xe44+ HCI was also observed in
+the simulations performed by Insepov et al. [27] using plasma
+model of space charge neutralization based on impact ioniza-
+tion of semiconductors at high electric fields. In the lower panel
+of the Fig. 3 upside-down 3D AFM images of the nanostruc-
+tures on Au 100 nm/Si nanolayer surface irradiated by 360, 420
+and 480 keV Xe35+ are presented to confirm crater like shape of
+the nanostructures.
+4.2. Analysis of the AFM images
+In many surface studies, a common data analysis strategy is
+to correlate the mean size of nanostructures (parameters like:
+diameter, depth, volume) with different ion parameters, e.g. ki-
+netic energy (ionic velocity) and potential energy (ionic charge
+state), nuclear and electronic stopping powers, etc. Following
+this strategy we have performed size analysis of the observed
+nanostructures.
+For this purpose, all observed images were
+6
+
+Xe30+
+Xe35+
+Xe40+
+20 nm
+360 keV
+420 keV
+480keV
+20 nm0.0
+1:Height
+500.0nm0.0
+1:Height
+500.0nm0.0
+1:Height
+500.0nm0
+1:Height
+500.0nm0.0
+1:Height
+500.0nm0.0
+1:Height
+500.0nm0.0
+1:Height
+500.0nm0.0
+1:Height
+500.0nmFigure 4: Example of the individual crater profile (black points) with fitted
+Gaussian curve (solid line). The sigma σ (standard deviation), FWHM (full
+width in the half of maximum) and crater depth d quantities are Gaussian distri-
+bution parameters. The crater diameter on the surface is assumed as 2FWHM.
+first carefully checked and optimized with NanoScope Anal-
+ysis software and, after extracting of the data from the AFM
+images (using STEP function of the software), analysed with
+Origin Pro data analysis software. From the individual profile
+of the craters we have extracted their size parameters, includ-
+ing the diameter on the surface. All unambiguously identified
+structures on the surface of the samples were analyzed indepen-
+dently. Example of the individual crater profile (black points)
+is presented in the Figure 4.
+The profiles were fitted by Gaussian curve (solid line), which
+reflected very well the shape of the crater. At this point, we note
+that the needle used for the AFM analysis of all samples had a
+tip curvature radius r = 2 nm, and the structures after HCI mod-
+ification were characterized by diameters of about 10-25 nm,
+so an incorrect tip contact was considered unlikely, especially
+in the diameters of nanostructures on the sample surface. The
+crater diameter on the surface was defined as double FWHM (2
+× FWHM). In the example presented in the Figure 4, the crater
+diameter at surface was fitted as 15.16 nm, while its depth as
+0.93 nm. An alternative way to determine the diameter is to
+take values of four standard deviations (4×σ). In presented ex-
+amples, it gives 4σ = 12.88 nm. In general, it was observed
+that crater diameter defined as 2 × FWHM was about 10-15%
+higher than 4×σ quantity.
+For the Au nanolayer irradiated by Xe ions in given charge
+state, for each sample around 50 to 150 craters were analyzed in
+the way described above. Finally, the mean values of the crater
+depth and crater diameters for each irradiated Au nanolayer
+were calculated. Based on the statistical analysis of the crater
+profiles, the dependence of the crater depth and crater diameter
+on the Xe ions potential and kinetic energy were studied. In
+the case of the crater depth no dependence on the ions poten-
+tial energy was observed. The crater depth was on the constant
+level of about 0.9 nm ± 0.15 nm. The linear fit to the data, in-
+cluding uncertainties defined by the standard deviation of mean
+value, gave a very week dependence (the slope is equal to 0.001
+nm/keV).
+The obtained mean values of craters diameter in function of
+the potential energy of the Xeq+ ions are plotted in the Fig.
+5. The uncertainties marked for experimental points were cal-
+culated as the sum of the mean value standard deviation and
+10% of the mean value (compensation of the difference be-
+tween 2FWHM and 4σ quantities within uncertainty). The Xe
+ions charge states marked by (*) denoted a slightly different ki-
+netic energy (290 keV), caused by difficulties in setting a given
+charge state and kinetic energy. As one can see from the fig-
+ure we have observed clear influence of the ionic charge state
+(expressed via initial potential energy) on the nanocrater diam-
+eter. For the lowest ion charge state (25+) the mean nanocrater
+diameter is 12.0 nm, next this parameter systematically grows,
+reaching for the highest charge state the value 23.4 nm.
+The results of the study of the crater diameter in function of
+the ions kinetic energy are shown in the Figure 6 for Xe35+. The
+nanocrater diameter is in the range 13-15 nm. The linear func-
+tion fitted to the experimental points showed a weak alteration
+of the dependence. In the Figure 6 the results of Donnelly and
+Birtcher experiment [39] for Xe+ ions are also presented which
+confirm small dependence of the created nanocrater diameter
+on the ions kinetic energy in the considered energy range. On
+the other hand, the nanocrater diameters for HCI xenon ions are
+much higher than for single ionized xenon.
+Figure 5: Dependence of the craters diameter on the Xeq+ ions potential energy.
+The Xe ions charge states marked by (*) denoted a slightly different kinetic
+energy (290 keV).
+7
+
+40
+Au 100 nm on Si
+35
+f craters diameter (nm)
+E,= 280 keV
+30
+Xe40+ (*)
+25
+Xe36+(*)
+Xe35+
+20
+T
+Xe30+
+15
+Mean of
+10
+2FWHM
+5
+0
+0
+5
+10
+15
+¥20
+25
+30
+35
+40　45
+Ep (keV)2FWHM
+-0.2
+-0.4
+Depth (nm)
+-0.6
+FWHM
+-0.8
+o = 3.22 nm
+W
+FWHM = 7.58 nm
+d = 0.93 nm
+-1.0
+-
+-1.2
+-3g
+-2g
+a
+2g
+3g
+(×c,c)
+experiment
+-1.4
+fit
+0
+10
+20
+30
+40
+50
+60
+Distance (nm)Figure 6: Dependence of the nanocrater diameter created by the Xe35+ ions
+impinging on Au surface on the kinetic energy (this experiment). For compari-
+son, the results of Donnelly and Birtcher experiment [39] for Xe+ ions are also
+presented. In the figure, also the nuclear stopping power S1/2 (solid line) is
+presented.
+5. Discussion
+5.1. Theoretical model of the crater formation
+In order to interpret the present experiments performed with
+Xeq+ ions of initial charges q = 25, 30, 35, 36 and 40 in the
+interaction with a gold 100 nm nanolayer deposited on Si (110)
+wafers (nMV-system) at velocity v = 0.29 a.u., as well as to
+examine the velocity dependence studied in the case of Xe35+
+for v = 0.29, 0.33, 0.36 and 0.38 a.u. we use the micro-staircase
+model.
+The formation of nanocraters we discuss from the stand-
+point of the energy dissipation into the surface, which consists
+both of the neutralization energy and the deposited kinetic en-
+ergy [41, 44, 12]. For velocities characteristic for the crater
+formation the neutralization is incomplete so that the corre-
+sponding neutralization energy represents only a part of the
+ionic initial potential energy. The remaining potential energy
+Ep − W(q,nMV) contributes to the charge dependent potential in-
+teraction (Eq.(3)) between the ion and the target atoms and thus
+it is converted into kinetic energy of the target atoms (stop-
+ping power calculated in micro-staircase model is charge de-
+pendent). We calculate the neutralization energy according to
+Eq. (1) for W(q,nMV) = W(q,MV); taking into account that the neu-
+tralization energy is weakly dependent on the solid work func-
+tion φ, we consider the neutralization energy for φ = 5 eV (work
+function of Au is 5.47 eV). For the calculation of the kinetic
+energy loss we employ Eq. (2) for the active interaction length
+∆x ≈ 5¯c ≈ 38.5 a.u., where ¯c is the mean lattice constant for
+Au-target; we note that the crater depth dmax ≈ 1nm = 18.9 a.u.
+To define the ion-atom interaction in solid we use the charge of
+the projectile Q = q fin(q, v) obtained in [12].
+Figure 7: Upper panel: neutralization energy W(q,nMV) and deposited kinetic
+energy Ek,dep versus ionic velocity v for Xeq+ ions, q = 25, 30, 35 and 40,
+impinging on the Au nanolayers (formation of the crater in the present exper-
+iment). Lower panel: the critical ionic velocity vc versus initial ionic charge
+q.
+In Fig. 7, at upper panel, we present the neutralization energy
+W(q,nMV) and the deposited kinetic energy Ek,dep relevant for the
+surface nanocrater creation by the impact of Xeq+ ions with core
+charges q = 25, 30, 35 and 40 on 100 nm Au nanolayer on Si
+(110) wafers as a function of the ionic velocity v. The neutral-
+ization energy W(q,nMV) decreases with increasing of the ionic
+velocity v; on the other hand, the deposited kinetic energy Ek,dep
+increases with increasing of v. The results indicate the interplay
+of these two energies in the process of the surface nanocrater
+formation. That is, we define [12] the critical velocity vc by the
+relation:
+W(q)(vc) = Ek,dep(vc).
+(4)
+For velocities v ≪ vc (very low ionic velocities) dominant
+8
+
+1800
+100 nm Au on Si
+V
+1600
+c
+V
+exp
+1400
+.40+
+△x = 38.5 a.u
+(a.u.)
+Xe
+1200
+1000
+.35+
+Xe
+800
+Xe40+
+600
+30+
+Xe25+
+Xe
+400
+ 25+
+Xe
+200
+0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+v (a.u.)35+
+25
+Xe
+E = 25.5 keV
+p
+Mean of craters diameter (nm)
+present experiment (Xe35+
+351
+20
+Donnelly and Birtcher (Xet)
+15
+10
+5
+/2
+n
+0
+0
+100
+200
+300
+400
+500
+600
+Ek (keV)100 nm Au on Si
+Vexp
+0.25
+C
+exp
+0.20
+(a.u.)
+x = 38.5 a.u.
+7
+0.15
+0.10
+≤ v. nanohillocks formation
+V
+0.05
+exp
+V. craters formation
+V
+exp
+0.00
+25
+30
+35
+40
+ionic charge (a.u.)Table 1:
+Critical velocities vc in the case of the surface nanocrater formation
+in the nMV-system by the impact of Xeq+ ions.
+100 nm Au nanolayer
+q
+25
+30
+35
+40
+Ek (keV)
+280
+280
+280
+280
+vexp (a.u.)
+0.29
+0.29
+0.29
+0.29
+vc (a.u.)
+0.07
+0.17
+0.22
+0.23
+role in the energy participation in the solid has the neutraliza-
+tion energy W(q,nMV), while for v ≫ vc (swift heavy ions) the
+deposited kinetic energy Ek,dep completely determines the pro-
+cess of the nanostructure formation [12]. The quantity vc we
+present in Fig. 7 at lower panel as a function of the initial ionic
+charge q. The values of the critical velocities are also given in
+Table 1.
+For all considered ionic charges the critical velocities vc are
+lower compared to the experimental value vexp = 0.29 a.u.
+(Ek = Mv2
+exp/2, M = 131 · 1836 a.u. for Xeq+ ions, where
+Ek denotes the initial ionic kinetic energy). For charges q = 35
+and q = 40, the critical ionic velocities vc are close to the ex-
+perimental one (see Table 1), indicating that both energies con-
+tribute to the crater formation. The values of vc for Xe25+ and
+for Xe30+ are much smaller than the experimental ones, so that
+the main contribution in the nanostructure formation gives the
+deposited kinetic energy Ek,dep. Concerning the type (shape)
+of the nanostructures, the appearance of the nanocraters in ex-
+periment is in accord with the prediction of the micro-staircase
+model. On the other hand, the hillocks have been obtained in
+experiment with Xe35+ ions [18] impinging upon the surface of
+the 50 nm gold layer at velocity 0.19 a.u., while the critical one
+is 0.22 a.u. [12]. In the case of 25 nm titanium nanolayers the
+experimental velocities for q = 20, 25, 30 and 35 were 0.144,
+0.16, 0.176 and 0.19, in a.u., respectively. The corresponding
+critical velocities are 0.06, 0.16, 0.22 and 0.24, in a.u. [12]. The
+results of the present experiment and the results of the previous
+ones [18, 52, 17] confirm a common conclusion: for the ionic
+velocities v < vc or v ≈ vc the surface modification leads to the
+nanohillocks formation [18, 12], while for v > vc the predomi-
+nant surface structures are the craters (rings) [52, 17, 12].
+The neutralization energy W(q,nMV) and the deposited kinetic
+energy Ek,dep can be also connected to the size of the formed
+nanostructures. The experimental results for the crater diame-
+ters show the significant increasing from q = 25 to q = 40, see
+5. For vexp = 0.29 a.u. and Xe25+ ion the diameter D = 12
+nm (226.8 a.u.)
+and for Xe40+ ion diameter D = 23.4 nm
+(442.3 a.u.). The ionic neutralization energy (and also the ini-
+tial ionic potential energy) exhibits the same increasing behav-
+ior (W(25,nMV) = 47 a.u. and W(40,nMV) = 162 a.u.), see Table 2.
+The q dependence of the deposited kinetic energy, obtained
+on the base of Eq. 3, is less pronounced (Ek,dep for Xe25+ is 557
+a.u. and for Xe40+, Ek,dep = 587 a.u.), see Table 2. For these
+reasons, it is convenient to present the experimentally obtained
+crater diameters as a function of the potential (or the neutraliza-
+tion) energy.
+The velocity effect on the crater diameter D for Xe35+ ions
+we study for the experimental values vexp = 0.29, 0.33, 0.36
+and 0.38 a.u. From the experimental results one recognize the
+weak decreasing of the quantity D with increasing of the ionic
+velocity (kinetic energy) see Fig. 6; (for v =0.29 a.u. diameter
+D = 15 nm (283.5 a.u.) and for v =0.38 a.u. diameter D = 12
+nm (226.8 a.u.)). On the other hand, the deposited kinetic en-
+ergies Ek,dep increase slightly with increasing of v (for v=0.29
+a.u. Ek,dep= 577 a.u. and for v =0.38 a.u. Ek,dep =635 a.u.),
+while the neutralization energy W(35,nMV) show a noticeable de-
+creasing character (for v =0.29 a.u. W(35,nMV) =127 a.u. and for
+v =0.38 a.u. W(35,nMV) =35.5 a.u.), see Tab. 3.
+The role of the neutralization energy W(q,nMV) and the de-
+posited kinetic energy Ek,dep can be more precisely discussed
+from the relation between the crater diameter and the total de-
+posited energy: Etot,dep = Ek,dep + W(q,nMV). Assuming that
+the energy Etot,dep is localized in the cylindrical region od the
+diameter D and depth ∆x, we get the relation:
+D = f
+�
+Ek,dep + W(q,nMV),
+(5)
+where the factor f reflects the target properties. Both energies
+Ek,dep and W(q,nMV) are charge dependent, so that the nanocrater
+size will express the same behavior. From Eq. 5 and Tables 2
+and 3 one conclude that the main contribution to the diameter D
+gives the deposited kinetic energy. However, the neutralization
+energy term in Eq. 5 must be taken into account in order to ob-
+tain the experimentally observed behavior of the crater diameter
+D discussed in Tables Tab. 2 and Tab. 3: pronounced increas-
+ing od D with increasing of q and weak decreasing of D with
+increasing of the ionic velocity v. The discussed significance
+of the deposited kinetic energy and the role of the neutraliza-
+tion energy is characteristic for the moderate velocity case used
+in the experiment. We note that for the very low velocities,
+the neutralization energy (close to the potential energy) plays
+a dominant role. The increasing of D with increasing of q and
+the v-dependence of the crater diameter obtained in the present
+experiment is in a qualitative agreement with the prediction of
+the proposed model.
+Within the framework of micro-staircase model, the mecha-
+nism of the nanocraters and nanohillocks formation at metallic
+surfaces is different. At velocities v < vc characteristic for the
+hillock formation, a dominant role has the neutralization pro-
+cess: the strength of the bonds between atoms decreases in-
+ducing their stretching. The rearrangement of atoms leads to
+Table 2: Neutralization energy W(q,nMV), deposited kinetic energy Ek,dep and
+crater diameter D in the case of the surface nanocrater formation for v = vexp =
+0.29 a.u. in the nMV-system by the impact of Xeq+ ions, for ∆x ≈ 38.5 a.u.
+100 nm Au nanolayer
+q
+25
+30
+35
+40
+W(q,nMV) (a.u.)
+47
+88
+127
+162
+Ek,dep (a.u.)
+557
+569
+577
+587
+D (nm)
+12
+13
+15
+23.4
+9
+
+Table 3: Neutralization energy W(q,nMV), deposited kinetic energy Ek,dep and
+crater diameter D in the case of the surface nanocrater formation for vexp =
+0.29, 0.33, 0.36 and 0.38 a.u. in the nMV-system by the impact of Xe35+ ions,
+for ∆x ≈ 38.5 a.u.
+100 nm Au nanolayer
+vexp (a.u.)
+0.29
+0.33
+0.36
+0.38
+W(q,nMV) (a.u.)
+127
+66.8
+47.2
+35.5
+Ek,dep (a.u.)
+577
+600
+608
+612
+D (nm)
+15
+12.9
+13.9
+12.15
+the rise of the volume above the surface and hillock forma-
+tion. The deposited energy is insufficient for melting the ma-
+terial (the hillocks are formed without melting). The predicted
+mechanism of the nanohillock formation on metal surface [12]
+is different in comparison to the thermal spike model used in
+the case of nanohillock formation on insulator [56]. In the case
+of crater creation (for v > vc) considered in the present paper,
+the neutralization (above the surface) induces the lattice vibra-
+tion and the first destabilization of the target. Inside the solid,
+the elastic collisions of the charged projectile with target atoms
+and produced recoils lead to the disordering of the target atoms
+generating the highly disturbed near surface area V. A large
+amount of kinetic energy deposits into the solid, resulting in a
+significant decrease in the target cohesive energy. The strength
+of the bounds between the target atoms inside the crater vol-
+ume tends to be zero and a number of atoms are ejected from
+the surface. In the intermediate stages of the craters formation,
+in the centre of the active volume V, it is possible that the tem-
+perature far exceeds the melting temperature. The deposited
+neutralization energy during the process above the surface has
+a small contribution to the nanocrater formation in comparison
+to the deposited kinetic energy during the collision cascade be-
+low the surface; however, the main q and v dependence of the
+crater size are governed by the neutralization energy.
+5.2. MD simulations
+We also compare the present experimental results with
+molecular dynamics (MD) simulations presented in [29]. We
+compare our results for HCI xenon ion with Xe+ ion after fit-
+ting the data on Fig. 5, and further normalization to the poten-
+tial energy equal to the potential energy of Xe+. The results
+of the comparison we present in Fig. 8. In the figure, the nu-
+clear stopping power S1/2 (solid line) and the ion energy E1/3
+(dashed line) curves are also presented. We obtain very good
+agreement with the experimental data of Donnelly and Birtcher
+[39] and MD simulations [29], which confirms the validity of
+our experimental procedure.
+It is important to mention that, very recently, molecular dy-
+namics methodology coupled with two-temperature model (2T-
+MD) [57], was used by Khara et al. to simulate the structural
+evolution of bcc metals (Fe and W) and fcc metals (Cu and
+Ni) following irradiation by SHI (electronic stopping power
+regime) [23].
+They found that number of material parame-
+ters (melting temperature, electronic thermal conductivity and
+Figure 8: Comparison of the results of crater radius, as a function of the ion
+kinetic energy, obtained by our group with the results of the experiment (Don-
+nelly and Birtcher) for single charged Xe+ ion impinging on Au surface [39]
+and MD simulations [29]. In the figure, the nuclear stopping power S1/2 (solid
+line) and ion energy E1/3 (dashed line) curves are also presented.
+electron-phonon coupling strength), and their electronic proper-
+ties temperature dependence, have a strong influence on the re-
+sistance of metals to damage induced by SHI irradiation. They
+also showed that high thermal conductivity and relatively low
+electron-phonon coupling of fcc metals render them relatively
+insensitive to damage, in spite of their relatively low melting
+temperatures. The strong electron-phonon coupling of the bcc
+metals (Fe and W) is primarily responsible for the sensitivity
+of these metals to damage [23]. The cited calculations are in
+contradiction with the experimental results for Au (fcc metal)
+- HCI systems, for which we obtain the surface nanocraters in
+the velocity range v ∈ [0.29, 0.38] a.u. and the nanohillocks
+for lower ionic velocities v ∈ [0.144, 0.19] a.u. [18]. In the
+case of nanocrater formation, both the deposited kinetic en-
+ergy and the neutralization energy participate in the process;
+the nanohillocks are formed predominantly by the participation
+of the neutralization energy. The calculations [23] showed a
+significantly different response of bcc and fcc metals to the de-
+position of energy in the interaction of SHI ions with surfaces
+and encouraged us to undertake such tests for HCI. At the mo-
+ment, similar calculations does not exist for HCI, where it is
+necessary to take into account the neutralization process. The
+model proposed here represents a theoretical approach of that
+kind, stimulated by the experimental findings.
+6. Conclusions
+Understanding of mechanism of the nanostructures creation
+on metallic surfaces is very important both from the theoret-
+ical and possible application point of view. In this paper we
+10
+
+10
+Xe->Au
+.1/3
+this experiment (extrapolated
+Donnelly and Birtcher
+Crater radius (nm)
+MD simulations
+H
+1/2
+8
+1
+1
+10
+100
+1000
+Ek (keV)have studied Au nanolayers surfaces irradiated by slow highly
+charged Xeq+ ions (q = 25, 30, 35, 36 and 40). For the first
+time, for such systems, well pronounced modifications of the
+nanolayers surfaces, due to impact of the HCI ions, in the form
+of nanocraters have been observed. This allowed for systemati-
+cal study of dependence of the size of nanostructures on poten-
+tial and kinetic energy of the ions. Analysis of the crater diam-
+eter D for different initial charge states q of the Xe ions showed
+a significant dependence of the quantity q (expressed via poten-
+tial energy in Fig. 5). Additionally, for interaction of the Xe35+
+ions with Au nanolayers the dependence of the structure forma-
+tion on the ion kinetic energy (280 keV, 360 keV, 420 keV and
+480 keV) was studied. Week alteration of the crater diameter
+(Fig. 6) with the ion kinetic energy was observed in the ana-
+lyzed energy range. Our results were qualitatively interpreted
+within the micro-staircase model for the neutralization energy
+combined by the charge dependent kinetic energy deposition.
+The experimental results are also compared with the available
+simulations and the previous experimental data. The results will
+be potentially of great importance for further development of
+modern technologies (e.g. single HCI nano-pattering [58], role
+of the HCI impurities in tokamak plasma-metallic wall interac-
+tion [59]) and will open up many application possibilities (e.g.
+DNA sequencing or water desalination [60]).
+Acknowledgments
+The equipment was purchased thanks to the financial support
+of the European Regional Development Fund in the framework
+of the Polish Innovative Economy Operational Program (con-
+tract no. WNP-POIG.02.02.00-26-023/08), the Development
+of Eastern Poland Program (contract no.
+POPW .01.01.00-
+26-013/09-04) and Polish Ministry of Education and Science
+(project 28/ 489259/SPUB/SP/2021). N. N. Nedeljkovi´c and
+M. D. Majki´c are grateful for the support of the Ministry of
+Education, Science and Technological Development of the Re-
+public of Serbia (projects 171016, 171029).
+References
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+nanostructures at surfaces, Nature 437 (2005) 671–679. doi:10.1038/
+nature04166.
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+MedSegDiff-V2: Diffusion based Medical Image
+Segmentation with Transformer
+Junde Wu1, Rao Fu2, Huihui Fang1, Yu Zhang2, and Yanwu Xu1
+1 Baidu Research
+2 Mind Vogue Lab
+Abstract. The Diffusion Probabilistic Model (DPM) has recently gained
+popularity in the field of computer vision, thanks to its image genera-
+tion applications, such as Imagen, Latent Diffusion Models, and Stable
+Diffusion, which have demonstrated impressive capabilities and sparked
+much discussion within the community. Recent studies have also found
+DPM to be useful in the field of medical image analysis, as evidenced by
+the strong performance of the medical image segmentation model Med-
+SegDiff in various tasks. While these models were originally designed
+with a UNet backbone, they may also potentially benefit from the in-
+corporation of vision transformer techniques. However, we discovered
+that simply combining these two approaches resulted in subpar perfor-
+mance. In this paper, we propose a novel transformer-based conditional
+UNet framework, as well as a new Spectrum-Space Transformer (SS-
+Former) to model the interaction between noise and semantic features.
+This architectural improvement leads to a new diffusion-based medical
+image segmentation method called MedSegDiff-V2, which significantly
+improves the performance of MedSegDiff. We have verified the effective-
+ness of MedSegDiff-V2 on eighteen organs of five segmentation datasets
+with different image modalities. Our experimental results demonstrate
+that MedSegDiff-V2 outperforms state-of-the-art (SOTA) methods by a
+considerable margin, further proving the generalizability and effective-
+ness of the proposed model.
+Keywords: Multi-rater learning · Optic disc/cup segmentation · Glau-
+coma diagnosis
+1
+Introduction
+Medical image segmentation is the process of dividing a medical image into
+distinct regions of interest. It is a crucial step in many medical image analysis
+applications, such as diagnosis, surgical planning, and image-guided surgery. The
+ability to better understand and track changes over time in these images is vital
+for medical professionals. In recent years, there has been a growing interest in
+automated medical image segmentation methods, as they have the potential to
+improve the consistency and accuracy of results. With the advancement of deep
+learning techniques, several studies have successfully applied neural network-
+based models, including classical convolutional neural networks (CNNs) [11] and
+arXiv:2301.11798v1  [eess.IV]  19 Jan 2023
+
+2
+J. Wu et al.
+the recently popular vision transformers (ViTs) [2,22], to medical image segmen-
+tation tasks.
+Very recently, the Diffusion Probabilistic Model (DPM) [9] has gained popu-
+larity as a powerful class of generative models, capable of generating high-quality
+and diverse images [18–20]. Inspired by its success, some researchers have at-
+tempted to apply DPM in the field of medical image segmentation [6,13,16,23,
+25]. One such method, called MedSegDiff [25], achieved great success and outper-
+formed previous state-of-the-art (SOTA) segmentation methods, such as nnUNet
+and TransUNet. However, these methods are all based on classical UNet back-
+bones. In a separate line of research, vision transformers, which have shown out-
+standing performance in vision representation learning on natural images, have
+also brought success in medical image segmentation and have quickly become
+a popular approach. Among them, transformer-convolution hybrid architectures
+have attracted the most attention and achieved the best performance.
+A natural next step is to combine the transformer-based UNet, such as Tran-
+sUNet, with DPM. However, we found that this straightforward strategy leads
+to subpar performance. One issue is that the transformer-abstracted conditional
+feature is not compatible with the feature of the backbone. The transformer
+learns deep semantic features from the raw image, while the diffusion backbone
+abstracts features from a corrupted, noisy mask. Additionally, the dynamic and
+global nature of the transformer makes it more sensitive than CNNs. Thus, the
+adaptive condition strategy used in MedSegDiff causes larger variance in the
+outputs in the transformer setting. This requires running the model more times
+for ensemble and makes it harder to converge during training.
+To overcome the aforementioned challenges, we have designed a novel transformer-
+based conditional UNet architecture for the diffusion process. The main idea is
+to use two different conditioning techniques to condition the backbone model
+with the source image segmentation features in the diffusion process. One is the
+anchor condition, which integrates the conditional segmentation features into
+the diffusion model encoder to reduce the diffusion variance. The other is the
+semantic condition that integrates the conditional segmentation embedding into
+the diffusion embedding. To effectively bridge the gap between diffusion noise
+embedding and conditional semantic features, we propose a novel transformer
+mechanism called the Spectrum-Space Transformer (SS-Former) that learns the
+interaction between them. This allows the model to have a smaller diffusion
+variance while also benefiting from the global and dynamic representation capa-
+bilities provided by the transformer.
+More specifically, in the anchor condition, we integrate the decoded segmen-
+tation feature of the condition model into the encoded features of the diffusion
+model. We design a novel Gaussian Spatial Attention mechanism to implement
+this integration. It relaxes the conditional segmentation feature with more uncer-
+tainty, thus providing the diffusion process more flexibility to further calibrate
+the predictions. In the semantic condition, we integrate the semantic segmenta-
+tion embedding into the diffusion model embedding using our novel SS-Former.
+SS-Former is an interlaced cross-attention chain with one part that enhances the
+
+Title Suppressed Due to Excessive Length
+3
+semantic embedding using the noise embedding and another part that enhances
+the noise embedding using the semantic embedding. We design a novel cross-
+attention mechanism over the frequency domain to eliminate the high-frequency
+noises in the noise embedding, thus aligning the noise and semantic features.
+We have verified MedSegDiff-V2 on a wide range of medical segmentation tasks,
+such as optic-cup segmentation, brain tumor segmentation, abdominal organs
+segmentation, and thyroid nodule segmentation. The images used in these tasks
+have different modalities, such as MRI, CT, and ultrasonography. MedSegDiff-V2
+outperforms the previous state-of-the-art (SOTA) on all the tasks with different
+modalities, which showcases the generalization and effectiveness of the proposed
+method. In brief, the contributions of this paper are:
+– The first to integrate transformer into a diffusion-based model for general
+medical image segmentation.
+– An anchor condition with Gaussian Spatial Attention to mitigate the diffu-
+sion variance and speed up the ensemble.
+– A semantic condition with SS-Former to model the segmentation noise and
+semantic feature interaction.
+– SOTA performance on sixteen medical segmentation tasks with different
+image modalities.
+2
+Method
+2.1
+Overall architecture
+The overall flow of MedSegDiff-V2 is shown in Figure 1. To introduce the pro-
+cess, consider a single step t of the diffusion process. The noisy mask xt is first
+inputted to a UNet with conditional integration, called the Diffusion Model.
+The condition sources are the segmentation features extracted from the raw im-
+ages through another standard UNet, called the Condition Model. Two different
+conditioning manners are applied to the Diffusion Model: anchor condition and
+semantic condition. Following the flow of the input, the anchor condition is first
+imposed on the encoder of the Diffusion Model. It integrates the anchor segmen-
+tation features, which are the decoded segmentation features of the Condition
+Model, into the encoded features of the Diffusion Model. This allows the diffusion
+model to be initialized by a rough but static reference, which helps to reduce the
+diffusion variances. The semantic condition is then imposed on the embedding
+of the Diffusion Model. This integrates the semantic segmentation embedding of
+the Condition Model into the embedding of the Diffusion Model. This conditional
+integration is implemented by the SS-Former, which bridges the gap between the
+noise and semantic embedding, and abstracts a stronger representation with the
+advantage of the global and dynamic nature of transformer.
+MedSegDiff is trained using a standard noise prediction loss Lnoise following
+DPM [9] and an anchor loss Lanchor. Lanchor is a combination of soft dice loss
+and cross-entropy loss. Specifically, the total loss function is represented as:
+Lt
+total = Lt
+noise + (t ≡ 0
+(mod α))(Ldice + βLce)
+(1)
+
+4
+J. Wu et al.
+Fig. 1: An illustration of MedSegDiff. For the clarity, the time step encoding and
+skip connection in UNet are omitted in the figure.
+where t ≡ 0 (mod α) control the times of supervision over Condition Model
+through hyper-parameter α, β is another empirical hyper-parameter to weight
+the cross-entropy loss.
+2.2
+Anchor Condition with Gaussian Spatial Attention
+Without the inductive bias of convolution layer, transformer blocks have stronger
+representation but also to be more sensitive to the input variance. Directly
+adding the transformer block to the Diffusion Model will cause the large vari-
+ance on each times’ outputs, as we show the experimental results in Section 3.
+To overcome this negative effect, we introduce the anchor condition operation
+to the Diffusion Model.
+The anchor condition integrates the anchor, which is the decoded segmenta-
+tion features of the Condition Model into the encoder features of the Diffusion
+Model. We propose a Gaussian Spatial Attention to represent the uncertainty
+nature of the given segmentation features from the Condition Model. Formally,
+consider we integrate the last conditional segmentation feature f −1
+c
+into the first
+diffusion feature f 0
+d. Gaussian Spatial Attention can be expressed as:
+fanc = Max(f −1
+c
+∗ kGauss, f −1
+c
+),
+(2)
+f
+′0
+d = Sigmoid(fanc ∗ kConv1×1) · f 0
+d + f 0
+d,
+(3)
+
+SS-Former
+Timestep t
+Scale & Shift
+X
+Scale & Shift
+MLP
+C
+FFT
+MLP
+W
+Timestep tTitle Suppressed Due to Excessive Length
+5
+where ∗ denotes slide-window kernel manipulation, · denotes general element-
+wise manipulation. In Eqn. 2, we first apply a Gaussian kernel kG over f −1
+c
+to
+smooth the activation, as f −1
+c
+serves as an anchor but may not be completely
+accurate. The mean and variance of. The mean and variance of kG are learnable.
+We then select the maximum value between the smoothed map and the original
+feature map to preserve the most relevant information, resulting in a smoothed
+anchor feature fanc. In Eqn. 3, we integrate fanc into f 0
+d to obtain an enhanced
+feature f
+′0
+d . Specifically, we first apply a 1×1 convolution k1×1conv to reduce the
+number of channels in the anchor feature to 1. Then, we use a sigmoid activation
+function on the anchor feature and add it to each channel of f 0
+d, similar to the
+implementation of spatial attention [24]. Gaussian Spatial Attention extracts
+a rough anchor feature from the Condition Model and integrates it into the
+Diffusion Model. This provides the Diffusion Model with a correct range for
+predictions while also allowing it to further refine the results.
+2.3
+Semantic Condition with SS-Former
+We propose a novel transformer architecture, called Spectrum-Space Trans-
+former (SS-Former), to effectively integrate the conditional segmentation em-
+bedding into the diffusion embedding. SS-Former is composed of several blocks
+that share the same architecture. Each block consists of two cross-attention-
+like modules. The first encodes the diffusion noise embedding into the condition
+semantic embedding, and the next module encodes the noise-blended semantic
+embedding into the diffusion noise embedding. This allows the model to learn
+the interaction between noise and semantic features and achieve a stronger rep-
+resentation.
+Since the Diffusion Model predicts the redundant noise from the noisy mask
+input, it will have a domain gap between its embedding and that of the condi-
+tional segmentation semantic embedding. This gap can lead to confusion when
+using matrix manipulations in a stranded transformer. To address this chal-
+lenge, we propose a novel spectrum-space attention mechanism. The key idea
+is to merge semantic and noise information in Fourier space, rather than Eu-
+clidean space. This allows for the separation and blending of components based
+on frequency-affinity in different spectrums. Formally, consider c0 is the deep-
+est feature embedding of Condition Model and e is that of Diffusion Model.
+We first transfer c0 and e to the Fourier space, denoted as F(c0) and F(e), re-
+spectively. Note that the feature maps are all patchlized and liner projected in
+accordance with the standard vision transformer method. Then we compute an
+affinity weight map over Fourier space taking e as the query and c0 as the key,
+as represented by the following equation:
+M = a sin(w(F(c0)Wq)(F(e)Wk)T ),
+(4)
+where Wq and Wk are the learnable query and key weights in Fourier space. We
+then employ a periodic active function to limit the representation spectrum in
+Fourier space, as a substitute of standard activation applied in Euclidean space
+
+6
+J. Wu et al.
+in standard self-attention. In the implementation, we use a sine function sin
+with learnable amplitude a and frequency w as the constraint.
+The affinity map is then transferred back to Euclidean space using inverse
+fast Fourier transform (IFFT) and applied to condition features in value,
+f = F −1(M)(c0wv),
+(5)
+where W v is the learnable value weights. We then apply the time embedding
+to an AdaIN normalization following the classic diffusion implementation [15],
+which normalizes the feature and then expands it using scale and shift parame-
+ters learned from the time embedding. This makes the transformer adaptive to
+the step information. We also use a Multi-layer Perceptron (MLP) to further
+refine the attention result, obtaining the final feature ˜c0. The following attention
+module is symmetric to the first one, using the combined feature ˜c0 as the query
+and noise embedding e as the key and value, in order to transform the segmen-
+tation features to the noise domain. The transformed feature c1 will serve as the
+condition embedding for the next block.
+3
+Experiments
+3.1
+Dataset
+We conduct the experiments on total four different medical image segmentation
+datasets. One dataset is used to verify the general segmentation performance,
+which is Multi-Organ Segmentation in Abdominal CT Images. We use public
+AMOS2022 [12] dataset, which we employ 200 multi-contrast abdominal CT
+from AMOS 2022 with sixteen anatomies manually annotated for abdominal
+multi-organ segmentation. The other three datasets are used to verify the model
+performance on multi-modal images, which are the optic-cup segmentation from
+fundus images, the brain tumor segmentation from MRI images, and the thyroid
+nodule segmentation from ultrasound images. The experiments of glaucoma,
+thyroid cancer and melanoma diagnosis are conducted on REFUGE-2 dataset
+[4], BraTs-2021 dataset [1] and DDTI dataset [17], which contain 1200, 2000,
+8046 samples, respectively. The datasets are publicly available with segmentation
+labels. Train/validation/test sets are split following the default settings of the
+dataset.
+3.2
+Implementation Details
+The primary architecture of MedSegDiff is a modified ResUNet [26], which we
+implement using a ResNet encoder followed by a UNet decoder. The specific
+network configuration can be found in [25]. All experiments were conducted
+using the PyTorch platform and trained/tested on 4 NVIDIA A100 GPUs. All
+images were uniformly resized to a resolution of 256×256 pixels. The networks
+were trained in an end-to-end manner using the AdamW [14] optimizer with a
+batch size of 32. The initial learning rate was set to 1 ×10−4.
+
+Title Suppressed Due to Excessive Length
+7
+3.3
+Main Results
+To verify the general medical image segmentation performance, we compare
+MedSegDiff-V2 with SOTA segmentation methods on multi-organ segmentation
+dataset AMOS2022. The quantitative results are shown in Table 1. In the table,
+we compare with the segmentation methods which are widely-used and well-
+recognized in the community, including the CNN-based method nnUNet [10],
+the transformer-based methods TransUNet [2], UNetr [8], Swin-UNetr [7] and
+the diffusion based method EnsDiff [23], MedSegDiff [25]. We also compare with
+a simple combination of diffusion and transformer model. We replace the UNet
+model in MedSegDiff to TransUNet and denoted it as ’MedSegDiff + Tran-
+sUNet’ in the table. We evaluate the segmentation performance by Dice score.
+The compared methods are all implemented with their default setting.
+Table 1: The comparison of MedSegDiff-V2 with SOTA segmentation methods
+over AMOS dataset evaluated by Dice Score. Best results are denoted as bold.
+Methods
+Spleen R.Kid L.Kid Gall.
+Eso.
+Liver Stom. Aorta IVC
+Panc. RAG LAG
+Duo.
+Blad. Pros. Avg
+TransUNet
+0.881 0.928 0.919 0.813 0.740 0.973 0.832 0.919 0.841 0.713 0.638 0.565 0.685 0.748 0.692 0.792
+Baseline
+(EnsDiff)
+0.905 0.918 0.904 0.732 0.723 0.947 0.738 0.915 0.838 0.704 0.677 0.618 0.715 0.673 0.680 0.779
+UNetr
+0.926 0.936 0.918 0.785 0.702 0.969 0.788 0.893 0.828 0.732 0.717 0.554 0.658 0.683 0.722 0.784
+Swin-UNetr
+0.959 0.960 0.949 0.894 0.827 0.979 0.899 0.944 0.899 0.828 0.791 0.745 0.817 0.875 0.841 0.880
+nnUNet
+0.951 0.962 0.939 0.889 0.843 0.962 0.870 0.958 0.865 0.835 0.801 0.768 0.835 0.832 0.836 0.876
+MedSegDiff
+0.963 0.965 0.953 0.917 0.846 0.971 0.906 0.952 0.918 0.854 0.803 0.751 0.819 0.868 0.855 0.889
+MedSegDiff
++ TransUNet
+0.941 0.932 0.921 0.934 0.813 0.946 0.867 0.921 0.880 0.821 0.793 0.528 0.788 0.813 0.837 0.762
+MedSegDiff-V2 0.971 0.969 0.964 0.932 0.864 0.976 0.934 0.968 0.925 0.871 0.815 0.762 0.827 0.873 0.871 0.901
+As seen in Table 1, advanced network architectures and sophisticated designs
+are crucial for achieving good performance. With regards to network architec-
+ture, well-designed transformer-based models such as Swin-UNetr outperform
+the carefully designed CNN-based model, nnUNet. The diffusion-based model
+MedSegDiff again outperforms the transformer-based models on most of the or-
+gans. However, network architecture alone is not the sole determining factor for
+performance. For example, the well-designed CNN-based model nnUNet consid-
+erably outperforms the transformer-based model TransUNet and UNetr in the
+table. This is also true for diffusion-based models. We can see that a straight-
+forward adoption of the diffusion model for medical image segmentation, i.e.,
+EnsDiff, achieves an unsatisfied performance. A simple combination of trans-
+former and diffusion model, i.e., MedSegDiff+TransUNet, obtains even worse
+performance than the standard MedSegDiff. This is because the transformer is
+more sensitive to adaptive conditions and extracts more delicate semantic fea-
+tures that diverge from the diffusion backbone. By introducing anchor condition
+and SS-Former in MedSegDiff-V2, the diffusion + transformer model overcomes
+these challenges and shows superior performance. We compare it with diffusion-
+based models, i.e., EnsDiff and MedSegDiff, using the same ensemble times (all
+set to five times), and it produces more stable and accurate results as shown in
+the table.
+
+8
+J. Wu et al.
+Fig. 2: The visual comparison with SOTA segmentation models.
+Figure 2 presents a qualitative comparison of MedSegDiff-V2 and other com-
+petitive methods. It can be observed that MedSegDiff-V2 segments more ac-
+curately on parts that are difficult to recognize by the human eye. Due to its
+ability to benefit from the superior generation capability of the diffusion model
+and the semantic representation capability of the transformer, it can generate
+segmentation maps with precise and accurate details, even in low-contrast or
+ambiguous areas.
+We also compare our method to state-of-the-art (SOTA) segmentation meth-
+ods proposed for three specific tasks with different image modalities. The main
+results are presented in Table 2. In the table, ResUnet [26] and BEAL [21] are
+used for optic disc and cup segmentation, TransBTS [22] and EnsemDiff [23]
+are used for brain tumor segmentation, and MTSeg [5] and UltraUNet [3] are
+used for thyroid nodule segmentation. We also compare to general medical image
+segmentation methods on these three datasets. The segmentation performance
+is evaluated using the Dice score and IoU.
+As seen in Table 2, MedSegDiff-V2 outperforms all other methods on three
+different tasks, showcasing its ability to generalize to various medical segmen-
+tation tasks and image modalities. Compared to the UNet-based MedSegDiff,
+it improves by 2.0% on Optic-Cup, 1.9% on Brain-Tumor, and 3.9% on Thy-
+roid Nodule in terms of the Dice score, illustrating the effectiveness of the
+transformer-based backbone. Additionally, when compared to MedSegDiff with
+TransUNet, it overcomes compatibility issues and significantly improves perfor-
+mance on all three tasks, demonstrating the effectiveness of the proposed anchor
+condition and SS-Former.
+
+Title Suppressed Due to Excessive Length
+9
+Table 2: The comparison of MedSegDiff with SOTA segmentation methods. Best
+results are denoted as bold.
+Optic-Cup Brain-Turmor Thyroid Nodule
+Dice IoU Dice
+IoU
+Dice
+IoU
+ResUnet
+80.1 72.3
+-
+-
+-
+-
+BEAL
+83.5 74.1
+-
+-
+-
+-
+TransBTS
+-
+-
+87.6
+78.3
+-
+-
+EnsemDiff
+-
+-
+88.7
+80.9
+-
+-
+MTSeg
+-
+-
+-
+-
+82.3
+75.2
+UltraUNet
+-
+-
+-
+-
+84.5
+76.2
+UNetr
+83.2 73.3 87.3
+80.6
+81.7
+73.5
+Swin-UNetr
+84.3 74.5 88.4
+81.8
+83.5
+74.8
+nnUNet
+84.9 75.1 88.2
+80.4
+84.2
+76.2
+TransUNet
+85.6 75.9 86.6
+79.0
+83.5
+75.1
+MedsegDiff
+85.9 76.2 88.9
+81.2
+84.8
+76.4
+MedsegDiff+TransUNet 82.1 72.6 86.1
+78.0
+79.2
+71.4
+MedSegDiff-v2
+87.9 80.3 90.8
+83.4
+88.7
+81.5
+3.4
+Ablation Study
+We conducted a comprehensive ablation study to verify the effectiveness of the
+proposed anchor conditioning and SS-Former. The results are shown in Table
+3, where Anc.Cond. denotes anchor conditioning. We evaluate the performance
+using the Dice score (%) on all three tasks. The models were run five times
+for ensemble. From the table, we can see that Anc.Cond. significantly improves
+the vanilla diffusion model, with an improvement of 2.4% on thyroid nodule
+segmentation, 1.6% and 1.8% respectively. SS-Former learns the interaction be-
+tween noise and semantic features with a vision transformer-based architecture,
+further improving the segmentation results. It promotes MedSegDiff-V2 by over
+1% on all three tasks and achieves new state-of-the-art performance.
+Table 3: An ablation study on anchor conditioning and SS-Former. Dice score(%)
+is used as the metric.
+Anc.Cond. SS-Former OpticCup BrainTumor ThyroidNodule
+84.6
+88.2
+84.1
+✓
+86.2
+89.4
+86.5
+✓
+✓
+87.9
+90.8
+88.7
+4
+Conclusion
+In this paper, we enhance the diffusion-based medical image segmentation frame-
+work by incorporating the transformer mechanism into the original UNet back-
+
+10
+J. Wu et al.
+bone, called MedSegDiff-V2. We propose an anchor condition to ensure the sta-
+bility of the model and a novel SS-Former architecture to learn the interaction
+between noise and semantic features. The comparative experiments were con-
+ducted on 18 organs and 4 medical image segmentation datasets with different
+image modalities and our model outperformed previous state-of-the-art methods.
+As the first transformer-based diffusion model for medical image segmentation,
+we believe MedSegDiff-V2 will serve as a benchmark for future research.
+
+Title Suppressed Due to Excessive Length
+11
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+

DNE4T4oBgHgl3EQf6A5m/content/tmp_files/2301.05328v1.pdf.txt

@@ -0,0 +1,524 @@
+TETRA-PENTA-DECA-HEXAGONAL-GRAPHENE
+(TPDH-GRAPHENE) HYDROGENATION PATTERNS: DYNAMICS
+AND ELECTRONIC STRUCTURE
+Caique Campos, Matheus Medina, Pedro Alves da Silva Autreto
+Center for Natural and Human Sciences (CCNH)
+Federal University of ABC (UFABC)
+Santo André - SP, 09210-170, Brazil.
+pedro.autreto@ufabc.edu.br
+Douglas Soares Galvao
+Physics Institute Gleb Wataghin (IFGW)
+State University of Campinas (UNICAMP)
+Campinas/SP, Brazil
+galvao@ifi.unicamp.br
+ABSTRACT
+The advent of graphene has renewed the interest in other 2D carbon-based materials. Bhattacharya
+and Jana have proposed a new carbon allotrope, composed of different polygonal carbon rings
+containing 4, 5, 6, and 10 atoms, named Tetra-Penta-Deca-Hexagonal-graphene (TPDH-graphene).
+This unusual topology created material with interesting mechanical, electronic, and optical properties
+and several potential applications, including UV protection. Like other 2D carbon structures, chemical
+functionalizations can be used to tune their TPDH-graphene properties. In this work, we investigated
+the hydrogenation dynamics of TPDH-graphene and its effects on its electronic structure, combining
+DFT and fully atomistic reactive molecular dynamics simulations. Our results show that H atoms
+are mainly incorporated on tetragonal ring sites (up to 80% at% at 300 K), leading to the appearance
+of well-delimited pentagonal carbon stripes. The electronic structure of the hydrogenated structures
+shows the formation of narrow bandgaps with the presence of Dirac cone-like structures, indicative
+of anisotropic transport properties.
+1
+Introduction
+The versatility in chemical bonding (different hybridizations) of carbon atoms allows the existence of a wide variety of
+different structures (allotropes) [1], such as fullerenes [2], nanotubes [3], and graphene [4]. Graphene is a 2D allotrope
+of sp2 carbon atoms tightly packed into a hexagonal honeycomb lattice. It presents high carrier mobility (5000cm2/V.s
+)[4, 5], high thermal conductivity (5000WmK−1) [6], and Young modulus value of 1 TPa [7], one of the highest
+values ever measured. It has unveiled new and unique physics phenomena, including the quantum Hall effect [8], the
+ambipolar electric field effect [4], and the massless charge carriers of Dirac fermions [9]. These remarkable properties
+have made graphene the subject of a large number of theoretical and experimental studies in different areas, such as
+catalysis [10], electronics [11], spintronics [12], twistronics [13], and gas sensors [14], to name just a few.
+However, graphene is a null electronic gap material, even exhibiting extraordinary electronic properties, which limits its
+use in some applications [4]. Chemical functionalizations, such as hydrogenation, are one viable mechanism for altering
+graphene-like structures’ properties (including opening the gap [15–17] or changing the Fermi level [18]). Structural
+and electronic changes are introduced when the chemical species form covalent bonds. The partial hydrogenation of
+graphene introduces unsaturated sp3 carbon atoms that can be used to attach additional functional groups.
+arXiv:2301.05328v1  [cond-mat.mtrl-sci]  12 Jan 2023
+
+Running Title for Header
+Figure 1: (a) Schematics of the unit cell of tetra-penta-deca-hexagonal-graphene (TPDH) and the corresponding
+carbon-carbon bond-length values. The different colors indicate non-equivalent carbon atoms. (b) A 2 × 2 supercell
+illustrating the TPDHG rings and the pores of the structure. The corresponding Unit cell vector values are indicated
+in the highlighted red rectangle. (c) The structural setup simulation used in the simulations. A TPDH membrane
+(indicated in blue) is deposited on a graphene frame (gray), and the TPDHG/graphene structure is immersed in a
+hydrogen atmosphere (yellow). See text for discussions.
+Despite these limitations, the advent of graphene created a revolution in materials science and renewed the interest in
+2D carbon allotropes. Among these structures, it is worth mentioning graphynes and biphenylene carbon networks [19].
+Graphynes are the generic name for families of 2D carbon porous structures containing hexagon rings connected
+by acetylenic groups and with sp and sp2 hybridized carbon atoms in the same lattice [19]. Graphdyines refer
+to the structural families where two acetylenic groups connect the hexagons [20]. They can exhibit metallic and
+semiconducting behaviors [21] and have been exploited in different technological applications [22].
+Biphenylene carbon networks (including biphenylene carbon and graphenylenes) are families of porous structures
+composed of mixed carbon rings (pentagons, hexagons, heptagons, octagons, etc.) [19, 23, 24]. Similarly to graphynes,
+they can be metallic, or semiconductors and have potential applications in catalysis [23], gas sensors [25], batteries
+[26], and energy storage applications [27]. Recently, new synthetic routes for graphynes [28, 29] and biphenylene
+carbon networks [30] have been reported increasing the interest in these materials. Bhattacharya and Jana [31] have
+proposed a new structure composed of two pentagons and a tetragonal ring called tetra-penta-octogonal graphene
+(TPO-graphene). It is metallic with a Dirac cone at 3.7 eV above the Fermi level. More recently, they proposed
+another structure belonging to the tetra-pentagonal graphene family composed of sp2 carbon rings with 4, 5, 12, and 6
+atoms (Fig. 1) named tetra-penta-deca-hexagonal graphene (TPDH-graphene). It possesses thermal and dynamical
+stability and exhibits elastic anisotropy with Young’s modulus value larger than that of graphene in a specific direction.
+Depending on the morphology, TPDH-graphene nanoribbons can exhibit metallic, or semiconductor behavior [32].
+In this work, we have investigated the effects of hydrogenation on the structural and electronic properties of TPDH-
+graphene (TPDH-gr). The hydrogenation of TPDH-gr sheets was investigated through reactive molecular dynamics
+simulations. Structural optimization, energy, and electronic properties were further analyzed using ab initio (DFT)
+calculations.
+In spite of graphene’s extraordinary properties, it is a null gap material, which Chemical functionalization is one viable
+mechanism to introduce specific modifications into graphene-like structures. Structural and electronic changes are
+introduced when the chemical species being introduced form a covalent bond. For example, graphite oxides can form
+oxygen groups in graphene sheets dispersed in water and organic solvents [33]. Stankovich et al. prepared graphite
+oxides functionalized with isocyanates that were later exfoliated into graphene oxides dispersed in an aprotic polar
+solvent [34] in a stable manner. Partial hydrogenation of graphene sheets introduces unsaturated carbon atoms sp3 that
+neighbor unpaired with electrons that can be used to attach additional functional groups. Chemical functionalization
+also allows one to change the electronic properties of the structure by opening a bandgap [15–17] or changing the Fermi
+level [18].
+2
+
+a)
+b
+c)
+C4
+C3
+C2
+1.41A
+1.50A
+C1
+1.43A
+A
+= 6.97
+-b
+4.94
+aRunning Title for Header
+Figure 2: Adsorption energies for TPDH-gr in a) the non-equivalent sites, b) with an H atom adsorbed in the C1
+site, c) two H atoms adsorbed in the C1 and C7’ sites, and d) tree H atoms adsorbed in the C1, C7’ and C5 sites. e)
+Non-equivalent sites in TPDH-gr. f) remaining sites in the tetragonal ring with the C1 site occupied. The top sites are
+indicated by the solid line, while the bottom sides are indicated by the dashed ones and prime labels. g) Top and side
+views of TPDH-gr with C1 and C7’ sites occupied. The side view also shows the buckling height. h) Top and side
+views of TPDH-gr with a fully hydrogenated tetragonal ring.
+2
+Computational Methods
+First-principles calculations were carried out within the Density Functional Theory (DFT) framework as implemented in
+Quantum Espresso code [35]. Electron-ion interactions were dealt with Projected Augmented wave (PAW) and Ultra-soft
+pseudopotentials for C and H atoms, respectively. They were obtained from the Standard Solid State Pseudopotentials
+library (SSSP) [36, 37]. Exchange and correlation potential were used within the Generalized Gradient Approximation
+(GGA) with the parameterization of Perdew, Burke, and Ernzerhof (GGA-PBE functional) [38]. Valence electrons were
+treated with a set of plane waves basis set with a kinetic energy cutoff of 680 eV. The diagonalization of the density
+matrix was performed with the Davidson iterative method with matrix overlap using the self-consistency threshold
+of 10−6 eV. In the ionic relaxation calculations, the convergence thresholds were set to 10−3 eV and 10−2 eV/Å for
+energy and forces, respectively. Brillouin zone (BZ) sampling was performed using a 12 × 12 × 1 (16 × 16 × 1) k-point
+grid for SCF (NSCF) calculations following the scheme proposed by Monkhosrt and Pack [39]. For electronic structure
+calculations, the k-points were chosen along the following path in the BZ: Γ(0, 0, 0) - M(0.5, 0.5, 0) - X(0.5, 0, 0) -
+Γ(0, 0, 0) - Y (0, 0.5, 0) - M(0.5, 0.5, 0) - Γ(0, 0, 0).
+We have also carried out fully atomistic molecular dynamics (MD) simulations using the large-scale atomic/molecular
+massively parallel simulator (LAMMPS) code[40]. Atomic interactions were treated with the reactive force field
+(ReaxFF) [41], with C-C interaction parameters developed by Chenoweth et al.[42]. All MD simulations were carried
+out in the canonical (NVT) ensemble, with a time step of 0.25 fs, and using a Nosé-Hoover thermostat [43]. The
+hydrogenation simulations were carried out considering a TPDH-gr membrane deposited on a graphene frame, as shown
+in Fig. 1.c. The TPDH-graphene membrane is a 24x15 supercell, in which only its central part (16x11) is exposed to the
+hydrogen atmosphere, resulting in a total number of 2112 available adsorption/reaction sites. The hydrogen atmosphere
+was composed of 500 atoms in a volume of 60 000 Å3 on each side of the membrane, constrained to the exposed region
+of the membrane. This methodology has been successfully applied to other systems, such as Me-graphane[16] and
+graphone[44].
+3
+
+a)
+b)
+)
+d)
+4.0
+4.0-
+1111
+3.5
+3.5
+3.5
+3.5-
+3.0
+3.0
+3.0
+3.0
+[eV/atom]
+2.5
+2.5.
+2.5
+2.5 
+[eV/ato]
+2.0
+2.0
+2.0
+1.5 -
+1.5
+1.5
+1.5
+1.0
+1.0
+1.0
+1.0-
+L
+0.5
+0.5
+0.5
+0.5
+0.0
+0.0
+0.0-
+0.0
+C6'
+C3
+C6
+C1
+C4
+C5
+C6'
+C2
+C5 
+C5'
+C6.
+C6'
+C7
+C7
+C5'
+C6
+Site
+Site
+Site
+Site
+C4
+g)
+f)
+h)
+e)
+C6
+-C6'
+h = 1.185 A
+h = 0.87 ARunning Title for Header
+3
+Results and Discussion
+3.1
+Ab initio Binding Energy and Hydrogenation Dynamics
+TPDH-gr has a Pmmm (space group #47) symmetry; the 12 carbon atoms in its unit cell are arranged in an
+orthorhombic lattice. The obtained optimized lattice parameters were: a = 4.94 Å, b = 6.97 Å with γ = 90o. There
+are three different bond lengths (1.41, 1.50, and 1.44 Å .) involving the C atoms, as shown in Fig. 1.a. Except for the C
+atoms bonded along the ⃗a direction in the tetragonal ring, the bond lengths are close to those sp2 in graphene (1.41
+Å)[45]. These results agree well with those reported by Batthacharya and Jana [32].
+The most favorable sites for H adsorption/reaction were investigated by evaluating the binding energy per adsorbed
+atom, calculated as the energy difference between the hydrogenated structure and its parts:
+Eb = −
+�ET P DH+nH − (ET P DH + nEH)
+n
+�
+where ET P DH+nH is the energy of TPDH-gr with n adsorbed H atoms, ET P DH is the energy of a TPDH-gr unit cell,
+and EH the energy of an isolated H atom. The negative sign means that high energies indicate more favorable sites for
+adsorption than others in the same structure. First, an H atom is adsorbed at each of the non-equivalent sites (Fig. 1.a).
+The site corresponding to the highest energy is taken as the most favorable. Then, a second H is adsorbed at each of the
+remaining sites, and the most favorable one is evaluated according to Eb. This process is repeated until the tetragonal
+ring on the TPDH-gr is fully hydrogenated. We present the binding energies and obtained structures in Fig. 2.
+The adsorption of the first H atom is more favorable on the C1 site, as seen in Fig. 2.a with Eb of 3.35 eV/atom. After
+C1-Cx adsorption (with x = 2, 5, and 7), the bond length values increased to 1.51, 1.55, and 1.53 Å , respectively,
+indicating a transition to sp3=like-bond in the C1 atom. It is worth mentioning that for sites located in the tetragonal
+ring, the top and bottom configurations (Fig. 1.d) were considered. Adsorption of a single H atom at each of these sites
+resulted in roughly the same results for Eb, as can be seen in Table 1S in Supplementary Material.
+The adsorption of the second H atom (resulting in 16% hydrogen coverage) is more favorable at the C7′ site (Fig. 2.c),
+with an Eb of +3.75 eV/atom. The resulting lattice distortions in the direction perpendicular to the structure plane
+lead to a significant buckling of h = 0.87 Å , as seen in Fig. 2.f. The distortions of the structure and the fact that two
+neighboring C atoms adsorb the pair of H atoms (but on opposite sides of the sheet) are in accordance with the results
+reported by Boukhvalov and Katsnelson for the hydrogenation of graphene sheets [46].
+Interestingly, the adsorption of a third H atom gives the same Eb for both C5 and C6’ sites, as seen in Fig. 2.e. In this
+case, the configuration in which the C1, C7’ and C5 sites are occupied was imposed, which will be justified later. The
+resulting structure presents an overall increase in the Cx-C bond lengths (with x = 1, 7, and 5). The vertical distance
+separating the C1 and C7 atoms is 1.02 Å versus 0.84 Å for the corresponding value between the C5 and C6 atoms.
+The adsorption of a fourth H atom (33% hydrogen coverage) is more favorable at the C6’ site with Eb = 4.0 eV/ Å and
+buckling of h = 1.185 Å(Fig. 2.g, h). It is clear that choosing the C5 or C6’ sites in the adsorption of the third H atom
+leads basically to the same configuration (C1-C7’-C5-C6’). Therefore, choosing C5 or C6’ for the adsorption of the
+third H atom is equivalent.
+These calculations reveal a pattern for the hydrogenation of the tetragonal ring, which consists of two lines of H atoms
+on opposite sides of the basal plane sheet, leading to the formation of well-delimited pentagonal ring strips along the
+direction of the lattice vector a. DFT calculations confirm that this configuration is indeed more favorable. Molecular
+Dynamics simulations, discussed below, produced similar results,
+Reactive molecular dynamics simulations were carried out to study the dynamics and temperature effects on hydrogen
+adsorption of bigger TPDH-graphene membranes (Fig. 1), which would be cost-prohibitive with DFT methods.
+Representative MD snapshots of both sides of the TPDH-graphene membrane during the hydrogenation process (at
+300K) are presented in Fig. 3 (a) - (c).
+The H atoms are predominantly incorporated throughout the MD simulations on the C1 sites. Analyzing the hydrogena-
+tion process, from Fig. 3 (a) to (c), we can see that the hydrogen-adsorbed C1 sites act as seeds to the hydrogenation of
+their C1 neighbors, forming lines through the structure surface, which is an expected result, based on the DFT binding
+energy ordering values.
+In Fig. 4, we present the number of adsorbed/bonded hydrogen atoms at each site of the TPDH-gr unit cell, as a function
+of the simulation time, for the different temperature values considered here. The hydrogenation occurs mainly at the C1
+sites for all temperatures. High rates of H incorporation indicate high reactivity for hydrogenation. At low temperatures
+4
+
+Running Title for Header
+Figure 3: Representative MD snapshots at different simulation times: a) 4 ps, b) 7.5 ps, and c) 200 ps of the
+hydrogenation of the TPDH-gr membrane. Results from simulations at 300k.
+Figure 4: The number of adsorbed/bonded hydrogen atoms at each site of the TPDH-gr unit cell as a function of the
+simulation time (at %) for 150, 300, 500, and 800 K. The color of the curves indicates the corresponding sites in the
+unit cell (left, upper).
+(150K), the C2 and C4 sites have approximately the same low adsorption rates, while the C3 sites exhibit insignificant
+or no hydrogen incorporations. Increasing the temperature, C4, C2, and C3 sites become more reactive, while above
+300K, the C1 site has a slight decrease in reactivity.
+3.2
+Electronic Structure
+In Fig. 5.a), we present pristine (non-Hydrogenated) TPDH-gr electronic band structure and the corresponding projected
+density of states (pDOS) (obtained from DFT-GGA-PBE calculations). We can see that pristine TPDG-gr exhibits a
+5
+
+Top
+Bottom
+4 ps
+7.5 ps
+200 ps
+a)
+b)
+c)
+C3
+C2
+C1
+C480
+088482
+Hydrogen
+bonded
+200
+(at.%)
+10
+150
+0
+800 K
+100
+500 K
+50
+Temperature
+Time (ps)
+300K
+0
+150 KRunning Title for Header
+Figure 5: Electronic band structures and the corresponding projected density of states (pDOS) for a) non-hydrogenated
+TPDH-gr, b) TPDH-gr with the tetragonal ring partially hydrogenated (C1 and C7’ sites occupied), and c) tetragonal
+ring fully hydrogenated. The total density of states is shown in black, while the blue and green curves represent the
+projected DOS into orbitals s and p, respectively.
+semimetallic behavior. The highest (lowest) valence (conduction) band is partially filled. These results are consistent
+with previous works published in the literature [32].
+The effects of H adsorption in the tetragonal ring were investigated for the cases with a pair adsorbed in neighboring
+atoms, in opposite sites of the sheet, and with all four sites of the ring occupied (Fig. 2.g,h respectively).
+The adsorption of two hydrogen atoms in the C1 and C7’ sites results in the opening of the direct gaps by approximately
+1 eV at k-points M and Γ, as shown in Fig. 5.b. Surprisingly, the valence and conduction bands overlap at the Fermi
+level, giving rise to a Dirac cone-like, between the k-points Y and M. Near this point, the electronic dispersion is
+unusually linear, and charge carriers behave like massless fermions, obeying the Dirac relativistic equation. It is expected
+that unusual transport properties arise from this pattern in the band structure, as predicted and experimentally observed
+for graphene [9]. The electronic band structure and the corresponding pDOS of the TPDH with full hydrogenation
+of the tetragonal ring are shown in Fig. 5.c. We can see the appearance of narrow gaps (0.5 eV ) between k-points Γ
+and M, and a very narrow direct gap at point Y . The Dirac cone-like is shifted near the Γ points with respect to the
+half-hydrogenated structure.
+6
+
+a)
+4
+Tot
+3
+S
+p
+2
+M
+e
+1
+0
+E
+1
+-1
+E
+-3
+4
+M
+X
+Y
+pDOS
+b)
+4
+Tot
+3
+p
+2
+M
+e
+1
+0
+E
+1
+-1
+E
+-2
+3
+4
+M
+X
+^
+M
+<
+pDOS
+c)
+4
+Tot
+3
+S
+p
+2
+M
+e
+1
+f
+0
+E
+1
+-1
+E
+-2
+4
+M
+X
+Y
+M
+pDOSRunning Title for Header
+4
+Conclusions
+This work investigated the effects of hydrogenation on the structural and electronic properties of tetra-penta-deca-
+hexagonal-graphene (TPDH-gr) sheets. Molecular dynamics (MD) simulations revealed that H atoms are mainly
+incorporated in the tetragonal ring (C1 sites) with up to 80% adsorption at 300 K (Fig. 4). The number of H atoms
+incorporated on C2 and C4 sites varies according to the temperature. Hydrogenation produces a pattern where H lines
+are formed on both sides of the sheet (Figs. 1.h and 3.c) generating well delimited pentagonal ring strips along ⃗a
+direction. DFT calculations further corroborate that the complete hydrogenation of the tetragonal ring is energetically
+favorable.
+Electronic structure calculations for the partially hydrogenated structure show the formation of gaps and the emergence
+of a Dirac cone-like between the points Γ and M. For the fully hydrogenated ring, narrow band gaps followed by wide
+gaps are identified, and the Dirac cone-like is translated near the Γ point. This electronic profile strongly indicates
+anisotropic transport properties, although these remain to be further explored in future works.
+Conflicts of interest
+There are no conflicts to declare.
+Acknowledgements
+The authors thank PRH.49 for funding and CCM-UFABC for the computational resources provided and CNPq
+(#310045/2019-3)
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+arXiv:2301.03582v1  [astro-ph.HE]  9 Jan 2023
+Mon. Not. R. Astron. Soc. 000, 1–5 (2019)
+Printed 10 January 2023
+(MN LATEX style file v2.2)
+Angular Momentum Transfer in QPEs from Galaxy Nuclei
+Andrew King1,2,3,⋆
+1 Department of Physics & Astronomy, University of Leicester, Leicester, LE1 7RH, UK
+2 Astronomical Institute Anton Pannekoek, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
+3 Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, Netherlands
+⋆ E-mail: ark@astro.le.ac.uk
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+A suggested model for quasi–periodic eruptions (QPEs) from galaxy nuclei invokes a
+white dwarf in an eccentric orbit about the central massive black hole. I point out
+that the extreme mass ratio allows the presence of strong Lindblad resonances in
+the accretion disc. These are important for the stability of mass transfer, and may
+trigger the eruptions themselves by rapidly transferring angular momentum from the
+accretion disc (which is likely to be eccentric itself) to the orbiting WD companion at
+pericentre. I consider the effects of von Zeipel–Lidov–Kozai (ZLK) cycles caused by
+a perturber on a more distant orbit about the central black hole. I show that ZLK
+cycles can change the orbital periods of QPE systems on timescales much shorter than
+the mass transfer time, as seen in ASASSN-14ko, and produce correlated short–term
+variations of their mass transfer rates and orbital periods, as recently observed in
+GSN 069. Further monitoring of these sources should constrain the parameters of any
+perturbing companions. This in turn may constrain the nature of the events creating
+QPE systems, and perhaps give major insights into how the central black holes in
+low–mass galaxies are able to grow.
+Key words: galaxies: active: black hole physics: X–rays: galaxies
+1
+INTRODUCTION
+X–ray observations of the nuclei of low–mass galaxies
+show that several of them produce quasi–periodic eruptions
+(QPEs: Miniutti et al., 2019; Giustini et al. 2020; Song et al.
+2020; Arcodia et al. 2021; Chakraborty et al. 2021; Payne
+et al., 2021, 2022). Typically these sources have outbursts
+by factors ∼ 100 in X–rays, which recur in roughly pe-
+riodic fashion. The recurrence times currently range from
+a few hours up to ∼ 100 d or ∼ 1 yr, and it is likely
+that these limits will widen as data accumulate from di-
+rect observations and archival searches. The X–rays have
+ultrasoft blackbody spectra and luminosities implying typ-
+ical radii ≳ Rg = GM1/c2 = 6 × 1010m5 cm, of order the
+gravitational (Rg) and ISCO radii of a black hole of mass
+M1 ∼ 105m5M⊙. These are consistent with the massive
+black holes (MBHs) one might expect in these low–mass
+galaxy nuclei.
+In (King, 2020) I suggested a simple model for the first
+known QPE source GSN 069. This postulated a low–mass
+star (with mass M2 ≪ M1) – found from the requirement
+of internal consistency with observational selection to be a
+white dwarf (WD) – on a highly eccentric orbit about the
+central MBH, transferring mass to it at pericentre passage
+via an accretion disc. In a second paper (King, 2022, here-
+after K22) I used the parametrization introduced by Chen et
+al. (2022), extended to the full WD mass range, to show that
+this model was consistent with the data for 5 of the 6 known
+QPE sources, together with a previously unassociated sys-
+tem HLX-1, where the periodicity is ∼ 1 yr. (I discuss the
+‘missing’ system ASSASN–14ko in Note 1 to Table 1 at the
+end of the paper.)
+In all cases, loss of orbital energy E and angular mo-
+mentum J via gravitational radiation (GR) is the ultimate
+driver of mass transfer, and observational selection effects
+mean that the orbiting star is in practice a low–mass white
+dwarf (WD) in detectable QPE systems (K22). This is the
+likely explanation for the presence of CNO–processed mate-
+rial found in GSN 069 (Sheng et al., 2021).
+This model requires that mass transfer is dynamically
+stable; the Roche lobe and the WD radius must move in
+step as GR reduces the orbital semi–major axis a and the
+eccentricity e on the same timescale. Since the WD expands
+as it loses mass, it must gain angular momentum from the
+accretion disc and move in a wider orbit, implying a larger
+tidal radius.
+This stability has recently been questioned, so I dis-
+cuss it further in Section 2. Orbital resonances within the
+accretion disc are likely to cause the required enhanced an-
+gular momentum transfer to the WD. The resonances may
+© 2019 RAS
+
+2
+Andrew King
+also directly cause the quasiperiodic eruptions themselves
+as the orbiting WD passes pericentre – a similar origin has
+been suggested for the superoutbursts of the stellar–mass
+SU UMa binaries (Osaki & Kato, 2013).
+During the orbital evolution of a QPE binary the peri-
+centre separation p = a(1−e) remains almost constant. This
+is reasonable, since significant GR effects only occur in an
+effective point interaction at pericentre. Unusually, the inter-
+action in QPE systems is strong enough that the resulting
+instantaneous mass transfer rate is close to the long–term
+evolutionary mean driven by GR. This is very different from
+the situation in accreting stellar–mass binaries. There the
+instantaneous rate oscillates widely around the evolution-
+ary mean because of unrelated short–term effects. It only
+converges to this mean when averaged on timescales long
+compared with that taken for the driving process (here GR)
+to move the critical (Roche) lobe by a distance of order the
+density scaleheight of the donor star. This means for ex-
+ample that it is generally unsafe to try to deduce the mass
+transfer rate of a stellar–mass binary by measuring the rate
+of change of its orbital period – see King & Lasota (2021)
+for a recent discussion – whereas QPE sources appear to be
+constrained to stay close to this mean mass transfer rate.
+Although the simple model of King (2020) and K22
+works well for QPE systems, there is clearly more complexity
+in the structure of these sources. The QPE system ASASSN-
+14ko (Payne et al., 2021, 2022) has a very nearly strict period
+of P = 114 days, and its mass transfer rate and luminosity
+agree with the predictions of GR driving. But the predicted
+GR period derivative ˙P ≃ −1.3 × 10−6 is in flat contradic-
+tion with the measured value ˙P = −1.7 × 10−3 (Payne et
+al., 2021).
+A second deviation from the expectations of K22
+emerges from the recent thorough observational study by
+Minutti et al. (2022) of the first QPE source, GSN 069,
+which gives a historical X–ray light curve. This shows an
+episode where the orbital period appears to increase by a
+factor ∼ 1.3 over a timescale of order 3000 d, and the mass
+transfer rate declines significantly on the same timescale.
+In this paper I suggest that the underlying cause of the
+unusual behaviour of both ASASSN-14ko and GSN 069 is
+that in each of these two galaxies the central MBH–WD bi-
+nary system is not isolated, but gravitationally influenced by
+a perturber which itself is in a wider orbit about the MBH.
+This object may be a star (or star cluster) which was part
+of the infall event causing the formation of the inner QPE
+‘binary’ itself. The interaction between the inner and outer
+binaries induces a variety of effects, now collectively known
+as von Zeipel–Lidov–Kozai (ZLK) cycles1, usually studied in
+the case where the secondary component in the inner binary
+(here the WD) has negligible mass compared with the pri-
+mary (MBH) and the perturber. In many cases one can treat
+the problem by expanding the combined gravitational poten-
+tial only to quadrupole order, effectively modelling the two
+binaries as wire loops with the masses spread around their
+1 The author ordering ZLK I adopt here follows the historical
+sequence in which the authors studied the interaction between
+two binaries in the contexts of the Solar System (von Zeipel 1910;
+Kozai, 1962), and artificial satellites in the Earth–Moon system
+(Lidov, 1961, 1962).
+orbits. As I explicitly noted in K22, it is inherently plausible
+that QPE binaries should be accompanied by other more dis-
+tant orbiting stars (or star clusters), as such satellites of the
+MBH are likely additional results of the tidal capture events
+which probably produce QPE binaries (cf Cufari, Coughlin
+& Nixon, 2022).
+I show here that ZLK cycles can account for large or-
+bital period derivatives, as seen in ASASSN-14ko, which are
+unrelated to the mass transfer process, and can also cause
+correlated rapid changes change of orbital period and mass
+transfer rate, as in GSN 069. In general the parameter space
+open to a perturber is large in any given case, but the pos-
+sibility of narrowing it down should encourage continued
+monitoring of QPE sources, as this may give insight into
+the capture events forming them.
+2
+MASS TRANSFER STABILITY IN QPE
+BINARIES, AND THE ORIGIN OF THE
+ERUPTIONS
+The QPE model discussed here requires that that mass
+transfer is dynamically stable, i.e. that the tidal lobe and
+the donor radius move in step as mass is transferred. Sug-
+gestions to the contrary have appeared in the literature, but
+several of these neglected the orbital expansion driven by
+the tidal coupling of the accretion disc angular momentum
+when mass is transferred from the less massive star (here
+the WD) to a more massive accretor (the MBH); see K22
+for a discussion. More recently, Lu & Quataert (2022) have
+argued that in a highly eccentric system such as the QPE
+binaries considered here, the tidal coupling would actually
+transfer angular momentum from the donor to the accretion
+disc. This would cause mass transfer from a donor which
+expands on mass loss (as here) to be dynamically unstable,
+and the binary would coalesce in a few orbits.
+The argument of Lu & Quataert (2022) implicitly as-
+sumes that the accretion disc is circular. In that case mat-
+ter at the outer edge of the disc moves more slowly than
+the donor in an eccentric binary, and indeed the angular
+momentum transfer is from the donor to the disc, leading
+to instability. But it is rather unlikely that the disc is at
+all circular in QPE systems. Since the mass ratio M2/M1
+is ≪ 1, the disc can easily grow large enough to contain
+strong Lindblad resonances (see e.g. Fig. 1 of Whitehurst &
+King, 1991). These make the disc very eccentric and cause
+prograde apsidal precession, and are the cause of the super-
+hump modulations at periods slightly longer than orbital in
+superoutbursts of the short–period (P ≲ 100 min) SU UMa
+cataclysmic binaries (Lubow, 1991). Early Lagrangian (e.g.
+SPH) simulations had readily found these effects, beginning
+with Whitehurst (1988). Eulerian simulations took longer to
+achieve this, as the use of axisymmetric coordinates tends
+to suppress the disc eccentricity which is the basis of super-
+humps, but with attention to this problem superhumps now
+appear in these simulations too (Wienkers & Ogilvie, 2018,
+and references therein).
+SU UMa superhumps are driven by the 3:1 commen-
+surability. In QPE binaries the much smaller mass ratios
+mean that the stronger 2:1 resonance is also accessible, so
+we can expect their discs to be strongly eccentric and pre-
+cessing. In the SU UMa systems, superhumps appear during
+© 2019 RAS, MNRAS 000, 1–5
+
+Angular Momentum Transfer in QPEs
+3
+superoutbursts, when the discs undergo outbursts which are
+longer and brighter than their usual dwarf nova outbursts. A
+suggestive possibility (Osaki & Kato, 2013) is that the tidal
+effects themselves actually cause the increased disc accretion
+of superoutbursts. By analogy, the presence of resonances in
+eccentric QPE sources may trigger the eruptions themselves
+when the donor is near pericentre. In addition, the preces-
+sion of the eccentric disc would naturally cause deviations
+from strict periodicity, particularly in short–period systems.
+(I note that in long–period QPE systems such as ASSASN-
+14ko and HLX–1 the eruptions tend to be more regular; K22
+discusses why in HLX–1 the disc may occasionally not re–
+form in time for the next periastron passage, and so miss an
+entire cycle). In this picture the angular momentum lost by
+the rapidly–accreting disc material is transferred to the WD
+orbit, maintaining orbital stability.
+Simulations (which are presumably easier with SPH)
+are needed to check these suggestions, and in particular to
+determine the size and direction of angular momentum ex-
+change between the eccentric binary orbit and the eccentric
+precessing disc. The presence of very significant CNO en-
+hancement in at least one QPE source (Sheng et al., 2021)
+strongly supports the suggestion of WD donors in QPE
+sources, and so the idea that mass transfer is stable in them.
+3
+ZLK CYCLES
+As remarked above, there is good reason to suspect the ac-
+tion of ZLK effects in QPE sources. The characteristic fea-
+ture of ZLK cycles is that the inner binary (the QPE system
+in our case) continuously exchanges its eccentricity e with
+the inclination i of its orbit to that of the outer (perturb-
+ing) binary. (The plane of the latter is almost fixed in many
+cases of interest, as the outer binary has the largest compo-
+nent of the whole system’s total angular momentum.) The
+exchanges are subject to the constraint
+(1 − e2)1/2 cos i ≃ C,
+(1)
+where C is a constant set by the initial conditions. This
+asserts that ZLK cycles have no effect on the angular mo-
+mentum component of the inner binary orthogonal to its
+instantaneous plane. This is precisely the angular momen-
+tum J being depleted by GR to drive mass transfer.
+For given initial conditions the inner binary plane ei-
+ther librates (oscillates between two fixed inclinations i) or
+circulates (revolves continuously wrt the outer binary). The
+characteristic timescale for these motions is
+tZLK ≃
+8
+15π
+�
+1 + M1
+M3
+� �P 2
+out
+P
+�
+(1 − e2
+out)3/2,
+(2)
+(Antognini, 2015), where M1, M3 are the MBH and per-
+turber masses P, Pout are the periods of the inner (QPE)
+binary and the outer one respectively, and eout is the eccen-
+tricity of the outer binary. We see that this timescale tends
+to infinity in the limit of vanishing perturber mass M3, as
+expected.
+ZLK cycles are quickly washed out if the binary pre-
+cesses too rapidly, as this gradually destroys the near-
+resonance allowing the exchange of inclination and eccen-
+tricity. The strongest precession in QPE systems with WD
+donors (cf Willems, Deloye & Kalogera, 2010) is the general–
+relativistic advance of pericentre, with period
+PGR = 4.27M −2/3
+5
+P 5/3
+4
+(1 − e2) yr,
+(3)
+so that
+PGR
+P
+= 1.28 × 104M −2/3
+5
+P 2/3
+4
+(1 − e2),
+(4)
+where e is the eccentricity, M5 = M1/105M⊙ with M1 the
+black hole mass, and P4 is the orbital period P in units of
+104 s. This ratio is given for the current known systems in
+Table 1, and is always significantly larger than unity, al-
+though only of order 22 and 36 for two systems. This sug-
+gests that ZLK cycles can appear stably in most QPE sys-
+tems, but may (if they appear at all) be fairly shortlived in
+some cases. I discuss this point further below.
+4
+EVOLUTION OF THE INNER BINARY
+DURING ZLK CYCLES
+ZLK cycles modulate the eccentricity e of the inner (QPE)
+binary. But we see from (1) that they have essentially no
+direct effect on its orbital angular momentum J. Since mass
+transfer is stable in a QPE binary, this system must evi-
+dently respond to ZLK changes of e by holding constant its
+tidal lobe R2: this must remain equal to the current radius
+of the WD donor, whose mass is unchanged. The lobe radius
+R2 is proportional to the pericentre separation (see K22)
+p = a(1 − e),
+(5)
+so that the ZKL effect makes the semi–major axis a change
+as
+a ∝ (1 − e)−1.
+(6)
+This in turn implies that ZLK cycles cause the period of a
+stably mass–transferring binary system to evolve as
+P ∝ a3/2 ∝ (1 − e)−3/2.
+(7)
+The GR–driven mass transfer rate must evolve in response
+to the changes in e and P as
+− ˙M2 ∝ P −8/3(1 − e)−5/2 ∝ P −1,
+(8)
+where I have used eqn (15) of K22 together with (6, 7).
+The constraint (1) implies that during a ZKL cycle the
+eccentricity e reaches a maximum as the inner binary plane
+crosses the plane of the perturbing outer binary at i = 0.
+From (7) the inner binary period reaches a maximum at this
+point. Logarithmically differentiating (1) we get
+e ˙e
+1 − e2 ≃ − tan idi
+dt.
+(9)
+From this equation and (7) we have
+˙P
+P = 3
+2
+˙e
+1 − e = −31 + e
+2e
+tan idi
+dt.
+(10)
+In all cases we expect di/dt ∼ ±1/tZLK.
+© 2019 RAS, MNRAS 000, 1–5
+
+4
+Andrew King
+5
+COMPARISON WITH OBSERVATIONS
+5.1
+Period Changes
+We have seen that ZLK cycles can produce very rapid pe-
+riod changes in QPE binaries (cf eqn 10), which may be
+accompanied by significant changes in the accretion lumi-
+nosity (cf eqn 8). Because ZLK cycles produce these changes
+by altering the eccentricity affecting the GR losses driving
+mass transfer, there is no paradox in changes more rapid
+than given by the timescale tGR for the latter. The appar-
+ently discordantly large period derivative of ASASSN–14ko
+is then a potential signature of this effect. There are sev-
+eral ways to explain values of order the ˙P = −1.7 × 10−3
+observed there as a result of ZLK cycles.
+If tan i ∼ 1 (i.e. the QPE plane is not close to the
+perturber plane) we must have e significantly smaller than
+unity. We see from Table 1 that this explanation cannot work
+for ASSASN–14ko itself, or any of the known QPE systems,
+which all have a much higher eccentricities.
+Future observations may reveal QPE systems with lower
+e, and these would have − ˙P ∼ 3P/2etZLK, so from (2) we
+find a value ˙P ∼ 10−3 would result if the perturber mass
+and period are connected by
+M3 ≃ 13 em5
+1 + e
+�Pout
+P
+�2
+(1 − e2
+out)3/2M⊙,
+(11)
+where m5 = M1/105M⊙. We need Pout > P = 114 d for con-
+sistency in ASSASN–14ko. This is evidently possible with
+perturber having a normal stellar mass, as e is very close to
+unity (see Table 1).
+So we must look to other candidates for the perturbers
+in known QPE systems. Other possibilities are that the per-
+turber is a star cluster rather than a single star, or that it is a
+single star with extreme eout approaching unity2. This latter
+case may be more likely if the QPE binary and its perturber
+result from the same tidal capture event. This is important
+in highlighting the potential the QPE sources have in sig-
+nalling these events, and their possible role in promoting
+black hole growth. Clearly, only further observational mon-
+itoring of the known QPE systems can distinguish between
+all these possibilities.
+5.2
+Light Curve and Correlated Period Changes
+Equation (8) shows that ZKL cycles can continuously mod-
+ify the mass transfer rate and so the luminosity of a QPE
+binary, as recently observed in GSN 069 by Miniutti et al.
+(2022). As the period is increasing here, and the luminos-
+ity decreasing, we must have increasing eccentricity, so the
+plane of the QPE binary is approaching the perturber plane.
+These events should eventually appear in time–reversed or-
+der. The 3000 d timescale is easy to accomodate (cf eqn 2)
+with a stellar–mass perturber and an outer period not much
+longer than that of the QPE binary.
+Presumably systems showing little change between
+eruptions, and no very rapid period changes, must either
+2 In this case we have the eccentric ZLK effect: this becomes
+considerably more complicated, as now there are octupole con-
+tributions to the gravitational potential of similar order to the
+quadrupole ones considered so far. See Naoz (2016) for a review.
+have no associated perturber, or a perturber period which
+is very long. Orbital changes induced by ZKL cycles might
+trigger other light curve effects, e.g. by cyclically altering
+disc accretion. These would add to the effect already noted
+by K22 that for systems with periods P ≳ 1 yr the accretion
+disc may have to re–form after a few outbursts, which may
+account for the missing outburst in HLX-1.
+6
+CONCLUSIONS
+I have argued that the presence of resonances in the ac-
+cretion disc makes it likely that in systems where a white
+dwarf orbits a massive black hole, mass transfer driven by
+the loss of gravitational wave energy is stable on a dynam-
+ical timescale. The resonances may also promote the rapid
+loss of disc angular momentum to the WD, and so directly
+cause the quasiperiodic eruptions.
+I have considered some of the effects that may appear
+in QPE systems because of von Zeipel–Lidov–Kozai (ZLK)
+cycles triggered by a perturber on a more distant orbit about
+the central massive black hole. The presence of perturbers
+of this kind appears likely, as they may be products of the
+same tidal capture events that formed the QPE binaries
+themselves. Evidently more observations are needed to check
+the validity of the ZLK idea. If it is tenable, the parameter
+space available to the perturbers is currently very large, and
+still more observations would be needed to narrow it down.
+QPE
+systems
+showing
+orbital
+period
+changes
+on
+timescales much shorter than the mass transfer time are
+obvious candidates for ZLK effects, and are very likely to re-
+ward further monitoring or archival searches. For example,
+the predicted timescale for the disappearance of the ZLK
+cycles in ASSASN–14ko is only of order a decade. Similarly,
+correlated short–term variations of mass transfer rates and
+orbital periods in QPE systems may result from ZLK cy-
+cles. Here we can expect the data and interpretation to be
+more complex than for period changes, because other effects
+can also modulate the mass transfer rates. But this kind
+of study can potentially give major insights into how the
+central black holes in low–mass galaxies are able to grow.
+DATA AVAILABILITY
+No new data were generated or analysed in support of this
+research.
+ACKNOWLEDGMENTS
+I thank Giovanni Miniutti for giving me early insight into
+important observational data and for many helpful and con-
+tinuing discussions, and Chris Nixon and the anonymous
+referee for very helpful comments.
+REFERENCES
+Antognini, J.M.O., 2015, MNRAS, 452, 3610
+Arcodia, R., Merloni, A., Nandra, K., et al., 2021, Nature, 592,
+704 Arcodia et al. 2021
+© 2019 RAS, MNRAS 000, 1–5
+
+Angular Momentum Transfer in QPEs
+5
+Table 1. Parameters of the Current QPE Sample
+Source
+P4
+m5
+(L∆t)45
+m2
+1 − e
+PGR/P
+eRO – QPE2
+0.86
+2.5
+0.8
+0.18 9.9 × 10−2
+1250
+XMMSL1
+0.90
+0.85
+0.34
+0.18 9.9 × 10−2
+2503
+RXJ1301.9
+1.65
+18
+1.7
+0.15 7.2 × 10−2
+36
+GSN 069
+3.16
+4.0
+10
+0.32 2.8 × 10−2
+130
+eRO– QPE1
+6.66
+9.1
+0.045
+0.46 1.4 × 10−2
+291
+ASSASN–14ko
+937
+700
+3388
+0.56 9.0 × 10−3
+22
+HLX–1
+2000 [0.5]
+1000
+1.43 1.2 × 10−4
+774
+Note 1 This table is adapted from Table 1 of K22, but now ordered by period P . We note the general tendency that the eccentricity
+e is smaller for shorter periods, consistent with the effects of GR losses. The very bright QPE source ASSASN-14ko (Payne et al., 2021,
+2022) was missing from Table 1 of K22. The difficulty in modelling it arose because it is (uniquely) extremely close to the limit
+m5(L∆t)1/3
+45
+≲ 104
+(12)
+required to avoid the model formally predicting that the WD pericentre distance a(1 − e) is larger than the innermost stable orbital
+radius, which is itself ≃ Rg = GM1/c2, where Rg is the black hole’s gravitational radius. Here m5 = M1/105M⊙, and (L∆t)45 is the
+total energy radiated at pericentre passage in units of 1045 erg. Equation (12) is the condition
+a(1 − e)3 >
+� GM1
+c2
+�3
+(13)
+written using the parametrization of Chen et al. (2022) followed in K22. Evidently the form (12) expresses the facts that the radiated
+energy is increased in a tighter orbit, but that a larger black hole mass increases Rg. For ASSASN-14ko we have m5 ≃ 700, requiring
+(L∆t)45 ≲ 3388, as compared with rough observational estimates (L∆t)45 ≃ 4000. Here I adopt the extreme value (L∆t)45 ≲ 3388 for
+this system. For all other currently known QPE systems the constraint (12) is very easily satisfied.
+Note 2 There is no secure mass estimate for the black hole in HLX–1. Here I adopt the minimum value m5 = 0.5 allowing the
+donor to be below the Chandrasekhar mass (i.e. m2 ≃ 1.4; see K22). Larger m5 values allow smaller m2 (see K22).
+Note 3 The Table also gives the values of PGR/P specifying the stability of possible ZLK cycles.
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+L40
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+© 2019 RAS, MNRAS 000, 1–5
+

H9E1T4oBgHgl3EQf_gbR/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf,len=341
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='03582v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='HE]  9 Jan 2023  Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' 000, 1–5 (2019)  Printed 10 January 2023  (MN LATEX style file v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='2)  Angular Momentum Transfer in QPEs from Galaxy Nuclei  Andrew King1,2,3,⋆  1 Department of Physics & Astronomy, University of Leicester, Leicester, LE1 7RH, UK  2 Astronomical Institute Anton Pannekoek, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands  3 Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, Netherlands  ⋆ E-mail: ark@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='uk  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  A suggested model for quasi–periodic eruptions (QPEs) from galaxy nuclei invokes a  white dwarf in an eccentric orbit about the central massive black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' I point out  that the extreme mass ratio allows the presence of strong Lindblad resonances in  the accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' These are important for the stability of mass transfer, and may  trigger the eruptions themselves by rapidly transferring angular momentum from the  accretion disc (which is likely to be eccentric itself) to the orbiting WD companion at  pericentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' I consider the effects of von Zeipel–Lidov–Kozai (ZLK) cycles caused by  a perturber on a more distant orbit about the central black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' I show that ZLK  cycles can change the orbital periods of QPE systems on timescales much shorter than  the mass transfer time, as seen in ASASSN-14ko, and produce correlated short–term  variations of their mass transfer rates and orbital periods, as recently observed in  GSN 069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Further monitoring of these sources should constrain the parameters of any  perturbing companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This in turn may constrain the nature of the events creating  QPE systems, and perhaps give major insights into how the central black holes in  low–mass galaxies are able to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  Key words: galaxies: active: black hole physics: X–rays: galaxies  1  INTRODUCTION  X–ray observations of the nuclei of low–mass galaxies  show that several of them produce quasi–periodic eruptions  (QPEs: Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Giustini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Arcodia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Payne  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Typically these sources have outbursts  by factors ∼ 100 in X–rays, which recur in roughly pe-  riodic fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The recurrence times currently range from  a few hours up to ∼ 100 d or ∼ 1 yr, and it is likely  that these limits will widen as data accumulate from di-  rect observations and archival searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The X–rays have  ultrasoft blackbody spectra and luminosities implying typ-  ical radii ≳ Rg = GM1/c2 = 6 × 1010m5 cm, of order the  gravitational (Rg) and ISCO radii of a black hole of mass  M1 ∼ 105m5M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' These are consistent with the massive  black holes (MBHs) one might expect in these low–mass  galaxy nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  In (King, 2020) I suggested a simple model for the first  known QPE source GSN 069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This postulated a low–mass  star (with mass M2 ≪ M1) – found from the requirement  of internal consistency with observational selection to be a  white dwarf (WD) – on a highly eccentric orbit about the  central MBH, transferring mass to it at pericentre passage  via an accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' In a second paper (King, 2022, here-  after K22) I used the parametrization introduced by Chen et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' (2022), extended to the full WD mass range, to show that  this model was consistent with the data for 5 of the 6 known  QPE sources, together with a previously unassociated sys-  tem HLX-1, where the periodicity is ∼ 1 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' (I discuss the  ‘missing’ system ASSASN–14ko in Note 1 to Table 1 at the  end of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=')  In all cases, loss of orbital energy E and angular mo-  mentum J via gravitational radiation (GR) is the ultimate  driver of mass transfer, and observational selection effects  mean that the orbiting star is in practice a low–mass white  dwarf (WD) in detectable QPE systems (K22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This is the  likely explanation for the presence of CNO–processed mate-  rial found in GSN 069 (Sheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  This model requires that mass transfer is dynamically  stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' the Roche lobe and the WD radius must move in  step as GR reduces the orbital semi–major axis a and the  eccentricity e on the same timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Since the WD expands  as it loses mass, it must gain angular momentum from the  accretion disc and move in a wider orbit, implying a larger  tidal radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  This stability has recently been questioned, so I dis-  cuss it further in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Orbital resonances within the  accretion disc are likely to cause the required enhanced an-  gular momentum transfer to the WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The resonances may  © 2019 RAS  2  Andrew King  also directly cause the quasiperiodic eruptions themselves  as the orbiting WD passes pericentre – a similar origin has  been suggested for the superoutbursts of the stellar–mass  SU UMa binaries (Osaki & Kato, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  During the orbital evolution of a QPE binary the peri-  centre separation p = a(1−e) remains almost constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This  is reasonable, since significant GR effects only occur in an  effective point interaction at pericentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Unusually, the inter-  action in QPE systems is strong enough that the resulting  instantaneous mass transfer rate is close to the long–term  evolutionary mean driven by GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This is very different from  the situation in accreting stellar–mass binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' There the  instantaneous rate oscillates widely around the evolution-  ary mean because of unrelated short–term effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' It only  converges to this mean when averaged on timescales long  compared with that taken for the driving process (here GR)  to move the critical (Roche) lobe by a distance of order the  density scaleheight of the donor star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This means for ex-  ample that it is generally unsafe to try to deduce the mass  transfer rate of a stellar–mass binary by measuring the rate  of change of its orbital period – see King & Lasota (2021)  for a recent discussion – whereas QPE sources appear to be  constrained to stay close to this mean mass transfer rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  Although the simple model of King (2020) and K22  works well for QPE systems, there is clearly more complexity  in the structure of these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The QPE system ASASSN-  14ko (Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', 2021, 2022) has a very nearly strict period  of P = 114 days, and its mass transfer rate and luminosity  agree with the predictions of GR driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' But the predicted  GR period derivative ˙P ≃ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='3 × 10−6 is in flat contradic-  tion with the measured value ˙P = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='7 × 10−3 (Payne et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  A second deviation from the expectations of K22  emerges from the recent thorough observational study by  Minutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' (2022) of the first QPE source, GSN 069,  which gives a historical X–ray light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This shows an  episode where the orbital period appears to increase by a  factor ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='3 over a timescale of order 3000 d, and the mass  transfer rate declines significantly on the same timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  In this paper I suggest that the underlying cause of the  unusual behaviour of both ASASSN-14ko and GSN 069 is  that in each of these two galaxies the central MBH–WD bi-  nary system is not isolated, but gravitationally influenced by  a perturber which itself is in a wider orbit about the MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  This object may be a star (or star cluster) which was part  of the infall event causing the formation of the inner QPE  ‘binary’ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The interaction between the inner and outer  binaries induces a variety of effects, now collectively known  as von Zeipel–Lidov–Kozai (ZLK) cycles1, usually studied in  the case where the secondary component in the inner binary  (here the WD) has negligible mass compared with the pri-  mary (MBH) and the perturber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' In many cases one can treat  the problem by expanding the combined gravitational poten-  tial only to quadrupole order, effectively modelling the two  binaries as wire loops with the masses spread around their  1 The author ordering ZLK I adopt here follows the historical  sequence in which the authors studied the interaction between  two binaries in the contexts of the Solar System (von Zeipel 1910;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  Kozai, 1962), and artificial satellites in the Earth–Moon system  (Lidov, 1961, 1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' As I explicitly noted in K22, it is inherently plausible  that QPE binaries should be accompanied by other more dis-  tant orbiting stars (or star clusters), as such satellites of the  MBH are likely additional results of the tidal capture events  which probably produce QPE binaries (cf Cufari, Coughlin  & Nixon, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  I show here that ZLK cycles can account for large or-  bital period derivatives, as seen in ASASSN-14ko, which are  unrelated to the mass transfer process, and can also cause  correlated rapid changes change of orbital period and mass  transfer rate, as in GSN 069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' In general the parameter space  open to a perturber is large in any given case, but the pos-  sibility of narrowing it down should encourage continued  monitoring of QPE sources, as this may give insight into  the capture events forming them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  2  MASS TRANSFER STABILITY IN QPE  BINARIES, AND THE ORIGIN OF THE  ERUPTIONS  The QPE model discussed here requires that that mass  transfer is dynamically stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' that the tidal lobe and  the donor radius move in step as mass is transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Sug-  gestions to the contrary have appeared in the literature, but  several of these neglected the orbital expansion driven by  the tidal coupling of the accretion disc angular momentum  when mass is transferred from the less massive star (here  the WD) to a more massive accretor (the MBH);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' see K22  for a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' More recently, Lu & Quataert (2022) have  argued that in a highly eccentric system such as the QPE  binaries considered here, the tidal coupling would actually  transfer angular momentum from the donor to the accretion  disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This would cause mass transfer from a donor which  expands on mass loss (as here) to be dynamically unstable,  and the binary would coalesce in a few orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  The argument of Lu & Quataert (2022) implicitly as-  sumes that the accretion disc is circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' In that case mat-  ter at the outer edge of the disc moves more slowly than  the donor in an eccentric binary, and indeed the angular  momentum transfer is from the donor to the disc, leading  to instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' But it is rather unlikely that the disc is at  all circular in QPE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Since the mass ratio M2/M1  is ≪ 1, the disc can easily grow large enough to contain  strong Lindblad resonances (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' 1 of Whitehurst &  King, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' These make the disc very eccentric and cause  prograde apsidal precession, and are the cause of the super-  hump modulations at periods slightly longer than orbital in  superoutbursts of the short–period (P ≲ 100 min) SU UMa  cataclysmic binaries (Lubow, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Early Lagrangian (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  SPH) simulations had readily found these effects, beginning  with Whitehurst (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Eulerian simulations took longer to  achieve this, as the use of axisymmetric coordinates tends  to suppress the disc eccentricity which is the basis of super-  humps, but with attention to this problem superhumps now  appear in these simulations too (Wienkers & Ogilvie, 2018,  and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  SU UMa superhumps are driven by the 3:1 commen-  surability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' In QPE binaries the much smaller mass ratios  mean that the stronger 2:1 resonance is also accessible, so  we can expect their discs to be strongly eccentric and pre-  cessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' In the SU UMa systems, superhumps appear during  © 2019 RAS, MNRAS 000, 1–5  Angular Momentum Transfer in QPEs  3  superoutbursts, when the discs undergo outbursts which are  longer and brighter than their usual dwarf nova outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' A  suggestive possibility (Osaki & Kato, 2013) is that the tidal  effects themselves actually cause the increased disc accretion  of superoutbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' By analogy, the presence of resonances in  eccentric QPE sources may trigger the eruptions themselves  when the donor is near pericentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' In addition, the preces-  sion of the eccentric disc would naturally cause deviations  from strict periodicity, particularly in short–period systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  (I note that in long–period QPE systems such as ASSASN-  14ko and HLX–1 the eruptions tend to be more regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' K22  discusses why in HLX–1 the disc may occasionally not re–  form in time for the next periastron passage, and so miss an  entire cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' In this picture the angular momentum lost by  the rapidly–accreting disc material is transferred to the WD  orbit, maintaining orbital stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  Simulations (which are presumably easier with SPH)  are needed to check these suggestions, and in particular to  determine the size and direction of angular momentum ex-  change between the eccentric binary orbit and the eccentric  precessing disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The presence of very significant CNO en-  hancement in at least one QPE source (Sheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', 2021)  strongly supports the suggestion of WD donors in QPE  sources, and so the idea that mass transfer is stable in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  3  ZLK CYCLES  As remarked above, there is good reason to suspect the ac-  tion of ZLK effects in QPE sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The characteristic fea-  ture of ZLK cycles is that the inner binary (the QPE system  in our case) continuously exchanges its eccentricity e with  the inclination i of its orbit to that of the outer (perturb-  ing) binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' (The plane of the latter is almost fixed in many  cases of interest, as the outer binary has the largest compo-  nent of the whole system’s total angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=') The  exchanges are subject to the constraint  (1 − e2)1/2 cos i ≃ C,  (1)  where C is a constant set by the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This  asserts that ZLK cycles have no effect on the angular mo-  mentum component of the inner binary orthogonal to its  instantaneous plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This is precisely the angular momen-  tum J being depleted by GR to drive mass transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  For given initial conditions the inner binary plane ei-  ther librates (oscillates between two fixed inclinations i) or  circulates (revolves continuously wrt the outer binary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The  characteristic timescale for these motions is  tZLK ≃  8  15π  �  1 + M1  M3  � �P 2  out  P  �  (1 − e2  out)3/2,  (2)  (Antognini, 2015), where M1, M3 are the MBH and per-  turber masses P, Pout are the periods of the inner (QPE)  binary and the outer one respectively, and eout is the eccen-  tricity of the outer binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' We see that this timescale tends  to infinity in the limit of vanishing perturber mass M3, as  expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  ZLK cycles are quickly washed out if the binary pre-  cesses too rapidly, as this gradually destroys the near-  resonance allowing the exchange of inclination and eccen-  tricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The strongest precession in QPE systems with WD  donors (cf Willems, Deloye & Kalogera, 2010) is the general–  relativistic advance of pericentre, with period  PGR = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='27M −2/3  5  P 5/3  4  (1 − e2) yr,  (3)  so that  PGR  P  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='28 × 104M −2/3  5  P 2/3  4  (1 − e2),  (4)  where e is the eccentricity, M5 = M1/105M⊙ with M1 the  black hole mass, and P4 is the orbital period P in units of  104 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This ratio is given for the current known systems in  Table 1, and is always significantly larger than unity, al-  though only of order 22 and 36 for two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This sug-  gests that ZLK cycles can appear stably in most QPE sys-  tems, but may (if they appear at all) be fairly shortlived in  some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' I discuss this point further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  4  EVOLUTION OF THE INNER BINARY  DURING ZLK CYCLES  ZLK cycles modulate the eccentricity e of the inner (QPE)  binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' But we see from (1) that they have essentially no  direct effect on its orbital angular momentum J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Since mass  transfer is stable in a QPE binary, this system must evi-  dently respond to ZLK changes of e by holding constant its  tidal lobe R2: this must remain equal to the current radius  of the WD donor, whose mass is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The lobe radius  R2 is proportional to the pericentre separation (see K22)  p = a(1 − e),  (5)  so that the ZKL effect makes the semi–major axis a change  as  a ∝ (1 − e)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  (6)  This in turn implies that ZLK cycles cause the period of a  stably mass–transferring binary system to evolve as  P ∝ a3/2 ∝ (1 − e)−3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  (7)  The GR–driven mass transfer rate must evolve in response  to the changes in e and P as  − ˙M2 ∝ P −8/3(1 − e)−5/2 ∝ P −1,  (8)  where I have used eqn (15) of K22 together with (6, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  The constraint (1) implies that during a ZKL cycle the  eccentricity e reaches a maximum as the inner binary plane  crosses the plane of the perturbing outer binary at i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  From (7) the inner binary period reaches a maximum at this  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Logarithmically differentiating (1) we get  e ˙e  1 − e2 ≃ − tan idi  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  (9)  From this equation and (7) we have  ˙P  P = 3  2  ˙e  1 − e = −31 + e  2e  tan idi  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  (10)  In all cases we expect di/dt ∼ ±1/tZLK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  © 2019 RAS, MNRAS 000, 1–5  4  Andrew King  5  COMPARISON WITH OBSERVATIONS  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='1  Period Changes  We have seen that ZLK cycles can produce very rapid pe-  riod changes in QPE binaries (cf eqn 10), which may be  accompanied by significant changes in the accretion lumi-  nosity (cf eqn 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Because ZLK cycles produce these changes  by altering the eccentricity affecting the GR losses driving  mass transfer, there is no paradox in changes more rapid  than given by the timescale tGR for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The appar-  ently discordantly large period derivative of ASASSN–14ko  is then a potential signature of this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' There are sev-  eral ways to explain values of order the ˙P = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='7 × 10−3  observed there as a result of ZLK cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  If tan i ∼ 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' the QPE plane is not close to the  perturber plane) we must have e significantly smaller than  unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' We see from Table 1 that this explanation cannot work  for ASSASN–14ko itself, or any of the known QPE systems,  which all have a much higher eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  Future observations may reveal QPE systems with lower  e, and these would have − ˙P ∼ 3P/2etZLK, so from (2) we  find a value ˙P ∼ 10−3 would result if the perturber mass  and period are connected by  M3 ≃ 13 em5  1 + e  �Pout  P  �2  (1 − e2  out)3/2M⊙,  (11)  where m5 = M1/105M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' We need Pout > P = 114 d for con-  sistency in ASSASN–14ko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This is evidently possible with  perturber having a normal stellar mass, as e is very close to  unity (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  So we must look to other candidates for the perturbers  in known QPE systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Other possibilities are that the per-  turber is a star cluster rather than a single star, or that it is a  single star with extreme eout approaching unity2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This latter  case may be more likely if the QPE binary and its perturber  result from the same tidal capture event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' This is important  in highlighting the potential the QPE sources have in sig-  nalling these events, and their possible role in promoting  black hole growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Clearly, only further observational mon-  itoring of the known QPE systems can distinguish between  all these possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='2  Light Curve and Correlated Period Changes  Equation (8) shows that ZKL cycles can continuously mod-  ify the mass transfer rate and so the luminosity of a QPE  binary, as recently observed in GSN 069 by Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' As the period is increasing here, and the luminos-  ity decreasing, we must have increasing eccentricity, so the  plane of the QPE binary is approaching the perturber plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  These events should eventually appear in time–reversed or-  der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The 3000 d timescale is easy to accomodate (cf eqn 2)  with a stellar–mass perturber and an outer period not much  longer than that of the QPE binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  Presumably systems showing little change between  eruptions, and no very rapid period changes, must either  2 In this case we have the eccentric ZLK effect: this becomes  considerably more complicated, as now there are octupole con-  tributions to the gravitational potential of similar order to the  quadrupole ones considered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' See Naoz (2016) for a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  have no associated perturber, or a perturber period which  is very long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Orbital changes induced by ZKL cycles might  trigger other light curve effects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' by cyclically altering  disc accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' These would add to the effect already noted  by K22 that for systems with periods P ≳ 1 yr the accretion  disc may have to re–form after a few outbursts, which may  account for the missing outburst in HLX-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  6  CONCLUSIONS  I have argued that the presence of resonances in the ac-  cretion disc makes it likely that in systems where a white  dwarf orbits a massive black hole, mass transfer driven by  the loss of gravitational wave energy is stable on a dynam-  ical timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The resonances may also promote the rapid  loss of disc angular momentum to the WD, and so directly  cause the quasiperiodic eruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  I have considered some of the effects that may appear  in QPE systems because of von Zeipel–Lidov–Kozai (ZLK)  cycles triggered by a perturber on a more distant orbit about  the central massive black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The presence of perturbers  of this kind appears likely, as they may be products of the  same tidal capture events that formed the QPE binaries  themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Evidently more observations are needed to check  the validity of the ZLK idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' If it is tenable, the parameter  space available to the perturbers is currently very large, and  still more observations would be needed to narrow it down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  QPE  systems  showing  orbital  period  changes  on  timescales much shorter than the mass transfer time are  obvious candidates for ZLK effects, and are very likely to re-  ward further monitoring or archival searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' For example,  the predicted timescale for the disappearance of the ZLK  cycles in ASSASN–14ko is only of order a decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Similarly,  correlated short–term variations of mass transfer rates and  orbital periods in QPE systems may result from ZLK cy-  cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Here we can expect the data and interpretation to be  more complex than for period changes, because other effects  can also modulate the mass transfer rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' But this kind  of study can potentially give major insights into how the  central black holes in low–mass galaxies are able to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  DATA AVAILABILITY  No new data were generated or analysed in support of this  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  I thank Giovanni Miniutti for giving me early insight into  important observational data and for many helpful and con-  tinuing discussions, and Chris Nixon and the anonymous  referee for very helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  REFERENCES  Antognini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', 2015, MNRAS, 452, 3610  Arcodia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', Merloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', Nandra, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', 2021, Nature, 592,  704 Arcodia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' 2021  © 2019 RAS, MNRAS 000, 1–5  Angular Momentum Transfer in QPEs  5  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Parameters of the Current QPE Sample  Source  P4  m5  (L∆t)45  m2  1 − e  PGR/P  eRO – QPE2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='86  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='18 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='9 × 10−2  1250  XMMSL1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='18 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='9 × 10−2  2503  RXJ1301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='65  18  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='2 × 10−2  36  GSN 069  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='0  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='8 × 10−2  130  eRO– QPE1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='66  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='4 × 10−2  291  ASSASN–14ko  937  700  3388  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='56 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='0 × 10−3  22  HLX–1  2000 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='5]  1000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='2 × 10−4  774  Note 1 This table is adapted from Table 1 of K22, but now ordered by period P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' We note the general tendency that the eccentricity  e is smaller for shorter periods, consistent with the effects of GR losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The very bright QPE source ASSASN-14ko (Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', 2021,  2022) was missing from Table 1 of K22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' The difficulty in modelling it arose because it is (uniquely) extremely close to the limit  m5(L∆t)1/3  45  ≲ 104  (12)  required to avoid the model formally predicting that the WD pericentre distance a(1 − e) is larger than the innermost stable orbital  radius, which is itself ≃ Rg = GM1/c2, where Rg is the black hole’s gravitational radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Here m5 = M1/105M⊙, and (L∆t)45 is the  total energy radiated at pericentre passage in units of 1045 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Equation (12) is the condition  a(1 − e)3 >  � GM1  c2  �3  (13)  written using the parametrization of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' (2022) followed in K22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Evidently the form (12) expresses the facts that the radiated  energy is increased in a tighter orbit, but that a larger black hole mass increases Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' For ASSASN-14ko we have m5 ≃ 700, requiring  (L∆t)45 ≲ 3388, as compared with rough observational estimates (L∆t)45 ≃ 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Here I adopt the extreme value (L∆t)45 ≲ 3388 for  this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' For all other currently known QPE systems the constraint (12) is very easily satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  Note 2 There is no secure mass estimate for the black hole in HLX–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Here I adopt the minimum value m5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='5 allowing the  donor to be below the Chandrasekhar mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' m2 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' see K22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=' Larger m5 values allow smaller m2 (see K22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  Note 3 The Table also gives the values of PGR/P specifying the stability of possible ZLK cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content='  Chakraborty, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
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+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
+page_content=', 2021, ApJL, 921,  L40  Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
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+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E1T4oBgHgl3EQf_gbR/content/2301.03582v1.pdf'}
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